The Dibner Library of the History of Science and Technology SMITHSONIAN INSTITUTION LIBRARIES Adopted for Conservation by Caroline Kenney in honor of all the women who came before, who broke down the wall and made the path easier for the rest of us. March 6, 2018 pi I à È Smithsonian Libraries and Archives Adopted for Conservation by Caroline Kenney in honor of all the women who came before, who broke down the wall and made the path easier for the rest of us. March 6, 2018 faro ANALYTICAL INSTITUTIONS. li i, SII Sor Lag HLA meet ar te tal = ik a cia A: Ul Di, ig > Thi h a oo È — A] hi E, | É ANALYTICAL INSTITUTIONS, IN FOUR BOOKS: ORIGINALLY WRITTEN IN ITALIAN, BY DONNA MARIA GAETANA AGNESI, PROFESSOR OF THE MATHEMATICKS AND PHILOSOPHY IN THE UNIVERSITY OF BOLOGNA. e ;;-.. {e TRANSLATED INTO ENGLISH BY THE LATE REV. JOHN COLSON, M.A.F.R.S. AND LUCASIAN PROFESSOR OF THE MATHEMATICKS IN THE UN!VERSITY OF CAMBRIDGE. cE ozio N Ow FIRST PRINTED, FROM THE TRANSLATOR'S MANUSCRIPT, UNDER THE INSPECTION OF THE REV. JOHN HELLINS, B.D. F.R.S. AND VICAR OF POTTER’S-PURY, IN NORTHAMPTONSHIRE» VOLUME THE SECOND, CONTAINING THE LAST THREE BOOKS. With AN ADDITION BY THE TRANSLATOR. LONDON: Printed by Taylor and Wilks, Chancery-lane ; @ AND SOLD BY F. WINGRAVE, IN THE STRAND; F. AND C. RIVINGTON, IN ST. PAUL’S CHURCH-YARD; AND BY THE BOOKSELLERS OF OXFORD AND CAMBRIDGE. 1601. 2 MESSO LA Se ale a i Val di è * ba Aa . EIA f 4 Th rd LI ni ty Do CORI La > a ue Se, » udire i a: i e non eS sec az pigri n ie sot ANALYTICAL INSTITUTIONS. BOOK . II. THE ANALYSIS OF QUANTITIES INFINITELY SMALL. HE Analyfis of infinitely fmall Quantities, which is otherwife called the Introdu&ion. Differential Calculus, or the Method of Fluxions, is that which is converfant about the differences of variable quantities, of whatever order thofe differences may be. This Calculus contains the methods of finding the Tangents of Curve- Lines, of the Maxima and Minima of Quantities, of Points of Contrary Flexure, and of the Regreffion of Curves, of the Radi of Curvature, &c.; and therefore we fhall divide it into feveral Sections, as the nature of the feveral fubje&s may regpire. SEGIGOI. Of the Notion or Notation of Differentials of feveral Orders, and the Method of calculating with the fame. 1. By the name of Variable Quantities we underftand fuch, as are capable of Variable continual increafe or decreafe, while others continue the fame. They are to be quantities, conceived as Flowing Quantities; Or as generated (as it were) by a continual what, motion. Vou. II, B For Conftant quantities, what. A fluxion or difference, what. 2, ANALYTICAL INSTITUTIONS. . BOOK Il. Figo. ua - For-inftance, in Fig. 1, let there be a right | | ao Jine ABC, which is conceived as generated by the F motion of the point A, and is produced im infi- nitum. Upon this, at any inclination, let another right line BD infift, and let it be conceived that, Whil the point B moves from B to C, carrying with it the line BD from the place BD to CE, always remaining parallel to itfelf, the point D fhall defcribe the line FE in fuch a manner, as A BC to pafs through all the points of the curve ADE. ~ Tt is plain that the abfciffes AB, AC, as alfo the ordinates BD, CE, and likewife the arches AD, AE, will be quantities conti- nually increafing and decreafing, and therefore are called Variable Quantities, or Fluents, or flowing Quantities. 2. Conffant Quantities are fach, which neither increafe nor diminifh, but are conceived as invariable and determinate, while others vary. Such are the para- meters, diameters, axes, &c. of curve-lines. 0 AO Aud | Conftant quantities are reprefented by the firt letters. of the alphabet, a, 4, ¢,d, &c. and variable quantities by the laft letters, z, y, x, v, &c. juft as is weal, done in the common Algebra, in refpe& to known and unknown quantities. 3. Any infinitely little portion of a variable quantity is called it’s Difference or Fluxion; when it is fo fmall, as that it has to the variable itfelf a lefs pro- portion than any that can be affigned; and by which the fame variable being | either increafed or diminifhed, it may ftill be conceived the fame as at firft. Let AM (Fig. 2, 3.) be a curve whofe _axis or diameter is AP ; and if, in AP pro- duced, we take an infinitely little portion Pp, it will be the difference or fluxion of the abfcifs AP, and therefore the two lines AP, Ap, may ftill be confidered as equal, there being no affignable proportion between the | finite quantity AP, and the infinitely little portion Pp. From the points P, p, if we raife the two parallel ordinates PM, pm, in any angle, and draw the chord mM pro- duced to B, and the right line MR parallel to AP; then, becaufe the two triangles BPM, MRwy, are fimilar, it will be BP. PM :: MR. Rm. But the two quantities BP, PM, are finite, and MR 1s infinitely little ; then, DSi SECT. IL ANALYTICAL INSTITUTIONS. — pe then alfo Rm will be infinitely little, and is therefore the fluxion of the dati PM. For the fame reafon, the chord Mm will be infinitely little ; but (as will be fhown afterwards,) the chord Mm does not differ from it’s little arch, and they may be taken indifferently for each other; therefore the arch Mw will be an infinitely little quantity, and confequently will be the fluxion or difference of the arch of the curve AM. Hence it may be plainly feen, that the fpace PMmp likewife, contained by the two ordinates PM, pm, by the infinitefimal Pp, and by the infinitely little arch Mm, will be the fluxion of the area AMP, compre- hended between the two co-ordinates AP, PM, and the curve AM, And drawing the two chords AM, Am, the mixtilinear triangle MA# will be the fluxion of the fegment AMS, comprehended by the chord AM, and by the curve ASM. 4. The mark or characteriftic by which Fluxions are ufed to be expreffed, is by How duxions putting a point over the quantity of which it is the fluxion. Thus, if the abfcifs are reprefent- ALA phen will it be Pp or MR = x. And, in like manner, if the ordi- °°» Quei Ko nate PM = y, then it will be Rw =y. And Su DE bi ES making the arch of the curve ASM = s, the | — fpace AVMS = 7, the fegment AMS = = 4, it will be Mm = 5, PMmp aia Vie sow And. 7 all thefe are called Firft Fluxtons, or Differences of the jr} Order. And it may be obferved, that the foregoing fluxions are written with the affirmative fign + if their flowing quantities increafe, and with the negative fign — if they decreafe. Thus, in the curve NEC, (Fig. 4.) becaufe AB = x, BF =x, BC=y, it will be DC= the negative fluxion of y. À That thefe differential sad are real things, and not merely creatures of the imagination, yeti what is manifeft concerning them, from the methods of the Ancients, of polygons infcribed and circumfcribed,) may be clearly perceived from only confidering that the ordinate MN (Fig. 4.) moves conti- nually approaching towards BC, and finally coincides with it. But it is plain, that, before thefe two lines coincide, they will have a diftance between them, or a, difference, which is altogether inaffignable, that is, lefs than any given quantity whatever. In fucha pofition let the lings BC, FE, be fuppofed to be, and then BF, CD, will be quantities lefs than any that can be given, and . therefore wil! ve Lao enable, or differentials, or infinitefimals, or, finally, fuxigns. Thus, by the common Geometry alone, we are affured that not only thefe infinitely little quantities, but infinite olio of inferier orders, reall y enter the compofition of geometrical extenfion. If incommenfurable quantities exift in Geometry,-which are infinites in their kind, as is well known to Geometricians B 2 wake . and 4 ANALYTICAL INSTITUTIONS. BOOK It. and Analyfts, then infinitefimal magnitudes of various orders muft neceffarily be admitted. | Fig. 5. For the fake of an example, let AB be the see etal AO GOLA fide of a {quare, and AC it’s diagenal or dia- di d meter; which two lines (by the lalt propofition F of the tenth Book of Euclid,) are incommen- furable to each other. Now it may be proved that this afymmetry of their’s does not proceed A E GH C_ from any little finite line CE, how fmall foever it may be taken, but from another which is infinitely lefs than it, and therefore of the infinitefimal order. Let it be fuppofed then, if poffible, that it is the finite line CE which is the caufe of the afymmetry or incommenfurability between the two lines AB, AC; confequently the remaining line AE will be commenfurable to the fide AB. - Let the right line F be their common meafure, which can never be equal to © EC, for then the diameter and fide would be commenfurable. It mutt therefore be either greater or lefs than it. 2 EE In the firft cafe, let F be fubtra&ed from CE as often as can be done, and let the remainder be CG. Now, becaufe F meafures AB, AE, and alfo EG, the two right lines AB, AG, will have to each other a rational proportion; and therefore it was not the magnitude CE that made the lines AB, AC, incom- menturable, but fome quantity lefs than it, fuppofe GC, which therefore is finite, the finite line F being once or oftener fubtra&ed from the finite line CE. Let F be bifected, and each part bifeAed again, and fo on, till there arife an aliquot part of F which is lefs than CG, and which being taken from CG, there will remain CH. But this, by the fame way of argumentation, is not the quantity that caufes the incommenfurability of the lines AB, AC. And as the fame way of reafoning obtains in all other finite magnitudes, we may thence fairly conclude that the incommenfurability proceeds from an inaffignable quantity, or which is lefs than any that can be given. The fame may be alfo proved in the other cafe, or when the common meature F is greater than CE. _ From hence I {hall proceed, further, to take notice, that the fquares upon the right lines AB, AC, which are to each other as one to two, notwithftanding that their fides are irrational, are neverthelefs commenfurable to each other 3 and that this proceeds from an infinitely little quantity of the fecond order. ‘The two fquares AB, AC, being propofed, (Fig. 6.) let the two quantities ED, FI, equal and infinitefimal, be thofe which render the fides AD, AG, AI, AH, incommenturable ; and the conftruction being completed as in the figure, it is known that the two rectangles DK, IK, are incom- I menfurable sECT. I. ANALYTICAL INSTITUTIONS, 5 menfurable to the fquare AB, But the whole {quare AC is to the other AB in a rational proportion: therefore the fquare AC is made fo by the infinitefimal fquare KC, a quantity of the fecond order, by which it exceeds the faid income menfurable gnomon. It may be obferved, that cubes upon the lines AT, AH, are incommenfurable, — although their bafes are rational; and it may be eafily proved, that they are made fuch by means of an inaffignable magnitude of the third order, and we may go on in like manner as far as we pleafe. 5. After the fame manner that firft differences or fluxions have no affignable How higher proportion to finite quantities ; fo differences or fluxions of the fecond order orders of have no affignable proportion to firlt differences, and are infinitely lefs than ioe they: fo that two infinitely little quantities of the firft order, which differ from Ù 1 each other only by a quantity of the fecond order, may be affumed as equal to each other. The fame is to be underftood of third differences or fluxions in refpe& of the fecond; and fo on to higher orders. Second fluxions are ufed to be reprefented by two points over the letter, third fluxions by three points, and fo on. So that the fluxion of x, or the fecond fluxion of x, is written thus, x; where it may be obferved, that & and x* are. not the fame, the firft fionifying (as faid before,) the fecond fluxion of x, and the other fignifying the fquare of x. The third fluxion of x will be x, and fo on Thus, ¥ will be the fluxion of y, or the fecond fluxion of yi and fo of others. But, to give a jut idea of fecond, third, &c. fluxions, the following The- orems Rett be convenient. THEOREM I 6. Let there be any curve MBC, and BC an Infinitefimals infinitely little portion of it of ia firft order, Proved 40 From the points B, C, let the right lines BA, CA, be drawn perpendicular to the curve, and meeting in A. I fay, the lines BA, CA, may be affumed as equal to each other. Let the tangents BD, CD, be drawn, and the chord BC. If the two lines BA, CA, be unequal, let one of them, as CA, be the greater, and to this let the perpendicular BH. È, DEC ae RENE i ee 6 ANA LYT.ICA LL INSTITUTIONS. BOOK If. be drawn. The difference ei edn the lines BA, CA, will be lefs than the intercepted line CH, which is lefs than the chord CB, becaufe of the right anele at H. But the chord BC is an infinitefimal of do firft order, the arch being fuppofed an infinitefimal; therefore the difference between BA and CA, at leaft, will not be greater than an infinitefimal of the firft order, and therefore thofe lines BA and CA may be affumed as equal. Coroll, 1. Therefore the triangle BAC will be equicrural, and thence the angles at the bale ABC, ACB, will be equal; and being fubtracted from the right angles ABD, ACD, will leave the two angles BCD, DBC, equal to each other, and confequently the two tangents BD, CD, will be equal. Coroll, II. The right line DA being drawn, the two triangles ADB, ADC, _ will be equal and fimilar; and that line will bife& the angles B::C, BDC. And, becaufe the two triangles AEB, AEC, are fimilar and equal, the fame line AD will be perpendicular to BC, and will divide it into equal parts in E. Coroll. TI. And the two triangles DAC, EDC, being fimilar, the angle DCE will be equal to the angle DAC; and the two angles DCE, DBE, being taken begsteht will be equal to the angle BAC. Corel. IV. From hence it follows, that any infinitefimal arch BC, of any “curve whatever, will have the fame affections and properties as the arch of a circle, defcribed on the centre A, with the radius AB or AC. Lia V. The two triangles AEB, BED, being fimilar, we fhall have AE. > EB.'ED. But AE isa finite n and EB an infinitefimal of the firft ade therefore ED will be an infinitefimal of the fecond order, and it’s EBg value will be = But the re&angle of twice AE into EI is equal to the Ak fquare of EB, from the property of the circle. Therefore EBg = 2AE X. EL = AE x ED, and confequently. 2AE.AE :: ED. EI. But the firt term of the analogy is double to the fecond, therefore the third is double to the fourth. Confequently the two lines EI, DI, of the fecond order will be equal. Corolì, VI. And therefore the difference between the femichord BE, and the tangent BD, is an infinitefimal of the third degree; for as much as from the centre B, and with the diftance BE, drawing the arch of a circle EL, a mag- nitude of the fecond ciafs, which edireides Wit Wi it’s fine; the two triangles BDE, EDL, will ‘be fimilar, which, -befides the right angles at E and ie havè “a common angle in D. Dia itwill;be.BD:;. DE :: DE. DL. Bat BD isa firtt fiuxion, DE is a fecond fluxion by the foregoing corollary, and therefore “DL will be a third fluxion, Wherefore the arch of the curve BI being greater 8 than SECT. I. ANALYTICAL INSTITUTIONS 7 than the femichord BE, and lefs than the tangent BD, it cannot differ from either of them but by a magnitude of the third order. THEOREM It 7. Let there be any curve whatever, DAE (Fig. 8, 9.), in whofe axis are taken two “equal infinitefimal portions of the firft order HI, IM; let parallel ordinates HA, IB, ME, be drawn, which in the given curve fhall cut off the little arches AB, BE, which are like- . wife infinitefimals of the firft order. Let meet the ordinate produced, ME, in the point C. I fay, that the intercepted line CE, be- | tween the curve and the chord AB produced, Fig. 9. pu fhall be an infinitefimal of the fecond order. Let the chord AE be drawn. If the right line IM were a finite and affignable quantity, then the triangle ACE would alfo be finite. But ME continually approaching, [from a finite diftance, | to the ordinate HA, [while IB remains fixed, | fo that IM may alfo become a fluxion, or may be an infinitefimal of the firft tise) order; the angle. ACE. always continuing Ha SONE the fame, the angle AEC increafes, making | the angle CAE always lefs and lefs, till at laft it becomes lefs than any given angle, that is, an infinitefimal. In this cafe, as the fine of an infinitely little angle of the firft order, having a finite and affign- able radius, is an infinitefimal quantity of the firft order; fo the fine of an ‘infinitefimal angle, CAE, of the firft order, with a radius AE or AC, which is | ‘an infinitefimal quantity of the firft order, Mall be an infinitefimal quantity of the fecond order. But in triangles the fides are proportional to the fines of the | oppofite angles, and therefore the right line CE fhall be an infinitefimal of the fecond order. ' Wherefore, calling DH = «, HA =y, HI = IM = x; thén FB = GC = y, and EC = —¥; the negative fign being prefixed, becaufe y does not ‘increafe but diminifh (Fig. 8.). And thus, on the contrary, it will have the pofitive' fign if y increafe, that is, if the curve be convex in this point to the axis DM (Fig. 9.). 3 | Corolla. there be drawn the chord ABC, which thal! - , 8 ANALYTICAL INSTITUTIONS BOOK II. Corcll. If from the point E the normal ES be drawn to BC, then alfo ES, CS, will be the fluxions of the fecond order; for each of them is lefs than EC. | THEOREM III, 8. If in the circle be taken an arch which is an infinitefimal of the fir& order, I fay, that it’s verfed fine fhall be an infinitefimal of the fecond order; and the difference between the right fine and the tangent fhall be an infinitefimal of the third order. Fig. 10. Let the arch DC be an infinitefimal of the firft order, DB it’s right fine, CE the tangent, and let DF be drawn parallel to AC. From. the nature of the circle, it is GB. BD :: BD . BC. But GB is a finite quantity, and BD an infinitefimal of the firft order. There- fore, as GB is infinitely greater than BD, fo BD will be infinitely greater than BC. There- fore BC or DF will be an infinitefimal of the fecond order. By the fimilitude of the triangles ABD, DEF, it will be AB. BD :: DF. FE. But AB, a finite quantity, is infinitely greater than BD, an infinitefimal of the firft order, and therefore DF, an infinitefimal of the fecond order, will be infinitely greater than FE, which is therefore a third fluxion, or an infinitefimal of the third order. g. Coroll. I. And whereas the tangent is always greater than it’s arch, the arch greater than it’s chord, and the chord greater than the right fine, the tangent and the right fine may be affumed as equal, they not differing but by an infinitefimal of the third order. Alfo, thefe following may be affumed as equal, the tangent, the arch, the chord, and the right fine. to. Coroll. II. If we conceive the radius of the circle AN to be an infini- tefimal of the firft order, the arch NO and it’s right fine OM will be infinite- fimals of the fecond ; and therefore the verfed fine MN will be an infinitefimal of the third order, È 11. Coroll. - SBCTs To MNAE VY PEC AL INSTDIUTIO. N, i 9 sg Fig. a1. : soo R bs rr. Coroll, III. Inthe axis DM (Fig. r1, 12.) let there be two firft differences HI, IM, equal ‘to each other, to which correfpond the two Infinitefimal. arches AB, BE, of the curve DABE ; and let be drawn the two chords BE, AB, of which this is produced till it meets in _.C the ordinate ME, produced alfo if neceffary. Let ES be drawn perpendicular to BC, and from i G ARE centre B, with radius BE, let the arch EO be a ita drawn. By the corollary of Theor. II. CS is Q:: an infinitefimal of the fecond degree, and, by ie | N the foregoing, OS is an infinitefimal of the PB. CIG . third degree. Then CO is alfo an infini- Q tefimal of the fecond degree, becaufe an VAN | | | infinitefimal of the third degree being added to, or fubtracted from, an infini- tefimal of the fecond degree, makes no alteration in it. Now, becaufe HI=IM, or AF = BG, and, becaufe of equal and fimilar triangles AFB, BGC, it will be alfo AB = BC. But the arches may be ‘affumed equal to their chords; then CO will be the difference of the two arches .-AB; “BÈ ;. and therefore, if the arch | Da 3, will be AB = BC's, and CO = — i with a negative .fign, ‘becaufe AB «decreafes when BE 1s lefs than AB, .as in Fig..a1. And, on the contrary, with a pofitive fign, as in Fig. 124 ia II RIINA ECAR 8 ERR SN DAT IST NOT PRIDE A AN L ; Kets 3 x :. sSCHOLIU™M. -12, In determining the fecond differences. (or fluxions) ‘of the ordinate, and of the arch of the curve, I have fuppofed, both in’ Theor. If. and in this lat corollary, that the firft differences HI, IM, are-equal; thats to fay, that the firft difference of the abfcifs does not alter, but remains conftant, inwhich cafe the fecond difference of the abfcifs is none at all. So that, calling the-ab{cifs #, it’s firft difference will be x, and it’s fecond xX =.0, ; Wherefore we.may further make thefe two other conclufions, one of which is, | that if the firft difference of the ordinate be conftant, thofe of the abfcifs:and of the curve will be variable. The other is, that if the’ firft difference of ‘the curve be.conftant, thofe of the abfcifs and ordinate -will be variable, | Von. II, | Cc Now, 10 ANALYTICAL INSTITUTIONS, BOOK IT. Now, thefe ‘things being premifed, we may eafily proceed to thefe two other hypo- thefes. Suppofing what has been already advanced, let BF (Fig. 13, 14.) be equal to EG; that is, let the fluxion of the ordi- nate be conftant; and let EP be drawn pa- rallel to BG, and PT perpendicular to it. Then will BF = PT, and therefore AF = Bi, AB = BP, and GY or EP - tr II I II [mm [_—__r_——trtst-- 20 THEOREM VIL. 21. Let the curve BEG (Fig. 20, 21.) be referred to a focus, that is, fuch, that all the ordinates proceed from a given point, which is called the Focus, and let this point be A. From hence let be drawn three ordinates, which are infinitely near, AB, AE, AG, which contain the two infinitely little arches of the firft degree, BE, EG; and draw the chord BE, which, pro- duced, meets the ordinate AG (produced if need be,) in the point L. With centre A let the LA arches BC, EF, be defcribed, and let BM, EN, N So) be their right fines. Laftly, make the angle NEP equal to the angle MBE. I fay, that the | 4 intercepted line GP fhall be the infinitely little difference of the fecond order of the ordinate AB, Let SECT. To ANALYTICAL INSTITUTIONS 15 ‘ Let the chord EG be drawn. Since the angles MBE, NEP, are equal by conftruction, and the angles at M and N are right ones, the triangles EBM, PEN, will be fimilar; then taking the fine BM for conftant, that is, fuppofing it egual to EN, the forefaid triangles will alfo be equal. Therefore it will be MEE. = NP. > Bot, fuppofing BM = EN. by the foregoing Theorem the difference of the verfed fines MC, NF, 1s infinitefimal in nretpect ofithem. . Therefore, allo, CE; FP, will be equal, and thence GP will be the difference between CE and FG. But the right lines EQ, QG, being drawn perpendicular to the curve in the points: E, G, the angie LEG will be equal to the angle EQG, by Theor.V. Coroll. II. _ {which is true whether the curve be referred to an axis, or to a focus.] And the angle EQG is infinitely little. ‘Therefore, alfo, the angle LEG will be infinitely little. And, becaufe the right lines EG, EL, are infinitefimals of the firft order, GL will be an infinitefimal of the fecond order; and much more GP, refpett being had to Fig. 20. | By Theor. III. Coroll. I. the line BM is equal to the arch BC. Then, in- ftead of the fine, taking the arch for conftant, and making it = x, AB.=y, CE = y, it will be GP = —¥. And with centre E, and diftance EG, de- {cribing the arch GV, it willbe VP — — 5, if BE =5. . | 22. Coroll, The angle LEP will be equal to the angle EAG. For the angle: EPA, by conftru&ion, is equal to the angle BEA; but the external angle EPA is equal to the two internal angles L and LEP; and the other, BEA, is equal to the two, L and EAG. Then, taking away the common L, there will remain the two equal angles LEP, EAG. Wherefore this will be true, whether the curve be concave towards the point A, (Fig. 20.) or whether it be convex, (Fig. 21.) as it is eafy to perceive. In the fame Fig. 21, the angle LEP will be an infinitefimal, and therefore LP is an infinitefimal of the fecond order.. But it has been feen, that GL is alfo an infimtefimal of the fecond. order.. Therefore the whole, GP, will be fo alfo, which will be = $; and with centre E, diftance EG, the. arch GV being defcribed, it. will be PV = s.. If we fuppofe y to.be conftant, with centre A, and diftance AG, let the arch GI be defcribed, and from the point T let the right line TOA be drawn. Becaufe FG = EC, by hypothefis, the triangle TEO will be fimilar and equal: to the triangle EBC; and therefore BC = EO, and BE = ET. Then. OF ==, and TV = 5, in Fig.20, But OF = — x, and TV =. —5, in. Fig. 2t. N Faking gf. ANALYTICAL INSTITUTIONS. BOOK If. Taking s for conftant, and drawing the right line VRA, it will be EG = EV = BE; and therefore the triangles EBC, EVR, are equal and fimilar ; thence is BC = ER, and Clie RV. Whence RE = x Vi = — ¥, in Fig. 20. But RF = — x, and VI =, in Hig. 21. If no firft fluxion be taken for conftant, let EG be greater than BC, (Fig. 22, 23.) by the fecond fluxion RF ; let the right line ART be drawn; with centre A, diftance AG, draw the arch GT ; and with centre E, diftance EG, draw the arch GV. There- fore, fince BC = ER, it will be alfo CE = RI, and BE = EI. Therefore TI will be the difference between CE and FG, and VI the difference between BE and EG. SC ioe AT eT 23. It may not be befide our purpofe to obviate a difficulty, which feems likely to arife. And this is, that in the foregoing Theorem the lines CE, FP, are affumed as equal, in virtue of Theor. VI.; which Theorem fuppofes as equal the fines BM, EN. Whence it may feem, that the determinations con- cerning fecond differentials can only take place in the cafe when we make a fuppofition of a conftant fluxion BC, and in no other. But, to remove this difficulty, it will be fufficient to confider, that, though BC be fuppofed variable, the difference will be an infinitefimal of the fecond degree, which does not hinder the equality of the firft fluxions BC, EF, nor of the fines BM, EN. SCHOLIUM (i. 24. In the foregoing Theorems are contained the principles, by which infi- vnitefimals of any order may be managed, and which prepare the way to make a right SECT. I, ANAEYTICAL INSTITUTIONS. 17 right ufe of the Method of Fluxions, whether direct or inverfe ; and befides, to apply the fynthefis of the ancients to infinitely little magnitudes of all _degrees; and to make ufe of the ftri&eft Geometry, which proceeds with a particular fimplicity and elegance. 3 Now, to avoid paralogifms, into which it is but too eafy to fall, it will be needful to reflect, that infinitely little lines of any order, (agreeably to what obtains likewife in thofe that are finite,) have two important circumftances to be confidered, which are their magnitude and their pofition. And as to their magnitude, I think they cannot be rejected except by thofe, who fancy fuch infinitefimal quantities to be mere nullities, Now, although quantities, by diminifhing ad infinitum, may pafs from one order to another, the proportions in every order continue the fame. And, be- caufe of three lines of any the fame order a triangle may be formed, it may be confidered, that if, by leffening proportionally the fides,. fo as to pafs from one degree to another, the angles are not thereby changed, the fides muft always preferve the fame ratio ta one another; that is, infinitefimals with the finite, and infinitefimals of the fecond order with thofe of the firft, and. with finite; and fo on. : , But if two magnitudes, of any order whatever, fhall differ by a magnitude which in refpe& of them thall be inaffignable, then with the utmo@t fecurity, and without any danger of error, one of them may be taken for the other; nor need it be apprehended that fuch a comparifon will introduce the leat error. | Therefore it is neceffary to be much upon our guard, when the pofition of lines and angles is concerned; for, to confound them when they ought tg be icely diftinguifhed, muft needs lead us into unavoidable paralogifms. sé ‘ 25: The principal foundations of this calculus being thus laid, I Mall pafs on to the methods or rules of finding the fluxions or differences of analytical formulas or expreffions. And, firft, let us take the differences of various quantities added together, or fubtraéted from one another; for example, of a+x%x+23+y—%. Asthe fluxion of x is x, of z is 3, &c; and as the conftant quantity 4 has no fluxion ; then, conceiving every variable to be in- creafed by it’s fluxion, according to it’s fign, the formula propofed will be changed into this other, 2 + x + % +2 +3 +y+9J —u — &; from which fubtra&ing the firft, the remainder will be x + % + y — 4, which is exactly that quantity by which the propofed quantity is increafed, that is to fay, ° at’s difference or fluxion. Hence we derive this general rule, that, to find the fluxion of any aggregate of analytical quantities of one dimenfion, it will be fufficient to take the fluxion of every one of the variable quantities with it’s fign, and the aggregate of thefe fluxions fhall be the fluxion of the quantity propofed. So, the fluxion of Bort, . è D | a 15 ANALYTICAD ‘INSTITUTION I. BOOK II. b—s— zx will be — ;— &, The fluxion of 24 — 4bz 4 by will be — 45% + by. And fo of others. 26. But if the quantity propofed to be differenced Mall be the produ& of feveral variables, as xy; becaufe x becomes x + x, and y becomes y + y, and xy becomes xy + yx + xy + xy, which is the product of x + x into y + y; from this produc fubtracting, therefore, the propofed quantity xy, there will remain yx + xy + xy. But xy is a quantity infinitely lefs than either of the other two, which are the rectangle of a finite quantity into an infinitefimal, But xy is the rectangle of two infinitefimals, and therefore is infinitely lefs, and muft be fuppofed entirely to vanifh. The fluxion, therefore, of xy will be LY | YX Let us difference xyz by this rule. The product of x + x into y + y into 2 + BIS MS A IBN + az) + 492 + wy + 9x3 + YS + xyz; which, fub- tracting the quantity propofed, will give the remainder yzx + xzy + ay + oxy + 9x3 + xy3 + xyz. But the firft, fecond, and third terms are each the product of two finite quantities and one infinitefimal; the fourth, fifth, and fixth are the products of one finite quantity and two infinitefimals, and therefore every one of thefe is infinitely lefs than any one of thofe, and therefore will vanifh : and much more the laft, which is the produ& of three infinitefimals. Therefore let all thefe terms vanifh, beginning at the fourth, and then y2x + x2y + xyz will be the fluxion of xyz. Hence arifes this rule, that, to take the fluxions of the produc of feveral quantities muluplied together, we muft take the fum of the products of the fluxion of every one of thofe quantities into the produ&s of the others. Thus, ihe fluxion of dxzti will be dxz/ + dxtz + dizx + xzi x 0; becaufe the fluxion of the conftant quantity 2 is nothing. That is, the fluxion of dxzt will be dazi + dxt3 + bizx. The fluxion of 4 +x X b—y will-be x X 2 —y — y xX 4+x, that is, 0 — yx — ay — xy. 27. Let the formula to be differénced be a fraction, fuppofe va . Hwe put *_ = 2, it will be then x = zy. And therefore their differences will alfo be eel si ® Poe equal, that is, X = By +} zy. Wherefore 2 = a But z'= an there» fore, fubftituting this value inftead of z, it willbe 3g = — —_ 4 EL, 2 i) bi | | 3 } sy JY ° Di è ° 5 x e Mon at a , then 3 will be the differential of vie and therefore the dif- ferential of On Wil Now SECT eke ANALYTICAL INSTITUTIONS, 19 Now the rule will be, that the differential of a fra@ion will be another fraction, the numerator of which will be the product of the difference of the numerator into the denominator, fubtracting the produ& of the difference of the denominator into the numerator of the propofed fraétion ; and the denomi- nator muft be the fquare of the denominator of the fame propofed fraction. Therefore the difference or fluxion of — will be — ft, The fluxion of ESE will be ceed Tit i = si The, fluzion: of gm will be by pri fet ees Wt io ge a that is, seach The fluxion of = Will: be sy ayn X Oma we X Bx _* 3400) + gaye — 3) z \& a x) Ra 28. Now let us find the fluxions of powers, and, firft, of perfe& and pofitive powers, that is, whofe exponents are pofitive integer numbers; for example, of x*. But xx is the product of x into x, and therefore, by the rule of products, it’s fluxion will be xx + xx, that is, 2xx. To find the fluxion of #3, Now this is the product of x into x into x, and therefore the fluxion will be-xxx + xxx + xxx, that is, 3axx. And, as we may proceed in the fame manner in infinitum, the fluxion of x”, 1 being any pofitive integer, will be Me I - Mx Xe’ If the exponent be negative, fuppofe ax~”, or —, the fluxion, by the rule of fractions, will be the product of the fluxion of the numerator into the deno- minator, fubtra&ing the produ& of the fluxion of the denominator into the numerator, the whole being divided by the fquare of the denominator. But . e . . ° om 2 “ the fluxion of the denominator is 2xx 3 fo that the fluxion of ex ‘ or - will 24%% . 2ax . —3 1 i Zak be — — , that is, — —. The fluxion of x °, or —-, will be — o, pa, ; cn 772 è MQ E x hi » DX a e b or #2, And, in general, the fluxion of , OF ——,; Will’be + die, È b ba” bb 27% e ae : . =~ IT ° Ma: that is, — ——_— . Let it be animperfe& power, and, firft, let it be pofitive; that is, let the | è | exponent be an affirmative fraction, as x/x", or wn, where — ftands for any pofitive fraction. Make x x = 2, and, raifing each part to the power 2, it È D2 will 20 ANALYTICAL INSTITUTIONS, BOOK II. . sui 47 F | e e e i Me tT willbe x. =, of which taking the fluxions, we fhall have maxx Ds mi T : ua "babo! n—I cts But, becaufe « == x, and thence z Pe . tT . tizi, whence B= UZ “~~, which being fubftituted, it will be 3 =.————-, that is, 3 = Sf . ga ani “DBA 4 ae 2 n - Pi ® Ad m . I È Pe COAL Se I If the exponent were negative, as as that:B,.& . 7, or elfe presse ‘ ò n x mato N 7 oe cea cen È LAS 9 OF me XX n n _fluxion, by the rule of fraQions, would be — , non Therefore the general rule is, that the fluxion of any power whatever, | whether perfect or imperfect, pofitive or negative, will be the produét of the exponent of the power into the quantity raifed to a power lefs by an unit than the given power, and this multiplied into the fluxion of the quantity. ; . : ae 4 See 1. : aa Let it be required to find the fluxion of #7; it will be 3x* x, that is, 3x*x, or elfe 1xV x. | a "REM R . Lu Li L, ì Let be given x*; it’s fluxion will be sy? ie” that! 1s, JE") Or ENN AR: I 3 — I ; age oe ; the fluxion will be — ixx 7, or Let be given ARE Lille pa coni ax csi 2x0 a OF, laftly, Mr ss: i 2x È The fluxion of ax + xx\* will be 2 x ax+tax X ax+2xx; that is, 242xX + 6ax°x + 40°x. The fluxion of xy + ax)? will be 3 x xy + axl? x xy+yX tax, that is, 3#9°9 + any + 340° + gy arn + gayax + ga’yx*x + jax. he fluxion of —— ra eine e re ay) TS The fluxion o rotaie will be — 2 X 4% — yy x am 24x + 499 | Ds iii | The SECT. Le ANALYTICAL INSTITUTIONS, 21 Se RARI I The fluxion of Varna, Or 0% — me, will be = 2 x DX cc = xx)? * x aim an, ib A ee _ 24% . ig that is = + mar 2 X ANAAO | | . i ees SIE ( TIVE GIU The fluxion of varia Or Ray, will be + x xe + xy)” * x -yw, that is, —— =" a eee 2x ate The fluxion of 44% XX, OF 4% == x)? , will be + PI ax — TT X ox — 2%X, Ax - QX that 1S "ae 2° ; 3 x ax — £23 I The fluxion of ===, or N ay + xy ay + xy\3 —+4 ; —————r ay te) + ye a hae ay +e X, CC — ——— y + xy X a rey yx, PE ree: > Or ay Py), will be — = x) ETS nas ezine e noe «I. . 2 (a The fluxion of a—xd4+x, or a—-a X 4+ 813, 15 — xX KX ates + 1 Al HK X G—_x X atal * Xx, or 1 T oes ate oro Pesci ae ary tar girs: = gee Se The fluxion ax + %% + » OF 2%, putting, inftead of yy, it’s value ax, given SECT. II. ANALYTICAL INSTITUTIONS . =e given by the equation of the curve. Therefore the fubtangent in the parabola is double to the abfcifs; fo that, taking AT = AB, and from the point T drawing the right line TD to the point D, it Mall be a tangent to the curve at the point D. Inftead of the value of x, given from the equation of the curve, if we fubftitute the value of y, or aie in the general formula = , It will be alfo 2 , as before; which may fuffice to obferve in this Example. a In the fame parabola, if we require the fubnormal BN ; the general formula of the fubnormal is =a . But, by the equation of the curve, itis x = a, fo that, making the fubftitution, the fubnormal in the parabola will be = ta, that is, half of the parameter; and therefore, making BN = by | the equation of the curve we have x = =. Then, fubftituting this value P (dA IVA dg dii Hy See lider ot inftead of x in the formula, we fhall have pot + epr re Ly 44) + aa = WV 455 + ax, (putting, inftead of yy, it’s value ax from the given equation,) which will be the tangent required. If we would have the normal DN, fubftituting the value of x = #2 in the IN 45) + aay _ V ay + aa wT. eS RT ee «PO ee na RI I IV kk FD x 2yy general formula s it will be V/ Fax + aa pei. aa I . » putting, inftead of yy, it’s value from the given equation. If we would have the right line BM; fubftituting the value of x = 2 yy yn Me ayy a | in the general formula ieee ot will be T=°-==" = 2 ERA ì xk +99 499.) + aajj V 459 + aa ayax V jax + da If we would have the right line BH; fubftituting the value of + in the general formula. Fs ne be pae eee ees MS à +9) 4) aay Vaytaa V4ax +aa EA Having 73° ANALYTICAL INSTITUTIONS, BOOK Il. Having found the fubtangent, there is no need of any formulas for finding the other lines, though here, by way of exercife, I have made ufe of them. For, when BT is known, the triangle TDB, right-angled at B, will furnith us with the tangent TD, and the fimilar seals es TBD, DBN, DMB, DHB, with ail the other lines. So that, in the following examples, I fhall apply the method to finding the fubtangents only. If we would have the angle which is made by t the tangent of the parabola with it’s axis; taking the fluxional equation ax = ayy, and refolving it into an analogy, it will be y . x 124. 2y. That is, that the right fine BD is to the fine of the complement BT, as the parameter is to the double of the ordinate; whence is determined the point D. And if we would determine the tangent to any certain point, for example, to the point D, to which correfponds the abfcifs AB = x = 14; from the equation of the curve finding the ordinate y, corre- fponding to x = 14, which, in this cafe, is y = 14, we fhall have the analogy, bak he ae cee a that is, the angle DTB will Re half a right angle, when it is JI = 14, OF x = 7a At the vertex A it is y = o, and therefore the analogy for the angle of the. tangent at the vertex will be y.x :: 4.0; that is, the ratio of y to x is infinite, which is as much as to fay, that the fine of the complement will be nothing at all, or that, at the vertex, the tangent is perpendicular to the axis. EXAMPLE IL 36. Let the equation be x = y”, which is a general equation to all parabolas of any degree whatever; where m ftands for any pofitive number, integer, or fraction, and unity fupplies any dimenfions that are wanting. By taking the fluxions, it will be x = myy” ‘5 and, fubftituting this value inftead of x in the general formula = , the fubtangent will be my = mx. Let-m = 3, that is, let it be the firft cubic parabola « = y°; it’s fubtangent | will be «9%. * Let m = i, that is, let it be the fecond cubic parabola xx = y’; the teen will be ix, &c. The fluxional equation of the curve & = my” 7! gives this analogy, y . x Lp ee a But, putting y = o, if m be greater than unity, the analogy will be y . 3: 1.0; or the ratio of y to x will be infinite, and therefore the tangent at ae vertex is perpendicular to the axis. And if m be lefs than unity, the SECT. II. ANALYTICAL INSTITUTIONS. 29 the analogy Will De Veet fl —; SALO Y 10s, Vin Len, which is as much as to fay, that the ratio of y to x is infinitely little, and therefore, at the vertex, the tangent is parallel to the axis. | enne EXAMPLE III Li 37. Let the curve be DCE, of which we de- fire the fubtangent, the equation of which is xy = aa, being the hyperbola between it’s afymptotes. By taking the fluxions, we fhall have wy + yx = o,orx = — # W here- fore, fubftituting this value of x in the formula of the fubtangent oa , the fubtangent will be A Bissix T as to fay, that the fubtangent BT muft be taken on the contrary part of the abfcifs. Therefore, taking BT = BA, and drawing the right line TC to the point C, it fhall be a tangent to the curve at the point C. — x with a negative value, which is as much Now, becaufe in the curve DCE, as the axis increafes, the ordinate y will decreafe, in taking the fluxion we might have put y negative; but becaufe, for the fame reafon, we ought to have taken the fame y negative alfo in the general formula, I have omitted to do it in both places, becaufe it comes to the fame thing, without incumbering ourfelves with changing figns; and what is now mentioned may be underftood on other like occafions. I . . + | Let x = — be a general equation to all hyperbolas ad infinitum, between their afymptotes, where m ftands for any pofitive number, integer, or fra&ion, 7 . 272 1 ì By taking the fluxions, we fhall have x = — __ = — i -» And, J fubftituting this value in the general formula =, the fubtangent will be mM Ae si ——, OF — mx, by the equation of the curve. 78 “Bx 30 ANALYTICAL INSTITUTIONS, BOOK II. EXAMPLE IV... 38. Let the curve ADF (Fig. 24.) be a circle whofe diameter is 24, AB=x, BD = y; the equation will be 24x — xx = yy, whofe fluxion is 24% — 2xx ce ey eae tHercLone ne > = È Then, fubftituting this value in the formula yy 20% — E: vw 22, the fubtangent will be — =, by putting, inftead of yy, that is, > a — x it’s value from the given equation, Therefore the fubtangent in the circle will _ be a fourth proportional to 4 — x, 24 — x, and x, Fig. 29. But if the circle Mall be denoted by this equation, 42 — xx = yy, in which the abfcifs AB = «x is taken from the centre; by taking the fluxions, we fhall have «x = — yy, and therefore x = Ye x Pare - Wherefore, fubftituting this value in the formula, the fubtangent will be = — =, that is, a third proportional to AB and BD, but negative ; that is to fay, it muft be taken from B towards T. EXAM PIE. N. 39. Let the curve ADF (Fig. 24.) be an ellipfis, with this equation ax — xx — 22. taking the abfcifs from the vertex A. The fluxional equation will be SR. al — 20x = =2-, and therefore x = — Now, fubftituting this value b xX @—2x : ' . x 2a : in the general formula a » then ; “J will be the fubtangent; or elle, Xam 2% 22223, inftead of =, putting it’s value av — xx from the given equation. a — 2% Making x = +4, half the tranfverfe axis, in the value of the fubtangent, it . will be e, that is, infinite. Therefore the tangent will be parallel to the tranfverfe SECT. Ila ANALY PUCAMINGTITUTIONS, 31 tranfverfe axis in that point, in which the conjugate axis meets the curve. And this we fhall find to be true alfo, if we inquire what is that angle, which the tangent itfelf makes with the fame axis. m+ 2 Let the equation, in general, to ellipfes of any degree be this; 5 penne b n . . x" X a — x, where mand x reprefent any pofitive numbers, whether integers or fraQionss The fluxion of this will be 7 = = X A = met x - npba—-1 n » 77 1m J . x Gy Ae — xe” x a = "7: and therefore x = ——a72 54 a bia” ~~! x a—x)\" ~ bax” x aa 7! And, fubftituting this value in the general formula, it will be m+n xX ayn tn (3 "a Then, inftead of 4 , putting it’s value om n mm . pas bad ara ie anal i ati mn Xam from the given equation, the fubtangent will be ee si : PO uc x eoxy ga ei Ù ® Y . è — — I e ° And, dividing the numerator and denominator by x x = » it will be, finally, EA a = MX — NX Make m = ta DOCET that is, let it be the ellipfis of Apollonius ; then the fubtangent will be 242 — 74" | as before. Make m = ER pe then the equation is a = =x a—n’, and the fubtangent will be i And fo of others. | i ; | ay +? mn n: If the equation were —=—~ = x X 4 + x), it would exprefs all hyperbolas of any degree, when referred to their axis; taking, in the fame manner, the beginning of the axis from the vertex A. Then, by a like operation, we fhould find the fubtangent to be Sk » which differs from the fore- going only 1 in it’s figns ; as alfo, the equation, from whence it is derived, differs only in it’s figns. Make m = 1, 2 = 1, which is the Apollonian hyperbola. The fabtangen 24% + 2x4 will be —————- , Make m= 3, m = 2, then the equation will be 4 La x a+ 2% x 4+?; and the fubtangent will be wa gs te 40. From Afymptotes. 3% ANALYTICAL INSTITUTIONS . BOOK II, 40. From this method of tangents may be further derived a way of difcover- ing whether curves propofed have afymptotes, and the manner of drawing them, when they are inclined to the axis. For, as to the more fimple cafes, in which they are either perpendicular or parallel to the axes, fufficient has been faid in the firft Part, Sect. V. : * os ANE Dae CI, 41. Let the curve be ADE, with the equa- m+n tion’ = 77 = x” X% 4+%\°, as above, the fub- m+n X an + ax tangent of which is TB = ——~— ais tlhe «> AR _ m+n X ax+ax Then the intercepted line AT = — eee nAX — X, that 15) EE It is plain that the tangent TD will become an afymptote, when touching the curve at an infinite diftance ; that is, when the abfcifs AB = x becomes Infinite, the intercepted line AT {hall remain finite. Now, putting « infinite in the expreffion of AT, the firft term ma of the denominator is infinitely lefs than the FE e e e . nNAx others, and therefore vanifhes. Whence, in this cafe, it will be agree m—, which is a finite quantity. Wherefore the curve has an afymptote, which will begin from the point M, making AM = = . Now, to draw it, let AH be raifed perpendicular to AB, and let it be, for example, MHP, This being fuppofed, if we take w infinite, it will be x.y :: MA. AH, and, ; i Pg: te in the fuppofition of x being infinite, the equation of the curve —;— = x x 4+ x), (4 being nothing inrefpe& of x, will be changed into this other, nl Aw Oty extraCting the root, and, for convenience, making m +. n == 7, it will be y ak + ax; it will be ¢ = 2, and therefore AM = ta, 2g Bon — x v= —1Vab. That is, AM is half the tranfverfe axis, and AMT half the conjugate, juft as it fhould be from the Conic Se&ions. EXAM PL Ed, 42. Let ADE (Fig. 30.) be a curve whofe equation is y? — x3 = axy: making AB = «, BD = y. By taking the fluxions, we thall have 3777 — 3x*x SY GO | The ol SED ME Telus = ax) + ayx, and therefore — = E and AT = 5 xe oa 393 — 3 - . . : Be ren 22) Or} inftead “of 39° — 34, putting it’s value 3axy from the 3xx + ay equation of the curve, it will be AT = —°—. And, making « infinite, 344 + ay that is, in cafe of an afymptote, in which AT becomes AM, the term ay is nothing in refpe& of 3xx, fo that ic will be AM = & — 2° i ì Zara 3x But, becaufe, in the propofed equation, the indeterminates cannot be fepa- | rated, nor, confequently, can the value of AM be determined; if we put AM = = = ?, (which expedient may alfo be ufed in other like cafes,) ca Co) will be y = di; which value being fubftituted in the propofed equation, it . 271343 2983 x : “ore 2 will be, Soe #3 ala”, QF DL pg gt. But, as « is infinite, the lat term will be nothing in comparifon of the others, fo that it will Bei a = o, or # = 4a. Taking, therefore, AM = 1a, the afymptote muft be 34. ANALYTICAL INSTITUTIONS BOOK ITs: drawn from the point M. Moreover, it muft be MA.AH :: x.y, and the propofed equation y? — x? = axy, or 9} = x? + axy, will be reduced to x = y?, or x = y, when « is infinite, and therefore x = y. Therefore,, making MA = AH, if from the point M, through the point H, a right line be drawn, it will be an afymptote to the curve. I add further, that the line AT muft neceffarily approach to a certain limit, beyond which it cannot pafs, and that the aforefaid limit is then an infinitefimal; or nothing. Here follows a plain Example of this. Let BCF be an equilateral hyperbola ;. andumacne Ab = 2,°AD = *«, DC. y, we fhall have the equation aa + xx = 9%, the fluxion of which is xx = yy. Thence gl LA we Fig. 31. Do the fubtangent will'be ED = va __. d04 xx , and confequently ED — AD» aa ve ape Ne aa Str + e AE, x Ww: D Putting x = o, AE will become infinite, and the tangent at the point B will: Be. parallel -t0 ine. agis AD. And, making sero ati will. be AE =o. Wherefore the point E defcribes the whole line AE infinitely produced, and finifhes it’s courfe at it’s origin A, beyond which it paffes not, though the ~ curve turns it’s convexity towards the axis. Therefore the afymptote AG proceeds from the point A, and makes half a right angle with the line of the abfciffes ; foralmuch as, in the equation of the Jocus aa + «x= yy, making x = co, the conftant quantity 44 will vanifh, and. it becomes xx = yy, or xv — De | | 43. Hitherto I have fuppofed.that the angle of the co-ordinates is a right angle; but, if it were obtufe or acute, making, as belore Dh tok, “CH toy C bsx OG = 7, (Fig. 32, 33.) the fubtangent will be neither more nor lefs than I for the two triangles GEO, EAC, will be full fimilar ; but the other formulas will have need of fome reformation. In the triangle EOG, the angle at O,. equal to the angle ACE, is fuppofed to. be known ; therefore, from the point G letting fall Gi perpendicular to AD, and producing SECT. IL ANALYTICAL INSTITUTIONS. 35 producing EO to H, if there be occafion, in the triangle GOH the angle GOH © _véavill be known, and the angle at H be a right angle. Wherefore the angie OGH is known, and confequently the triangle OGH is given i /pecie, that is, the ratio of GO to GH is given. Let this be the fame as @ to i and therefore it will be a.m 3: ly. GH = es Alfo, the ratio of GO to OH will be given, which may therefore be as 4 tov; and confequently 4.7 ::y. OH = mr Then EH = x + —, (where the fign muft be affirmative in Fig. 32, | aa 2 n + mm; and negative in Fig. 33.) Wherefore EGq = pro coi & ani) + mal) + mi) ; But1f OG be expreffed by a4, GH by 2, OH by 2, then it will be aa = mm + an, and aayy = mayy + nuyy, which, being fubftituted in this value oi DE I oul make e A ai ‘. ,,0x2 th 21%) + aj? my printing i This being fuppofed, by the fimilitude of the triangles EGO, AEC, it will be GO’. GE's! EC . EA; that le, fer +22 ty LEA; or EA = da ot one + ayy a = È A expreffion, of the element or fluxion of the curve. > which will be the formula of the tangent. | Let TE be perpendicular to the curve, and ES to the diameter AT. De by fimilar triangles GOH, ECS, we fhall have ES = —, and CS = =. And, by the fimilar triangles GEH, EST, we fhall have EH . HG : TELO, + Ò ae : ‘ White. ig fe On Chat Be ai mn And rcieire CT = a a a aXaxt ny j ‘ : ) ayy + anyi avy nji oe ef se INT SRS eS OEE re DS Shae ae the Word ot A + ARL” Spera ny a # axaktn. << È the fubnormal. my In.a like manner, the other formulas may be reduced, which it is fufficient only to take notice of here. | 44. But the curves, aint nati we dle may be Tranfcen ten or Tangents te Mechanical, that 1s, are not expreffible by any Algebraical equation, but may tranfcendent depend on the rectification of other curves, which: are not re&ifiable, Let the °!"Y° Fa, curve 36 ANALYTICAL INSTITUTIONS. BOOK Ife. curve be APB, whofe tangent PTK we know how to draw, at any given point P.. Then, producing to M the line QP per- pendicular to AQ; let the relation of MP to the arch PA be exprefled by any equation, to find the tangent MT of the curve CMA, defcribed from the point M.. - Draw gm infinitely near to QM, and MR. parallel to PT; and fuppofing the rectifi- cation of the’ arch APs make AB; PM = y, andit willbe Pp = x, Ru = y, and the two triangles mRM, MPT, will be fimilar, and therefore mR. RM ce MP .PT, that is, y «x soy, PT = n° the formula for the fubtangent of the curve CMA, by taking it on the tangent of the curve APB. From the given equation of the curve AMC is found the value of x or y, to be fubfti- tuted in the formula. All the reft is to be done as ufual. EXAMP.L E. 45. While the circle DPC revolves uniformly upon the right line AB, be- ginning at the point A; the point © of it's periphery, which at the beginning of the motion fell upon A, leaves an impreffion in the plane of it’s motion, which it continues till the point C ar- rives again at the right line AB. It will ) defcribe a curve ACB, which, from it’s. generation, 1s called a Cycloid, It will be the ordinary cycloid, when the circle fo moves upon the right line AB, as that it fhall meafure out the whole exactly by it’s periphery, after that the point C fhall have paffed from A to B, fo that AB may be equal to the periphery of the fame circle. It will be a prolonged cycloid when the motion is fuch, that the right line AB is longer chan the periphery of the circle; and a contracted cycloid when the fame AB is Morter than the periphery. 7 From the defcription of this curve it plainly follows, that,. drawing from any point the right line MQ. parallel to AB, the intercepted line MP, between the curve and the circle CPD, will have to the arch CP the fame ratio as the line AB has to the whole circle. | rn N, B. The chord ME is omitted. in Fig. 35. : Suppofe SrerT: TI. ANALYTICAL INSTITUTIONS. 37 Suppofe the generating circle to be in the two pofitions EMF, DPC; draw the chords ME, PD. Now, becaufe the arches EM, DP, are equal, the chords EM, DP, will be equal and parallel, and. therefore MP = ED. But, by the nature of the curve, it is AE. EM:: AD. EMF :: AB. EMFE, And in the fame ratio is alfo ED... MF, And MF = PC, ED = MP; there. | fore it will be MP. PC:: AD. EMF :: AB. EMFE. Therefore, if we call \ the right line AB = a, the periphery of the generating circle EMFE = 4, and any areh or abfciffla CP = x, the ordinate PM = y; the equation of the by : è a curve of the cycloid will be x = Having therefore the equation of the curve, in order to find the fubtangent, it’s fluxion will be x = =: and, inftead of x, fubftituting this value in the id ir b ; formula ee + 1b, Whe Dew Lo = = x. Therefore, taking, on the tangent of the circle, PK, (Fig. 34.) which is fuppofed to be drawn, a portion PT equal to the arch of the circle AP, and drawing the right line TM to the point M, it fhall be a tangent to the cycloid in the point M. Now, befides, if the cycloid be the ordinary one; becaufe, in this cafe, we fhall have 6 = a, and therefore y = #, it willbe PM = PT, and the angle PYM = PMT. But the external angle TPQ is double to the angle TMP, and the angles TPA, APQ; are equal, by Euclid, ill. 29 and 32, therefore the angle APQ will be equal to the angle TMP, and therefore the tangent MT is parallel to the chord PA. : | 46. Without the affiftance of the tangent of the curve APB, (Fig. 34.) we may have the fubtangent of the curve AM, taking it in the axis KAB. Make AQ.= #, QP =, the arch AP = s, QM =.=, and Jet the relation of the arch AP to the ordinate QM be exprefied by any equation whatever. Let gm be infinitely near to QM, and MS parallel to AB. It will be MS = x, Sm = %, and the fimilar triangles mSM, MQN, will give us 2.% 3: z.QN = = , a formula for the fubtangent. Inftead of taking for the ordinate QM = x, if we take PM = 4; drawing MR parallel to the litle arch Pp, it will be mR = 4, RS = po = y, and therefore mS = % + y. And the fimilar triangles mSM, MQN, will give us. ut+y.x;siu4ky,,QN = a , another formula for the fubtangent. È 3h ANALYTICAL, INSTITUTIONS, | BOOK II. EXAMPLE I. 47. Let the curve APB be a circle whofe diameter is 27, and let the ratio of PM to the arch PA be that of 4 to 8; that is, let the curve AMC be a cycloid. Make AQ = #, QP = y, QM = 2, the arch AP = s; then drawing mg infinitely near to MQ; MR, parallel to Pp; MS, Po, parallel to AB; it will be m5 = 8, RS = po = y, Pp = 5; and mR, the difference or fluxion of MP, will be % — y. But, becaufe, by the property of the curve, we have MP, to the arch PA, a asatod; in the fame ratio, alfo, will be their differentials wR, pP; and therefore it will be 3 — y.s 114.2; that is, as z—J= 7. Buts= Yxx +yy, and, by the property of the circle, JRE HiT PRE A 4 Leo i jh 72% — OL HAL? DI = A200 — xe. Therefore y = suli. and Wis nS o 214 — ax 20% — XX ì rà whence s = ————-. V2rx — xx Wherefore, thefe values being fubftituted inftead of s and y in the equation Mede . . arx brx — bxx usi : + = 2—yJ, we hall have 3 = a, the differential equation 27K — XX of the cycloid. Therefore, the value of 2, given from the equation, being fubflituted in the ZX GON are. 0 dhe la f ana Milas ap pi be formula for the fubtangent i VEE fhall have QN = aii aioe be Now, if the cycloid be the ordinary one, it will be 4 = 4, and ee PIVA TH — XX ON = = gione s that is, ar —a.V2rx—-xx 1:2.QN;or2r— x.y ja SI se by the property of the circle, it is 27 — w“.y iti y.. erefore it will be y.. x 1! z.QN; that is, QP.QA:: QM .QN. There- fore MN will be parallel to PA. PR SA E X- SECT. IL ANALYTICAL INSTITUTIONS, | 39. EXAMPLE. IT 48. Let the curve APB be a parabola, the equation of which is px = yy. Make AQ*= %, QP =:y, and letthe arch AP = 5s, PM: = w; and the ratio: ot MP to the arch PA be that of ato 5. Therefore it will bemR. Ppiia.d Fhat is, «7.5 a. 6, and therefore — = uw. But, in the parabola, it is- y = Mpx, and J mer da Therefore 5 = SA, And this value being. fubftituted inftead of 5. in the equation — = u, the equation to the curve’ AMC will be See ean = & Wherefore, taking the formula of the fub- uby X « Tani inftead of i and j, it willbeQN = St. But y = Vpa, by the ie QI an ape + pp - bp J p ) property of the curve APB, and — = 4, by the property of the curve AMC;: tangent which is proper to this cafe, and making the fubftitutions: } wherefore QN = ——— 49. From the different manners by which many curves may be generated,. arife different formulas of their fubtangents, though the method of finding. them is alike. It will be enough to fhow it in one, to give an idea of the manner, and of the artifice, which is to be ufed on all other occafions. Where- fore, two curves AQC, BCN, being» given, having a common diameter. TF, whofe tangents can be drawn; let there be another curve MC fuch,. as that the relation of the ordinates. PO; PM, PN, in refpe& of any, point at pleafure, M, may be ex- preffed by any equation whatever ;. and let the tangent MT be required,. at any point M. Let pS be drawn. infimtely nearto PN, and the lines: NS, MR, QO, parallel to AB, and. | I make: Big, 36 40 ANALYTICAL INSTITUTIONS. BOOK If, take PE & PF = 4, known by fuppofition, PQ_ = x, PMz=y,PN= È Becaufe of fimilar triangles QPE, gOQ, it will be QO = — = MR co NS; and, becaufe of the fimilar triangles mRM, MPT, it will be PIT = a È ; Vy a formula for the fubtangent. Now, by differencing the equation of the curve MC, in order to have the value of x, to be fubflituted in this formula, it will - be given by y and 3; but the fubtangent itfelf is not to be had in finite terms. It is to be confidered, then, that the fimilar triangies NSv, NPF, will give us NP.PF::28.SN, thatis, 2.232 + %.SN= + È. (Thatis, & muft 3 have a pofitive fign, if, when x and y increafe, z will increafe allo; and a negative fign, if, when x and y increafe, z will decreafe.) But it is alfo sà tz, 5% : sà ; SN = +; then + — = —, and therefore 2 = + 7. Therefore, in- ftead of 2, putting this value in the fluxional equation of the curve MC, we fhall have the value of x exprefled by y, which, being fubitituted in the formula for the fubtangent A , will make the fluxions to vaniih, and the fub- tangent will be expreffed in finite terms. EXAMPLE T | 50. Let «z = yy be the equation of the' curve MC, the fluxion of which will be zx + «3 = 29/5 and, inftead of 3, fubftituting it’s value + =, i : . ssi . . 2tyy | will become 2% + — = 2yy, and therefore 4 = - =: Wherefore, inftead of x, fubftituting this value in the formulagfor the fubtangent, it will be dite ea ae eg : : PI = fax + ssa tks the curve AQC be a parabola whofe parameter is 6; the curve BCN a circle whofe diameter is AB = 24. If, therefore, the point N falls in the periphery of the firft quadrant beginning at A, in which 2 is pofitive ; the formula of the , when, inftead of yy, we put it’s value xz. Now let fubtangent PT will be mi , and the fubtangent of the circle will be ta avV_ {making AP = g,) and that of the parabola will be 2g = s. Therefore, thefe dui t values of ¢ and s being put in the expreffion "A we fhall have PT = Bag — 499 | 4a — 39° 7 51. But SECTS ti, ANALYTICAL INSTITUTIONS. O 51. But if the point N falls in the periphery of the other quadrant, & will be negative, and the formula of the fubtangent will be PT = == In this cafe, ‘the fubtangent of the circle 1s = 4, and that of the parabola continues 2aq — 97 ee mee to be 29 = s. Therefore, inaking the fubftitution of the values of ¢ and s in t i 8aq — the expreffion =, we [hall have PT = an re the fame as before. | 52. Let AP be denominated as before, AQ being a parabola; it will be PQ = « = Vbg. And BCN being a circle, it will be PN = z= Vv 247 — 99. Then the equation yy = zx of the curve MC will be yy = gy/ 246 — og. And thus, the equation being given by the two co-ordinates AP, PM, the fubtangent PT may be found by the ufual and ordinary formulas 2% » There- Sg differencing the equation yy = 91/24 — dg, it will be w= sini, | Now, multiplying the numerator and denominator of the formula “2 by y, tc will be DI, and fubftituting the refpective values inftead of yy and yy, i will be VÎ = S40 — PT, as before. me TO a 44-94 53. Let the equation of the curve MC be more general, thus, x”z" = Pe the fluxion of which is mz”xx"~* + mx"32""* =m tn xt And, n. Mat n. m—tI imz xx + sn xx LI ae MN inftead of 2, putting it’s value + 3 it will be è ie ; PETRI ARDA i pian ag m+n xX DS all "; and therefore x = TUX Whence PT = Tl Hiwe{ mt E ns xz Ke st, if we put it's value x" inftead of gr. ye _ mtn x sty t* mtn ar o a "0, ee a mer ns X Bw TE See 54. If the two curves AC, BCN, become right lines, in the cafe of the fimple equation xz = yy of the curve MC, it will be one of the Conic Seétions of Apollonius, as is to be feen in Se&. HI. of Vol. I. $ 135. Yc will be an ellipfis, when the ordinate CD falls between the points A and B: an hyperbola, when it falls either on one fide or the other: and laftly, a parabola, when the points A, B, are infinitely diftant one from the other, that is, when one of the Vou. II, G right AZ © RONDE a aes Pa: ae Oe Ae a iy Oe Sige Sv a Bs BOOK IF. right lines AC, BC, is parallel to the diameter. Hence it is manifeft, that, in the {ame circumftances, the fame curves will be conic fections, but of a fuperior degree in infinitum, when the equation to the curve MC fhall be this general Hom wan one, Xx 3. — } ® Fig: 3%» nee gg. If the curve AP be given, pres Ape it’s origin in A, of which we know how dai to draw the tangent; let there be another ig curve CMD fuch, that, from a given point ‘ek. I drawing the right line FMP any how, MLAM the relation of FM to the portion AP may di be exprefled by any equation: we are to find the tangent of the curve CMD. is 8 0 Let PH be a tangent to the curve APB i 4 La in the point P, and let FH be drawn per- pendicular to FP, and Fo infinitely near ; and with centre F let the infinitely little arches MR, PO, be defcribed ; and let MT be the tangent required of the curve CMD. Make Ba eet EE ve FM = y, FP. = a; ard the: arch AP = #.\~ Becaufe, inftead of the Infini» tefimal arches, their right fines may be aflumed, the triangle MRm will be right-angled at R; and, becaufe the angle MmR is not different from the angle TMF, but only by the infinitefimal angle MF, the two triangles MR, TEM, may be confidered as fimilar; and, for the fame reafon, the two tri- angles POp, HFP, are fimilar. Therefore it will be mR. RM :: MF.FT; that is, y. MR ay SET bd me. of FT, it is neceffary to have that of MR firft, which we might have if PO «were known. Now, by the fimilar triangles PFH, POp, it will be PH.FH tr Po PO; that is, 1.511 #. OP = = . And, by the fimilar fectors FPO, So that, to have the value —. FMR, it will be FP. PO :: FM. MR; thatis, x. È ::y.MR = %, Whence Fil. a, the formula for the fubtangent. Now if, inftead of x, we fubftitute it’s value, which may be obtained from the fluxional equation of the curve CMD, we fhall have the fubtangent expreffed in finite terms. SECT. Ey ANALYTICAL “INSTITUTIONS, ee 43 EXAMPLE I. ind 56. Let there be a circle ABCD defcribed with centre H, and radius HA; and whilft— the radius HA, with one end fixed in the centre, moves uniformly round, and with the other extremity A defcribes the periphery ABCD; let the point H move uniformly upon the radius HA, fo that when the radius returns to it’s firft fituation HA, the point H, in the mean time, fhall pafs through the radius, and fhall then be found at A. The point H will then defcribe the curve HEcA, which is called | the Spiral of Archimedes. From the generation | of this curve, it is eafy to perceive that any arch of the circle whatever, as AB, fhall be to the correfponding portion of the radius HE, as the whole circle is to the whole radius, Therefore, making the radius = 7, the periphery of the circle = ¢, the arch AB = x, and the ordi- nate HE = 9s inwilbbe way 2t è. and therefore » = —, an equation to the fpiral, in which the ordinates proceed from the fixed point H. This being premifed, if we would find ET, the tangent of the fpiral; becaufe, in this cafe, FP (Fig. 37.) is the radius HB of the circle, it will be z = r, and the two lines, PH the tangent, and FH the fubtangent, (in the fame Fig. 37,) are in this both perpendicular to the radius HB, (by the nature of the circle, ) and confequently parallel to each other, and alfo equal; whence it will be Wx =,» Lhen,-chf- ry s = #, and therefore the general formula, in this cafe, will be i 5 PORRI . rx ; gia. = — eyam —j the val - being ferencing the equation y = ——, it will b z —; and the value of x o fubftituted in the formula, it will be a2 = HT. Or elfe, putting, inftead of y, it’s value — , it will be = — HT. - Therefore, with centre H, and radius HE =», defcribing the arch EQ; and taking HT equal to the arch EQ, it Mall. be the fubtangent. For, by fimilar fectors HEQ, HBA, it will be HA. AB :: HQ. QE, That is, r.xitiy.QE= =. Ge CI If 4t ANALYTICAL INSTITUTIONS, BOOK IT. . ai ° TH . ° i 74 If, inftead of making the equation y = — It were, in penaral, y. = 9. | i +, —— 3 that is, the periphery to the arch AB, as any power integral or fractional of the radius, to a like power of the ordinate: Then taking the fluxion of the ° 7 <= I * 7 . © è E 772 ) vd MC e equation, it would give us y= _, and ya = i. Then fubftitut- Tr r | giù met! ing this in the formula of the fubtangent gol would be a iis. Bat r mm HONE —; therefore ~— ene oF Spree) Re Go PE 57. We fhall have the formula of the fubtangent more fimple, if the equation of the curve APB were given from the relation of TM to FP. For the fimilar triangles pOP, PFH, will give us PO = —, and the fimilar fe&ors FPO, FMR, will give us MR = as and laftly, the 2% fimilar triangles MRm, TFM, will give us PT a dt... | SOI URI BAAM PLE i. Fig. 39. 58. Let the curve CMD be ite the line APB, which makes no alte- ration, and let APB be a right line, | and let FM, FP, always differ from | each other by the fame quantity, that q is, make the conftant line PM = a, Then willy — z = 2 be the equation of the curve, which is the Conchoid of Nicomedes, whofe pole is the point F, and afymptote AB. Taking the F | fluxions. SECT. II, À AMALYTICAL_.INSTITUTION Se - 4.5. fluxions of the equation, it will be y = 2, and thence the fubtangent FT — BS f Drawing, then, ME parallel ta PA, and MT parallel to PE, MT will be a tangent to the curve in M. For it will be FA = i PES =, and FT yy 59. Any curve AM being given, to the axis EAT of which curve we know how to draw the tangent MH, at any point M; and a point F being given out of the curve, from which let be drawn the right line FPM; if we conceive the right line FPM to revolve about the immoveable point F, making the plane PAM to move upon the right line ET, always parallel to itfelf, the intercepted line PA always continuing the fame : ‘hen the point M, which is the common | interfection of the two lines FM, AM, by this motion will defcribe a curve CMD, the tangent of which is required. | Let the plane PAM move, and, in the firft inftant, let it arrive at an infinitely near pofition pam, and let SRm be drawn parallel to ET. The fimilar triangles MRm, MHT, would give the right line HT, which determines the tangent required, if the fides MR, Rm, were known. Therefore, to obtain them, let us make FP, or Fp = x, FM, or Fm = y, Pp = è, and the known lines: PA = 4, HM =¢, PH =s. It is plain, by the conftruction, that it will be Pp = Aa = Rm = 2; and, by the fimilar triangles FPp, FSm, it will be Ep. Pp: Fm. Sm That is, e. 222 yp. Sm = 2, ob hén SR ca den And, by fimilar triangles MPH, MSR, it will be HP. HM :: RS. RM. That RA (i «MR. pacata Laftly, by the fimilar triangles. MRm, MHT, it will be MR Rm :: MH.HT. Thatis, =". s :: }.HT = i È yaa From the point F draw FE parallel to the tangent MH, and taking HT = PE, draw TM, which fhall be a tangent to the curve at the point M. ‘For, becaufe of fimilar triangles PMH, PFE, it will be PM . PH :: PY. PE; that is, pee west: Ae = PEs n VY. 60. It 46 ANALYTICALAINSTIITEUTILIO ka BOOK Ife . 60. Tr has been already demonftrated, Vol. I. Sect. III. $ 136, that, if the line AM were a right line, the curve CMD would be an hyperbola, which would have ET for one of it's two afymptotes. If AM were a circle with centre P, the curve CMD would be the conchoid of Nicomedes, the pole of which is F, and its afymptote ET. And laftly, if AM were a parabola, the curve CMD would be the companion of the paraboloid of Cartefius, that is, one of the two parabolical conchoids. | Fig. 41, 61. To the diameter AP let there be [ any curve AN, whofe tangent we know how to draw, and a fixed point F out of it; and jet there be another curve CMD fuch, that, drawing, as we pleafe, the right line FMPN from the point F, the relation between FN, FP, FM, may be exprefled by any equation whatever. It is required to find the tangent MT, at any given point M. Through the point F draw HK perpendicular to FN, which meets the «diameter AP produced in K, and the given tangent NH in H. Let FQ be infinitely near FN, and with centre F let the arches MR, Po, NQ; be de- igibedse Maker Pics ai EH oo BIPD EM ey, TIN: = “then ié will be mR = y, po = x, Qu = — 3. And, becaufe of like triangles NQz, tz NFH, it will be NQ= — É. Alfo, becaufe of like fedtors FNQ, FMR, it will be MR = — 2. Laftly, becaufe of like triangles MRm, MFT, it DA willbe FI = + 2 ‘ , the formula required for the fubtangent. But here it 2%) might be fufpe&ed, that, taking the fluxion of the equation of the curve, the value of y to be fubftituted in the formula will be given by x and 2, by which means the fluxions would not vanifh. Yet, however, the fimilar fectors FNQ» FPo, will give us Po = — ey and the fimilar triangles Pop, PFK, will give us the analogy, v.— —-i:a.s. Whence the equation szzv¥ = — /axz, and therefore — 2 = on Therefore, fubftitute the value af y in the formula for the fubtangent, which value is to be obtained from the fluxional | equation of the curve, and then this value inftead of 2; by which the fluxions will vanifb, and we fhall have the fubtangent in finite terms. If \ Seth «— |»; ANALYTICAL INSTITUTION: ay If the line AP were a curve inftead of a right line, drawing the tangent PK, by the fame way of argumentation we fhould find the fame value of the fub- tangent FI. ) : BoA AP LE, 62. Let the curve AN be a circle which paffes through the point F, and is fo pofited, that, from the point F drawing the perpendicular FB (produced). to AP, it may pafs through the centre G of the fame circle; and let PN be always equal to PM: the curve CMD of the foregoing figure, that is, FMA in this, will be the ciffoid of Diocles, the equation of which will be = + y = 2x. Then we (hall have, by taking the fluxion, 2 + y = 2%, or y = 2% — 2; and fubftituting this e I eee yytz value of 7 in the formula — = of the fubtangent, it will be — —_; : By LEB — BBS ee and laftly, putting, inftead of — 2, it’s value mati WS fhall have eee ores, = FT, the fubtangent required. Here it is plain, that if the point M, at which the tangent is required, fhould fall upon the point A ; in this cafe, KH being perpendicular to FA, it would be BW = FP... FM —*FASS FR cer et y- and therefore PI — ce hs eee “a et 63. Perhaps we might find the fubtangent of the ciffoid more fpeedily, by | means of the ufual formula, at § go. For, drawing NE, ML, perpendicular to FB, and making: FB = 24, FL. = x, LM = y; ‘by the property of. the curve FMA, it will be BE = FL = «; and, by the property of the circle, ic will be EN = 2axv—xx ; and the fimilar triangles FLM, FEN, will give FL.LM:: FE.FN, and therefore FL. LM :: EN. EB; that is, x.y: Re 7 x3 i ~~. the equation of the » or yy miti qua the af 2ax — XX + %, whence y = o; 204% — XX curve FMA. Therefore, by taking the fluxions, we fhall have ayy = Caxxk — 23% x 24 — x)" ; and taking the ufual formula sf , and making all the neceflary 8 | 3 ai fubftitutions,, ld 48 AMAL YTICAL INSTITU TAO WK. BOOK IF. 24 — x)? = Lo — 24% — XX i by putting, fabhididene. ill be Oe one gee E i 3 y yy x 3axt — x3 3a £ x » 3 inftead of yy, it’s value 5 A — 64. Let there be two curves ANB, CPD, and a right line FK, in which are three fixed points A, ©, F.. Further, let the curve EMG be fuch, that, drawing through any of it’s points, M, the right line FMN from the given point F, and from the point M the right line MP parallel to FK; the relation of the arch AN to the arch CP fhail be exprefied by any equation at pleafure. It is re- quired to find the tangent of the curve EG at the point M. | Let MT be the tangent required, which meets in T the right line FK, produced if need be, and from the point T let there be drawn TH parallel to FM, and through the point M let be drawn MRK parallel to the tangent in P, and MOH parallel to the tangent in N, and let FwOx be infinitely near to FN. Radke Ni Pie Moena, andthe arches AN =v, GP. x’; and therefore Nu = y, Pp = x. By the fimilar triangles FNz, FMO, it will be C By FN .Na:: FM. MO; that is, ¢ -¥ Heth MS 2 . And, by the fimilar triangles MmR, MTK, and MOm, MHT, it will be MR . #M :: MK. MT, and Mm. MO :: MT. MH; and it will be allo MR. MO :: MK. MH, Tal iS da Li oy es VEE be . Wherefore, by taking the fluxion of the given equation, we fhall have the value of y given by x; and, by making the neceflary fubftitutions, we fhall have MH expreffed in finite terms. Taking, therefore, MH equal to the value now found, and parallel to the tangent in N of the curve ANB, and drawing HT parallel to MF; if from the point M be drawn the right line TM to the point T, it will be a tangent to the curve EMG in the point M. Noi The letter 7 has been put, by miftake, for the letter , in Fig. 43. E ae “SECT. IL | ANALYTLOAL INSTITUTIONS, | 49 EXAMPLE, 65. Let the curve ANB be a acdevatt of a circle, whofe centre is F ; and let CPD of Fig. 43 be the radius APF of Fig. 44, which is perpendicular to the right line FKTB, and let the tangent AR be drawn. Let the radius FA be conceived to revolve equably about the centre F, and, at the fame time, the tan- gent AR to move equably upon AF towards FB, always parallel to itfelf; fo that, when — the radius FA falls upon FB, the tangent AR may coincide with FB. By this motion, the point M, which is the interfeftion of the radius and the tangent, will defcribe the curve AMG, called the Quadratrix of Dinoftratus. It is plain, from the SR of this curve, that the arch AN will be sto the intercepted line AP, as the quadrantal atch AB is to the radius AF. Therefore, making AN = +, AP = mr AD = ¢,. Al sar i wall be 27 or, ow and y = os ; then, fubftituting this value of 7 in the formula ae it will be MH = — ; but, in this cafe, FN is the radius of the circle, and MK = AF | cenare AP; Let pa La U = pix; whence MH _ a ask a aoe, putting, inftead of ax, it’s value ry from the given equation. From the point M raife MH perpendicular to FM, and equal to the arch MQ defcribed with centre F, radius FM, and let HT be drawn parallel to FM ; then MT will be a tangent to the quadratrix in the point M. For, becaufe of fimilar fectors ae FMQ; it willbe FN.NB:: FM. MQ. That i 18).F 0 a Aer MQ — sy = — MH. 66. Let there be two curves BN, FQ, — of which it is known how to draw the tangents, and which have the right line . BA for a common axis, in which are tivo" fixed points A, E. And let there be another curve LM, fuch, that, drawing the right line AMN through any of it’s points M, and with centre “A and radius AM detcribing the arch MG; and from the point G letting fall GQ_perpendi- cular to AG; i the relation ot the fpaces Vou. Il. H ANB, 50 PMWALYTICAL INSTITUTIONS, | | BOOK If. ANB, EFQG, and of the lines AM, AN, OG, may be given by the means of any equation. The tangent of the curve LM is required at the point M. Drawing the rig ht line ATH perpendicular to AMN, let there be another Amn infinitely near to AMN, and the arch mg, and the perpendicular gq. Then, with centre A defcribing the little arch NS, making the given fubtan- pents ELA a: OR A make AM =, a cum pi QG = = and tie | fpaces EGQF = 5, ANB = 4, it will be Ra = = And, becaufe of the fimilar triang] es POR Q0g, It will be 0 = Te uT AA And, by the fimilar triangles HAN, NSw, Pope SN =. Fhe {pace GQqg may be taken for the peo GQ0g, becaufe their acià Og | is an Infinitefimal of the fecond order. Whence it will be GQ ge = 4 = —s. Thus, there- fore, it will be ANa = tAN X-NS = 4ez-— i. Manele tore thefe values. | being fubftituted, inftead of ‘ 4, 5, Î, in the fluxion of the PaoRolsa equation,. we thall have an equation from whence may be deduced the value of 2 given Day) Now, becaufe of fimilar fectors ARM, ANS, it will be MR = on a by the fimilar triangles #RM, MAT, it willbe AT = ayy% he DR oy formula for the fubtangent ; in which, inftead of z, if we fubftitute it’s value- given by y from the equation of the curve, the fluxions will rig a eit and the: fubrangent will be given in finite terms. EXAMPLE. 67. Let the fpace EGQF be double to ABN, that is, s = 2%; then 5 = 27s. But s = — wy, and } = — 443; therefore it will be wy = az, and 2 = ~~. ‘Then the fubtangent is AT = oe a ZS Let the curve BN be a eircle with centre: A, radius AN = c; whence 2 = c; and: Jet the curve FQ _ be an hyperbola with the. equation wy = ff; the fubtangent will be ATE a, that is, the ratio of AM to AF will be conftant. The curve LM (Fig. 46.) will be called, in this cafe, the Logarithmic Spiral. ; và, Here n Vee ti. ANALYT EGAL':IN STITUTIONS. — 5I ti it is ‘manifett, that the curve LM. will IA an inGlile number of i ns before it arrives at the point A; forafmuch as, when the point G (Fig. 45.) coincides with A, the fpace s will be infinite, as may be feen from the Inverfe Method of Fluxions. For then, alfo, the {pace ¢ muft be infinite, which cannot be but after infinite revolutions of the radius AM. LS 68. It remains, laftly, to confider a particular cafe belonging to Tangents. It has been deen that, the reno nase of any curve being x and y, the general x) R : ‘ cere of the Fig Ses will bette or a , according as y or x fupplies the Ala place of the ordinate. WI herefore, the fluxion of the equation of the curve being taken, if from thence we deduce the value of x or y, this value, being fubftituted' in the general formula, will give us a fraction in finite terms, which is the expreffion or value of the fubtangent for any point of the propofed curve. Now, if we defire the fubtangent for any determinate point of the curve, nothing elfe is required to be done, but to fubftitute in this fraction, inflead of » x and y, their values which they have at the point given. But it may fometimes happen, that, by fubftituting, inflead of x or y, a determinate value in the fraction which expreffes the fubtangent, or otherwife, in the ratio of # to y deduced from the fluxional equation af the curve, all the terms in the nu- merator and denominator may vanifh of themfelves, and that there will only. O © SS Ae = = —, and thence, alfo, the fubtangent will be mek from whence, however, we are not to infer that the fubtangent is nothing in this point. For an NERO let us take da curve belonging to this equation 3° — 8ay? — 1207) + 10aayy + 48aaxy + 4aaxx — 64a°x = O) and let y be the abfcifs, and «x the ordinate. Therefore -£ will be the formula for the fub- tangent. Therefore, by taking the fluxion of this equation, we fhall have y vi y — 12447 — 2aax + 1643 cho e I II A ad dhe ila be + y3 — 6ayy — 6aay + Saay + 12045 x 3axyy — 1240xy — 2aaxx + 1643x cai da dai Now, if we would have the fubtangent to that point of the curve, which correfponds to the abfcifs y = 22, it being alfo in this. cafe « =. 24, by’ the given equation ; make the ubficutibhis in the fraction which exprefies the ratio of x to y, and we fhall find it to be 12a3 — 2403 — 403 + 1603 Bai — 2403 — 2403 + 1643 Li - 2403 9 that js, = , becaufe all the terms deny one 6) another ; and therefore the fubtangent i at this point, is —s which informs us of nothisg, although one or more fubtangents may belong to that point, Hz 69. This ANALYTICAL INSTITUTIONS | BOOK If ___ 69. This cafe will always happen, when. R* ever the curve has feveral branches which ‘interfe& one another, and when we would have a tangent at the point of concourfe, And, indeed, the curve NOPQMR (Fig. 47.) of the propofed equation has two fuch © branches, which cut one another in the point G, to which exa&ly correfponds y = 24, OT being the axis of the y’s, and it’s begin- nip atO. Allo, « = 24, taking. the x°8 in the axis OQ, P 7 V O M ey | To give a reafon for this cafe, it is enough to take notice of two things. The firft is, that, at the point of concourfe of the different branches of the curve, feveral roots of the equation become equal to one another. Thus,:as to the propofed equation, in the point G the two values of x are equal, and alfo, two are equal of the four values of y. The fecond is, (what is demonftrated in Des Cartes’s Algebra,) that if an equation which contains equal roots be multiplied, term by term, into any arithmetical progreffion, the produ& will be equal to nothing, and will contain in it fewer by one of the equal roots. And if this produ& be again multiplied by an arithmetical progreflion, the product - will, in like manner, be equal to nothing, and will contain ftill fewer by one of the equal roots, than were contained by the firft product ; that is, fewer by two of the equal roots, than were contained by the firft equation. And thus on fucceflively to that product, which fhall contain only one of the equal roots. If, therefore, any equation of a curve, treating x as variable and y as con- ftant, fhall be multiplied by an arithmetical progreffion which terminates in nothing ; in the cafe of equal roots the produ& fhall be equal to nothing; and it will alfo be fo, if the product be divided by x, which divifion will fucceed when the laft term is multiplied by nothing. The fame thing will obtain alfo. by treating y as variable and x as conftant, and multiplying the equation by fuch an arithmetical progreffion as has nothing, or o, to put under the laft term. ) This being fuppofed, it is eafy to perceive that fuch an operation as thiS performs the very fame thing as taking the fluxion; that is, if it treats x a° variable, and multiplies the equation by an arithmetical progreffion, the fir& term of which is the greatelt exponent of x, and the lat term is nothing, and produces a product multiplied into x. Then, if it treats y as variable, and multiplies the equation by an arithmetical progreffion, the firft term of which is the greatelt exponent of y, and the laft is nothing, or o, and produces a product multiplied into y. Bat, in the cafe of equal roots of x, and in that of | equal roots of y, as well the product multiplied by x, as that by y, are equal ie ; a O i n x \ è to nothing. So that the ratio + = 7 ought to arife, in that pdint wherein two branches of the curve interfe& each other. 3. That. SECT, IT. ANALYTICAL INSTITUTIONS, 53 | That this may be feen more fully, I here fet in order the equation of the propofed curve according to the letter y, and multiply it by an fee pas hes progreflion, the laft term of which is o, gt Bay? — 120%)° + 48aaxy + 400X% Lai + 16aay* — 644°% : 3, 2; I, The produ& will be | 4y* — 24ay? — 24axy* + 3200)° + 48aaxy = o. _ That is, dividing DI 4%» I ym bay" — 6axy + 8aay + 124aax = Os Then I fet the fame equation in order according to the letter x, and mulgoly it by the arithmetical progreffion, the laft term of which is o. 4400x° + 48aayx + y* — 64000% — Bay? — 122))x -+--16a*y’ 2, ts QO, The produ will be Baax* + 48aayx — 64054 — 12ayyx = That ts, dii by 44, 2aax + 1240) — 1623 — 3ayy = 0. This being done, I take the fluxion of the propofed equation, which is 493) — 240)°) — 240%)) — 120)°x + 3240y) + 48aany + 48a%yx + 8a°xx — 640°x = 0; that is, dividing it by 4, and tranfpofing the terms belonging. to 3° — 6ay’ Vino + 82% + 120°x hake: = 309° — 1240y + 244% ch 1643 into x. Noiy here the multiplier of y is the fir® produ& into the arithmetical pro-. greflion, and confequently = o in relation to the point G, in which y has two. equal values. And the multiplier of x is the fecond produ& into its arith+ metical progreffion with it’s figns changed, which does not hinder it being = 0, in relation to the fame point G, in which x has two equal values. Tee. It Po. o will bey xo, MOO. that is, = = in the point G. But, if to multiply any equation by an arithmetical progreffion, or to find it’s fluxion, (which is the fame thing,) bring it to pafs, that, on the NERO | of iG; 54 si ANALYTICAL INSTITUTIONS. BOOK IH, ; . ‘ D . e D tak sia . v of equal roots, that cafe will arife of which we are treating, that is, -— = SG area it alfo brings it to pafs, that, inthe equation derived from thence, there will be one lefs of tl hole equal roots. Wherefore, ifthe equation propofed fave Iwo equal roots when differenced it will have but one of thofe equal roots. And, if the propofed equation have three, by differencing again that which was differenced before, (atfuming as conflant the differences or fluxions x, y,) the equation thence arifing will have only one; and fo on, Therefore, if we affume as conftant the fluxions He Ws: ES well the terms multiplied into x as thofe multiplied into y, will mutually ge irey.. each other, in i fuppofition of fuch a determinate value of wand y; alfo, the terms muli iplied into x and y will deftroy one another. 2 proceeding in this way of operation, equations will be reduced to contain only one of the number of equal roots which ney had. at firft; and therefore, finally differencing the laft, to obtain the ratio of y to x, there can no longèr È ) lo) arife the cafe of — 2 —. x O Therefore I refume the foregoing equation whofe fluxion was found to be sy Cay) — Caxyy — 300°x + Saayy + I2¢axy + 1204)x + 204Xxx — 164% = o. But, becaufe, by fubftituting, inftead of y, it’s value 22, and, inftead of x, it’s correfpondent va value 24, in order to have the tangent at the point G; I find only —- = —: I go on to difference that already differ- enced, taking always for ies the fluxions x, y, and I fhall obtain 399° = 1247)" — 6axy° + 8aay* — Izayyx + 24440)x + 240x* = o. Inftead of y and «, lubtnie their values 24, in relation to the point G; and I find x = +yW8. Then, in the general formula for the fubtangent Lo A putting the values of «x = 24, and x = + yV8, I fhall finally have the a | fubtangent = + Ja? 9 to {peak more properly, the two fubtangents corre- {ponding to the point G, one pofitive, the other equal to it, but negative. If the curve fhall have three equal roots at the point in which the tangent is required, that is, if the curve fhall have three branches which meet one another in that point; becaufe, after the equation has been differenced once, it will {till have two equal roots; it muft be differenced again, that we may have the ratio of yto.x: It will give ts, notwithftanding, by what has been already faid, the ‘ Ni O ; i i ratio. 8 => eg and therefore it will be neceflary to take the difference or Xx fluxion a third time. And, in general, the equation muft be fo often differ- enced as is the number of equal.roots, or the branches of the curve; and from the lat difference muft be obtained the ratio of y to x. And fo many will .be the tangents as are the branches of the curve, which cut one another in that point, Let x oe \ AR ALY ULC SE, INSTITUTIONS. ae > Let the curve be QADH Aba Al, ; whol È equation is at — ayax + by? = 0, and which has three branches QAD, “TAd, LATI, which cut one another in 0 And Ter AP be the axis belonging tov, and AB perpendicular to AP, the axis, belonging to y, and the point “A. their common origin. By differencing the equation, it will be giù ice Zaye — ax) + gui = e; that.18, aif any — 2b | vo ca ae a Bat, if we would have- Avr — 20yx. the tangent at the point A, becaufe there i it is WO, FS Ors it will be 2 > as ia We muit therefore go on to. fecond fluxions, and the equation will be 12xxxx — 20yax — 4axXj + 6dy; iy \ O. x oo | Bat from this we fhall on] obtain. ak =. every term being multi- y é I y Di bi =.95 by fappofition, or by y= 0. Therefore; differencing for the third time, it-will be 24xx3? — ayn” + dota io Here, manne 193 the firî term vanifhes, and. therefore It is ayx° = by*, from whence we have three values of 'y; that is, y = 0, and y — + = , which give ds three ratios: of x to y; that is to fay, three tangents at the point A, One of them will be infinite, which coincides with the axis AP, and ferves for the branch 2AH.. The other, taking any line AS, and desing ST perpendicularly in fuch a manner, as that. it may be ST .SA : «4/83 the lines TA will be tangents in the point A, one of the branch Se sage other of the branch IAd.. ro. The truth of thefe conclufions may alfo be demonftrated after another manner, and; as. they fay, è pofferiori. The differentials of finite equations,; which are found by the foregoing rules of differencing, are not really the com-. plete differentials, ‘the rules giving. us only thofe terms which contain the firft differences, or of one dimenfion only ; and omitting, for brevity-fake, and for greater convenience, the differences of other degrees, ‘or of greater dimenfions : which, by the principles of the calculus, would make thofe terms in which they. are found to be relatively nothing. eda a the SO JT — Bay? — 120xy? + 484°%yx + 40°%x° — + 16a°y° es i, | i it’s fluxion or difference will be 4yÎY — 24ay°y — 12049x — 240%YY + 320°y) + 48aaxy + 48407 + 8aaxx — 64a°x = o. But here, if y be confidered as. increafed by it’s fluxion or difference, and likewife x; and that in the propofed: equation, inftead of y.and it’s powers, we fhould write yy and it’s correfponding. powers; and .fhould do the fame by writing x; a x and it’s powers inftead Py ‘ thofe. 56 PRESSA SI BOOK If, 7 thofe of x; we fhould then have i). terms as they are fet in order in the following Table. I. ; i, aot AL, FX. V. eg E ayy" ot = 82; — 2440) a4ayyy Bays — 120xy° — 24axyY — 1240%)Y =m 124xy° “+ 1640y° — 124)JX — 240)%Y + 48aaxy + 324°y + 16aayy = 0. + 402% + 482° ‘x + 48aaxy — 60445 + 482%xy + 400XxX + 8a7xx — 644°x Now the fum of all thefe columns, excepting the firft, which is the peapeles equation itfelf, will be their complete and entire fluxion. But, becaufe the laft or fifth column is infinitely little in refpe& of the fourth, and the fourth in refpe& of the third, and the third in refpect of the fecond; we affume the fecond column alone for the fluxion of the propofed equation, which compen. dium proceeds from the common rule of differencing. But it can be fo only when the columns after the fecond are abfolutely nothing. If, therefore, a cafe fhall arife, in which the fecond column is abfolutely nothing, the third may not be nothing in refpe& of it, and therefore ought not to be omitted, but will itfelf be the differential of the frft. And the fame DI be faid of the fourth, _ when the fecond and third are nothing; and fo of the refl, But this cafe pre= cifely happens, when we feek the relation of x to y in the propofed equation, in that point in which it is y = 24, and x = 24; becaufe, making the ne- ceffary fubftitutions, we find the fecond column itfelf to be nothing; and therefore we go on to make ufe of the third. And this is exactly the fame thing as to difference the equation twice, as appears from hence. a. ey the fame principles, and after the fame manner, a like cafe may be refolved, which arifes in the conftru&tion of curves, when the ordinate is ex- prefled by a fraction, the denominator and numerator of which become each equal to nothing, when a determinate value is affigned to the abfcifs. Now, to remove this difficulty, it is enough to confider the fraction as if it exprefied the ordinates of two curves, which meet in fome point of their com- mon axis. And becaule, in this point, their ratio cannot be exprefled otherwife than by = , it is neceffary to find what may be their ratio in a point infinitely near it, that is, when they are increafed by an infinitefimal. That is to fay, we muft proceed to differencing the numerator, and then the denominator of the faid fraction, and that once, twice, or oftener, till at lat, REDS, the deter- minate value of the abfcifs in the fraction, it may no longer be —, for the fame reafon mentioned before, concerning the columns of differentials. Let SECT, IL ANALYTICAL INSTITUTIONS. © 57 | PO MS ae A : Let the equation be y = —° ora aax . «Taking x = a, and mak- . i i > 4 lo) 1 b ing the fubftitution, it will be y = = from whence we cannot therefore infer, that when the abfcifs x = 2, the correfponding ordinate will be y = o. For, by differencing the numerator, and then the denominator of the fraction, it x 4 — larxa 3x i x N ax — 203% X 203% — 24 MERONE, = will be y = . Then, dividing both — danxx x a above and below by x, and making « = a, it will be y = +e, ad 403 +403 — ax — aa Let the equation be y = , in which, if we put v¥ = 4, A/ 20A + 2% — % = A ‘ . We i É | it will become y = —.- Wherefore, differencing, firft, the numerator, and 2 . È © 3 i 7 A i sd 3 : 3 er 7 CE then the denominator of the fra&ion, it will be y = 4axe X 403 + 443) <9 | 20 X z2aa+-2xx re — I omitting x, which fhould be ‘in both the numerator and the denominator. But now, in this fraction, if we put x = a, it will be ftill y = — ‘Therefore, proceeding to difference this fecond fra&ion alfo, we fhall have y = I ure grata x 403 + 443) 73 DI RRO TAMA, =» Omitting the x. And now, making x = a, it will be 404 X 24a +-2xx) poo ye 24., Vow II. AMES Ti, oe I SEC. 58 ANALYTICAL INSTITUTIONS, BOOK Ile ode TL, II, The Method of the Maxima and Minima of Quantities. ine _ nar LTR eo, A 72. IN any curve whatever, whofe or» dinates are parallel, if, the abfcifs BC (Pig. 49, 50; §1,..42;) continually. in- creafing, the ordinate CG increafes alfo to a certain point E, after which it decreafes, or is no longer an ordinate of any kind; or, on the contrary, the abfcifs increafing, the ordinate CG goes on continually de- creafing to a certain point E, after which it either increafes, or elfe is no more: In this cafe, the ordinate EF is called a Maximum or a Minimum, In the curve GHF, let EF be the greateft of the ordinates, (Fig. 49.) or the leaft, (Fig. 50.) taking any abfcifs BC, and drawing the ordinate CG; let GA be fuppofed to be a tangent at the point G, and DH to be infinitely near to CG. Make BC = x, CG = y, and drawing GI parallel to BC, it will be GE = CD: =t&, card IT 9... New, becaule “the: triangles ACG, GHI, are fimilar, in Fig. 49, it will be AC. CG :: GE. THs And, becaufe the triangles ATG, GHI, are fimilar, in Fig. 50, it will be AT.TG GI. IH. ‘Fhis being fuppofed,- let the ordinate GC, being always parallel to itfelf, N. B. The letter A is omitted in Fig. go. sECT, MI, AWALYTTORL INSTITUTIONS. 59 riva, | itfelf, be conceived to approach to the x preatelt or leaft ordinate EF. It is plain, * that, as CG approaches to EF, the fub-. tangent AC, or AT, will always become greater and greater; fo that, when CG falls upon EF, the tangent will become paralleli to BC, and confequently the fubtangent will be infinite. Therefore, in this cale, GL | or + we fhall have AC to CG, or AT to TG, mio ton | an infinite ratio, CG full remaining a finite quantity. But, fince it is always ACCO ; Co, or Aa’ TG =! Gr: IH, GI to 10 will alfo have an infinite ratio. Therefore it will be as nothing in refpedt of x, that is, y = o in the point of the greateft or leaft ordinate. Let: the: Curvel:be GHE, (Fig. Br, 52.) ER the leatt of the ordinates; (Fig. 51.) or the greateft (Fig. 52.); therefore, taking any abfcifs BC, and drawing the ordinate CG, the tangent GA, DH infinitely near to CG, and GI parallel to BC ; and making BC = ee ee oe it wil be Ges Cis x, IH = y. Now, becaufe of the fimilar triangles ACG, GIH, it will be (Fig. 51.) NOY CG 33 GI edH andj obecautle of the finan triangles ATG; GIH, it will be (Fig. 52.) AT. TG :: GI.IH. Now, the ordinate CGI always remaining parallel to itfelf, and continually approaching towards the greateft or leaft ordinate, the fubtangent AC or AT will always become lefs and lefs ; fo that, when CG falls upon EF, the tangent will become perpen- dicular to BC, and confequently the fabtangent will be nothing. Therefore, ‘ ain this cafe, we fhall have AC to CG, or AT to TG, in the ratio > of nothing to a finite quantity ; and therefore, GI to IH being in the fame ratio, x ue be nothing in refpect of y, that is, y = oo, in the point of the greateft or leaft ordinate. Wherefore the general formula for the greatelt and leaft ordinate will peli o orcelie We sa 43. Therefore, the equation of the curve being given, of which we would find the greateft or leaft ordinate, we mutt difference it to find the value of the | fraction or ratio at ; then making the fuppofition of y = 0, or elle of x ma, that is, y =: 00, we fhall have the value of the abfcifs x, to which belongs the greateft or leaft, y; and this value, being fubftituted in the propofed equation, will give us the greateft or leaft ordinate, as required. Qaly here we muft obferve, that, in the cale of the fuppofition of. yi nre, that is; Ob aero, x will fupply the place of the ordinate; if in the other fuppofition, it is y that does the fame, That, if neither the firft fuppofition of y = 0, nor the fecond of y = co, will fupply us with any real value of y, it is then to be concluded, that the propofed curve has no greateit or lealt ordinate, | . WE 2 | 94. This ba: ANALYTICAL INSTITUTIONS, BOOK tt 74. This method will help us to acquire a complete and exact idea of curve-lines; to find in what points the tangents are parallel to their conju- gate axes, &c. Befides which, it may be applied to an infinite number ‘of queftions, which we may want to have refolved, whether geometrical or phyfical. Such it would be to inquire, among the infinite parallelopipeds of a given folidity, which is that which has the leaft furface: as it would be to in- quire, among the infinite different ways along which a moving body may pafs, to go from one point to another not in the fame vertical line, which is that which may be defcribed in the fhorteft time, according to fome given law of motion : and many others of a like kind. In fuch queftions muft be found an analytical expreffion of what we would have to be a maximum, or a minimum, which may be put equal to y. Then taking the fluxion, we muft proceed according to the rules here given. {a A RE A AL TIT TE ET sbirri tici EXAMPLE 1: 75. Let there be a curve with this equation 24% — x = yy, and let it be required to know, to what point of the axis, or of the abfcifs n the greatelt ordinate y correfponds, and what that ordinate is.” The fluxional equation of this will be dii — 2xx = 29), that is, — Bees Gee 4 . Making the fuppofition of y = o, the numerator of the fraction ought to be nothing, or 4 — x = o, whence x = a. Therefore the greateft ordinate belongs to that abfcifs which is equal to 4. This value being fubftituted inftead of x in the propofed equation, it will be 244 — aa = yy, that is, y = +a. Therefore the greateft ordinate, pofitive and negative, will be equal to a. Making the fuppofition ory = = co, the denominator of the fraction ought to be nothing, and therefore it will be y = o. Wherefore, fubftituting this value inftead of y in the propofed equation, we fhall have x = 0, and x = 24; which is as much as to fay, that ax = o will be the leaft, and x = 2a the greateft : Or, more properly, that, when « = o, and x = 2a, then y being infinite in re{pect of x, the fubtangent will be nothing, or the tangent will be parallel to the ordinate y. E X- “SECT. DIL ANALYTICAL INSTITUTIONS, EC ae EXAMPLE: IL »6..Let it be the curve of this equation xx — ax = yy. By taking the. 22 Lue ioppondon of y.—- Oo gives here 2y i x = ta. But this value being fubftituted inftead of x in the propofed equation, ‘—y will be found imaginary ; fo that the curve has no ordinate correfponding to fuch an abfcifs, and therefore much lefs will it have a greateft or a leaft. The fuppofition of y = co, that is, of x = 0, will here give y = 0: which de- clares that the tangent will be perpendicular to the axis of the abfcifs x in the point in which y = 0; which correfponds to the two abfcifles x = 0, and x = a. For, inftead of y, fubftituting o in the propofed equation, it will be iM nen ON O and therefore x =O. ande ma. fluxions, it will be — Si EXAMPLE Hk 77. Let the curve belong to this equation 2axy = 4% + axe — dax, ino which x is the abfcifs, and y the ordinate. By taking the fluxions, it will be È i : i : ) ax — ba — a 20%) + 2ayx = 24xX% «= 2bxx, and therefore "= " Li, The fup- pofition of y = o SIV ES Soe 2; and this value being fubftituted in the ey REI 2aay ay? — arby? sì propofed equation, it will be — Jag + 2 that is, yw = @ x oer: a-—b? A bb, andy = + aa — ab, the greateft or leaft ordinate. And, fince we have x = 2, fubftituting this in the value of y, it will be x = + a, na the abfcifs, to which belongs the greateft or leaft ordinate now found. The fuppofition of y = o, or x = 0, gives us ax = o, that is x =o. And making the fubftitution in the propofed equation, it will be a3 = 0; which implies that a given finite quantity is as nothing : fo that the curve will have no other maxima or minima but thofe found in the firft fuppofition, which, be- caufe of the ambiguity of the figns, are two, and thofe equal; one of which is pofitive, and correfponds to the pofitive abfciffes, the other negative, and belongs to the negative abfcifs, 4 aan: 28. ‘This 62 ANGINELCALINSTDICUTIO N BOOK IT. 78. This method, indeed, gives us the maxima and minima, but ambiguoutly ann indiferimi. nately ; ; nor by this can we diftinguifh one from the other. But they become known when the progrefs of the curve is known. But, without fuch knowledge, we may proceed after this manner, Let there be afigned a value to the abtcifs in the piven equation, which is either a little greater or a little lefs than that which anfwers to the greateft or leaft ordinate with which we are concerned, and the value of the ordinate which arifes from thence will determine the queltion. For, if it fhall be greater than that which the method difcovers, the queftion is about a minimum; but, being lefs than that, the queftion is about a maximum, Therefore the curve of this Example will have two leaft ordinates. | EXAMPLE. 1 79. Let the curve MADEAN belong to this equation x3 + y? = axy; make AB x, and BE —=y. By differencing, we fhall have -— = rile a and there- | A 349 — an fore, making the fuppofition of y = o, it will be y = =. Then fubftituting dig. 53. / | this value in the equation, we fhall find taXa. Wherefore, fincé y= 2, . a it will be y == 1494 = BE, the. greateft ordinate in the curve, which correfponds to the abfcifs x = Java = AB. The fuppofition of x = o will give us x = a , and making the fubfitution in the given equation, it will be y = 1442, whence « = ia%4, the greateft AC, to which correfponds Sie OD See which is the tangent in the point D. So. But, before we SIE to more Examples, it will be convenient to panes for a cafe, which fometimes is wontsto happen; and that is, that as well the fuppofition of y = o, as that of y = oo, will furnifh the fame value of the:ordinate, or of the abfeits ; in which cafe, no maximum or minimum will be determined, but only a point of interfeétion or the meeting of two branches of weer, § “NATE Pea I NEPI PONI, | 63 of the curve, And the reafon of this is plain ; forafmuch as, = being equal to a fraction, if from the numerator we derive the fame value of x, for example, as from the denominator, this value or root being fubftituted, will make each of them equal to nothing, and therefore in fuch a. point of the curve it will be o re) . = = >. But it has been already fhown before, at § 69, that when p O . roe ; | —= >» I always: indicates the meeting of two branches of the curves. Therefore, &c. AD ag dh A Pa Sr. Let the curve GFM (Fig. 51.) be the cubic parabola with the equation: yr-a= — Va — 240% cf dite, BAS bl oa, BC = %, CG=y:. Taking. 2AX = 200 the duzions it will be — Sie, dhe fuppolition of 4 0 È oe 3: X 43 — 2aax + ann) 3 PP TR will give us x = 4, and the fuppofition of y = oo will give, in like manner, “x = a. Therefore the curve has a point of interfection F, which correfponds to the abfcifs x = a, and to the leaft ordinate y=4; which is derived. from: the propofed equation, by fubftituting it’s value in the place of x, Let us take the fame: equation, but freed from radicals, that Is) 7? — 399° di + 3aay — dì = aì — 200x + axx. By taking the fluxions, it will be = < — x eine] ZAX 240 399 — bay + gaa — value in the sii equation, we have y == a. The fuppofition of y = co will alfo give y = 4, and therefore x = a; and y = gives us the point Fy. | which is a point of meeting or conta& of the two branches O), Biv) URI ar. the fame time, the leaft ordinate y. . The fuppofition of y =o will give x = 4, and putting this But, if we fhould operate upon the equation y— a = 43 X a—x)F, which. exprefles the branch GF alone, (the other branch FM would be expreffed by Bitvg eh ety Pica) — 243 ui ym a= 43 KX x — a|)?,) we fhould have <- dre The tuppofitioh. ‘3 3.x vt 3 gigi o. informs è us of nothing. The fuppofition of y = co gives us x = a, and therefore y = 4. And the point F, in this cafe, A us. with a. maximum in refpe& of x, and a minimum in refpe of Y.. | 82.1: 64 | ANALYTICAL INSTITUTIONS. ‘BOOK Il. . - . . . . L . 82. I faid that the fuppofition of y = 0, which here gives 24: = o, informs us of nothing, meaning in refpect of finite maxima ; for, taking in the infinite alfo, it fupplies us with two of them. If 2a: = o, it will be then a = 0; and fubftituting this value in the propofed equation, it will be —- = Vue, that ; 3 Lie Ta . is, ¥ = + VE; and therefore x and y are infinite. The maxima are two, one belonging to the branch FG, the other to the branch FM; for, putting a = 0, the equations exprefs them both. This café will generally arife, as often as the fuppafition of y = 0, or of y = ©, exhibits a conftant finite expreffion, or a conftant divifor, to be equal to notbing; which value, being fubftituted in the propofed equation, does not bring us to an imaginary quantity, or to a contradition. And the reafon of it is this, becaufe a finite quantity cannot be taken for nothing, but only in refpect of an infinite quantity. Batata VIE 83. Let the curve belong to the equa- tion x* — 24x + aaxx = y°. Make AB = a4, AC or AP = x, CM or PM = y. Taking the fluxions, it will be y — 493 — 6ax? + 200% — = -_———.. The fuppofi- Pi x 4y° i 4 è; A i a n ; ; . a ii tion of y = o will give us three values LEG PSN Old CUR dS, x Cio Nara. E VASTA A The value # = o, being fubftituted in pica iit A the propofed equation, makes y = o. The value x = 4, makes y = o. The | : value x = ta, makes y — +g. The fuppofition of y = co gives us y = 0; fo that y has the fame value in both the fuppofitions, when x = o and x = y. Whence the points A, B, will be points of meeting of the branches of the curve, and « = 34 = AC will give the greateft ordinate y = + +4 = CM, or Cm. The /ocus of the foregoing Example may be called a double /ocus, which arifes from one or other of the «two fimple formulas, (ax — xx = yy to the circle, and xx — ax = yy to the hyperbola,) being raifed to it’s fquare. Whence it would not be fufficient to reduce the equation to a fimple circle, or to a fimple hyperbola ; but it will be neceflary to have a view to the complication of the two /oct or curves with each other. | pe % SECT. Ill. ANALYTICAL INSTITUTIONS. 65 EXAMPLE VII 84. Let it be the curve of Fig. 55, the aax — 2axx + x3 equation of which is yy = ne . Make. AP, iv, PM ay, AD =.24 : . ; 3 + 40°x+ 4ax® — x3 The fluxions will be fi = ;gx i x y X2a~—x? | . Y _ @—gaax + 4axx — x3 that is, — = tto kt aa X feX 2a—a I proceed, I fhall here obferve that both the numerator and the denominator of | the fraction are divifible by 2 — x; there- fore, in the fuppofition of y = o, and in that of y = co, we fhall have a —- x = 0, or x =a. And this, being fubftituted, will give y = 0, and therefore the curve will have a node in the axis at the point B, making AB = a. The fup- . Before GA = 3ax | xx 2a—% xX rana pofition of y = o will give x = Pil, But the value « = srt wi cannot be of ufe, becaufe, being fubftituted in the propofed equation, it makes the ordinate imaginary ; and this, in general, is imaginary, when x is affumed greater than 24, as may be plainly feen, Wherefore, fubftituting the other Valucs?x= se = oS, it gives y = + due i Making, then, AP = Etat PM, Pm, will be the greateft ordinates, one pofitive, the other 2 negative ; as above. Therefore, making tbe divifion, it will be = = . The fuppofition of y = co will give x = o, and x = 2a. Thefe values being fubftituted in the propofed equation, we fhall have y = O Any es that is, taking x = o, or in the point A, the tangent wiil be parallel to the ordinate PM. And taking x = 26 = AD, the ordinate will be Infinite, that is, will become an alymptote to the curve, in relpet of the branches Bri, DI N. B. By miftake of. the Wood cutter, a Roman M has béen put in the lower part of Fig: 56; inftead of an Italic m, i ba LI, Kia | EX 66 ANALYTICAL INSTITUTIONS, BOOK Il. Box A MP feeb VAL. 8s. Let the curve be the conchoid with’ the equation yy = aaxn — xt + 2aaba — 2b03 — bbxx + aabb io NRE ARE SE SATTA IMRE ONCE PIRA taiaheslgasiieaniprialetinianstianiaaiiaisiibin. an y Xx . Taking the fluxions, it will be II cesso e ao + sal aaza — 24 + 2aaba — 2bx3 — ba* + ab confidered three cafes of this curve. The firft is, when 4 = d. The fecond, È when @ is lefs than 4, The third, when d Fig. 56. aa is greater than 4. As to the firft cafe, the curve will be that of Fig. 56, and the equa- a4 + 203% — 2043 — 24 . In Vol. I. $ 239, I have already Loawy = pe apro Making a n, LN VI and, taking the fluxions, it is — = x — «4 — 0x3 — a3x — at of Ii of y = o will give the numerator equal to P A nothing, that is, x + aX + 43 = 0; | and therefore x = — a, which value, fub- ftituted in the equation of the curve, gives y = o. The fuppofition of ¥ = co » Thefuppofition gives the denominator equal to nothing, that is, “x + 4 x 22— xt = 0, and therefore x = 0, x = — 2, and x = 4. But the value x = — a was alfo found in the fuppofition of y = o. Therefore, when it is x = — a, that is, taking GP = a, the curve will have a point P, where two branches meet each other. The value x = a, being fubftituted in the equation, will give us y = 0; and therefore the fame x will be = 4 = GA, to which correfponds yes The value x = o, being fubftituted, will give y = co . Lherefore; through the point G, where « = o, if a line be drawn parallel to the ordinates, it will touch the curve at an infinite diftance, that is, it will be an afymptote. As ‘SECT: cI. — ANALYTICAL INSTITUTIONS, 67 bg As to the other two cafes, Fig. 59; 98. epr M Let:GA = GK = 2, GP = 4, and the xB reft as above. The fuppofition of y = o A Will give — at — dx} — aabx — aabb bi 'o;sthatissvti x — x? © gab 20, E " and therefore x = — 4, x = Y— aab. EDI e cin LD The fuppofition of y = co, will give KS ax ome 4° + 20° bd n 2hx8 bu + ah? mt 0, that is, xxVx+5 x aa— xx =0;. more thence x = 040 9, x | a, The value x = +. 2, whichis the . fecond cafe, being fubftituted in the equation, makes y = o, and is exhi- bited by both the fuppofitions. Therefore (Fig. 57.) taking GP on the negative fide, and equal to — è, the point P fhall be a meeting or an interfection of two branches of the curve. The fame value. « = —d, being fubftituted in the equa- : b n tion of the curve a pS aa — xx, | in the third cafe, gives the radical nega» tive, becaufe of d greater than a, and therefore the curve is imaginary, and of no ufe. i The value « = Y— aad, fubftituted in the equation of the curve, gives us Mi — aa — bb x 3/ab5 + Zab — aab + 3abb sh haa! ‘ ‘ yore ra ae which is therefore imaginary when 2 is greater than 4, (Fig. 58.) and therefore, in like manner, ferves to no purpofe in this third cafe. But it gives y real when 4 is lefs than 4; and there- fore, (Fig. 57.) making GI = #— aab, IN will be the greatelt ordinate, or y, as above. The value x = o here gives y = co, that is, an afymptote. The value x = + @ gives y = 0; that is, the tangent in the points A, K, is parallel to the ordinate. 68 ANALYTICAL INSTITUTIONS BOOK If. PMA NRT, Bx. ‘ cycloid. 86. Let AMF be half the contracted Make AB’ 22, BE =, AP = «a, PM = 3, the femiperiphery ANB = c; the arch AN = g; it will be PN = v/2ax-xx, NM = 2 — Y/20x—x%; and, by the property of the curve, it is ANB .BF:: AN.NM; that is, c.d: pedi ev , > defierciore a = 2 — Ss) BS /2ax—«e. By differencing, it 1s L= AK n KX Fred Now, drawing mp infinitely near to MP, it will be Nz = Max — xx sis ax ; aoe! ; : dio masi Whence, making the fubftitution in the equation, we fhall 24% — XX bo ab -4- ac — cx ‘ È ; : : have — = ===. The fuppolition of è = o will give here x = A Cc V 24% = NI — + a. Therefore, if H be the centre of a circle, taking HE equal to the. fourth proportional of the femiperiphery ANB, of the right line BF, and of the radius ; the correfponding ordinate will be the greateft, as was required. _ The fuppofition of & = co gives us x = 0, and # = 24; which is as much as to fay, that, in the points A, F, the tangent will be parallel to the ordinates. PROBLEM I. 84, A rectangle ADCB being given, the Îeaft right ONE DI © line QH is required, which can be drawn through the de | | point C in the angle QAH. ~ \ . | Make AB =a, BC = 4, BH = «; it will be | CH = 4865 + xx; and, becaufe of the fimilar triangles HBC, HAQ, we fhall have HB. HC :: HA. HQ; that is, x, 4/00 Fax itix + a, HO EV bb + en. Wherefore, SECT. 11%, ANALYTICAL INSTITUTIONS 69 Wherefore, fuppofing HQ = y, as if it were the ordinate of a curve, we fhall _ a ba oO i Pg RR y 2° #3 abb have y = vd + xx, Ags Py. differencing, it will be T= ar: The fuppofition of y = o will give a = Wabb; and therefore, making BH = Wabb, and drawing HCQ; it will be the leaf line, as required. The fuppofition of y = co will give x = 7 — 25, and « = o, which anfwers no purpofe ; it not being meant that the right line drawn through the point C, which, in this cafe, would be BC infinitely produced, fhould be a maximum, for that reafon becaufe infinite. Wherefore, in fuch cafes as thefe, it will be fufficient to difference that expreffion, which we would have to be a maximum or minimum, and afterwards to fuppofe the numerator equal to nothing, and then the denominator. | | : PROBLEM Il Rit... | 88. The right line AB being divided into three given parts, AC, CF, FB, the Bi E 43 point E is required, in which the middle | ortion CF is to be divided, fo that the rectangle AE x EB to the retangle CE x EF, may have the leaft poffible ratio. i Tg I | Make AG c= 2 CE = 7,,GB.= gi and CER x; (en AR =.0 + 4, Ax Bb FBmec— — b—«; herefore th to wi e ——— = EBs oe ws EE = è x; and th e the ratio will b CE x EF Km AN — NA . las h . . Cee , which muft be a minimum, The fluxion, therefore, will be Comm XX i x— ann = bux + 2acK—abe ; ¥ | aks cenare = Niet eee x x; and making the numerator equal to nothing, we bx — ax) — A alabec — abbc — aah a ‘ fhall have x = med vee cal cc . One of the values is pofi- tive, which gives the point required, E, from C towards B. The other is negative, which would give us the point E, from C towards A. Making the denominator equal to nothing, we fhall have « = o, and x = 4, in which two cafes the ratio of the rectangles will be a maximum ; for, taking # = o, the point E falls in C; and taking x = 5, the point E falls in F ; and therefore, in each cafe, the rectangle CE x EF is nothing. PRO. 70 ANALYTICAL INSTITUTIONS BOOK II, - PERSO Bi ENT. 89. The given right line AB is to be fo cut in the point C, as that the produ& ACg x CB fhall be the greateft of all fuch produéts. Wake ab, AC = mena lx, Therefore AGgi4/ CB = axx — 43. The differential will be 2anx — 3xxx, which, compared to nothing, WH GENE ea, ANG — O, NEI ton, takine” AO @ oo the bre. du& will be the greateft poflible ; and taking x =o, the produét will be a kind of minimum, becaufe it will be nothing, the point C falling in A. The differential not being a fraction, the other ufual fuppofition cannot take place, of the denominator being made equal to nothing. But if we will confider the expreflion of the product axx — x* as an ordinate of a curve, by the laws of homogeneity that product may be divided by a conftant plane, and thus the differential will be a fraction with a conftant denominator. But that conftane quantity can never be nothing, but only relatively in refpe& of x being affumed infinite ; and furely then the product mult be a maximum, when it is AC = # wan Oe I faid that the product ACg x CB is a maximum, when it is AC = 3a; ? Axx — x3 Fot eG For all the ordinates between A and B are lefs than that which correfponds to the abfcifs x = 34. The other value, x = o, being fubftituted, it will be y = o, from whence it may be concluded, that this value will be of no ufe. which will be plainly feen by defcribing the curve of the equation oe go. In the foregoing Problem, and in all others of a like nature, this method may be made ufe of to difcover, whether the queftions propofed are concerning a maximum or a minimum. PROBLEM IV. gt. Among all the parallelopipeds that are equal to a given cube, and of which one fide is given ; it is required to find that which has the leaft furface. Let the given cube be a’, and the known fide of the parallelopiped = 4. ai Let one of the fides fought be «, and then the third will be 7—, becaufe the | 9 product ee”, ae nl $ heh SECT. III AWALYTICAL INSTITUTIONS, 91 product of the three makes the Biven cube a, The products of the fides, taken two and two, that 1s, dx, = ; and — , form the three planes which are half the SIBA of the parallelo piped, Di therefore the fum of thefe, that is, dx HE e + da, muft be the minimum required. ‘Therefore, taking the a3x bxx — as. ; me 3 fluxions, we fhall have 4x — at The fuppofition of the nu- ® s 3 î -merator equal to nothing gives x = V Là Therefore the three fides of the ‘required parallelopiped will be 4, “—- È , and de ae ees =: Therefore by — pe. b the two fides dalla will be equal. The fuppofition of the denominator, being equal to nothing, ferves to no purpofe ; for then « = o, which contra- dicts the Problem. | If we would have a Sdtallelopined with the conditions affigned, but without | afluming any fide as given; making one fide = x, the two others will be equal, aid each = W =. The fum of the three fides or planes, which is to be a s e © È * a3 | a3 Ld e es pa 3 minimum, will be 2xV — + —, which, by differencing, is —— — ©; or Pi x a3 gu. ~ Î È ti xe cnc si - x as alzi — Bx N{ — thus, ——_ e Here, making the numerator equal to nothing, we say — fhall have x = 2, and, in like manner, the other two fides will Db ag fo that the cube itfelf will be the parallelopiped Du PROBLEM V. 92. Among the infinite cones s that may be inferibed in a fphere, to determine that whofe convex fuperficies is the greateit ; the bafe being excluded. In 72 , ANALYTICAL INSTITUTIONS. BOOK II. Fig. 62. Ep : In the femicircle ABD let there be the triangles ABC, AEH, and let a femicircle revolve about it’s diameter AD. At the fame time that it defcribes a fphere, the triangles will defcribe fo many cones. But, as it is demonttrated by Archimedes, that the re fuperficies of the infcribed cones will be to A Hs er eaen “other as the crectangles; AF, <_KEh. | AB x BC; the queftion is reduced to this, to determine fuch a point C in the diameter AD, that the produ& AB x BC may be a maximum. Therefore make AC = x, AD =a; by the property of the circle, it will be CB = Vax — xx, AB = Wax, and AB x BC = Vax x Vax — xe = Vaaxx — ax. Therefore, taking the fluxions, we (hall have 2202348" a ‘ hg - 2VAUKK — ARI And making the numerator equal to nothing, it will be x = 24, and x = o. Making the denominator = o, it will be x = 4, and x = 0, Taking, there- . fore, AC = sAD,; the-fuperficies.of.ithe cone defcribed: by the triangle ABC will be the greateft, as required. ‘The other two values x = 0, and x =a, can be of no ufe in this Problem, as is evident. LI O n TT sae ea PROBLEM I 93. The angle FDG being given, and the point A being given in pofition, to find the leaft right line, which, in the given angle, can pafs through the point A. Let CB be the line required, and let AQ be drawn perpendicular to FD, FAP perpendicular to DG, and CK perpendi. cular to FP. Becaufe the angle FDG is given, and the angle FPD is a right one, the angle AFQ will be known. But the | point A is alfo given in pofition; then i the lines QA, QF, FA, QD, will alfo be known. Therefore make QF = a4, QA= QD = 4; and QC = x. Therefore it willbe FA = Yaa + ce, CA = Wee + xx, FD = 6 + a, and FC = @— x But, becaufe of fimilar triangles FAQ, FDP, it will be FA 4 © SECT. HI, AN ALY TICA By i NSTITUTIONS. oe #7 a+ ab ab — cc FA .FQ: FD.FP. Veri EP, = e: and AP = e Now, becaufe of fimilar triangles Behe Abe, itt will be AK. CA :: AP .AB. Ia Therefore AB = Bo aiid thence CB = Vee + 4 KK + cod ax ab — ce e ax V cc Hy which i is to be a minimum. Therefore, taking the fluxions, Gu è a ax — a x ab =e x co + 88 Sa Saale ee Ea 3 And Vee + ox ; Cd X co} aa , putting the numerator = =o, (frft reducing to a common denominator,) it will tas 14 Cipe oe ee a o, hich isa folid equation. ed : ; ts ; it will be 20" x be #° + canone ae # # © é To conftru& it, I take the equation to the parabola xx = ay ; making the | | } fubftitution, it will be ay 4 sc + i dia. 0, a locus to the aa. da. di hyperbola between it’s afym ee | This eee on the right line QD i is taken Qu = na = , and drawing the fight line MN = “£ from the point M, ‘and parallel to AQ; NS is en | parallel to QD, a do the afymptotes NS, NT, the hyperbola HOV 1s 2bc* + aabcc — act defcribed with the conftant rectangle = . And, on the right line QF, from the point Q let the x’s be taken, and the y’s perpendicular to them. ‘Then, with the axis AQ; vertex Q; and parameter = 4, let the parabola a QO of the equation xx = ay be defcribed.” From the point O, in which the parabola cuts the hyperbola, let OC be drawn parallel to AQ; and from the - point C let the right line CAB be drawn A the POI A. This hall be the minimum required. And, ‘indeed, BI the stiro it iS NS= Sg Ap di SO=y + hela ada And, by the property of the hyperbola, It eS to i NS. x SO, equal to the conftant rectangle. Therefore ay + a= + << + : a = si aa But CO = gd = so s by aa of the ver Therefore, inftead of beca © bee ct abi tutine thi al l La bite ee ATI Da a, i e this value, we fhall have i ora Io x 1 i 2CCxXX bee LA SIN ; “that j Is, 4° +. —— E — | —_— hit = o, which is the very equation from whence the sue of x was to be derived. Therefore, &c. <. Vox. DI a: da ha | 1 have e gd ANALYTICAL INSTITUTIONS: | BOOK 17, I bave here made the fuppofition, that the numerator of the fra&ion, which expreffes the minimum, îs to be nothing. The other fuppofition, that che deno- minator muft be nothing, will give cc + ax\* x “cc + xx = o, that is, Nec + kx = 0, € + az =O. But Wee + xx =o givesusx=V— cc, 2? ns ita È ? S ta G which is imaginary, and therefore of no ufe. cc + ax = 0 givesusa = — £, But, taking Qe=x = — £, and drawing Ac, the triangle QAc will be fimilar to the triangle QFA, or PFD, and therefore the angle QcA will be equal to the angle FDP. Whence cA will be parallel to DP; which is as much as to fay, that a line drawn from the point c, and through the point A in the given angle FDG, will be infinite, which is a kind of maximum. It may be fhown {till in a fhorter manner, that the right line here fought will be infinite. For, in the expreffion Vcc + xx + oo “oc + ax = CB, a di Cw, 10) ° CEO: È e, inftead of «, if we fubftitute it’s value — wi the denominator becomes no- thing, and therefore the line is infinite, SECT 1: Of Points of Contrary Flexure, and of Regreffion. 94. In Se&. VI. Vol. I. it has been faid already, what are Contrary Flexures and Regreflions of Curves. Suppofing, therefore, that to be already known, let ADEM. be a curve whofe ordinates are. parallel, and which in E has a contrary flexure or regreffion. ‘Taking any abfcifs, AB = x, and it’s ordinate BD = y, and drawing CF parallel and indefinitely near to BD; it is plain, that, affuming x =BC as conftant, that, as the abfcifs AB = « conti- nually increafes, the fluxion GF of the ordi- 9 nate Pi PO è eteri DIO = ; O FS < E TERA IT AT E a TSO NEO RIA IIS a ee LIDIA ILE * SECT. IV. ANALTITIEBAL INSTITUTIONS, 75 aa 5 | , nate BD, that is, y, will always become lefs and lefs, till the ordinate becomes HE, which correfponds to the point of contrary flexure or of regreffion : after which point, in both cafes, the fluxion y will go on continually increafing. Therefore, in the point of contrary flexure or regreffion, y will be a minimum, Whence, by the Method of Maxima and Minima, j = o, or elie ¥ = ©, will be the formula of contrary flexure or regreffion. a | Fig. 65. » Ù ‘ eS) ; ‘ * ‘ » < Ù È nf i fj : ; If the curve fhall be firft convex, and afterwards concave to the axis AH; the abfcifs increafing continus'ly, the fluxion or difference of the ordinate will increafe to the point E of contrary flexure or regreflion, after which it will go on decreafing. There- fore, in this point, y isa maximum, and, for that reafon, we may put $ = o, or elle A, net OO : i AT DOCH The fame thing may alfo be inferred from this confideration, that, in a curve firft concave towards it’s axis, the fecond fluxion of the ordinate y, that is, ¥, is negative to the point E of regreffion or contrary flexure, after which it becomes pofitive. And, in curves that are firft convex, that fecond fluxion is pofitive as far as the point E, after which it becomes negative. But no quantity from pofitive can become negative, or from negative can become pofitive, but it muft pafs through either nothing or infinite. Therefore, in the point E of regreffion or contrary flexure, it ought to be j = o, or elfe j = cc. Let the right line DT (Fig. 64.) be a tangent in the point D to the curve AEM, which is firft concave towards the axis; and alfo, the right line EP at the point E. As the abfcifs AB increafes, the line AT, intercepted between ‘the tangent and the origin of the abfcifs will always increafe fo far till the point B falls in H, after which, in the cafe of contrary flexure, the abfcifs ftill in- creafing, that intercepted line will decreafe. Therefore, in the point E of contrary flexure, that intercepted line AP = 2+ — x ought to be a maximum. Wherefore, by differencing, taking x for conftant, it will be PT, equal to nothing, or to infinite; that is, by reducing, and dividing by — yx, and multiplying by yy, it will be, finally, j = 0, or #== co. In cafe that the point E be a point of regreflion, if the intercepted line AT increafe, the abfcifs AB will alfo increafe, till the point T falls in P, and the abfcifs fhall be AH ; beyond which point T the abfcifs will go on decreafing. Therefore AH will be a maximum, and it’s difference will be equal to nothing, or infinite. Therefore, relatively to fuch a difference, the difference of AP will be infinite, ‘or nothing. Therefore ¥ = cc, or j = 0, as before. 76 ANALYTICAL INSTITUTIONS: BOOK If; If the curve be firft convex to the axis; ‘the intercepted line AT will be = x. — È, : Sai e so ery xy ; À and the difference n AI that 18, sa : es : Ail gut: therefore; dividing by yx, and’ multiplying by yy, we fhall have neither more nor lefs than j = o, or elle ¥ = co. A *: DR os rear ees : se dr vezzo ape O SR e II PRI isa une ia pe SENTE: FER COR A RICO et E SERA REI SOR SEME SRI) A ON ii SE RARE oe na IL, NOOO SOLITI oe ser In the curve DEM, the origin of the Fig. 66%. abfciffes x being A, and E the point of con-. D - trary flexure, the intercepted line AP will be- equal to AH + HP. But, in this cafe, the fubtangent HP is negative, that is, — A € Therefore-it will be AP —xa 3 Hence we fee, that in no cafe the intercepted line. A. Apes, FP ap can be xa + fo ur: Fare oS te ie Bey n BIRRE, SER RE ee “ALII e NA Mer eT LR SIEMPRE La PAIR RA TEA Ie LI I ioe IE I TINI 95. The formula here found will ferve for curves which have parallel ordi-. nates, or fuch as are referred to an axis or diameter. But it is different in. curves that are referred to a focus.. Let the curve be ADE, (Fig. 67, 68:) it’s focus Q» from whence the ordinates QD: proceed ; and let Qd be infinitely near to. QD. Draw QT perpendicular to QD, and. and Q? perpendicular to Qd Draw DT a È tangent to the curve in the point D, and ds i a tangent in the point di Let Qé (pro-. duced if need be,) meet DT in the pointo. = Now it is plain, that, as the ordinates ins. i creafe, if the curve be concave.towards the - { focus Q, (Fig. 67.) Qi will be. greater ‘ than QT. But, if the curve. be convex. : towards the focus Q;.(Fig. 68.) Q will be i lefs than QT. ‘Therefore, as the curve : | changes from being concave. to convex, or vice verfd, that is, in the point of contrary. flexure or regreffion, the line or quantity 07, , È from, i SOOT Te. > AUTOR LT INSITE Tio wa 77 | fore muft pafs through nothing or infinite, Wherefore, make QD = y, DM = x, and with centre Q let the infini-. tefimal arches DM, TH, be defcribed.. The two triangles dMD, dQT, will: Be fimilar, as alfo, 4Qo, THo, and therefore it will be ZM-. MD :: dQ. (or. DO). QT. Thatis, j.x:ry.QT = 22. But the two feétors DQM; TQH, are alfo fimilar ; whence QD.DM:: EL bat is, gx se ‘22. TH = È. And, becaule of the fimilar triangles 406, THo, it will be y (or DQ) . Qo (or QT) :: TH. Ho. That is,.y. 22 n 22. Hos Sy. dQ_(or DQ) . Qe (or QT) | ee eet Stes _. from being pofitive, ought to become negative, or the contrary, and there» ee | ‘ È sid ‘ oe di ; 3 pi ; ae ayy — yxy È x But He (Fig, 67.) is the difference of QI, that is, He = oa ae taking x È "a } n È 55 . as vi ad o ù " for conftant. Therefore to = *H + Ho = DITE, which. mu be- ‘equal to 0, or to co. And therefore, alfo, multiplying by yy and dividing: by x, it will be yy — yj + xx, equal'to nothing;. or infinite. = +9 — 43° yy i Therefore, dividing by — x,,and multiplying by yy, it will be xx + JJ. — yi. | equal to 0, or to o.. I i In Fig. 68, the line of becomes negative, and therefore = Fig. 69. - Wherefore, if any curve be referred to a focus Qì MY whofe ordinates are QB = y, and the little arches. BE =x, and ‘fhall have a contrary flexure or’ re-. ereffion ; the general formula to determine it will be. YUE ee 0,00 ma Oa: Here, .if we fuppofe y. infinite, the two firft terms. of the formula will be nothing in re{pect of the third,. and therefore it will be — yy, equal to nothing, or Infinity ; and dividing by — y; we hall have j =o,. HE | or j = co; which is the formula of the firft cafe of curves referred ‘to a diameter, as it ought to:be. For, fuppofing y infinite, the ordinates become parallel to one another. di 96. The nature. of a curve being given by means of an equation, and x. being fuppofed conftant ; by differencing twice, if the curve be algebraical, or once, if it be a differential of the firf degree, that we may have the value of j exprefled by .x ; this, compared to o or co, will give thofe values of the abfcifs x3 to which will correfpond that ordinate y, which meets the curve in the points of contrary flexure or regreflion. . Wherefore, if thofe values be fubftituted in. | | the. ng: ANALYTICAL INSTITUTIONS, BOOK IX the equation of the curve inftead of x, we fhall have y either real or imaginary. If y be imaginary, or fhall involve a contradiction, then the curve will have no fuch points. 97. To diftinguifh the points of contrary flexure from thofe of regreffion, becaufe this method gives us each of them indifcriminately, it will be fufficient to fee the progrefs of the curve, by taking an ordinate very near. And this will afford light enough to remove any doubt about it, | 98. Curves may have another kind of regreflion, different from this which ‘bas been confidered. And that is, when the curve returns backwards towards it’s origin, turning it’s cavity the fame way as it did before it’s regreffion. After I have firft treated on the Radii of Curvature, I fhall give a general formula, alfo, for regreflions of this fecond fort, at the end of the following Section, ERAMVIE LL 99 - ne, Fig. 51. | | 99. Let there be a cubic parabola with | the equation y= 4 + Va? — 244x + axxs which, in $ 81, has been found to have a point of interfection. Now, by differ- : x ; — 2aax + 2axi encing, it will be y = — nei, 3Xa—2aax+axz)$ and differencing again, taking x conftant, i 2axx —r_—_——_—————-. The gxai-240r+axx\7 fuppofition of 7 = o will give us — 2axx z= 0, which is of no ufe; making, therefore, the fuppofition of j = ce, it will be 9 x 4° — 244% + axx\® = o, that is, aa — 24x + xx = o, and therefore x =a. This value being fubftituted inftead of x in the propofed equation, it will be y = a, and therefore the curve has a contrary flexure, or regreffion, which correfponds to the abfcifs x = a, to which belongs the ordinate y = a, it willbe j= — CD iE. A And, becaufe we know otherwife, that this is alfo a point of interfection; it cannot therefore be a point of contrary flexure, but muft be a regreflion. In È 5 ak È x a A seat teens (pico È SLA POR pera = pr NR TNR TARE RE IE OE Te gg ot ROOD e Regi eee re ee Sara, ae TIR TIR a Te a pe Si oi eso lei nia rissa oe ce IRE E ite i eine. + di ae rt tenace > VIa a i ee I Bach. IV. . ANALYTICAL INSTITUTLONS. 79. Vele, Joe a ee In the. fame. cubic parabola, taking the SL eae -abfcifs: AB = «x from the vertex A, and the ordinate BC = y; the equation is axx=y?, the fluxion of which is 24xx = 3yyy. And taking the fluxions again, making x con- — byyy + 2axx ile. galt the equation, it Is 3yy = 3x¥%cax, and, by the firlt differencing, j = 2. There 3h aan” ftant, it will be j= But, by A È \ ia i : . (a 200% Li fore, making the fubfticutions, it will be 7 = dara The fuppofition of j = o has no ufe. The fuppofition of j = co will give ox¥aax = o, that is, x = 0; which value, being fubftituted in the equation,. gives y = o. Therefore the curve has a regreffion at the vertex A, EXAMPLE Il 100. Let the curve be DFM, commonly called the Witch, the equation of which is y = a/ ——, AB = x, BF = y, AD=a; è aa =; and | 24M ax — N taking x conftant, and differencing again, it by differencing, y = — » 308xx — 40AxX% MRO eee ar © 4x X ai xx os The fappofition of ¥ = o will give 30° — 4aax = o, that is, x = 34; _ which value, being fubftituted in the equation of the curve, gives y = a7 “ Whence, taking AB = 4a, the ordinate BF = av + will meet the curve in the point F, which will be a contrary flexure. The fuppofition of j = oo gives us 4% X ax — XA E = o, that is, x = o, and x = 4. The firft value fubfti- tuted in the equation makes y = co, the fecond, y = o, But neither the one | nor the other cafe infer a contrary flexure, but only that the afymptote AQ; as alfo the tangent in the point D, is parallel to the ordinates, i EX. 80 ANALYTICAL INSTITUTIONS. BOOK il. - : EXAMPLE. UL 101. Let AMF (Fig. 72, 73, 74.) dea arx + brx — bad cycloid with the equation = ———, BM aren $ 47. By differencing, it will be 2 = ATE — arr — brr oe KOEN * bX 2ra—wxa\e The fuppofition of % = o will give Arg — brr — arr = o, that is, x = # + 4 . If a be greater than 4, it will be the protracted cycloid. Whence, taking CE trom the centre, @nd equal to the fourth proportional of BF, the femicircle, and the radius, and drawing the ordinate ED; (Fig. 73.) it.will meet the curve in the point of contrary flexure D. If 4 Le lefs than 2, (Fig. 74.) the cycloid will be conumeted, But when 2 < d; the’ hae exert n will be greater than 27, that is, greater than AB, in which cafe the ordinates are imaginary; becaufe there is no part of the curve under the point F. Therefore the curve has no point of con- trary flexure or regreffion. If it be 4 = 4, it will be the common cycloid, (Fig. 72.) and therefore x = 7 + — = ar = AB, and y = BE ; which gives no contrary flexure or regreffion, but only informs -us that the tangent in F will be parallel to the abfcifs or diameter AB. : The fuppofition of & = ce gives us 4 x 2rx — xa) È = o, that is, #a=0, and x = 2r. The value x = o, in all the three cafes, gives the tangent in the point A parallel to the ordinates. The value x = 27, in the firft and fecond cafe, gives the tangent in the point F, in the fame manner, parallel to the ordinates. But, in the third cafe, it gives us a contradiction. For, the equation 8 rf 3 SECT. IV. ANALYTICAL INSTITUTIONS 81 : ak è Vor =e | | equation being 2 = È > ‘3 inftead of x fubftituting it’s value 27, it will be 3 = o. But it cannot be è = o, and at the fame time £ = so 3 therefore fuch a value ferves to no purpofe in this cafe. 3 i ENAMPLE LV. «102. Let the curve be the conchoid, of Micbomedes, confidered above at i $ 85; the equation of which is yy = «TTT TIT — se Dire + gal! AT di i È i : ME ite Ps oe or y= be x Vaa = ae | Taking PIA it will be y = età — aabs Vv 2045 — 42x73 — 30° bx ; San x on a x3 XK da — xx\t As to the three ufual cafes, which this curve may have, I begin with the firft, and taking them again, making conflant, j= when a = 4, (Fig. 56.) This fuppofed, it will be 7 = oe ce seer JF Note | x3 X da — Kx|T | The fuppofition of j = o will give 20° — aax? — gala = o, that is, | x3 a: 3ax7, — 24° = 0; and, relolving the equation, it is * = x344— 2, «= — /3aa—a, and x = —4. The firft value gives us sl abtcifs GE = x = VW 344 — a, to which belongs the ordinate EM = y = a/3aa X V 24/344 ~ - 34a ee cf 300 —a flexure ; the fecond value is of no fervice, becaufe it makes the equation of the curve imaginary ; the third gives us a regreflion in the point P. s which meets the curve in M, the point of contrary As to the other two. cafes, the fuppofition Of 20 gives 2006 — x4 — 3bxx = 0, or x + 32x° — 2446 = o. Now, to have the roots of this equation, I make «x = dz, a locus to the Apollonian parabola ; and, making the fubftitution, there arifes the fecond locus xZ + 302 — 244 = 0, which is to the hyperbola. Blog: | Between the afymptotes AQ AD, take AC=2a, © | ‘°°’ the perpendicular CN = 4, AD= :85; and taking the abfcifs x from the point D on the afymptote AD, let the hyperbola GNF be defcribed, with the conftant rectangle = 244; it will pafs through the point N. Then raifing DM perpendicular to DA, on the axis DM, with the vertex D, and parameter = b, let the parabola of the equation xx = bz be defcribed. Vai. gg If, $2- ANALYTICAL INSTITUTIONS, BOOK It. If, BE, we affume @ greater than a, becaufe AD = DA AC: = aa | CD will be greater than 2. Now, taking in the parabola the abfcifs 2 = = CN, the ordinate will be x = “ad. But if 4 be lefs than 4, allo V 45 will be lefs than b, and thence alfo lefs than CD. Therefore the parabola will cut the hyperbola between N and D, fuppofe in the point I. Now, if we afume x = — a, it will be in the spasso effe > » and mn the hyperbola z= ———; but — 7 is greater than therefore the 2a e b? er 4? parabola will cut the ih vperiolan in fach a point I, as je it will be HI = —yw | lefs than a. Therefore this abfcifs will Best A | have in the conchoid a real ordinate, in the point N, for example, of the lower branch KN. The line GM, drawn from the point G, another interfection of the parabola and hyperbola, will neceflarily be greater than a, and therefore to fuch an abfcifs there can be no correfponding real ordinate in the conchoid; fo that this 7 value is of no ufe. Laftly, the third pr: | value TF will give us an abfcifs, to which an ordinate belongs in the epper branch, which meets the curve in the point of contrary flexure M. Fig. 76. | Let 4 be lefs than 4; then CD will be o lefs than 4; and in the parabola, taking Apri? or CN, the ordinate will be x = y/4b, that is, greater than 2, and therefore greater than CD. Whence the parabola will pafs between N and C: fo that it will either not cut the hyperbola, and the two negative values. of x in the equation #°* + 30x" — 24ab = o will be imaginary; or, if it cut it, they will always be greater than a, to which, in the conchoid, (Fig. 57») ima- ginary ordinates correfpond, and therefore are of no fervice. Wherefore the parabola will certainly cut the hyperbola, on the pofitive fide, in the point F for example. Whence TF, which is lefs than @, will be the value of #, to which the ordinate correfponds in the branch AM of the conchoid, which it meets in M, the ee ef contrary flexure. I faid that if the parabola cut the hyperbola between N and O, the two negative values of x would be greater than 4. For, taking x = — 4 in the parabola, which here determines the contrary flexure SE aS I MES Oe NE IER oe a VS AE Oe IS Sa RT OE AE ATI Naga PROZIA ; Vu n } ; 5 Pi VBRErA», ANALYPIGAL INSTITUTIONS. 83 it will be z = =, and in the hyperbola x = n But = is lefs than , 200 gi than a, the parabola would not cut the hyperbola; fo that it wili cut it in a point in which x fhall be greater than a. Taking x pofitive equal to a, it will , for è is lefs than a. Now, if fo be that x negative be not greater be in the parabola z = > and in the hyperbola z = 2° 2. But 3 is ; fo that the parabola will cut the hyperbola in fuch a greater than ~—— 306 + a | point F, that TF will be lefs than a. The fuppofition of ¥ = = 00 gives us 4° Xx aa — aa\? è = o, that i Jey o, .and x = +45 which is as much as to fay that the afymptote and tangent in interfection has alfo been ae ee at § 85, 2923 — bbyz — 24 + dbzz at will be ¥ = A are parallel to the ordinates in all the three cafes, as likewife the tangent in K, in the fecond and third cafe: and ia the fir&, that in P there is a point of interfection, (as the regreffions alfo intimate,) becaufe thè fame value x = — a has alfo been already fupplied from the fuppofition of 7 = 0; which point of rs 103. The fame after another manner. I take the fame conchoidal curve, but with all it’s ordinates proceeding from a fixed point, or from the pole P. Therefore make PM = y, (Fig. 56, 57, 58.) and draw PF infinitely near to PM. Then with centre P defcribe the little arches MB, DH; make MB =x, (AGIs ay GPi band make Pie ay PIO ce iz: By the property of the curve, the equation will be y = z + 4; thatis, y = 2 + @ in refpe& of the curve above the afymptote GR, and y = 2% — 4 in refpe& to the curve below it.’ È | Therefore, finding the fluxions, it will be in both cafes y = 2. Becaufe of fimilar triangles PGD, DHO, (for the angles GDP, DOH, do not differ but — by the infinitely little angle DPH, and the angles at H and G are mi angles, ) we fhall have PG. GD :: DH. HO; that is, 4. J 2z— bb — db: 3 2... è} and i iN za — bb si i x — bb : therefore 2 = _ 5 But, ey therefore PIE roi and taking the fluxions again, making x conftant and: putting % inftead of y, j = 2byzz — b3y — bz3 + b3z ; i i PIZZI 22 £ 2%: and then putting the value of 2, we fhall have j.= bbyyVaz — bb tr RO awe J yi x xx; and laftly, fubftituting the value of y = 2 + a, n zt + 2023 2423 È Su) x x 1a | Ob xX we xa ta Mz | ore | dhe 84 | ANALYTICAL INSTITUTIONS, BOORIR The formula of curves referred to a focus has been found to be xx + yy — Jy = o, or elfe = co. Therefore, putting the values of y, of y, and of ¥, aabb + gabbz Fo 2423 bb x z È al the formula being equal to o, will give 425 + 3552 F 22° = o. In the firft place, let it be 4 = 4, and let us confider the upper branch; it will be 23 ini $A mm a 2: 003° and. the three values of z.are‘z-= — 4 = = qt will be atte, i a ve, eV ga aes SVI. Beg it. is y = 3 + a; therefore it will be LA x eae duc and ys eave - The third value ts of no ufe, becaufe it gives the ordinate lefs than 24, where there is no curve. The fe- cond gives the ordinate y, which meets the curve in the point of contrary flexure, for example, at M. The firft is alfo fupplied by confidering the lower branch, and determines the point of regreffion P ; and, in refpe& of the inferior branch, will be z? — Zaaz + 323 = o. Hence the three values, LA igs Mh ae ETNIA, But, in this cafe, y = z — 2, fo that we fhall — 3a + V344. 2 o have y = 0,9 = The two laft values ferve to no purpofe, becaufe they give y negative, where there is no curve. As to the other two cafes, (Fig. 57, 58.) it will be 2? — i5l2 = ja = o, To obtain the roots of this equation, I put zz = 15, a locus to the Apollonian parabola; and making the fubftitution, there arifes a fecond /oc4s which is to the hyperbola, pz — 342 = + ab; that is, the homogeneum comparationis is _pofitive in regard to the upper branch of the curve, and negative in regard to | the lover. Between the afymptotes PQs NM, perpendicular in A, are defcribed the oppofite hyperbolas (Fig. 77.) in the angles PAN, MAO; if the bomo- geneum be pofitive, and in the angles PAM, NAQ: if it be negative. And,. fuppofing 4 to be greater than a, make AB = 6, BC = a; the hyperbolas will pafs through the point C. And taking AM = 34, from the point M in the afymptote MN let the p’s proceed. Then at the vertex M, with axis MN, and parameter 45, let there be defcribed the parabola EMD of the equation ee ee top, Then takinpop jy MB == 2b, the ordinate in the parabola is = = 6, greater than 2, that is, than dc, the parabola will pafs without the points ‘C, and will cut the hyperbolas DC, x oe = 0, corel 00. ‘The fuppoftion of © $HOT OLY. ANALYTICAL INSTITUTIONS. 85 CT, in the points D, T, I; from which the right lines DH, TV, IO, Pene drawn parallel to the afymptote QP, will be the three roots or values of = the equation 2° — 2442 — 5455 = ©, that is, in refpect of the upper each ef the crac But y= zz +a, then DH + a Mall be the ordinate y, which meets the curve in the point of con- trary flexure, for example in M, (Fig. 58.) The other two roots VT, Ol, ferve to no purpofe ; for, being negative, and 4 ad- Joined .to VT, the eiference: or y, will be negative; and a, adjoined to OI, the difference will be pofitive, but lefs than 4; and, in this cafe, the curve will not corre- fpond to y negative, or lefs than 2, As . to the inferior branch of the conchoid, | that is, inthe equation 2* — 30523 + tabb == O, the three roots will be OG, VK, HE; but if from the firft, and from the dia a be fubtraéted to have y, the dif-fence will be negative, that is, y ne- gative, to which the curve does not correfpond, and therefore they will be of no ufe. If @ be fubtracted from the fecand, VK, the difference LK will be aN ys Una meets the curve in the point ‘of contrary de La that Is, in Fig. $7. SETA Suppofing & lefs than a, the SEA o A | will pafs between the points e, C, of the hyperbolas GcK, ICT ; and. therefore the two negative values of z in the equation — BF — ibbz — zabb = o, by adding a, will f give y lefs than 4, to which the curve does “ not correfpond. The third, by adding a, ., will-give y, which will meet the curve in wo RIE contrary flexure, as at M, (Fig. 57.) UA to the inferior branch, that is, to the equation 2% — 3d4z + igbb=o, from the. two pofitive roots, which are lefs than 4, fubtra& 4; and alfo, being fubtracted from the negative root, we fhall always have negative y greater than PK, to which the curve does not correfpond. Therefore the inferior branch of the couion when 4 is lefs than a, has neither contrary flexure nor regreffion. : | The fuppofition of the formula being = ©, dive, in all the three cafes,. 2 = Fa, and thereforey = o. In Fig. 58, the value y = o ferves to no purpole, becaufe there is no curve. do Fiew 90.7 59, 10 gives the tangent in P, which 1s alfo a point of regreffion in Ee. 56, but not fo in Fig. 57. 9 | E X« 86 . ANALYTICAL INSTITUTIONS, BOOK Il. EXAMPLE VV. 104. Let the circle AED be defcribed with centre B, and let AFK be fuch a curve, thar, drawing any radius BFE, the fquare of FE may be always equal to the rectangle of the corre- fpondent arch AE, into a given right line 4; and the contrary flexure of the curve AFK is re- quired. | Let the arch AE be called z, BA = BE = a, BF = y, and FG = x. Drawing Be infinitely near to BE, and with centre B, radius BF, de- B fcribing the little arch FG; by the nature of the - curve, it will be dz = 44 — 20y + yy. Then taking the fluxions, it is 53 = alii 2ay + 29), whence 2 = ST = Fe. But, becaufe of fimilar feAors BEe, BFG, it will be BE, BF :: Ee. FG; that is, a.y 1 = 2@ | x, Whence x = PL . And taking the fluxions again, making x conftant, ; la pe: e ag nl CITTA » . ay — ayy it will be 4yyy + 257 — 2ayy — 2ayy = O, whence yy = “Rigo Oy In the general formula of curves referred to a focus, xx + yY — yj = 05 fubflitute the values of xx and of yy given by y, and we fhall have. aytin Bay ds ati ata aabb yra 495 — 12ay* + 124053 — 4a3yy + 3aabby.— 203bd nominator, will be = 0, Or = bo, aabb x y — a Wherefore, this equation being conftructed, one of the roots will give the value ef the ordinate y, which meets the curve in the point of contrary flexure. DELI 3; which, reduced to a common de- ps! SECT, Vy © © AWAUYPTICAL INSTITUTIONS, — a D SEAEGI.. V. ' Of Evolutes, and of the Rays of Curvature, 105. Let the curve be BDF, and let it be involved or wound about by the thread ABDF ; that is, the thread being faftened by one of it’s ends in the fixed and immoveable point F, let it be con- ceived to be ftretched along the curve BDF, fo that the portion AB may fall upon the tangent of the curve AR in the point B. Let the thread move or un- wind by it’s extremity A, continually evolving the curve, but in fuch a manner that it may always have the fame degree of tenfion. By this motion, the point A will defcribe the curve AHK. The curve BDF is called the Evo/ute of the curve AHK, as has been already faid before, at § 16. And the curve AHK is called the Involute of BDF, or ‘the curve generated by the evolution of BDF; and the portions AB, HD, KF, © of the thread are called the Aays of the Evolute, or Rays of Ofculation. 106, Now, becaufe the length of the thread ABDF always continues the fame, it follows from thence, that the difference of the rays of ofculation AB, HD, will be equal to BD, the correfponding portion of the curve. As alfo, the other portion DF is equal to the difference of the radii HD, KF, and the whole curve BDF is equal to the difference of the radii AB, KF. And if the radius AB fhould be none at all, that is, if the point A fhould fall in B, the radius HD would be equal to the portion BD, and the radius FK. to the whole curve BDF, — | pio. 107. From = 38 ANALYTICAL INSTITUTIONS. BOOK 11. 107. From the generation of the curve AHK, by the unwinding of the thread, it may be clearly feen that every radius HD, KF, at it’s extremities D, F, is atangent to the evolute BDF, 108. Let the arch HK of the curve AHK be an infinitefimal ; ee -alfo, the arch DF of the evolute will be an infiniteftmal; and, as it -has been demonftrated in Coroll. 4. Theor. I. $ 6. that any infinitely little arch of a curve . has the fame properties as the arch of a circle: and in Theor. IV. § 15. that the radius HD being produced, fo that it may meet the radius KF in 3, the lines SH, SK, differ from each other only by an infinitely litle quantity of the © third degree; therefore thofe lines ‘SH, SK, may be aflumed as equal: and therefore they are perpendicular to the curve AHK de ons PER But the two lines HD,. HS, differ from each other by DS; an infnitefimal of the firft order, and HD is finite; ; thefefore they may be aflumed as equal, Where- fore, to determine any point D in the evolute, that 1s, to determine the length of any ray of ofculation or of curvafare FD, it will fuffice to have given in pofition the perpendicular HS of the given curve AHK, (which is done by the ‘Method of Tangents;) the point S may be determined, in which it is cut by the infinitély near 6 pe taping KS. This may be done in the following manner. 3 109. Firft, let the curve DABE be referred to it’s axis; let the two infinitel little arches be AB, BE, the perpendi- cular BQ; and the other EQ: which meets it in the point required, Q. Make, as ufual DH = +, HA = y; draw AF, BG, parallel to DM, and the chord PABC which meets ME produced in C, and draw the other chord EBR. Now, with centre B, and diftances BE, BP, the little arches ES, PO, being defcribed, it will be AF = x, FB = y, AB= 5 | V xx yy... But, by. Coroll.. 2. Theor. V. § 19, the fectors QBE, BES, are fimilar. Therefore we (hall have QB. BE :: BE . ES, that is, QB.5 3: 5. ES, (calling the element of the curve 5,) and therefore QB = a. Now, becaufe the little arch PO may be exprefled by it’s right fine, (Car 1...Iheor. LI, Yroe) the triangles RPO, BEG, will be fimilar, and therefore BE. EG :: RP. PO; that is, 5 vid RP.PO = 2X2°. But the feGors BPO, BES, are alfo fimilar ; and there- fore it will be BP. PO :: BE. ES; that is, z : = ISO PENE dea! And laftly, QB = | Vij + me + xx 3 OF elle, ubticring di the value x, Gb = SECT. Ve ANALYTICAL INSTITUTIONS, 89 HS | CRE ax RP? 3 general formula for the rays of ofculation, or the radii of curvature, in which nothing elfe remains to be done, but to fubftitute the value of RP, the fluxion of DP = + — %, according to the different hypothefis of the firft fluxion which is to be taken for conftant. If no firft fluxion be taken for conflant, it will be RP = aS, and therefore QB nidi JE J* — i) | a x a afte as comhatie it will be RP = i , and therefore QB = xd +52, Tf. y be affamed as cl eae it will ie RP = a, and therefore QB = xx +4) ga > If es s be aflumed as conftant, that is, SW xx tI + yy, it will be xx ~ Sf, and ie j= ba pier RP EXE ii ; and therefore OB" = AT o piso as. no Givion is taken YH — 5 fore, in the apre _ OB = for conftant, it will be fufficient to expunge the term yx, in the fuppofition oe ae conftant; to expunge the term xj, in the fuppofition of y-conftant; and to put, inftead of — }j, it’s value — , in the fuppofition of s conftant. 110. The curve may be referred to a diameter, or the co- as may be inclined to each other in an oblique angle. Make the. abfcifs DV = x, VK = x, the ordinate VA = y, and the reft as above. Becaufe the angle DKB is known, the angle BNF will be known alfo. Wherefore, it being NB = y, NF and FB will be given, and therefore AB, or 5. But the triangle RPO 1s fimilar to the triangle ABF, for the angles at O and F are right ones, and the angle ORP does not differ from the angle FAB but by an infinitely little angle RBP. Wherefore there mest be given RP PO; ei thence ES, __and finally, QB. Wet. | n N | 5 tr1r. From | Subofculatrix, what. 90 sn ARALNITICAL IIMSITITUTIONA . BOOK IT, 111, From the extremity of the radius of curvature BQ_ is drawn QT parallel to the axis DM, which meets in T the ordinate Bi produced ; the right line BT is called the Subo/culatrix, or the Co-radius. The radius BQ_being given, the co-radius BT will, in like. manner, be given alfo; for, by the method of tangents, the normal of the curve Bm is given, and therefore BT will be given by means of the fimilar triangles Bal, BQT. | But if we would have an expreffion for the co-radius independently of the radius, we may make BT = z. The triangle BTQ is fimilar to the triangle BCG, or BAF; for, the two angles TBG, QBC, being right ones, take away the common angle QBG, and there will remain the equal angles TBQ, CBG, and the angles at T and Gare right ones. Therefore it will be x. 5:12. BQ conc wash i But, by Theor. IV. $ 15, BQ is equal to EQ, be- Bas . mes x 14 caufe they differ from each other only by an infinitefimal of the third. degree ; therefore the difference of QB fhall be nothing; and, by differencing, without afluming a conftant fluxion, ee IO need « ) | | wa xk + yy But 3 = y, becaufe TB and IB have the fame difference. Therefore x = PA = BT, a formula for the co-radius, in which no fluxion is yet affumed as conftant. If x be conftant, the term yx fhall be nothing, and therefore the formula, on this fuppofition, will be =e = BR Repl he conftant, the term — xj will be nothing, and therefore the formula, on this fuppofition, will be ae = BY. If the element of the curve be con- = ftant, it willbe —y = =, and therefore the formula, on this fuppofition,. will be 2 = BT, the value of ¥ being fubftituted : or elle — = = BI, tie value of x being fubftituted. The co-radius being given, by the fimilitude of the triangles BmI, BQT, the radius QB will be given in a like manner. ag i 112. If the co-ordinates fhall be at an oblique angle to each other, in the analogy x .s :: 2. BQ: inftead of x and s, it will be enough to put the refpective values, which in this cafe agree to AF, AB, and to do the reft as above ; and then you will have the formula of the co-radius BT, in that cafe when the co-ordinates are at any oblique angle. 113. After ? 4. ere ES TO oe = agi ee È OE LOI MOR LOE ARIE i LIT IAT RN RE SENI III RI RIONI PRT EM "sect. V. ANALYTICAL INSTITUTIONS. ‘gi 113. After feveral other manners the fame formula of the radius of curvature may be had. As, with centre Q. diftance Qm, defcribe the little arch mu, | Affaming the infinitefimal arch mn by the tangent at 7, the two triangles BCG, wing, will be fimilar, and therefore BC. BG :: mq . mn; that is, xx + yy mq X x —_ ® > OR bY vini a WN ae But mq is the fluxion of Dm, that is, of the falitiormal Im, with the abfcifs DI or DH; that is, of «x + Po, Therefore, _ by differencing in the hypothefis, that no fluxion be conftant, it will be mg = 6 es o è 76 Ce . PURI ‘ie et è ce ety + 9 =D. Therefore ms = TI 2%. But, becaufe ake AVak+Jy | Dy ah È IVEIRETO of fimilar fe&ors Quan, QBE, it will be BE — mr. BE :: Bm (2) - QB; i i e °° 3 : that is, fubftituting their analytical values, QB = ee Which formula, being modified according to the fuppofition of fome conftant fluxion, will give an expreffion for the radius QB, correfponding to that fuppofition. 114. In another manner, thus. Let EM be produced to ¢, and BG to L. Becaufe the triangle EGL is fimilar to the triangle Blm, the angles GEL, IBm, being different from each other only by the infinitefimal angle BQE, it~will be Gr = n i piererore BL. = TEA. But it has been feen, that mg = se... LG. And the fimilar triangles QBL, Qn4, give BL — mg + BL :: Bm. BQ. Therefore, fubftituting the analytical values, we fhall have no = FEDI SE = Ky" Fig. 81. 115. Now let us refume the curves which are referred to a focus. Therefore let the curve be BEG, the focus AL Ando taking the two infinitely little arches BE, EG, and drawing the ordinates AB, AE, AG, with centre A let the little arches BC, EF, be defcribed, and to the chords GE, EB produced, let AI, AD, be per pendicular, Laftly, let the chord DE, produced, meet the ordinate AG in L, N 2 7 and 9% ANALYTICAL INSTITUTIONS, . BOOK IL, «and with centre E let the little arch GR be defcribed. Make AB = 7, sei y, BCom xD, ed. ocDhediule arch DH «being defcribed “with centre A, it will be HI = p. But HM is an infinitefimal quantity of the fecond degree ; Theor. III. § 8. Therefore we may take as equal HI, IM, and thence it will be MI = p. The triangles EBC, EAD, are fimilar, which gives ED = ae SS dds gdo DOLO, different only by an infinitefimal. And, affuming the little arch GR by it’s tangent, the triangles EIM, EGR, will be fimilar. Hence’ GRI= ti, Now, drawing EQ», QG, perpendicular to the curve in the points E, G, the fectors QEG, EGR, are fimilar; fo that QE = 2. The fimilar triangles EBC, EAD, will give us p = E. P * Vick AOR and, by differencing, without afuming any conftant fluxion, fp = I ee Whéate, x Bt: te +5} ne fubftituting this value inftead of p in the expreffion of QE, it will be QE = PE Sa ate 2 Cho , a general formula for the radius of curvature of curves OMAV e Tos" resi referred to a focus, without taking any fluxion as conftant. If we would have x conftant, taking the value of p in this hypothefis, and” fubftituting; or, without any thing elfe but expunging the term yyx in the yX xe +4) general formula, it will be QE = +,

ANAL YI INsrituriona | 93 116. If, in any of thefe formula, we fhould fuppofe y infinite, all thofe terms would vanifh in which it is not found, and the formula will be the fame as thofe found for curves referred to an axis; which ought to obtain, becaufe, if y be infinite, the point A will be at an [fante diftance, and therefore the ordinates will be parallel. Fig. 82. 117. After another manner. In the 3 R point E let ER be a tangent to the infi- : | nitely little arch EG, and let QE, QG, be the two radu of curvature, and pro- duce QG to R. From the focus A draw AN perpendicular to QG, and AK per- pendicular to QE, mac make EK. =}; thenis KM —= Becaufe.the triangle AKM is fimilar to the triangle QNM, and this is fimilar to the triangle QER, ic will be QE. ER:: AK. KM hi. But, becaufe of the fimilar triangles ELC, or EGC, EAK, it is AK = ti , and ER may be affumed for EG. Then it will be QE. di = . i, and therefore QE = LA But:hk. ana nia Rar doing the reft as before, that is , differ- -encing the value de i, and fubRituting i in the expreffion of Sh we thall obtain the fame formule as above. 118. Making QP perpendicular to EA produced to P, the triangles EAK, EQP, will be fimilar, and therefore EA. EK 3; EQ. EP. . But it has been fhown, that EQ = A , Then y 833 vp pa ZL. And, infiead of 7, EIA x +9 ; fubflituting it’s value a, and, inftead of 7, the differential yas s xii bye — gay | ya A- yay yp 43 + ayy bye — yay -, a general formula for the co-radius, in which no fluxion is made conftant; from which, being modified, we obtain the other formula, which correfpond to the fuppofition of a conftant differential. And if inthefe we - fhould fuppote y to be infinite, that is, if we fhould cancel the’ terms in which it is not found, we fhould have the fame formule which have been found for curves referred to an axis or diameter. I 10. Now, 94 ANALYTICAL INSTITUTIONS, 0! BOOK . Now, whatever the curve may be, as we find but one expreffion only for È radius of curvature, and for the co-radius ; and that as well in curves referred to an axis, as in thofe referred to a focus; it follows from hence, that, - whatever the curve be, it can have but one evolute, 120. Therefore, any curve being given, expreffed by any equation whatever, of which curve the ragius of curvature, or the co-radius is required ; it will be 7 neceflary to difference the equation, in order to Fig. 83. Rie have the values of y, yy, and ¥ given by x; or ES thofe.of x, &c. given by y; and to fabftitute ay them in the formulas now found, by which we thall have the expreffion in finite terms, and quite free from differentials, of the radius of curvature, or the co-radius of the propofed curve, | 121. If the value of the radius of curvature, or of the co-radius, be pofitive, they ought to be | taken on that fide of the axis DM, (Fig. 80.) or \p of the feeus, (Fig. 81.) as has been hitherto fup- pofed, and the curve will be concave to this axis or focus. But if it fhall be negative, they ought to be taken on the contrary fide, and, in this cafe, the curve will be convex. Hence it follows, that, in the point of contrary flexure or regreffion, if the curve have any, the co-radius, from pofitive, will become negative ; and two radi of curvature that are infinitely near, from being convergent will become divergent. But this cannot be, without they firfl become parallel, that 1s, the radius of the evolute muft be infinite in this point; or elfe they muft coincide one with the other, and thus make the radius of the evolute nothing. It is evident, that when the evolute is fuch, as that the wd o) "an eonemee ha 34 Fic. 05. È radii go on always seria. as they approach to É the point B (Fig. 83, 84.) of contrary flexure or €: og regreffion, to pals from being converging to be fat hi tt ut dA come diverging, they mult Art becoine parallel, being then AD, FE, the evolute of the curve ABF. But if the evolute of the curve ABF, (Fig. 85, 86.) fhall be DBE, the thread, un- winding itfelf from the point B, and proceeding towards A in refpect of the portion BA of the curve, and going on towards F, in refpect of the | portion ANALYTICAL INSTITUTIONS, 95 portion BF; becaufe, as the radius is always lefs, the nearer it is to the point B, it muft of neceffity become nothing before it paffes from being pofitive to become negative. EXAMPLE I. 122. Let the curve AB be the Apollonian parabola of the equation ox = yy, of which we would find the radius of curva» ture at any point B. By taking the fluxions, it will be ax = 2yy; and taking the fluxions again, making, if you pleafe, x conftant, it will be 27y + ay = a. But > = ne, therefore j = — Wherefore, thefe values being fubftituted - SE. in the formula for the co-radius oath 4y? + ag _p i sa | it will be eee orelfe, by putting, _ inftead of y, it’s value given by the equation of the curve, it will be BE = par fia i + fax. From the point B let the tangent BT be drawn, which meets the axis in T, and from the point T is drawn TE parallel to the perpendicular BM : this will meet BP produced in the point required, E. For, becaufe of the right angle BIE, it will be BP. PT :; PT. PE; that is, by the property of the para- bola y/ax i20015:2% PE. ir ron pui, Therefore BP + PE = BE è Li 96 ANADYPEOAL INSTITUTIONS, BOOK II. = et + “ax. Now, BE being determined, draw EQ parallel to the axis AP; the normal BM, produced, will meet EQ in the point neh will be a point in the evolute. Or elfe, becaufe of the fimilar triangles BPM, BEQ, it will be BP. PM: BE . EQ. But, by the property of the parabola, it is PM = ja. Then I ca GRA aE 4 RETTA I seen OF OR ai oe Vigil 1 Whence IO: a og: tase (PR ang MK = 2x. Wherefore, taking MK double to AP, or PK = TM, and drawing KQ |. parallel to PB, it will meet the perpendicular BM produced in the point Q, which will be in the evolute. And, becaufe it is BP. BM :: BE. BQ; and BM = ao 2, it will be Vaw . EE i: see a 2 3 ax + da 3 . dar: BO = 2, the radius of ‘ciivature: 244 x : Taking the formula Et? of the. radius of curvature, and making the sas AC, 244 2aa | 3 fubftitutions, it will be QB = ayy + aa\z _ 4ax + + aa\e Proceeding to the fecond fluxions of the equation ax = yy, without making any conftant fluxion ; becaufe ax = ayy, it will be ax = ayy + 2)» or $ = Wheref = 40 n; nerefore, taking the formula for the radius of curvature ica which belongs to this cafe, and making the fubftitution of the value of ¥, it oe Tix; 3 È Jo Ra a X Ar -b yy jz ? | : si ) Vode wi | be QE previ and laftly, putting the values of y and y, it 40% + adi is ORE “ngi MII above. ax — IY > The fame thing will be found in the other fuppofitions of y or s conftant; which, confulting brevity, I Mall here omit. If we would have the radius of curvature at any determinate point of the curve, it will be fufficient to fubftitute, in the finite expreffion already found for the radius of curvature for any point, the value of x agreeing to that deter- minate point. ‘Thus, if we would have the radius of curvature in the vertex A, or in the point N in which the axis AN of the parabola touches the evolute NQ; fince, at the vertex A, it is x = o, by expunging the term 44x in the 4ax + aa\z 2G aed expreffion of the radius of curvature, we fhall have AN = ta; which. . SECT. V. ANALYTICAL - (INSTITUTIONS, i 97 which cannot be otherwife, the radius AN in this cafe being the fame as the “fubnormal, which, in the parabola, is known to be equal to half the parameter. 123. Now it will be eafy to find the equation to the evolute NQ, after the manner of Des Cartes, or the relation of the ordinates NK, KQ, in the follow» - Ing manner. hale ho, KO a hance PE = v=, we fhall have the equation ¢= ©“, Bur AK = AP + PK = 3x + 34, and AN = ta, Then NK = 3x = a, and x = 34; therefore, putting, inftead of x, this value in the equation # = 4, we fhall have ¢ = 43%, and, by {quar- | 4 3a | ing, 274f = 164°, which is an equation to the fecond cubic parabola, with a parameter = ae which expreffes the relation of the co-ordinates NK, KQ> and is the evolute of the propofed Apollonian parabola. Pig. 88. It is evident that the whole fecond cu- bical parabola will be the evolute of the whole Apollonian parabola ; that is, that the branch NQ will be the evolute of the upper part AB, and the branch Ng of the lower part Ab: and that the two branches Ng, NQ; change their convexity, and have a regreflion at N. 124. It is alfo evident, that if the propofed curves be algebraical, their - evolutes alfo will be algebraical curves, and that we may always have an equa- tion in finite terms, expreffing the relation of the co-ordinates; and that, befi des, thofe evolutes will be rectifiable, or we may find right lines equal to any portion of the fame; for example, to QN. For, if the propofed curve AB be alge- braical, we may always have the radii of curvature BQ; AN, in finite terms; and, from BQ fubtraCing AN, the remainder will be the arch NQ. Vot, II O EX 98 . ANALYTICAL INSTITUTIONS BOOK II° EXAMPLE IL Fig. 89. — 125. Let the curve MBM be the hyper. Q bola between the afymptotes, whofe equation R kr Ex od 6 | 1s aa = ay. By differencing, it is xy + yx n \ GAS { : = o, and by differencing again, and taking \ È VA by ae { NET 2xy i \ ci ind x as conftant, it is j = — —. Subfti- V|/D | ~ | tuting thefe values of y and y in the formula J hoe. zk by i PARSO ——— for the co-radius, we fhall have | de di | La Rea DMI cand PREIS 5 SS ao Te te A bi i a eee is BE — “ty , a negative value. If, uns ar, ie (heretone, dt.is A oy, F > 7, IN Ae Sid | i produced, taking BN = st: fac IV xw-Lyy, and raifing the perpendicular NE, which may meet the ordinate BP, produced in E, the co-radius will be BE, as was required. For, becaufe of fimilar triangles BPA, BNE, it will be BP. BA :: BN . BE, that is, y. xe + yy EV we oy BENS Doe, and therefore, on the negative fide, it muft y oo 7 bé ot = . Wherefore, drawing EQ_ parallel to AP, and producing to Q the perpendicular to the curve FB in the point B, the radius of curvature will be BQ; and the point Q_ will be in the evolute. To determine the radius of curvature at the vertex of the hyperbola D, ; } ° LI make x = AH =, and therefore y = HD = a. Then the co-radius = at the vertex D will be equal to — 4, and the radius equal to — w 244, If we do but confider a little the figure of the curve MBM, we fhall find that the evolute will have two branches, with a point of regreflion at L, in which the radius DL will revert, and will be the leaft of all the radi BQ, the difference or fluxion will be nothing, or infinite ; that is, fuppofing x to be — Bi? + ee Ee _ Niue Wherefore, by differencing the formula of the radius of curvature : conftant, it will be o, or co. And, di- Hayy viding by Vxx + yy, and multiplying by xjj, it will be xxy + WI — 3499 5 A PI = Qs, als kai a n; See Sl oth as It * Ei PR, E — ae, Se SS AU ER ee A ae i again, fuppofing Ng conftant, it will be mm —m X yy SECT. V. ANALYTICAL INSTITUTIONS. Rd, 99 I : SERI RAND "7 O ir = 0, or se. But, by the equation of the curve, tis y = — —.,y = QIAXXkX + 6aax3 | é tae i — Jr — —~. Therefore, making the fubflitutions, and fuppofing the we Ce : faid quantity to be equal to nothing, we fhall have x = 2 = AH. That isto fay, the regreffion will be in the radius of curvature at the vertex D of the curve. But it has been feen, that that radius is equal to — W2aa; therefore it will be DL = — Y2aa = DA. In the formula of the radius of curvature, fubftituting the values of y and | +. | , and therefore, differencing, that % . ; 3 i we fhall have BQ. = = ago E sou 2XY i “we may have the leaft radius, that is, the point of regreffion L, it will be 34% + 39) X Max + yy = 0; and, inftead of y, putting it’s valtie, it will be ZXKX — Byyn x Mixx + yy = ©, that is, x = y = a. And fubftituting this value in the expreffion for the radius of curvature, it will be = — V 244 = DL, as found above. The radius BQ may alfo be conftructed in another manner. For, becaufe jz- coi inftead of x and x, fubftituting their values by y, it will be j = = , and therefore the co-radius BE = I . And, becaufe of fimilar triangles BPF, BEQ, we fhall have EQ = — AT -- - Now draw the 2% tangent BT to the point B, and from the point T the line TS perpendicular to BT, or parallel to BQ, and make BE = +BS, or PR = tFT. Now, if E _ be drawn parallel to AT, or KQ perpendicular to it, they will meet the line BQ in the point of the evolute Q. For it will be BS =*"42¥ | then BE = ty, it will. be. allo FP... PT — FT = ZL 2, and there. — 29 Fo = fore FQ = —% fons ll DA 25 If the equation be y” = x, which expreffes all parabolas ad infinitum, when m denotes an affirmative number, and confequently the parabola of the fir& example: (and it exprefles all hyperbolas between the afymptotes, when m ftands for a negative number, and therefore that of the prefent example.) By taking the fluxions, we fhall have mjyy”~* =»; and taking the fluxions 2 eae + my ssi nae Qe O 2 Now, IOO ANALYTICAL INSTITUTIONS, BOOK IT. Now, dividing by my”~", it will be — f~=m—1x ra Wherefore, taking the forinula for the co-radius i , and making the fubftitution of the value of ¥, we fhall have BE = LITIO , and therefore EQ; or PK = Mm Ly "ye ; ype or nn È + li e Mm = 1} MIT 1% Fig. 87. | From the point T (Fig. 87, 89.) in which the tangent BT meets the axis AP, is drawn, in like manner, TS parallel! to meets in S the ordinate BP produced, Then take BE = , on the negative 7 — È fide, if m be a negative number, as in the hyperbolas which are convex towards the the axis AP, (Fig. 89.) that is, to the Q afymptote. But BE muft be taken on the pofitive fide, if m be a pofitive number, and greater than unity, as in the parabolas (Fig. 87.) that are concave to the axis AP; and on the negative part, if m, being po- fitive, be lefs than unity, in which cafe the parabolas are convex to the axis AP. To determine the point in which the axis of the parabola touches the evolute, I take the formula of the radius of curva» «x we FIS ture, which is gr from whence, by fubftituting the values of x = myy™~*, and of — j = ni , we fhall have | , BQ: = aes 2-. It is here underftood, that unity may fupply any. mx m—1 XJ | powers required by the law of homogeneity. Whence, fuppofing m to be. greater than unity, for that reafon the parabolas will be concave to the axis AP; if 1 be lefs than 2, the y in the denominator will become a multiplier in the- numerator, and therefore, making y = 0, as the prefent cafe requires, it will be BQ = o, that is, the axis will be a tangent to the evolute in. A, the pia BQ, a perpendicular to the curve, which - % . a v tirate asi en e n I le a A NA an cà a O er na, n a Pià E, ea, oo cree ee SS eS SECT. Ve - ANALYTICAL INSTITUTIONS. Mega "0 of the parabola, as it would be (for inftance) in the fecond cubic parabola ED e Tie. Fee | Fig. 90. Now, if m be greater than 2, the y of the denominator would be raifed to a pofi- tive power, and therefore, making y = o, BQ would be infinite, that is, the axis of _ the parabola will be an afymptote to the Q evolute;.as in the firft cubical parabola AB, (Fig. go.) whofe axis AP is an afym= ptote to the evolute LQ. B ‘A a The evolute CLQ of the cubical femiparabola: ABM of the equation aax= 9’, has a point of regrefiion L, and therefore two branches LQ, LC; by evolving. the branch LQ, the portion BA will be generated, and by evolving the branch LC, the infinite portion BM will be produced. To determine the contrary flexure L, take the value of the radius of curva- ture, which in this curve 1s eae which ought to be a minimum; and theres 2a i (Lx 3.x 18atyty x gyt + at) — otf x Qyt + ai fore, by taking the fluxion, it will be AO II i 4. j f ero thatis, 46)" —“a* = 03° Whelce = tia And this value, being a* griz5° , and drawing the: ordinate PB, the point fubftituted inftead of yin the equation aax =, y*,.we fhall have a =. &/ at ‘Faking, therefore; AP = Vine of regreffion L will be in the perpendicular to the curve at the point B. And, in the expreffion of the radius of curvature, putting %/ > inftead’ of y, we. fhall have the value of BL. nt: After another manner. By differencing the equation 44x = 5; or y = 2 ri O Sg ‘ ae Li Zi ee Kate gel atx, it will be y = fatxw 3, j = — Zatxxn 3; ¥ = Zpaixx 7%; fuppof- ing x to be conftant. Whence, taking the formula xx) + yy) — 37) = 0, and. fubftituting thefe values, we (hall have AP = <{/ ‘ QI125 >. as-before.. * ‘TOR ANALYTICAL INSTITUTIONS, BOOK IT, EXAMPLE II. 126. Let the curve ABD be an ellipfis or hyperbola, the axis of which is AH = a, the parameter AP =~ J, AP\= +7, PB = y, and the x bx Fd ; : 7 equation y = yor. By differencing, it eee i abe ae abs dai — a3bbix will be y = 2N aabx = baxx’ a LS x aaba Fabxdi” taking x for conftant. Making the fubftitutions I erp mi ees in the formula ~*~” of the radius of curva- mig pi e RELL EE TE O Le a LL eae scorge I ] ae | b b aabb = 4qabba + abbax\x ture, it will be BGQ= #4 F 4 eee. But the normal zaaba + 4abax + aabb x gabbx + 4bbxd È 20 \ will be found to be BG — Therefore i ° . _ 4BG cub. i the radius will be BQG = prep firft term, the normal BG for the fecond, and continuing the geometrical pro» portion, the quadruple of the fourth term will be the radius of curvature BQ. fo that, taking the parameter b for the Making x = o in the expreffion for the radius of curvature, it will be BGQ = AM = Th. And making x = AO = fa, we fhall have in the ellipfis. | /a BGQ = DOQ = e: a , that is, equal to half the parameter of the conjugate axis; andin Q will be a regreffion; and the evolute of the portion AD = DH will be MQ,—of the portion DH, will be RQ. But, in the si the radius is extended ir infinitum. In {he ellipfis, if we make 2 = >, the radius of curvature BGQ will be = 74, wherever the point B be fituate, Therefore the radii will all be equal to one one another, and the evolute will become a point; that is to fay, that the ellipfis, in this cafe, degenerates into a circle, having the centre for it’s evolute. ‘ OROBIE CINESI LEI NITTI EI I n BLN, SIRO NOR RE SETT TO SE I RO IO ENTI MP IRR REV A a n a SO ae a pe me RR Re ERS n ae (RO DR TAR, SPO der 4 Ni ‘ SECT. Ve CCANMALYTICAL INSTITUTIONS: T03° EXAMPLE IV, 127. Let the curve ABD be the com- mon logarithmic curve, the equation of which is È — x, J < By taking the fluxions, making x con- fant, it willbe j = È = =, by fub- | ftituting the value of y. Making the ufual fubftitutions in the formula =? of the KF Ft co-radius, we thall have BE = —“=, 9 © and becaufe, in the logarithmic, it is. found that the fubnormal PH = =. it will be EQ = —4 — 2, Therefore,. a taking PK = TH, and raifing KQ at right angles, it will meet the normal. HBQ in Q, the point of the evolute required. If we would determine the point of greateft curvature in the logarithmic,. that is, the point where there is the leaft radius-of curvature ; making the fubfti-- i So A 3 : . - 3 tutions in the formula +22" of the radius of curvature, it will be “ es ua. | ee | = ys IE bei and taking the fluxions, it will.be —222~ “41 y* To A act” ra Os valid: aayy 3 therefore PB. = y = W2aa, Or, taking the formula of §.125, xx} + JYj — 377 = 0, and making the: ry ay 3 so e se PX x 3 fubftitutions of y = Ss I= — IRE: 1090 fp a we fhall come to the fame: conclufion of PB = y = Wiaa.. EX 104 ANALYTICAL INSTITUTIONS, BOOK II, EXAM PLE , making NPD = 4, NP c= @,.. He wadins: AP = 2, and ARI: Then, by taking the fluxions, it will be I + . 272 «== I a. x 4 7 e e myy = <<. Now, drawing the radius Ap infinitely near to AP, and making BR = x; becaufe of fimilar feGors APp, ABR, it will be è = ni - Wherefore, putting the value, inftead of By VoL TI. iP : ; in 106 in the fluxional equation, it will be myy” = ANALYTICAL INSTITUTIONS, BOOK lf. M1. 27.3; and theref kine 3 3issn herefore, taking the fluxions again, making x conftant, we fhall have mianyyy + my ij m= Oy yy = — myy. Wherefore, making a fubftitution of this value, and of the value of x, in the formula of the co-radius, it will be BE = , that is, 9X mmbby”” + a mmbby*" 21m 4-2 2m+ 2 + m+-Il Xa ‘« Make TAC perpendicular to AB, and draw BT a tangent to the curve in B, and BC perpendicular to it; it will be AT = mbe * 1 gti N y X mmbby?” anmbby?” m-+-1 abb a quit 2 oe —__, and therefore TC = —— —— — . Whence mby ve quet2 + mI x gaet2 aha + ba I the fourth proportional to TA + 2 + 1 x AC, to TC, and to AB, will be — = BE. And cir Dane EQ parallel to fe it will meet the perpendicular BC in i the point Q which will be a point in the evolute. EXAMPLE VII. VN 131. Let the curve ABD be half of the common Seo the equation of which is 7 = x = i; making AC = 24, AP = a, PB=y. By differencing, and taking x for ©. >. conftant, it will be y = = XV 24M — LK and fubftituting thefe values in. Cale fore ak + yy lt mula for the radius.of curvature tee > 3) it mq be BQ = 2V 444 — 24x. But the normal BG = \/ 44a — 244, which is equal to the chord EC. ..T herefore: the radius of curvature BQ = 2BG = 2EC, | 3 Making gangs >. e Te EG Sa AP Scipio a pho aap Lea Sees tea clin tina dI errata EA AEREE SENIO mi AR BECT. ¥. ANALYTICAL INSTITUTIONS, © oe . Making a = 0, to have the radius of curvature in the. Da A, it. will be BQ'’= AN = 4a, and therefore CN = CA = 24. Making x == 22, the radius of curvature in the Rothe D will beve, and therefore the dvatute begins in D, ‘and terminates in N. pee the tangent of the cyclotd in B is parallel to the chord EA, (§ 47.) the normal BQ will be parallel to the chord EC. This fuppofed, complete the rectangle DCNS, and with the diameter DS = CN = AC defcribè the femi- circle DIS, rand sa the chord DI parallel to BQ, or to EC. The eS ae Ei DCE, will be equal, and confequently the arches DI, CE, and the chords. face DI, GQ, are equal and parallel; and drawing IQ, it will be equal and: parallel to DO. But, by the property of the cycloid, DG 1s equal to the arch EC, and therefore tothe arch DI. Then the arch DI = IQ; and the femicircle DIS =-SN. Whence the evolute DON is the fame cycloid, DBA, in an inverted fituation. 132. The radius of curvature and it’s formula being now fufficiently ex- plained, it will not be difficult to find the formula for the regreffions of the fecond fpecies,. mentioned before at § 98. Let the curve be BAC, with a contrary Beas at A, and let it be evolved by the thread beginning at any point D, different n the point of contrary flexure A; Fhe evolution of the portion DC generates the curve DG, and that of the portion AB ge- nerates the curve EF; in fuch manner, that the evolution of the whole curve BAC will «form the entire curve FEDG, which has two regreffions ; one at D of the ufual form, becaufe the two branches DE, DG, turn their convexity; the other at E of the fecond fort, becaufe the two branches ED, EF, are concave towards the fame parts. Let NM, Num, be any two rays infinitely near, of the evolute DA, and NH, aH, two perpendiculars to the fame; the two infinitefimal fectors NaM, HNa, will be fimilar, and therefore HN . NM :: Na. Mm. But, in the point of contrary flexure A, the radius FIN ($ 121.) ought to be either infinite, or equal to nothing, and the radius NM, which becomes AE, con. tinues finite. Therefore, in the cafe of contrary flexure A, that 15, in de point of regreflion E, of the fecond fort, the ratio'of Nz, Mw, that is, the ratio of the differential ci the radius MN to the element of the curve, ought to be either ney great or infinitely. little. But the formula of the radius — Pa : MN Fig. ae 108 ANALYTICAL INSTITUTIONS BOOK 1% wae a I ; . vi a x e MN is TE, taking x for conftant; the differential of which is come sii ae +0 4 + + ~ ij X_ * X kx + + + ple DOG TR 4; MA Na date Sg; enni , and Mm = xx +yy. Therefore 7 = KAY + Dj — 355 5; = ©, or co, the formula for the points of regreffion of the fecond fort. This formula is the fame as that already found, $ 125; but in that place it ferved for the regreflions of the firft fort of evolutes, and here for the re- greffion of the fecond fort of curves, derived from evolutes ; x and y, in both, cafes, being the co-ordinates of the curves fo produced. © END OF THE SECOND BOOK. ANA. ANALYTICAL INSTITUTIONS. BOOK TE OF THE INTEGRAL CALCULUS. ui Gas Integral Calculus, which is alfo ufed to be called the Summatory Introdu&ion. Calculus, is the method of reducing a differential or fluxional quantity, to that quantity of which it is the difference or fluxion. Whence the operations of the Integral Calculus are juft the contrary to thofe of the Differential ; and therefore it is alfo called Zhe Inverfe Method of Fluxions, or of Differences. Thus, for example, the fluxion or differential of y is y, and confequently the Fuent or integral of y is y. Hence it will be a fure proof that any integral is juft and true, if, being differenced again, it {hall reftore the given fluxion, or the quantity whofe integral was to be found. Differential formule have two different manners, by which their integrals are inveftigated. One is, by the help of finite Algebrdical expreffions, or by being reduced to quadratures which are granted or fuppofed. In the other, we are allowed the ufe of infinite feries. In this firft Section, I fhall deliver the rules required in the firft manner. In the fecond Seétion, I fhall treat of the fecond manner; to which. I fhall add a third Section, to fhow the ufe of thefe Rules in the Rectification of Curve. lines, the Quadrature of Curve-fpaces, &c. And laftly, I fhall add a fourth, which fhall teach the Rules of the Exponential Calculus. Soaks ilo ANALYTICAL INSTITUTIONS, BOOK III, ode aa Grd li I. The Rules of Integrations expreffed by finite Algebraical Formule, or which are | reduced to fuppofed Quadratures. 1. As in fimple quantities raifed to any power, their differential or fluxion is the product of the exponent of the variable into the variable itfelf, raifed to the fame power leflened by unity, and multiplied by it’s fluxion or difference ; fo the fluent or integral of the produ& of a variable raifed to any power, into the difference of the fame variable, is the variable raifed to a power the exponent of which is increafed by unity, divided by the fame exponent fo increated, And this obtains, whatever the exponent hall be of the power of the variable, whether pofitive or negative, integer or fraction, Thus, for example, the a 772 fl PIE SETA mee ES mn È or. Mais uent of me x will be = -, or # . The integral of x x will be ‘ m Pi 74 salata 772 an ; and fo of others. 2. Any conftant quantities, fimple or complicate, by which the fluxions may be multiplied or divided, will make no alteration in the rule; for they remain in the fluents juft as they were in the fluxions. Therefore the fluent of dn DR —— willbe 2 ; MD. GC n+I XK mb—ce 3. Thus, if the differential formula were a fra&ion, the denominator of | which were alfo any power of the variable, multiplied (if you pleafe) by any M . #2, x XX 3 OI Sree es aax” — bha" aa — bb X x conftant quantity; as the formula , which will be ca” : the fame as ———., and therefore fubject to the general rule, 4. But: fe LI | RL Li BEM SR SPE SEAT RI SECT. I. ANALYTIGAL INSTITUTIONS, I3I. 4. But here we are to obferve, that, in order to have the integrals complete, we ought always to add to them, or to fubtra& from them, fome conftant quantity at pleafure, which, in particular cafes, is afterwards to be determined as occafion may require. Ot this we fhall take further notice in it’s due place. Thus, the complete integral of x, for example, will be x + 4, where @ fignifies fome conftant quantity. That of x°x will be tx? + a’; and fo of others. The reafon of which is, that, as conftant quantities have no differ- entials, but x may as well be the differential of x + 4, or of x — 2, &c. as ot x; fo x, or x + 4, or x—d, &c. may be the integral of x. The fame obtains i in any other formula. ) 5. The fame rule of integration ferves tor complicate differential formula, or thofe compounded of many terms; whether they have a denominator, whether that be wholly conftant, or contains the variable in it, whether it be | fimple and of one term, or whether it be reducible to fuch. For, in thefe cafes, the complicate differential formula may be refolved into as many fimple ones, as are the terms of the complicate, and then each of thefe 772 è #71 — e bx x + aax i comes under the given rule. Let the formula be ———_-—, 3 this will t } fa a aan” x : : | be equivalent to thefé two, ———— and ———, and therefore the integral of aa — bb aa = bb | a be tt thefe two formule will be the integral of the firft; that is, ——— 3 vl, mo i X aa— bb 77 GAax pf m xX aa—bb ~~ 533; A mena 4 bust ote fos it be Ree AI this is the fame as thefe two, —— — —— | ALR — CHX a@a—c xX x* a-C X ae? ; bux ate x | + ae ba > i or as thefe, = ————., and therefore the integral will be —- — ™ ¢ anc , p Z XA i sati. bxx 7 if nde oy ee 4 fi = %, the equation of the curve. Vou. II, Q From 114 ANALYTICAL INSTITUTION & BOOK Ill. From this conftruétion we deduce alfo this, which the firft fuppofes ; that is, the primary property of the logarithmic curve, that the ordinates are in geome- trical proportion, which correfpond to the abfciffes in arithmetical proportion. For, fuppofing the equal portions of the axis to be infinitefimals, the little arch OC, produced, will be the tangent NO, the little arch CD, produced, will be the tangent BC, the little arch BD, produced, the tangent KD; and fo of all the others. ‘Therefore the triangles OIN, CAN, will be fimilar, and therefore ~ it willbe OI. CA :: NI. NA. Thus, alfo, by the fimilitude of the triangles CAB) DEB ie willbe CADE BA. BE, But NI = BA, NA = BE; therefore it will be OI. CA :: CA . DE; and fo fucceflively. Therefore the ordinates will be in continual geometrical proportion. Hence, alfo, if we con- ceive the axis to be divided, not into infinitely little parts, but into finite and equal parts; becaufe the intermediate proportional ordinates, for example, between IO and CA, are neither more nor fewer in number than the inter- mediate between CA and DE, and thus of others; therefore IO, CA, DE, will be in geometrical proportion, correfponding to the abfciffes in arithmetical proportion, ‘Therefore, taking any two ordinates at pleafure, and other two alfo where you pleafe, provided the diftance between the firft and fecond be the fame as the diftance between the third and fourth, as would be IO, CA, RG, SH; then the firft will be to the fecond, as the third to the fourth. ‘The logarithmic curve cannot be defcribed geometrically, but only organi- cally, and therefore it is called a mechanical curve; and the impoffibility of _ being geometrically defcribed is the fame as the impoffibility of the quadrature of the hyperbolical fpace, as will be feen in it’s place. Wherefore the integrals of fuch differential formula as belong to the logarithmic curve, are alfo faid to depend on the quadrature of the hyperbola. - Hence, in the logarithmic curve, the portions of the axis, or the abfciffes taken from fome fixed point, correfpond to the ordinates juft in the fame manner as, in the trigonometrical tables, the logarithms correfpond to the natural feries or progreffion of numbers. ; Fig. 99. 10. This fuppofed, let DC be the | / logarithmic curve, the fubtangent of which is equal to unity, or, if you pleafe, is equal to the conftant line 4; and let the ordinate AD be equal to the fub- tangent, that is, equal to unity, or to the conftant line a, which is in the place of unity. Taking any abfcifs AB = x, make BC =. But the .cquation ist 3 ay 4 4 the curve Is “a Land tnerciore The integral SECT. Ie ANALYTICAL INSTITUTIONS, tro integral or fluent of a will be x. But « = AB, and AB is the logarithm of BC, or of y. Now, to make ufe of the mark / to fignify the integral, fum, or fluent, all which mean the fame thing; and of the mark /, which means the logarithm, it will be iui = 7, in the logarithmic curve, the fubtangent of which is 4. After the fame manner, it will be f a — #9, ty tiie losarith nic whofe fubtangent = 1 fe = Ly, im the logarithmic whofe fubtangent is 4; ce Ge = /b +, in the logarithmic whofe fubtangent is equal toa, That Is, . taking, in the logarithmic, the ordinate BC = AH = y, if to it we fhall add HK = 4, and if we draw KG parallel to the afymptote, and draw GE parallel to AD, it will be GE = y + 4, andthen AE=/5+y. 11. From the nature of the logarithmic it is plainly feen, that whenever the quantity is infinite, of which we would have the logarithm ; which quantity will be reprefented by an infinite ordinate in the. logarithmic ; ; then the line inter» cepted in the axis, between that ordinate and the point A, will alfo be infinite, that is, the logarithm will be infinite. And if it Mhall be equal to. the firft ordinate AD, that is, to the fubtangent, the logarithm will then be equal to nothing. And if it thall be lefs than AD, as if it were OA, the logarithm will be QA, and therefore negative. And. lattly, if the ordinate were = o, the logarithm would be negative and infinite.. If the differential formula were: Co HE sal > the integral would be — / yi And if it were — TTM the integral: . would be — Ja a+ y we - [fit were — ae the eed would be ia sy y > and if it were — | va the integral would be —/a—y. Thefe logarithms are to be STR in the logarithmic of which the fubtangent is unity. The reafon of this is, that as the integral of = =— is /y, fo the differential of /y is a Essi) . And, to-fpeak in general, the el ia of a logarithmic quantity is. that fraction, the numerator of which is the product of the fubtangent into the: differential of the quantity, and the denominator is the fame quantity. Thus, the differential of — /4+y will be — —4—. The differential of Ja —y will be — 2. The differential of —/a—y will be 2, fuppoling the Q2 fubtangent. 116 ANALYTICAL. INSTITUTIONS. BOOK Ilf. o fubtangent of the logarithmic = 1: and whenever it is not fo, the numerators of the differentials muft be multiplied by the given fubtangent. 12. But, becaufe the logarithmic has no negative ordinates, it would feem that we cannot find the quantity which correfponds to the expreffion / a — y, that is, what is the logarithm of 4 — y, when @—¥ is a negative quantity, or when y is greater than 4. But, in this cafe, it may be obferved, that | la—y and /y — a are the fame thing; and that in fuch a fuppofition, when y — @ is pofitive, it may be the ordinate in the logarithmic; and, indeed, if De — y? and if we difference we difference the firft logarithm, we (hall have — the fecond, we fhall have 3 - 73 and changing the figns of the numerator and denominator, it will be — ear , the fame as the firft. 13, Other properties concerning logarithmic quantities may be deduced from thefe of the logarithmic curve; and firft, that the multiple or fubmultiple of a logarithm fhall be the logarithm of the quantity raifed to the power of the given numbers! Tia ieee lati gle let stl co de igo pe dee gone lx; | CRETE | I VELA. e e © o . e mini. il = lx ; and the reafon of this is, becaufe, in the logarithmic curve, if we take any ordinate whatever, OP = y,. Fig. 99. | (Fig. 3.) whofe logarithm is AO; if ! 3 AO, OS, SV, &c. be equal to each other, then AO, AS, AV, &c. will be arithmetical proportionals, and the ordi- nates AD, OP, ST, VI, &c. will be geometrical proportionals. Wheretore, putting AD equal to unity, OP = y, it Will (bes fo ge y? V Lis 9°, 8c. Rat AS, the double of AO, is the logarithm N D of y°, or Jy? ; and AV, the triple of AO, n is b*. Sothat2y = sb = ty, Bec: nae ee Thus, aifo, making AO = Jy, and bi- re . crt ; IPAS DO DA By . ° ° ° - LL AMOQS BVE feding ir ac M, it will be MN =y3, | and .theretore AN" = TAO, that=15; sly = ly?, In the fame manner, making QR = y, and dividing AQ into three equal parts in M and O, it will be MN = #¥y =y7. But AM = 35, and therefore 2/y-= lys ; and, in like manner, of all others, We SECT. Ia ANALYTICAL. INSTITUTIONS. oe We mutt here obferve, that the integral ope - is not only — è, as was ‘ feen before, but may be thus expreffed allo, la, or ly" s for, taking in the logarithmic any ordinate OE and ia A= oe it will be, by the nature of the curve, OR AD yD. OA; that is, y.1 : SOA = But OA is the negative logarithm of OP, that is, of sa and is alfo the e of QA. Therefore it will be +f = ar = ly°°; that is to fay, the nega- tive losarithm of any quantity whatever will be the fame with the pofitive logarithm of the fraction, of which the fame PELO the denominator, or of the fame quantity with a negative oo Thus it will be — my = — 713 I= ph y | 14. Moreover, the fum of two, three, &c. logarithms will be equal to the logarithm of the product of the quantities, of which they are the pofitive loga- rithms ; and the difference of two, three, &c. logarithms fhall be equal to the Jogarithin of the fraction, the numerator of which is the product of the quan- tities, of which they are the pofitive logarithms, and the denominator is the product of the quantities, of which they are the negative logarithms. For, becaute it. 1 OP = y, OR Booz ito wilh be. AO = fy, AG Sie. Lake QB = AO, it willbe AB = ly + /z. But AB is alfo the logarithm of BC, and, by the property of the logarithmic, BC is the fourth proportional to AD, OP, QR, that is, = y2; therefore it will be AB = Fy + lz = ay. Let there be another ordinate MN = p, and take BV = AM; it will be AV = AM + AB = /p + yz; but AV is the aap OF vin aud: VP — pee. Therefore /p + :/y + & = Dyz. | | Now make QR = z, OP = y, and take QM = AO; it will be AM = AQ — AO = /z— ly. But AM is the logarithm of MN, and, by the fame property of the logarithmic, it is MN = ie . Therefore AM = lz — ly ve + .. ;Let.there be another ordinate BC = p, and take ZA.= BM. It will be YA = — AB+AM=—p+ i. But XA is the logarithm of II, and 2M = —, (becaufe it is the Li proportional to BC, MN, AD >) shblefore lz — si a tay n i, 15. As 118 ANALYTICAL INSTITUTION & BOOK III, 15. fis in other cafes, fo alfo in thefe integrations by means of the loga- rithms, fome conftant quantity fhould always be added, that is, the logarithm of an arbitrary conftant quantity, which is to be determined afterwards as particular cafes may require, 16. But when the differential formula propofed to be integrated are fractions with a complicated denominator, fome cafes may be given in which it is eafy to have their integrals by means of the logarithmic, and this will be as often as the numerator of the fraction fhall be the exact differential of the denominator, or as often as it is proportional to it. And, in this cafe, the integral of the formula. will be the logarithm of the denominator, or it’s multiple, or fubmultiple, or proportional to that logarithm. 24% 2x —~ will be / aa + «x; the integral of — ad Tr XX Thus, the integral of AA — MI sata PNR will be 7a? + x; the integral of will be / 24 — xx; the integral of ' cepa A , 2. : xx will be 2/24 + xx, that is, / aa + xx) ; the integral of re will be 3/ a + x3, or Axx a+ Xx be :/ aa + xx, orlaa + xx)? the integral of will xx as + 43 | SERE ERIN CONTI ° e FAX di « 1/73 È i¥a? + x*; and, in-general, the integral of ——— = will be + — la +b x’; mM rr enews sei De N. n n 20K that is, + mla +x ,>otlate ati cot » Thus the integral of a | Ale NIC: will be Jax — xx; the integral of tico will be /4/ ax — xx ; and thus si AX —- XX all others whatever, taking thefe logarithms from the logarithmic, the fub- tangent of which is = 1. 17. But if the numerator of the fraction be not of the form we have now confidered, though the denominator may be fuch ; and that no one of its linear components is imaginary; that 1s, when all the roots of the product from whence it arifes are real ones ; then we may proceed in the following manner, 18. And, firft, the roots of the denominator are all equal to each other, or MI è. they are not. If they be all equal, as in the formula —— , make x + @ sal a = %, and therefore x = 2, « = 2 Fal, a+" = 2°; and fubftituting thefe SECT. I. ANALYTICAL INSTITUTIONS IQ 3 ——m : _thefe values in the formula, it will be —t“—-~= a Wherefore, actually raifing a % Fato the power m, each term can be integrated, either algebraically, or, at leaft, tranfcendentally, by means of the logarithmic. Whence, inftead of 3, reftoring it’s value given by x, we fhall have the integral of the formula propofed 271 è x Xx «x +a)” : | 13% , ce è , Let it be, for example, => Put.” — 4 22, and therefore x = z, a = 23 + 342° + 3003 + a3, x — a\? = 23; and, making the fubftitutions, 3aa 733 22 O 3% As x Se See de ela dia © by! dieceration, ‘9° LL sie a ade EA Zz3 it will be cu i and, inflead of z, reftoring it’s value given by x, we fhall have at lat 903% gaa a3 sai i “fa piego — ——— ; which integral, bein IT 1 x — A 2%. oak S differenced again, will reftore the formula propofed to be integrated. 19. Now, if the roots of the denominator fhall not be all equal, but either all unequal, or mixed of equal and unequal ; then it will be neceffary, firft, to prepare the formula, by making the term of the higheft power of the variable in the denominator to be pofitive, if it fhould happen to be negative, and then to free it from co-efficients, if it have any. Then, if the variable in the | numerator, when there is any, be raifed to a greater or equal power to the higheft in the denominator, the numerator muft be divided by the denominator fo long, as that the exponent of the variable in that may be lefs than in this. Laftly, the roots of the denominator are to be found algebraically. Take this Gax formula — rpms for an example. Changing the figns, and dividing by 4, veo Zaaù ‘ Laax = | it will become —*——.,, that is, .; Again, let the formula wav 444 e—taX x + tia | , aax +e ge Paar 340% propofed be (PRA ade E de ae dividing by 2, IL will be Trooper A Laax x +24 Xx +% to a higher power than in the denominator, we muft make an aual divifion, by which we fhall have both integers and fractions. ‘The integers muft be aie in the manner before explained; the fractions in the manner fol- lowing. that 1s, 5; If the variable fhould be in the numerator, and raifed 3 20, Let 120 ANALYTICAL INSTITUTIONS, _ BOOK Ill. 20. Let the fraction be — e +24 Xx + d fractions, the numerators of which will be the fame as of the firft, and the denominators will be thefe : Of the firft, it will be the produét of one of the roots into the difference of the conftant quantity of the other root, and of the conftant quantity of the fame root: Of the fecond, it will be the product of the other root into the difference of the conftant quantity of the firft root, and ; I fay, this will be equal to two Laax A of the conftant quanti ofcthisy feconda xoot. 1 us, = _L'a e +24 Xx a + d . And if the roots fhall be three, four, &c. Laga; doo. 5 4Ax Qaax eee Oa eee ea Se. x + 24 X db 2a x+ dx 24 — d proceed always in the fame method. And if the fractions found after this manner fhall be reduced to a common denominator, they will reftore the firft fraction from which they were derived. : | Now the ‘integrals of fuch fractions fo fplit, which will always be in our power to find, fuppofing the logarithmic curve to be given, will be the integrals la of the formula DEE ae it will be /$———— LESS GRES Se 3%! x+8 x +24 Xx + x +6 4G if È ta a+b a e+b . ER game > + 24; that is, = ici SINO, I: rere Tl the logarithmic whofe fubtangent = a. : Faax . “ e Laax Let it be ——== ; this may be fplit into thefe two, —_-__ «+34 X x — 24 e+ia x —ja-tla dana dad 1.° zA40Xx Ax z4X e e i, OF — ——~, and therefore it will be Ig PE x — 74 x + 14° i 17% — Lo ii f 338 — sy or ly , in the logarithmic of which “a+iaX «—7e Rah ae ta the fubtangent = 4. ; this may be fplit into three, ax a3x ax + pen rm] s+ax = b-axc-a x-bXxa+bxc+8 x+cxa-6cx —b—c a3x aa and therefore SEEM onesie ae |e Pa 2 Nc AT gine emer eg? Geb a SC meal aa pi eee xX lx — bh — x /x +6, in the CE whofe fubtangent — Ge Let 5) sa Ae SECT. Ie ANALYTICAL INSTITUTIONS, 321 ® aad a3% e Gc as DI ; o . i Let it be ——— , that is, —— °—; this may be fplit into ELIA *«taXatT-aXaxto — ae Li 3 — ax Ghefe three: se cere as eee en x +a X — 24 X 0-Td4a X_- a X 24.X 0 + x+o X a-0X —a—0 9 1 e ; int ax ax ~ di . 0) hg a ax that-ispprdic tthe bb AL and therefore it will bel ft età 2Xaxdka 2Xa—-da | — GAN — Il xx — aa, that is, / — , in the logarithmic of fubtangent = a. 4% — GA E Po. 22. If the denominator of the formula fhall be mixed of equal and unequal dai then the formula mult be confidered roots, as, for example, ———— Cae 4) xX | c ele 34 e ‘. 3 È arx as, if, it; were —— , and being fplit as ufual, ic will be 3 CmoxX wpe a—bX x +e ax arx ee —————===; and then, multiplying the denominators v—bxXct+b * x + cX — dec | a3x x — be Kate by x — è, the other root of the propofed formula, it will be a3x a3; | = = + ; but the firft term of the Zomoge- x — db) X c+d abe xX ed xX —b—c è neum comparationis has all the roots of it’s denominator equal, and the fecond term confills of roots all unequal; fo that, both of them being managed as a3k before, we may have the integral of ii which will be partly alge= x + c a3 braical, and partly logarithmical, that is hi gra ? P y ‘05 ; i x — b ad x bc taking the logarithm from the logarithmic, whofe fubtangent = 4. If there fhall be a greater number of equal roots, the operation muft be repeated in the fame manner, as often as fhall be neceflary. . ‘23. That cafe remains to be confidered, in which the fraQions have alfo in the numerator the variable raifed to any power; always meaning, as has been already obferved,' that the power of this variable in the numerator be lefs than the greateft which is in the denominator ; and not being fo, it muft be made fuch by actually dividing. In thefe cafes the formula muft be treated in the fame manner, as if in the numerator there were no power of the variable, fplitting it, in the manner before ‘. Xplained; into fo many parts, as are the roots of the denominator. Then, if the exponent of the variable inthe numerator of the given formula be an odd Vou. II, 7 SV R number, 122 ANALYTICAL INSTITUTION Se BOOK I1f- number, let the figns be changed in the numerators of the fractions found ; and if it be an even number, their own figns mult remain to the numerators, After which, every numerator muft be multiplied by fuch a power of the conftant quantity of that root, which is in the denominator, as is the power of the variable in the numerator of the propofed formula, prefixing fuch a fign to ‘that conflant, raifed to that power, as it’s natural fign requires, which it has in the denominator, | bhxx Let the example be —————-——. This being confidered as if there were “+axXx#—a@ no variable in the numerator, it will be fplit into thefe two, n + Vv +La X — 24 bb | i: | i —=——— ; but, becaufe in the numerator there ts the variable raifed to the x — a XK 24 power denominated by unity, or the firft power, the figns are changed in the numerators, and are multiplied relatively by the conftant of that root which is ‘in it’s denominator, that is, the firft by a, and the fecond by — a, and we {hall : sees bbe x — 6 : bi fect blbxx * bbx x a To RR E a that is, bbx 4. etal oe ag igen Ra x + a x — 24 *X — a X 24 2Xata bbx È : bbxx e een Ue ——— ; and therefore it will be /———_——- = UV x + a + blyYx — a, 3 XX — A «La X a — a or bl4/xx — aa, in the logarithmic of the fubtangent = 42. Or otherwife, it will be 22/4/ xx — aa, in the logarithmic of the fubtangent = 1, But it was needlefs to reduce this formula to two fraGions; for, as it was . bbax VY — aa therefore, without any other operation, the integral will be 22/4 ax — aa, (as is faid at § 17.) in the logarithmic whofe fubtangent is unity. ; the numerator is exactly half the differential of the denominator, and x 2 e atic es e e Letit.be —— that is, — Re Ei ee 5 te nu — bx3c + aaxrx + aabxi merator by the denominator, we fhall have xx + —; eo bob dividing again the term -~ dx3x by the denominator, we fhall have i . aaxric + bba%x — aabba bes —— = x — dx i diri ILL sv — 4a X «+b xa — da X x + db are integers, and the laft has not the variable in the laft term of the nume- rator, and therefore may be managed; fo that there only remains the term . Now the two firft terms At {till to be reduced. This being confidered as not having the Ksfeda Xx d : CUOR variable form, SECT. IL. ANALYTICAL INSTITUTIONS, — 123 Sini aa + bb x & Ga + bb Kk wariable in the numerator, will be —_———_—_- = iat las dieta | xe—aa X x + db a+b x — aa + bb aa + bb x & aa + bb x x st —— —————t— È ee eg and therefore it will be ata Xx — 2aò + 244 a — a X 20b + 24a. aa + DI x att aa + bb x bei + aa + bb x aax x+ 65 X xx — aa Mote Boia e OP xt a. X .— 2ab + 244 bb x ai 4; i ; aabba - ie cala e AL hence, laftly LA = XX -— DX — x — 4 X 2ab + 2a? KA GA X e+ dI wtb X xK«% —aa aa + bb x bbx ; aa + bb x aax aa + bb X aax ; and if we e nee «+0 x — aa + bb x + ax — 26b << 204 a — a X 200 + 204 aossa , in order to have, finally, the integral a+ K #x—aa would ftill {plit the term — ; i 4 b4x of the propofed formula, it will be —— © TX — DX = Xx — aa XK x + db wbx —aatbb 4: 4, 4 2 4 ah, hues ee cea ae 0a: Then, by integration, we fhall have x +a X 2aa—2ab x — a X 2ab +2aa dx = te — oe a xX le fb +" Xl ta + XE — GA X &£ + b at 2aa + 2ab tangent = I. x Za == a; taking fuch logarithms in the logarithmic of the fub- Now in this, as well as in all other integrations that can be made, we are to conceive a conftant quantity is to be added, though, for the fake of brevity, I here omit it; but it will be enough to mention it here. È 24. But differential formule may have, and often have, fuch denominators, of which we cannot find the roots algebraically ; yet, notwitbftanding this, we may make good ufe of the Rule of Fra&ions in thefe cafes alfo. For we may treat the denominator as if it were an equation, and, by means of the inter» feGtions of curves, may be found geometrically, in lines, the values of the variable, juft after the fame manner as folid problems are conftructed. And fuch values or lines may be called A, B, C, &c. with pofitive or negative figns, according as they come out pofitive or negative. Every one of thefe, being fubtra&Qed from the variable, will form a root of the denominator in fuch manner, that the propofed differential formula will be.converted into one of this LA VA a- AXKE4B ER al &C , and with this we may ftidaeetl in the fame manner, as the operation has been performed in the cafe of algebriiical roots. R 2 25. 4 124 ANALYTICAL INSTITUTIONS BOOK IIT. 25. It may be eafily obferved, that the rule here produced ferves only in fuch cafes, when the roots of the denominator are real ; for when it is otherwife, the formula being fplit into other fractions, fo many of thefe will be imaginary, (and confequently the integrals will be imaginary,) as are the imaginary roots in the denominator of the differential formula propofed. 26. Therefore, when the denominator of the propofed difserential formula is compofed of imaginary roots, either wholly or in part, there is a neceflity of having recourfe to other means. And, in the firft place, let the given formulz have their denominators of two dimenfions only, that is, of two imaginary bbc: ; roots; and let it be, for example, meee es: The integral of this formula, and of all others like it, depends on the rect- fication or quadrature of the circle ; I fay re&ification or quadrature, becaufe, one of them being given, the other is reciprocally eiveo alfo. K Wherefore let ACG be a quadrant of a circle, the fags A — Beatie poco CO — ri it will be AB = D aa o Fig. 100. Gr | ‘CP CA Ai RO Db cre. i WE oe ; Vga Aan NV aa + xx Drawing AK infinitely near to AD, then EO will be the fluxion or difference of the arch CE. And from the point O drawing the right line OM parallel to EB, and . EH. parallel to AC, then will HE be the differential © of CB, and HO the differential of EB, and therefore ro EH = — and HO = —££ -. Thence the little arch EO = aa + xa \i i aa + xi ; ‘esi Ann + arene pra aax | La be | a / HE? + OHg, will be = v CELL di tit; Whence the integral will be the arch CE of the tangent CD = x, and of of the formula ——~ aa | xx radius CA = a. - bba ì . . ° Now I refume the formula n; multiplying the numerator and denomi- at xx —~— ; but the integral of = is the cir- bb nator by aa, it will be eo x ia "i ant xe bbx ab xe = to the fourth proportional of 44, of 3, and of the arch of He circle with radius = a, and tangent = x. cular arch, which has for it’s tangent x, and it’s radius = 4; therefore /- 3 Lect se ANALYTICAL INSTITUTIONS, | 125° Let the formula be ii as, by multiplying the numerator and deno- : ’ 3 ‘ ; È am aby ihe 7 minator by 4, it will be equivalent to this other, —- X Py it will be p patter = to a fourth proportional to nb, to am, and to the arch of a nxx + nab | circle, with radius = Wad, and tangent = x. And fo of all others of a like kind. i 27. On the contrary, therefore, the differential of any arch of a circle is the produ& of the fquare of the radius into the fluxion of the tangent, divided by the fum of the fquares of the faid radius, and the fquare of the tangent. And, as a conftant quantity is always to be joined to other integrals or fluents, fo alfo to this of the rectification of the circle; to have the integral complete, we muft add a conftant arch of the fame circle ; for the difference by which the arch, thus compofed of a variable and a conftant, can increafe or diminifh, can never be any other than what belongs to the differential of the variable arch ; fo that to the fame differential may belong, by way of integral, the fum of the variable arch, together with any conftant arch of the fame “circle, Let us fuppofe that x is the tangent of an arch of a circle whofe radius is 4, and that 4 is the tangent of another conftant arch of the fame circle; we know that the tangent of the fum of thefe two arches (Vol, I. $ 108.) will bed aab + aax forni . But the differential of this, multiplied by the fquare of the aa — bx radius, and the produ& divided by the fquare of the radius, adding the fquare aax aa it’s integral will be 4 x ax—ax + 2612. a } Vax —xu + bb ux n 2aAKX recs Eg 3 — x 6/08 — ax’; Let it be xxx — 2axx x 4/45 — ax’, that 1s, make 4/4° — ax? = 2, and therefore 2* = a° — ax’, and gara — 24xx 4tetln | ANALYTICAL INST TO Tron. — oy = 422%; and making the fubftitutions, we fhall have +2*z, and by integrating, a - @ pass e i ®. 3 : e . 3 «425 ; and, inftead of z, reftoring it’s value, it will be 4, x «° — axx)*, ZGIKK — 248% If the formula were —————--— 3 V 23 — axe % i Di ‘ ; 3° » the integral would be 4 x x° — axx)*. | | : als 2 di x Let be’ 2047 &/ xx + ad)”, that is, aux X xx +aa7; put sw -+aa\¥ = 2, Pa DS nee % i i e stg Zeasf- ° ° and therefore «x + aa = 2*; and 2xx = #2% ‘3; and making the fubfti= ae . . La tutions, we fhall have 4272, and by integration, 323 ; and, inftead of z, re- ftoring it’s value, 3 x wr + aa X Wn + aay’. È) DES rmula were — Tf the fomniula were rn the integral would be 39 xx + aa. | | a L 4 | : fim TE. 12 Vu. nes È or ‘And, in general, let the formula be pu x Xx + a” '” , in which p and . n da ‘m may allo be fractions; put #° + a" — %; and therefore z* =e’ +4”, % and mx” = — a %; and making the fubftitutions, we fhall have u : u-+n DIL Reni. ° 0) ® pa gti A e # : ® Pig x %, and by integration, Tm X3 ” 5 and, inftead of z, reftoring ; 3 ani È pu mt m Me LL a a . it’s value, the integral will be — eager a, +a Xx + i | pe lx If x were negative, or if the formula were —-, in which # is now a i m\ 4 Ue 2 773 UA “. E bg x ° pofitive, we fhould have the integral ae = Hence we may form this general rule, that the integral of fuch a formula will be the quantity under the vinculum, the exponent being increafed by unity, and dividing it by the exponent fo increafed ; or the integral will be a proportional — to this, according to the proportion which the differential quantity out of the - vinculum will have to the precife differential. a su Vou ii. | S | _ 32 But 130 ANALYTICAL INSTITUTIONS. BOOK III, ve : i oT Tiiwnm Ts 32. But ftill in a more general manner: Let the formula be px’”” x _x UA cers aed ia ete > ° if ° ° x” 4.4”), fuppofing r to be a pofitive integer. -It will be equivalent to this n n, 13 «e 173 bi i : r #7 27 ua other, pa Xx do 1% x + a”; make, as ufual z=x +4 | 5 % 7 u I o eni el. 74 —— mm J, and therefore #° + 4” = 2", and mx" '% = a Be 23 and, becaufe uv ; “. Ng] ULI SE 773 Ti == 773 FIERO mM e eo Bt a, it will be « = 22 —4' ., Therefore, making ania: Se ie u the fubftitutions, we fhall have px zn — 4% X —z nz. Now, fuppof- ing r to be a pofitive integer number, then alfo r — 1 will be a pofitive integer % number ; and actually raifing x “ — a” to the power r— 1, each term will be algebraically integrable, in which integral reftoring, inftead of 2, it’s value given by x, we fhall have the integral required. If x were negative, that is, if the formula were da ——, in which # is now pofitive, making the fubftitutions, it will be p x z* — a x BS mae A : sai ES an n %, which is likewife integrable, In all thefe cafes, if the quantity under the vinculum, inftead of being Mm 772 i . #° + 4°, had been x” — a”, or a” — x”, we might proceed after the fame manner, without hindering the operation. By this method we may find likewife, that it will be fasts x af e + fr = mp + felt, SS 2k e Xx e 4 fia. Mc + fa nf Saw N do + È RCN) : è di. SECT.I © ANALYTICAL INSTITUTIONS. 131 a Sax di xf e + fx ee poner) ny x q x e + fa : aes a ae ze i fa” x axe + fed 2 ‘59 er Li m fg? i 1 fan tee e Fe ae AE th Lala me Xe + ae 105f*m 3 774 1 mn 2m ms axre __ 160 — Befa + Offx fer . mm dl de e + fai. N e + fa ey And fo we might go on as far as we pleafe. 33. Likewife in the cafe, in which thé variable out of the vinculum fhall be in the’denominator, the formula will be algebraically integrable by the help of two fubftitutions, provided the exponent of that variable out of the vinculum : ; i {hall have a certain condition; thus, let the formula be SO i, 3 Ad API 5 aad ° ‘Then make =, N= eS xy mn | | ; ye Then making the fubflitutions, the formula will be VA vent Te eee aay a tide ee oe te bei —_ — XxX ——— x J 7 D aS eee 3 —— rien 3 MAL I ym ty yy pa” a amn > “) SS 5 2073 È + 2 : È arm + 2110. | a U a formula which has the conditions here required, and which may be integrated algebriically, by means of the fubftitution mentioned at $ 32. ; î » 54 ° ie g If the formula propofed were —=——, that is, —!——; this having : | x4N ax + xv Py: Ja + % i ; the conditions required, will be algebraically integrable; which is alfo to be obferved of others. 9 2 | ov ee But 32 ANALYTICAL INSTITUTIONS, BOOK IIf. 34. But here it may be cbferved, that, in the general formula, it may alfo be # — 1, in which cafe the power of x" + @ will be rational, that is, inte. grable. [qu. integral. ] Alfo, in this cafe, fuppofing # to be a negative number, (for when it is affirmative there will be no difficulty,) we may make ufe of the fame fubfti- tution, and of the fame method, by which the integrals may be found of fuch formule, the integrals of which will not always be algebraical. For very often they will depend in part upon the quadrature of the hyperbola, that 1s, on the Jogarithmic curve. , Therefore, by a known method, we fhall find that Bis pee ee genes K I 71 VA ei SIE Hi pa a one DE Mel + G. J 24 nl ni\® m ne i x + a | 171 = J ut x ie I m mt a Se ia +s + Py 772 mm YA 778 x ba i mxa ka E 271 nt n. m 2m x3 a +x 9a. dali a 278 mh \* M Mi m Mh: Aas! moxa ba a mmmmaraiel e 7 m\® amXa +3 Ly Q JPR + Sa x I tats ; e” Ta m \8 m m\* a +x mx aaa amxa tax Hi wm SI A RR a 774 2 ; x x ni 722 m 2a a i ae — aor la + x + Ta. E ne pie tn, e &c. m n 7 772 mn a +bw mxa +e 27 Xa +e ” 35. But the manner of° proceeding will be very different when the propofed differential formula containing the radical, are not fuch as that the quantity out of the vinculum fhall have thofe conditions before mentioned. Thefe formule may always be delivered from their radical, provided they contain but one, which is that of the fquare-root, and that the variable under the fame does not exceed two dimenfions. Now, for thefe there will be occafion for fome caution in the choice of fuch fubftitutions as are to be made, that they may be freed: from radical fignss When this is done, we may go on to integrations, either algebraical, or {uch as depend on the quadrature of the circle or hyperbola, after the manner already explained, if they come under the given rules. If not, we muft have recourfe to other methods, which are to be given hereafter. g If È) sect. 1. . © ANALYTYOAL INSTITUTIONS. | di; If the radical of the propofed formula were /ax + wx, or ax + 4X3 this radical may be made equal to = , Meaning by za new variable, and by è any conftant quantity whatever, | i | If the radical were \/aa + xx, make it + %, OX Z If the radical were 4/aa — xx, or fp — xu, put the radical = W/p + — ,or = vfp — 25. From fuch equations the values of x and x may b be derived, expreffed by z and conftant quantities; which values are to be. fubflituted in the given formule, and we fhall have other formule free from radicals, and given by z. In the integrations of which, if they can be had, the value of z by x being reftored, we fhall have the integrations of the propofed formule. | | 36. If the quantity fhould have three terms, that is, the fquare of the variable with the rectangle of the fame into a conftant, and befides, a term which is wholly conftant; then either the fecond term muft be taken away, after the ufual manner, as in the common Algebra; or, if the conftant term be pofitive, asinVaxx + ax + aa for inftance, however the others may be pofitive or negative, provided the quantity be not imaginary ; make g/ xx + ax + aa = a+ —-; and if the conftant term be negative, as, fuppofe ax + aw — aa, it may be made Vxx + ax — da x +2 From hence it may be feen, that the whole artifice confifts in. comparing the radical quantity to fuch other quantity compofed of the given variable, and of a new one wit) conitant quantities, as that an equation may refule from thence, from whence we may have the value of x and of x, free from radical figns. Let there be propofed to be integrated the differential formula x3x/ax — xx. ORRIN HR ) HE : Ib ut 4x — xx == —-, and therefore ¢— ¥ = —_, th: Licia DE zabbzz . ab Tr xR abz «e Mei Mai BX + 55) 3 zz + bb)” * i; è zz + db Make the fubftitutions in the propofed formula, and it will be — 20/2, ‘ age a formula which, though free from radical figns, yet, as to it’s integration, will not fubmit to the ufual methods. and: x, = | Let tab ANALYIICAL, ANSTITUTIONS.. > BOOK Ill. - Let it be ipa . Make Wax + ax = n, and therefore it will be BV AX HE ; abb 20, . 2abbez Az ue scr sorse ago PA 33 — bb zz — bb aan È Pea ‘24% È i fubflitutions in the propofed formula, it will be — 3 and by mp Vax + se = = E - Making the zz — bb 24% x , grea, den and, inftead of 2, reftoring it’s value by +, 1678 ie Il x“ N ax ae + xx 2ax/ ax + BN (ei —__—6@—@—€€@ ( Tite : A ae aa sé za pice oe gp performed by the foregoing rules, it is ; = =e 74+ 12 + tala — az, in the logarithmic with fubtangent = 1. And, inftead of z, reftoring it’s value by x, it will be, laftly SEI a 33S an Ig ee di ENTI eee da +8rm BV danza 0 | + iV we + cx — aa + tala + 2% 2V&% + ax — aa, in the logarithmic whofe fubtargent is unity. | 93 37. As Fig. 101. _M ia a n n re a ‘4 È é ; a SECT. 3, ANALYTICAL INSTITUTIONS. 135 37. As to fome radical differential formule, the trouble, indeed, would be - -fuperfluous to tranfmute them, by means of thefe fubftitutions, into others that are free from radical figns, in order to prepare them for integration ; and fuch are all thofe which of their own nature require the quadrature or rectification of the circle. Wherefore let there be a femi- circle GMD, (Fig. 101.) it’s radius AD=a, © AB = #, whence BF = / 4a — xx; and drawing CH infinitely near to BF, it will be BC = x, EF = ———. Therefore MV aa — x the expreffion of the infinitefimal rectangle GC api Be D BCHE will be xy/44 — xx, and therefore SXV aa — xx is equal to the {pace ABFM. Alfo, a will be the expreffion of the infinitely little arch FH, and AA = XX therefore / a = arch MF. And if the little arch FH be drawn into aax half the radius, then papas will be the expreffion of the RR little fector AFH, and therefore / — È = to the fector AFM. Naa— x In the fame circle let it be now DC = x, and CB = x. It will be CH = x RATES ATE IERI ax ever is 4 SS pa: V 24% — xx, EF = = - Wherefore /xx/2ax — xx will be equal to ; ZAK — KA i the fpace HCD. — And thus HE = = arch HD, and / be fector N 20x — xx aN 24x— ax AHD. Infuchasthefe, therefore, the trouble [of transformation] would be needlefs ; ‘ 4 . li PAIR ee eae % à for, in the fir& cafe, we fhould make VW aa — xx = 4 — —, and therefore x = b 2abu è . _ 2abz — 2abzzz renne i wz __ abb — azz le ca a aa wee gii ù making thefe fubftitutions, it will be —— = = s a formula for the . dA «n XX rectification of the circle, the tangent of which is equal to z, as has been {gen already at $ 26. | Alfo, ig CANALE PECA LO) INSTETUTFONA.. BOOK II! Dai . aa aabs ; ‘ ; | POS ECT Cire Seng aM formula which requires the fame Vara zz + etna; pigli DE rt _ 2aab& x Th — 37 rectification. In like manner, it will be xWaa — xx = SUE. : 4 f ZZ formula which, though at prefent we cannot manage, yet afterwards we fhall find ro depend on the fame circle. | ~ » , | Le ELL Te Do 2abb : : Did ayer Te! herefore x — -——_~ In the fecond cafe, I put 4/2 57» and therefore « 20 4abbzz in az 2abz ? ; — ro» id. "a ee eae Te eT. “Ino » di mr and VS 20K — Xie ; nes ag Making the fubftitu ti it will be = = Pas the rectification of the circle tions, It wl "== Fees È caXx arbz ; È » erate tro i ee he recuficationi of the ‘ci before, In like manner, it will be x/2ax—xx = a Rial which includes tl : 3 GR in VA TT E ch includes th Cay zz + 60)3 d eee fame circle. È 38. If our differential formule fhall be compofed of two radical quantities, in this cafe the operation will be double, but ftill it will fucceed as well. For, in the radica] quantities, the fecond term may be wanting, or it may be taken away, and the formula may be multiplied by an odd power of the variable; and that by putting one of the radical quantities equal to a new variable. And thus the propofed formula will be reduced to another, which will contain one radical only, and which confequently may be managed in the ufual manner. Let it be, for example, —~——-——. I put Waa + xx = y, and therefore xx yy — aa, xx = yy. Making the fubftitutions, it will be ee ak NV yy — aa + bb : yy aayyy . i thats, (ieee ae gg — , each of which we know how to Vyy — aa + bb N yy — aa + bb | Manage. 39. If we confider a little this manner of operation, we may eafily perceive, that, in thefe radical formule, it will not fucceed in general, that we fhall be able to free them from their radical vinculum, except when it is a fquare-root, and the invariable under the vinculum does not exceed the fecond dimenfion. I fay in general; becaufe, in feveral cafes, it may fucceed, whatever the radical | may following formule, the fir& of which is this, pina miti - SECTS, ANALYTICAL INSTITUTIONS. 3837 may be, and whatever the power of the variable may be, which 1s under the vinculum. And certainly it will, in all cafes, be comprehended in the two i | E ° #3 Uci # be b : o iaia #, t, are pofitive integers, and may alfo be nothing; and this obtains, by I n m aa | Wi. a nm. Pa cast” 1: making y + = 2 gwhencey 3S 2 db, Ja a5 and making my i si i n 1. +1 n pina dh I i+ P a . NZ 5 X 2 _ the fubfitutions, it? will bee cao eae tts 9s, Sot + But | . tam +1 Xm my è my = 2° — 3" ; and when ¢ is an integer, the power ¢ + 1 will be an integer, fo that the propofed formula will be free from radicals. If ¢ were negative, the formula would be the cafe confidered above at § 32, which has an algebraical integration. In other cafes, the integral will depend on the quadrature of the circle, and. of the hyperbola, as will be feen in it’s place. _ 2 i aa ise The fecond formula is yy xy +5" ?, which, when nil is a whole number, may always be freed from it’s radical figns, either in the whole, or, at leaft, from radicals of the complicate quantity, which will be fufficient. Where- | P fore, make y” +2”'? = x, and then it wi be y° =z‘ —S", y = : eB — J f. a eta e» = Vf: falle #6 122 % x x 32 atto (DI di ¢ ib id SESIA E; d Prc EY Seam a ni CT Dee A Se a Pp ar. Di 2" — 51": and making the fubftitutions, we fhall have the formula | sa na | ties ne ica n+t. i RT 7 | - But when —— Isaninteger, tne | the power De — 1 will always be an integer, [or o, | fo that the formula will have mon II: DR; | only 138 ANALYTICAL INSTITUTIONS | BOOK IIe. only radical figns of the complicate quantities. And therefore, when LI — 1 is a pofitive integer number, the integration, at moft, will depend on the quadra- ture of the byperbola, or on the logarithmic, and may be had by the given rules. And when - ri 2 quadrature of the circle, and of the hyperbola, and may be had by the rules which will be given in due place. : — 1 is a negative integer, the integration will depend on the 40. Now let us go on to fuch formule, which being fractions free from radicals, the variable is raifed to any power in the denominator, which I wilk fuppofe to be compofed of imaginary roots, becaufe in thefe only there is any difficulty. I fay, that as often as the denominator is reducible to real com- ponents, in which the variable does not exceed the fecond dimenfion, the formula may always be fplit into fo many fractions, as are the forementioned real components, each of which will be integrable, fuppofing the quadrature of the circle and hyperbola; and confequently the propofed formula will always be reducible to the faid quadratures, To do this, let there be propofed this aax formula, ————--———__—--———.._ Take a fictitious equation, we + ax + db x ae + cx + bc . A e Bi G. ° DI ; Mi ili, A a RSI + a ——, in which formula ax + ax + bb X xx + ca + be wx + ax + bb li nic the capitals A, B, C, D, are conftant arbitrary quantities, which are to be deter» mined by the procefs. Thus, if the formula were Ae: MEN La SEA PA EI we fhould make se + ax + Db x xx + aa x ate ù * Axx + Be Cxx + Dx Hx e it equal to Tian Se roy voga eee And thus we may proceed in the fame order, if the components în the denominator were more in number. When this is done, the terms of this equation are to be reduced to a common denominator, and laftly, by tranfpofition, the equation muft be made equal to nothing. Then, by comparing the firft terms to nothing, the value of the affumed quantity A may be found. And fo, by comparing the fecond, third, fourth, &c. terms in the fame manner, the values of the other capitals B, C, D, &c. may be found, expreffed by the given quantities of the propofed formula; which values, being fubftituted in the places of the affumed capitals A, B, C, D, &c. in the equation, will fupply us with fo many fradtions as are equivalent to the propofed formula ; and which, being reduced to a common denominator, will exactly reftore the formula at firft propofed. an | Of n SECT. I. ANALYT VOGAL INS TERT UT1LON5, 139 Of this we will take an example. Let it be propofed to find the integral of aan this foriinla - . Therefore I affume this fictitious equation xx + 20% — aa X xx + ada — aax at Axx + Be Cre + De Ae fo hen: | yeduce: the XX + 20% — da wu + aa mx + 20x — aa X x% + aa equation to a common denominator, and, by tranfpofing the term aaa’, I reduce it to o, and find it to be. | Axx + Bx°x + Aaexx + Baax + Cex + Dax + 2Daxx — Daax Sys + 2Caxx — Caaxx — daro Wherefore, from the comparifon of the firfl terms with o, we fhall have A +C—=o, or A = —C, From the fecond, B+ D + 2Ca = o, that is, putting — A iInftead of C, B= 2A4 — D. From the third, Ae° + 2Da — Ca* = o, that is, C= 4a 4+ ni From the laft, Baa — Daa — da = o, that is, putting, inftead of B, it’s value given by D and A, ir will be D = Aa — +, and therefore it will be C = 2°£2!; but C=- A, els SE i The. tas a and therefore A = pra Dit Be ay OS Ors whence we fhall eee ar tate aax | ae xe + ga Lal xXx ax i xXx + 24% — aa XK Xx + ad 44 X xx + 20x — da 4a X xx + aa” But, by making the fecond term of the denominator to vanifh, where there is occafion, the Lomogeneum comparationis is integrable by the quadrature of the circle and hyperbola ; the integral of which, by the given rules, will be found I to be qa IV na + Bax — ad Sh INa+a=y244 — rag Vita+y 200 4/ a4: oan ay lM He + aa, fubtracting, befides, from thefe logarithms the fourth proportional of 444, of unity, and of the arch of the circle, the radius of which is 4, and the tangent = x. Therefore the integration of this formula depends on no higher quadratures than thofe of the circle and hyperbola. 41. If, befides, the fraction fhall be multiplied into any power of the variable, He which power is pofitive ; as if the formula were Ms SPERA AN ee pani XX << 204 n aa X KK + aa i Atti: a By Gatta a Dai i it equal to Oa pe? and let the values of the capitals A, B, C, &c. be found in the fame manner as above, or you may work as if the faid power were not there ; and the refulting fractions may be multiplied by the Jeo ea faid È ite - 140 ANALYTICAL INSTITUTIONS, BOOK IIT. faid power, and we fhall have, in like manner, fo many fractions, which wilf not require any higher quadratures than thofe of the circle and hyperbola, and which may be managed by the rules already given. | 42. And if the power of the variable fhall be negative, that is, if it fhall be pofitive in the denominator, all the denominators of the refulting fractions. may be multiplied by this power, and they will acquire the form following. | =" b As, for example, see. This being refolved as if ax tax + bb x xa kaa x xt tir i Le: i x — 1. o x” were abfent, and then multiplying every term by x”, it will be a x / Axi + Bi Cxi+Dîi Tre ÉTAIT TITANI. I TE san I nn Aoi i IE > ARTES seer. RR STARE = 7% a ritira (OI ae RRND UE riparata! n pane e A: DER Ai sx + ax + bb x sx taa X xa bc cx + ax + bb xx sataa x a" x Lex a underftanding now by the capitals fuch values, as, being found by the foregoing method, fhall make the fum of thefe fractions equal to the propofed formula, The laft fraétion will have no occafion for any particular artifice, becaufe it’s integration is known by the common rules. | As to the firft, to clear up the example, let it be A = aa, and B = add, aanx + abbi whence it will be thus expreffled, =» Which is to be made equal to. xx +ax+bb x « Maxi + Nx were. + ft + Ex" 73 5, &e. Bx + ax + bb ee fh . And thus we muft go on till nw the laft term becomes conftant, that is, the laft power of the variable x muft have it’s index = o. When thefe fractions are reduced to a common denomi- nator, and all made = o, we fhall have the values of the capitals, as was done before. The fame thing muft be done in regard to the other fra&ion Cat + De , and thus, finally, the integral will be found of the propofed se aa X x formula. Wherefore generally, fuppofing only the quadratures of the circle and hyper- bola, we may always have the integral of the foregoing formula, if the com- ponents of the denominator be real, provided in them the unknown quantity do not exceed the fecond dimenfion. | 43. But if the denominator of the propofed formula, or fraction, may not be refolvable into it’s real components, in which the variable does not exceed two. dimenfions, nor can be reduced to fuch by the common rules of Algebra; yet At may always be reduced:to fuch by a little further artifice, as often as it is a | convertible Pi | SECT. I AMALYTIICAL: LNSKITUTION 141 convertible formula, or the product of feveral convertible terms. I Mall call A convertible that a convertible formula, in which the variable has the greateft exponent "ei it’s dimenfions an even-affirmative number ; as, fuppofe » were fuch, then the!" laft term would be e*, and the terms equidiftant from that in the middle muft have the fame co-efficient, and be affected by the fame fign, fupplying the di- menfions by that conftant quantity, of which the la& term is formed. Such would be the formula x° + 4°, or this, x* + da° + cexe + aabey + at, or this other, 2° — dx5 + b3x? — atdx + a°. Now, if it were a + dx4 + 2% + atd, it would be written in this equivalent form, #* + 44 x «x + 2, in which x* + a* is a convertible formula, and x + 5 is linear, which does not increafe the difficulty. The fame thing is to be underftood of infinite others. RI , . . 44. Therefore now let us have x” — a” to be refolved into it’s real com- ponents, in which x may not exceed two dimenfions, and which fhall not have fractions for their exponents ; and, in the firft place, let m be an even affirm- i : / i vi : 27m. Tm 72 ative whole number. In this cafe, it will be divifible into x?” + a?” and x — a* , without any fractions in the exponents, becaufe of # being an even whole number. The firft divifor may be refolved by the’ rules which will be 00 ihe 1 dg foon given for the binomial x” + 4”. The fecond, x?” — 4°”, if tm thall Li . | by im _ ies Im lig be an even number, may be again refolved into ~*” + a?” ‘and «*” — a*”, without a fraction in the exponents. But, if tm fhall be an odd number, it will be refolved by the rules that will be prefcribed for the binomial x” — a”, when m is an odd number. | In the fecond place, let it be x” 4 a”, and let m be an even affirmative whole number, in which cafe the formula is convertible. Let us fuppofe Nr; RE . . . x + a = oy and then let there be formed a convertible formula, in which the greate(t exponent of x may be m— 2, and which may have all it’s terms, and the laft term may be a” ”, and the co-efficient of the fecond term may be 8, for example, that of the third cc, that of the fourth 43, and fo on; and: let this be compared to o, whence refults an equation. | Let this equation be multiplied by xx + fa + 44; the produ& will be another convertible equation, in which the greateft exponent of x will be = m. Let this equation be compared, term by term, with the fiditious equation eg o, in which the co-efficients of the intermediate terms ‘are = 0; and, by the comparifon of the fecond terms having the value of the affumed quantity è, from the comparifon of the third terms the value of cc, from that of the fourth rerms the value of 4’, and fo on to the middle term, taking this in alfo; now, from that of the middle the other e 142 ANALYTICAL INSTITUTIONS, BOOK III, other equations will become the fame, becaufe of their being convertible equa- tions which are'compared. From this laft term will be found the value of f exprefled by an equation, which will have tm for the number of it’s dimenfions, of which all the roots will be real, and will give us the values of f; which being fubititured in the trinomial xx + fx + aa, will give us fo many trinomials, the produ&s of which will reftore the propofed binomial a” + a”. Let the example be x* + a*. I take a convertible equation of the fecond degree, xx + bx + aa =o, which I multiply by xx + fx + aa = o, from whence I have another convertible equation, xt + bx? + 200% + aafx + a* nd, + fa + bf + cabx “let I compare this with the fictitious equation a* +- a* = 0, and from the compa- rifon of the fecond terms | find 44 f= 0, or b= — f. From the compa- rifon of the middle terms I find 224 + hf = o, and, inftead of J, fubftituting it's value — f; 1t will be ff — 244 = 0, or f= + Y 200. | Let it be x° + a°. I take the convertible equation x* + dx + e°x° + abe + a* = 0, which I multiply by x* + fv + aa = o, and the refulting equation is xo 4 ba + coxt + 2a0bwé + atx® + atfe + a + fa + bfxt + feca8 + abfr® + atta + a’x* + a'c°x° eee I compare this with the fictitious equation x° + a° = o, and from the compa» rifon of the fecond terms I find 4 + f = 0; from the comparifon of the third terms I find cc + 2f + aa =o, that is, fubftituting the value of 2, cc — ff + aa = 0; from the comparifon of the middle terms I find 2448 + fcc = 0, that is, inftead of è and cc, fubftituting their values, ff — 3aaf = o, Now, by actually performing thefe operations, we fhall find that If m = 4, it willbe ff — 244 =o. | If m = 6, then ff — 3aaf = o. If m = 8, then f* — 4aaf* + 204 = o. If m = 10, then fi — saaf? + satf = o. If m = 12, then f° — 6aaf* + ga*f? — 20° =o If m= 14, then f? — 740f° + 140°f? — af = o. And fo we might proceed to the other even values of m. Inftead _SBECT: Ii ANALYTICAL: INSTITUTIONS: 143 Inftead of «+ a a*, let it be x* + 24x3 + 2aaby + 24, which is alfo a ‘ convertible formula. I multiply the convertible equation xx + bx + aa = 0 by wx + fx + aa = 0, and I fhall have, as above, wt 4 bx? + 2440x° + aafx + at VU _- + fx® + bfx? + aabx } a ay I compare this with the fictitious equation «* + 26%? + 2aabx + at = 0, and from the comparifon of the fecond terms I find 6 + f = 28, that is, b = 26 — f; from the comparifon of the middle terms I find 244 + bf = 0, and, inftead of 4, fubftituting it’s value, we (hall have 244 + 2bf — ff = o, that is, ff — 2bf — 24a = O. Let it be x° + a’x? + 4°. I take the convertible equation a* + dx? + cox’ + gabe + a* = o, which I multiply by «x + fx + aa, and I Mall have this produ&, + fe + bfe* + ccfa* + a’ bfx* + a°bx _ «+ aax* + ac an) È; x° + bai + cent + 2000x8 + a‘ + atfe + a 1 | This being compared with the equation x° + a3x3 + 4° — o, I find, from the comparifon of the fecond terms, è + f = 0; from the comparifon of the third terms, cc + 4f + aa = 0; and, inftead of 4, putting it’s value, it will be cc — ff + aa =0; from the comparifon of the middle terms, 2408 + ccf = a’; and, inftead of 6 and ce, putting their values, it will be f? — 3aaf — 4° = 0. And fo for as many others as you pleafe. | Pb Now let us have x* + 2643 + 2442x + a* to refolve into it’s real compo- nents, in which x has no fraction for it’s exponent, and does not exceed the | fecond dimenfion. ‘The equation which fhould give us the values of fis there- fore ff — 20f = 2aa, from which we obtain both the real values of f, that is, {= + W240a+4 bb, and f=% — WV244+0b. Wherefore, fubftituting each of thefe values inftead of f, in the trinomial xx + fx + aa, we fhall find that #* + 204% + 2aabx + a* is the produ& of the two real components «x + be + av aaa + bb + aa, and xx + bx — xv 204 +66 + aa. Thus, if it were a° + aax* + 44° + 4° = o. The equation which gives the values of f being /* — azaf = o, from thence we fhall have the values of f all real, that is, f = 0, f.= #24, and.f = —' 4/208 She tie x° + dax* + atx? + a° is the produ& of the three real components xx + @a, we + x 202 + aa, and xx — xV 244 + aa. | | DI US Hh x°° + a", The equation which ought to give the values of f is f° — saaf*? + 5a*f = o. From whence we derive the values of f all real, | chat 144 ANADYFICAL' INSTITUTIONS, BOOK III, I f=z— avis, Wherefore, fubftituting every one of thefe values inftead of f in the trinomial xx + fx + aa, we fhall find that x" + e°° is the pro- duct of thefe five real components, xx + @a, xx + af PS + 4a RA ” ai ag A / KX - any SENS + 4a, xx + axv-—® + aa, and we — any PS + aa. 3 Whence it is to be concluded, that the integral of any differential formula, whofe numerator is x multiplied into any conftant quantity, and the denominator is of a like nature with thefe here confidered, will not depend on quadratures «higher than thofe of the circle and hyperbola, and may be had from the rules here given. MH ° 45. Now let «” + 4° be given to refolve as above, and let m be any affirm- ative integer, but odd, The formula may be divided by x + a, and the quotient (which in the firft — I cafe will be x 7! — ax”? + ga” 73 — 4353774, &c. to the laft term, which will be + 75; and, in the fecond cafe, it will be a”T! + ax”7? + a’x” 3 + "77, &c. to the lat term, which will be + alin: ) may be fup- pofed = 0; and let this fictitious equation, which is a convertible one, be compared, term by term, with the produ& of ‘a convertible equation, in which the number of dimenfions of the variable x is # — 3, into the trinomial xx + fx + aa; and, from the comparifon of the fecond terms, we fhall have the value of the affumed quantity, for example 4; from the third the value of cc, from the fourth the value of 43, &c.; and laftly, from the comparifon of the middle terms, we may derive the values of f, expreffed by an equation 7 — I of which the number of dimenfions will be + All the roots of which will be real, and will determine the values of f all real; which, being fubftituted in the trinomial «x 4- fa + aa, will fupply us with fo many trinomials, which, multiplied together, and alfo by x + a, will reftore the propofed formula x ta’. By this method we may find the following equations, which will ferve for the . e e 7 . . ° refolution of the binomial x° + 2”, when m is an odd, integer, and pofitive number. ‘ If SECT, È ANALYTICAL INSTITUTIONS, 145 pat will be f + a=o. THRE WI 28 Spe nl then /f + af—— aa = O. If m= 7, then f! + aff — 2aaf — è = o. {ma 9, then f* + af? — gaaff — 20°f + at = o. K{m= 11, then ff + af* — 4aa/? ie) galf° + 34% + di cuni e | If m= 13, then f° + eft 50°f* — 40°f° + ba*f* + 305f — a = 0, And thus we might proceed to find the other values of /, if m be an odd number. i If the propofed formula were x” — 4%, and m were an odd integer affirm- ative number, dividing by x — @ as before, the fame equations would be had, only changing the figns in the fecond, fourth, and fixth term, and in all others in even places. 46. If, inftead of x” + a”, fuppofing m to be any odd affirmative integer, the formula were any other, but fuch, as that, dividing by x + fome conftant quantity, that which refults fhould be a convertible formula; as «* + bx* — aax? — aabx’ + atx + ath, which, being divided by x + 4, will give x* — 44° + a*; this lat being managed as ufual, and the values of f found and fubfti- tuted in the trinomial xx + fx + 44, we fhould have fo many trinomials, which being multiplied together, and alfo by « + 4, would reftore the pro- pofed formula. Let it be required, fot example, to refolve x° + 45 into it’s real components, in which x may have no fractional exponents, and may not exceed the fecond dimenfion. The equation which is to give the values of f (according to what goes before) will be /f + af — aa = 0, from whence we derive thefe values SPIRE piso . Thefe being fubftituted, inftead of f, in the trinomial xx < fx ++ aa, we fhall have the two real trinomials xx — tax + Lax 5 +44, and xx — tav «e 12x45 + aa, the product of which, together with x + a, will reftore the formula propofed. Let it be required to refolve into real components the formula x5 + dx — aax? — aabx* + atx + ath, which, being divided by * + 4, will give xt — aax’ + a*. The equation that gives us f will be /f = 344, and the values of f will be f= + 344. Thefe being fubftituted inftead of fin the trinomial xx + fx + aa, we fhall have thefe two real trinomials «x + «1344 + aa, and xx — xW 344 + aa; the product of which, together with x + 2, will reftore the formula propofed. da; Vou. II, U 47. From, 146 ANALYTACAL INSTITUTIONI italia 47. From hence I conclude, that the integral of any differential formula whatever, the numerator of which is x into any conftant quantity, and the denominator of a nature like to thefe here confidered, will not depend on qua- dratures higher than thofe of the circle and hyperbola, and which-may be obtained by the rules here given. | da. 48. But, beeaufe in higher dimenfions the value of f cannot be obtained by actual feparation, from the equations before cited; in fuch cafes it will be enough to have recourfe to the geometrical conftruction of the fame equations. 3 MRI to find the components of x7 + a’, and thence the integral ef the ; xv 1 . Va formula —-——., the denominator being divided by x + 4, the quotient will a + a be x° — ax5 + aax* — a°x + atx? — a5x + a°. The values of f for the refolation of this formula muft be furnifhed by the equation f* + af* — 2aaf —- a> = o. Wherefore, by the ufual methods of Algebra, by means of the interfeGions of two curves, or by any other way, baving found the values of f affirmative and negative, which are to be all real; for example, let one be À, another — B, the other — C; the quantity «7 + 4° will be the product of x + a into xx + Ax + za into xx — Bx + aa into xx — Cx + aa; and the quantities A, B, C, will be real and given. Then we may proceed to the integration of the formula ar Do , by the quadrature only of the circle and. hyperbola. 49. By the fame artifice by which we find the equations for the refolution of the binomial x” + a”, we may find them for the refolution of the trinomial oo” + 2aax” + aa, fuppofing 2m to be an even affirmative integral number. And thus, in general, as often as it is propofed to refolve a formula which is convertible, or is the product of a convertible into a linear quantity, and which has not a fraction in the exponents ; toy may always be reduced by the method. here explained. The cafe of the product of a convertible formula into a linear, we thal have when. m is an odd number, and otherwife. Let this be an example, & + x + 2x4 oe Ott CL gh that 18} kT SEE Real 4% Or et + bt OX xx XX Ee Ai aa X% xx — aa. Wherefore, the divifor x* + 54 being refolved into it’s real com- ponents of two dimenfions, which may be, for example, x + Ax + 25, and xx + By + 25, it will be x* ye bt X e a = se + xx + Ax + db X xx + Be + bd x CS, x xe — aa. And if it had been xt + “xt + bt 30 Aaa", then, by the refolution of «* + 44 into xx + Cx + aa, and. xx + De + aa, it would be xi + bf Mx tat = xe + Ax + 06 a sx + Be + 20 x ae + Ca + aa X ex + Da + aa SECT. I, ANALYTICAL INSTITUTIONS. way ms ma x mn mh * tka affirmative integer number, let A, B,C, &c. reprefent the feveral values of f 50. To have the integral of the formula » in which # denotes any “ut . . i i ° pu i A . n 12 with their figns, which ferve for the refolution of the denominator x ta. And it muft be obferved, that of thefe values one may fometimes be = o, which will obtain as often as # is a term in this feries 4, 8, 12, 16, &c. it being x” — 4” in the given formula. And as often as m is a term in this feries 2, 6, 10, 14, 18, &c. when itis x + e". This being fuppofed, the in- i ri A ica By tegral required will be x —/WVax + Ax + aa 4 —/Vax + Bu + aa + iW xx + Cx + aa, &c. taking thefe logarithms from the logarithmic curve, the fubtangent of which is = a; adding to, or fubtracting from this ° ° f e We è aggregate of logarithmic terms, (according as the fign of the term @ in the denominator fhall be + or —,) twice the fum of fo many arches of a circle, as are the values A, B, C, &c. of which arches thefe are the radii in order, N44 — :AA, N aa — +BB, V 44a — 2CC, &c. and the tangents are in the fame order, x + 1A, x + 1B, x + +C, &c. Such will be the integral of ULI ‘ si — , if m fhall be an even affirmative number. But in the x +a : fame formula, if m fhall be an odd affirmative number, it will be neceffary to add to the whole the logarithm of x + 4, becaufe the denominator has alfo the the formula real:root x + 4. And if the formula fhould be i m being an odd af- — da _ firmative number ; inftead of the logarithm of « + 4, that of x — 4 mutt be a 778 è added. ‘And laftly, the formula being ——, and m being an even affirm- PM pte’ ative number, it will be neceffary to add the logarithm of x — 4, and to fubtract that of x + 4; full taking thefe logarithms from the logarithmic with fubtangent = 4. | | a 51. Butif in the propofed formula _— the number # fhould be a ne. E Ae i gative namber, that is, if it were ———-——, it would be expreffed thus, 148 ANALYTICAL INSTITUTIONS, BOOK III, ——, which, reduced to a common denominator, is equivalent to this, ; and dividing the numerator by the denominator till the greateft power 2 1 a La of the variable is lefs in this than in that, we-fhall have at lat + e”x 27. + » in which # will be a pofitive number. And what has been faid cone” senti a” + a” before will alfo take place, in the formula a: when # is an integer nega- x ta tive number. 52. Moreover, if the fra&tion be fuppofed to ipa multiplied by x, i ft tg x being an integer number either affirmative or negative, the denominator being refolved into it’s real components, in which x does not exceed the fecond dimenfion; this will be the cafe already confidered by me at § 41, 42, and is therefore reducible to the SS ery of the circle and hyperbola. 53. But when # is negative, it may be reduced more expeditioufly thus. Firft, let 2 be lefs than m. The formula may be thus expreffed n mi mn aS a Ke x 772 H è ° by equivalents, — ges —_—. And likewife, the formula = yee ae ie ris whe vam ah, x Pi m_ Nn. by — 7 - Secondly, let x be greater than m The formula: la, TS ) € be expreffed by the equivalent feries —— — — i NL rm Ps x ali Pe Es dal 772 ——_T_ —_ &c. till we come to that term, in which the exponent ga wee are Ra i of x is but juft greater thanm; + =: » Here the fign mufl be i at m ri +a xax + or —, according as the alternate cioe of the figns {hall require; and 7 is the fame exponent of the quantity 4, as in the antecedent term, and £ 1s the remainder of. the divifion made of the number # by the number m, taken as ‘often as it can be done, Now SECT. I. ANALYTICAL INSTITUTIONS. ig Be i itl a i | I Now if it were ———_——, Àfuppofing # to be greater than #; all the mt 7 n cn Poe sok terms of the feries ought to be affected by the negative fign, and the term out of the feries, that is, » ought always to have the affirmative fign m 73 ¢ È x = @ X ax prefixed. ‘Thus, if the formula were » it would be equivalent to x3 + ad xX x5 x x . x a e But we know that — 138 equal tio — eg arn #3 + a> X atx? x3 +a? x a3x° ta RX ° . (001 i ui, Therefore it will he — i -L+t_ ao x x3 + a3 x3 + 43 XK x5 ata a’ x a° x 28493 all which are quantities that may be managed by the given rules. 54. But if m fhall be a fraction either affirmative or negative, let ¢ be the numerator of the fraction which is equal to m, and reduced to the fimpleft terms, and let p be the denominator of the fame: fo that the given formula may be thus expreffed, i . Pata = yt LAT Va » and the for. xp + ap py Ty Do: 1) nents, and may therefore be refolved by the given rules. mula will be converted into this, » which has no fractions for it’s expo- Let the formula be, for example, ps dEi ‘make x = @-=2'dd, and it x? + a? \ will be x = 2yY; and making the fubftitutions, the formula will be changed: into —, which has no fractions for it’s exponents. 27) yt b He CX 55. Now if the given formula be —, im which m and x are broken: i, bigs, numbers ; making 7 the numerator of the fraQion #, and p the denominator of the fame; and thus making ¢ the numerator of the fraction m, and ¢ it’s deno- minator, (fuppofing thefe fractions to be reduced to their fmalleft terms,) the- r Ceti x pù È formula will be oy. in which 7, p, 9, ¢, will be integer numbers, #5 Ge, Te Gn Qe. pofitive or negative. Now 150 ANALYTICAL INSTITUTIONS. BOOK IIT. Now let it be made x = yÎ5, and a = 34; the formula will be converted py” +P9- Ts into this, prora gf £d , which has no fractions in it’s exponents. Let it be, for example, the formula ————; make x = yy", @= 8°; it will be x = vai E a v IO 9, KE Fare. 15 i 7 3 vs) 3 9 4 . . a ° i * =; and making the fubftitutions, the formula will be changed into | x vi waste x x 74 i? which has no fractional exponents. Paco gh era m\t pofitive integers, we may always have it’s integral, fuppofing only the quadra- tures of the circle and hyperbola. And the integral will be compofed of algebraical quantities, and of one fluential quantity ; which will be done in the following manner. 56. Laftly, if the formula fhall be , the exponents #, m, #, being Rc Suppofe the formula ina ae mai Tha x a perte SAI e as far as to a conftant term, or to that term in which the exponent of x is o, and let this be K; then oto x muft be adde ; u added As Ae. that is, it muft be made OC vi vie i (he + Cx n-+-um—2m Da nA UM 2M I ke + K we ee ee ae ed ge ata x cha Difference the equation, make it == o, and fet the terms in order, From making the firft terms = o we fhall find the value of the aflumed quantity B. Making the fecond terms == o, we fhall have the value of ©. And fo, one by one, the values of the others ; ; which values being fubftituted inftead of the. dt capitals, as the fluent of will depend only on the quadratures of the Pe de n a a circle and hyperbola, and the other terms in the Zomogeneum comparationis are . purely algebiaical, fo the propofed formula will require no higher quadratures. 57. Sometimes it may happen, that fome one of the co-efficients B, C, D, &c. may come out arbitrary, or to be determined at pleafure ; but it will he only ~~ SECTOR: ANALYTICAL INSTITUTIONS, | Ist only when # is greater than m— 1. And it may alfo be obferved, that as often as it is m == + 1, the co-efficient A will be found = o, and confequently the integral of the propofed formula will be algebraical. 58. But if, in the propofed differential formula, the exponent # fhould be a x ° . ° . negative integer, fo that it might be reduced to ————<— ; in which it is 5 5 5 nl i ge m\# 3 XKR now politi; ; the integral would be aa pete A f—_—= ». Which tot x Piga pete at ae x ai me ane co-efficients B, C, D, &c. will be determined in the fame manner as before. As, for example, —“=— in which cafe we have x = ogg oe D 3 Pp tna seta an a See Motta taller È e" Ba? + Ca + K car . Wherefore it will be f= = n na AS Fia od And taking xXx dati 2Bxk + Cx x x3 +a3 — 30°% x Ba? + Cw + K © Axx the fluxions, ———- = —__ ——_—_—— +. ° x? Fad)? a+ ad)" a3 + as” Then reducing to a common denominator, fetting the equation in order, and making it equal to o, it will be 3 : 2Butx + Cox — 3Kx%x + 2Baex + Caix — 3Bxtx — 3Cxx + Adîx - i" 'o% + Ax*x corsa wx Now making the firft, fecond, third, 8c. terms = 0 fucceffively, we fhall find, A B= 0,01 Bo Ay Co, Ki ton 29 4+ Ags. — 1 — 0, Or Aa = 1 — 2Ba*; and putting A inftead of B, it will be A = — cs Ba gi | oe Whence, laftly, it is È ri de i SAI È Ys 13% = 303 X 23403 È 343 > fon as” =r acre PO eC nar — ax + 44 — [rego +4; together with a multiplied into the ~ Bur fi. 3aa arch of a circle with radius = Pari and tangent = x — 14. So that it will be = ——— Hi i Gach nice) anes eer + = x arch of a circle with radius “aa, and tangent = x — ta: taking the VO from the logarithmic with fubtangent = a. 595 Bur. 132 ANALYTICAL INSTITUTIONS. BOOK I1T. 59. But if the exponent m be negative, the formula muft be changed into another that 1s equivalent to it, in which the exponent is pofitive ; according to the manner fhown at § 51 of this Book. Pad And if both # and a fhould be fractions, the fubRtitutions mutt be made according to $ 55 of this Book. 61. Again, if the exponent 4 were not an integer, but a fraction either affirmative or negative, it will fuffice that the formula be one of thofe cafes confidered at § 39. Forafmuch as it may be tranfmuted into another form, which is capable of being managed by the given rules. LE , the PECANS 11, m, u, being pofitive or nega- x x Ri e ie tive integers, or elfe rational fractions of any kind, with the figns + and — at pleafure ; it will be integrable, or, at leaft, may be reduced to known qua- - dratures, as often as the faid exponents fhall have fuch a relation to one another, vana ° J 1 that one of thefe two quantities compofed of them, that is, 4 — Pf or 22 —, or — TL —, fhall be equal to any integer number. If this integer number fhall be pofitive, the formula will admit of an algebraical inte« gration, except the cafes in which the power # — "x fhall intrude, which obliges us to recur to the logarithms, If this integer number fhall be negative, the formula will be reduced to the quadrature of the circle, or of the hyperbola. I # È To obtain our purpofe as to the firft cafe, in which « — pi = gia | tt n mt n ali a equal to an integer, make x +4 = zx; then x = —, a = —s xz — Tm Pi a+ ' 241.00 n a n+I a ss a % rasi Z— ; and therefore xx = — tome ee a > x san? erore pa x z= 1) m fa «e I ULI ine m mu_t | Mm m m a 3 miu a Z— 1 SER seh BOC e e eas , anda +e”! — 5 si % — If ; Therefore, making the neceflary fubftitutions in the propofed formula, it will be n — U. mi pri TU » n X z= » which is plainly feen to be alge- alge inci op 5 7 equal 3 | to braically integrable, (except the excepted cafe,) when t meri wl ANA Te UN srt Td 6 WS, 153 ° ° ° = 7 om T ò é . to a pofitive integer number. And that if ———- — 1 + % is an integer number, but negative, by what is advanced in the foregoing articles, the inte- - eration of this formula will depend on no higher quadratures than thofe of the circle and hyperbola. 4 I fl e e I come now to the fecond cafe, when —- — 1 + — is equal to an integer 2 7 m : ; “n mM i e . A m rca number, Make x + @ = 2, and then it-will be « = z-a,& = : I n i IHx min n mm 2-4-3 wm) me 2. z S— a fF HOG Pee geen eRe | x = 2—a4 AAA a inn #88 s' mR 141 i \ UL: 7 772 Wu LA gg i ». Butx + a = 2, anda +a< = 2’ ; therefore, ‘mak+ e . . e e % . % ing the fubftitutions in the propofed formula, it will become RX aa morti TI} 3 = Ss “i ia ti) 78 . . È vo. e. seen ay 508 Chie = i ee » which is algebraically inte- Ha I grable, (excepting in the cafe excepted,) when = oe gi equal to a pofitive integer, or a negative; for then the integration will depend on the known quadratures of the circle and hyperbola, as appears by the foregoing articles. % 62. Now if the denominator of the propofed fraction, raifed to any integral power, thould not be a binomial, as has been confidered hitherto, but fhould be any multinomial whatever ; provided it be reducible into it’s real compo- nents, in which the variable does not exceed the fecond dimenfion ; either by means of convertible equations, or fome other manner; the formula may always be reduced to known quadratures. ' Let it be, for example, —————---———; raifine aQually the powers i P ° we + be +aa x x +03 5 ) P of the denominator, make a fi&itious equation thus : 2iolinigat 2 Asti + Bate + Cab + Di Boh 4 Gos: poli wetbxtaa? x £ +3 x* + 2bx3 + 2aax* + bbe? + 2aabx + at x3 + 304% + 3ccu + 03 : Fiere are fo many terms taken in general, as are the components of the denomi- nator ; and in thefe terms fo many capitals, as is the higheft power of the variable in it’s refpective denominator, multiplying alfothe firft capital in each term by the higheft power, leffened by unity, of the variable in it’s denomi- nator, the fecond capital by the fame power diminifhed by 2, and fo on to cr You. II. ‘ 154 ANALYTICAL INSTITUTIONS, BOOK 1II, laft conftant quantity. Thefe.affumed conftant quantities are to be determined in the ufual manner, and the firft term will furnifh fo many fra&ions divided by xx 4+2x 4+-aa\?; in which denominator making the middle term to vanifh, the | va 4 ° ° i i “AN A fractions will be a particular cafe of the general canon = . And the i Rea fecond term will give us fo many fra&ions divided by x + ©}, which may be reduced to the ufual rule of denominators compounded of equal roots. 63. Moreover, if the numerator of the propofed formula be multiplied by a pofitive or negative power of the variable; having found the values of the capitals, and operating as if the fraction had not been multiplied by any fuch power ; the refulting terms may be multiplied by the faid power, and the reft may be done as ufual. 64. I (hall finifh this Section by fulfilling my promife made to the reader, concerning the Method of Multinomials, of Sig. Count Fames Riccati, which is as follows. By the name of Differential Multinomials I call fuch fractions, as have for their numerators the fluxion x, and for denominators an aggregate of powers, the exponents of which conftitute an arithmetical progreffion, which proceeds till it terminates in nothing. And till this condition is fulfilled, the abfent terms muft be fupplied, and their co-efficients made equal to nothing. Suppofe = eV ° ° o . » we had this expreffion -———_—. At firft view it might feem to be a trino- gi + ne -+ @ mial, but is really a quadrinomial, and is thus to be compleated : ——— wo +a5+ox5+a In any multinomial expreffed by a fra&ion, the denominator of which is raifed to the power p, being a pofitive integral number, there is a method which would be general, if it were not frequently made ufelefs by the inter- vention of imaginary quantities. But there are fome particular artifices, which often come opportunely to our affiftance. x I begin with the trinomial —_—=- = y, becaufe to fuch an expreffion a eg 1. x + ax + 8 - as this every trinomial may eafily be reduced. Make x” = % + A, where 2 is a new variable affumed, and A is a conftant to be afterwards determined. The necefiary computations being made, to arrive at the fubftitutions we fhall have as follows, | 27% di SECT. I. | ANALI TICAL'INSTITUTION Ss 155. x” = zz + 2Az + AA, and confequently an? ee az + aA ) bi a | b | x + ax ap? — 22 + 2A ha es PF AA + GAL 7, It ought to be contrived in fuch manner, that the quantities AA + aA + è may difappear, by putting them = o, and in cafes in which A is no imaginary quantity, this reduction fucceeds very well. It is therefore x” = z= + A; I and taking the fluxions, mx” ‘x = %, anda = z+ mS CRS Oe ae % % Mm — { Meo I ‘i MA > mXZ+A\) m In proceeding to the neceffary fubftitutions, in our principal formula, inftead of x and it’s powers, are to be fubftituted the affumed variable z, with it’s functions ; and we fhall find —— =< = TST IRE we” La + 5 p (sa —— : - RELA xzz+z2Ataxz? and freeing it from the quantity 2, which multiplies the binomial z + 2A +4 / : i ; =D under the vinculum, it will be seen oo LA i POGGIATO) è me 1 mx FA\# xzt2A+at The moft fimple cafe is, when the exponent p 1s equal to unity, the other being when m is any number, integer or fraction, affirmative or negative; and, for brevity, making 2A + a = g, the general expreffion, [when p = 1,] will become this la peers m—t oe gxzt+A # «Lg X Z4A) I make a fir& divifion by dividing the numerator of the fraction by it’s am J. Z % particular one, denominator, and the firft quotient will be —— 3 and making the ex aba multiplication and the fubtra&ion, according to the ufual method, the re- mainder will be — ome to be divided by the denominator; and therefore . — 1. "4 wm T, + n % B Bye vA % eae omen «= oR =“ Mi — 1 171 = I 7 «= | m—T 23 1 gxxtA) m +3xX 31 A) #@ EXZTA) m EIXZ+A) m +gr xt Al mo ra, Ra aly ate 8 eni er 156 ANALYTICAL INSTITUTIONS. | BOOK 111, The firft term of thé fecond member is already reduced to known quadra- tures, and the other term may eafily be reduced, by making 2 PA = a, and performing the neceflary fabftitutions. © For then we fhall have —7-+ 1 — 5 u 7; "1 MI gg -£A+gw° ge x + Al m + gz x z+A) To purfue our inquiry, let the exponent p be equal to any pofitive and integer number; to obtain our defire it will be fufficient fomething to produce Dal the operation. Refuming, then, the general formula ———————- = logi ni bo ve eee = y. And, for example- ake, making p = 2, mXxz+ A mn Xz +e this af a reduced to the following, Jie bi Tm J ui I). gg XztA\|m + 2g2 xZPFA\m + 2% geal acy Then, as before, I divide the numerator of this fraction by it’s denominator, wre, A and the firft quotient will be ——-—— 4 ge x z+A mm ; and, after the neceflary opera» mm J. 22 Z tions, we fhall have the remainder — ——, to be again divided by A 88 the whole denominator. Then I make a fecond divifion with the fra&ion - EA aa Ze Zz J i ua pee nor RR Here, after the necef- ex xt A) m + 29g X 2+ A\ mn m + g2% XK 3 z4b+A) m ea Rs fary operations, we fhall have the remainder -1© + —,, to be divided by the SS whole denominator. Whence there will arie the following equation, ; LIE see eet 3% eet ta Pt gyn ge. m— I +, 1-1 RR z+A\ m x z+g)° E X Z+A)\ m 23; ik 78 Yi LIK St A) m° x zs) ZV, TAO TL a m=-1 83 xa PAL m x +9)" The SECT. I. ANALYTICAL INSTITUTIONS ‘ig The two fir terms of the bomogeneum comparationis are two binomials, and the other two may eafily’be reduced to the form of binomials, by making Z2+ A= 4, orz#+eg =. In cafes more compounded, in which are made Dida, OF rey OF By ec dhe tecdioutnels of calculation will indeed. increafe, but the method will {till be the fame. This method may be extended to all multinomials ix infinitum, fuppofing p to be a pofitive integer ; for, if it were a negative integer, the matter becomes fo eafy that there is no need to mention it. To apply the method, ae elfe is required but to repeat the fubftitutions * = z + A, 2 = u.+ B; &c. always making thofe terms to vanith, in which only conftant quantities are found ; by which means quadrinomials (for inftance) may be reduced to trie Moena 18; and thefe to binomials. It will alfo be needful, from time to time, to make ufe of a partial divifion, that we may not be interrupted by negative ‘exponents, which will often intrude in the numerator of the fraction. After all, the manner of operation will be better sn by examples than by precepts. Let us take the quadrinomial —_——————__—_—_—_—t__—_c The conftant | 23” 4 axe” fol a AP quantities 4, 6, may be Soro. E tuppore «° =z + A; then we hall have 037 dad” 4 be” Peo + 3A2° + 3AAz + A} + az’ + 244% + aA’ - + 62 +A +e. I make A? + aA* + Ad + e. 0, and thus I determine the value of the aflumed conftant quantity A. Then repeating the operations as in the trinomial, gz Px I find —_—-_—_--____—.._ The letters g, 4, denote conftant quantities, Ara ztA sm XxX 3% +e +f which are fubftituted in the place of others more compounded. And, fup. pofing p to be a pofitive integer, I raife LI trinomial 22 + gz » h to the power pi: si After this, I make ufe of as many divifions as are neceflary, to make the exponent of the variable in the numerator to be negative; and in the deno= Rica minator, that no other quantity Mall enter but the binomial z+ a| 7 . And I fet afide fuch fractions, as, neglecting the co-efficients, [hall ul eve to awe 7] è —— ; fuppofing x to be any pofitive integer. The other terms are PAL mam | reprefented this, 158 ANALYTICAL INSTITUTIONS | BOOK III, ————-. Then I repeat HI — I z + A) m x 2B -+ox+h\P the operation, making z = w + B, making the laft term to vanifh as ufual, and raifing the hibolnial i co B to any power x + 1, and fubitituting, inftead of z and it’s powers, their values exprefled by the new variable w; all the parts will appear under the afpe& expretled by the following formula, x WD, m—I utA+B) m m x u+hP When p is greater than #, fo BG the expanede # — p is negative, then the divifions muft be put in practice, and the formula thence arifing will be reprefented by the general formula da e è by = ; then a — p, being pofitive, we fhall have —_ _—— —. u + A+B mm “+A+B) m_ mx 24+- pd And laftly, making « + & = w, and, as well # as p being integer numbers, the binomials that will arife from the forementioned operations will always be reducible to more fimple quadratures. . It is true, that, upon the account of imaginary quantities, this method ree mains limited; but very often the roots, either in the whole or in part, are real; and befides that, in many particular cafes, thefe imaginary quantities may be eliminated. Nor ought we to defpife the much we may have, becaufe we cannot obtain all. Let us take, for example, the trinomial ia . Make x: =z+A, x + 24/% + 2\P | then x + 24/%x +2 = ez 4+ 2Az + 22 + AA + 2A + 2. By making AA + 2A + 2 =o, we find A =vV- tr 1, Now here we have a magnitude made up of real and imaginary quantities; therefore, proceeding sala ei 3+ A) ra x 2+:A49? pena; Now, that the imaginary quantities may be avoided, let us according to the method, we fhall have Ax hx z+ 2/~ TP change our manner, and in the magnitude 22 + 2A +2 x z+AA42A +42, ~ let us bring it about, that the middle term 2Az + 2% may be Bags ei by putting it = 0; whence itis A = — I, and AA +- 2A +2 =1. So ‘that % LZ ipa Ge aii a a Cone And now, in the two binomials of the bomezeneum comparationis, which are equivalent to the two others already confidered, we fhali meet with no difficulty. the formula will be as follows, SECT. II ANALYTICAL INSTITUTIONS, 159 Si a Se Waa oF Of the Rules of Integration, having recourfe to Infinite Series. 65. Now, to proceed to the other manner of Integration, or of finding fluents, which was mentioned at the beginning, that is, by means of infinite feries; it is neceflary to premife thefe Rules following. | Rute I. To reduce a fra&ion to an infinite feries. Divide the numerator by the denominator, according to the ordinary method of divifion, and let the remainder be again divided, and thus from term to term in infinitum ; and you will have a feries confilting of an infinite number of terms, which is equal to the propofed fra€tion. Therefore it muft be obferved, to make that term the firft which is the greateft, and that as well in the nume- rator as in the denominator of the fraction propofed. Wherefore, by operating after this manner, we fhall have.as follows: Fe e ee wa i POTE ge ST FARE AR SO E Sl ae OF af oa afn* __ afn® afn* nba 28 Pont tnt tn mio SC Here the figns of the feries muft be alternately +- and —, when the fecond term of the denominator is pofitive ; and all the figns muft be pofitive when it has a negative fign, | In like manner, it will be ee n tà Leni fi i fn apart fas ea these eae te ee 1 &c. Tae Let St te Lia See ai 2: 3 : - 247 — gt EA 3 : 3 peewee 2% — 2% + 74% — 13K" + 34x37, &c, ip 2? — 3% f cdi 3fr 6fn* — 10fn? I fas | Feriali die ir ali pi Or 160 ANALYTICAL INSTITUTIONS: BOOK IIT. Let there be a fra&ion, of which the numerator and denominator are each én infinite feries ; for example, this following : 4 2 I 4 î A, 6 4 3 sata e pe 1 ELE 4.00% SR ON eC ‘The quotient will be I + Ton BB at ee DPW Ot? Lan? + 2abx* + aged ex? + ab 6 e SY mm i 4%x 66. Rue II. To reduce a complicate radical quantity into an infinite feries. Take, for example, “aa + xx; let the fquare-root of the firft term be extracted, and then let the operation be profecuted im infinitum, in the ufual manner of the extraction of the fquare-root, and we fhall have - Per a? at i 5x3 Ga na —= a + snesszi croce See Go, v ad “tg 843 — 1645 128a7° x; 5; si 9 : CAT Bist Tsk ye xe a 5x” Vax bee = Oe Li3CHE 1, &c. 2a* 8a? 16* 1284” It may here be obferved, that in each of thefe two feries, if the numerator and denominator of each term be multiplied by 3, beginning at the fourth, the numerical co-efficients of the numerators will be in order, 3, 3 x a 3% 5 X 7, &c. arifing from the continual multiplication of the odd. numbers. Then in the denominators, beginning at the fecond, they will be 21,2 se a4, SA Oe Ke OC Ok Os ote, dllino from the continual multiplication of the even numbers, 67. Rune Ill. All this may be done more generally by the help of the following canon : | mac E I cene Se ar P+ PO" =P” +2 AQ +75*3Q + 7S" cQ + DQ ke. ° ° . ° 772 ‘o ‘ In which P + PQ isthe given quantity, — is the numeral exponent, I reprefents the firft term, Q is the quotient of all the other terms divided by the firft, and every one of the capitals A, B, C, D, &c. fignify the preceding terms | 3 refpectively SECO. ANALYTICAL. INSTITUTIONS. 26% refpectively ; fo that by A is underftood pe , by Bis meant — — AQ, by C, 2“ BQ, and fo on. Let the formula 44 + xx be propofed to be reduced into a feries; then it will be P = aa, Q= —, Mm = 1, n = 2; therefore a Her ot, un xezdk — ee + — |, &c. Vi aa xx pi 24 8a3 de 1645 12841? 3 Let it be 3/25 + atx — x5, that is, a + atx — x°\;; it will be P = a’, ata — 95 a pr pesa Pegi a ee : atx — x Se ee ————— , m 21, n = 5; therefore 4 + atv — 15 = @ + 2a = = Rec Wo da CA &c. 25a? Let it be —— = 8 x pp — aay; it willbe P= y, Oo = — % d 53 — aay È i 31°, ” i = —1,% = 3; therefore —I b aad aah 1405b TI \ 3 — 7 LL start buca ir +. die & RARO y ty Tg Big? 0 Let it be sen which would be expreffed thus, d x @ + aS, and a %| the relt would be done as before. Let itbe è x 7 +5; thenP=a,Q=4,m= — 3,n= 1; x hi; b b 6hx% 3 therefore 2 X 4 + x) soa Mod ne. - — LE si SEC: 68. Let us have a complicate quantity to raife to a given Paste, or let a +x (for example) be raifed to the power m. Then P=4,Q=—,m=yn, nice s therefore ra m mata mx m i de mx m= xX maa. ds eno = SS he GA t x "qua a + PO + esere re: vt manna + eet -|- —@EP_uxz—===- n° sc; I EE. ix 2568 Let us have an infinite feries to raife to a ELISA power. For example, i y + ay? + by? + cyt + ay’, &c. be raifed to the he: power pi. Then will P = y, | Q= ay + by” + 95 + dy*, &c. m= m, mn = 1; wherefore Vou. II, i Y ra | 162 ANALYTICAL INSTITUTIONS. BOOK III. mi y+ ay + by? + cy* + dy®, &c. * o + SARI I mX m—-tI ato DA mXm—1 X m—2 aay? HS IX 2 REZ KS si mby™ t 2 we mx m—! aby 3 I ae | mey +3 I m xX m—t X m—2xX m—3 arty” t4 IX 20% 35% "% cc. Fri MX Mel X m—2 ar by +4 ‘ LX ERI mx m—t acy” IxI mx m—t1 pay t4 1 x 2 may” * 4 I + bx ata . . b . : to be integrated. The fraction irs being reduced to a feries, and every 69. This being now fuppofed, let the differential formula be propofed la 2 £ Î bx bx by ® b 2° numerator being multiplied by x, we fhall have — = — — © Sed i a aa as b 3% ba4; ‘ x o — ra &c. And by integration, ee | 2 23 4 bx5 af be. ‘ba he bx bx bx ® Gel ata a 240 Ras 4a* 54° no. Let the formula be —. Making x = 3 -+ 2, where 4 denotes any 4974 . » . . ax conftant quantity at pleafure, and z a new variable ; it will be — = ; rage The fraStion aia being reduced to a feries, and multiplied by 2, it will be | SECT. II. ANALYTICAL INSTITUTIONS, 163 "es ax azz az*% 13:33 az4z 4 ala pupe nego. er i er n &c. And by integration, az sc az® az3 az4 azs . ax ax«e—b pax =: | axa—ds axxmb\ Re Lo ane 25? ae gfe es 71. Let the formula be TA bi: ; this, reduced to a feries, is ——— —= sla +a? vata) a a; tes oe De Ba Sf i rs Lilo __ 52da =, &c. And by integration, / =; = a a sa 2a 12545" : ane a 42 ba but | Aas Laz + — — 22°, &c. And the fame may be done by any other 1045 9545 50045 propofed formula. ea. 42. If the feries thus found, which exprefs the fluents of propofed differential formule, and which are compofed of an infinite number of terms, fhall be infinite in value; the fluents or integrals of the propofed fluxions will be infi- nite. And if thefe feries fhall be finite in value, and alfo fummable, that is to fay, if we know how to find the values of thefe feries, though compofed of terms infinite in number, and which very often may be done; we fhall have them in a finite quantity, and therefore the algebraical integral of the propofed differential formule. But, if the fertes fhall be finite in value, and yet not fummable, the more terms fhall be taken of the feries, fo much the nearer we fhall approach to the true value of the formula; but we cannot arrive at the exact value, except we could take in the whole feries, #2. In order to know what feries are infinite in value, what are of a finite value, and which are fummable ; the treatife of Mr. Fames Bernoulli de Seriebus. infinitis, may be confulted, and other authors who have written exprefsly on this fubje&. i ® ; 74. But whenever the differential formula fhall be compofed of two terms only, we may, in general, and with expedition, make ufe of the following canon ; in which the exponents m, 7, #, may be integers or frations, affirm- ative or negative; and which may be continued to as manyterms as we pleafe; for Di thefe four terms fet down, the law of continuation is fufficiently manifett. Yz fay 19. aa 164 AGERE Y DIO AL ASIA Ot DO Se: BOOK IIl f—I. mai MR E vee aE oe R's ay {Pun + n , a i+n é a an: CSS a eae nce [fy yRobo+ ey b 4 6} into = CE “~ 4 t+4oan+n t+mn4 2 ach #42 rimase t-+-ma+2n pil ite btm t-+ 2” 103 È ita tz t+3% a3 4434 À The manner of finding this canon is this. Take the fictitious equation i—I. SE Lee eee LE FS PI > ft i-p 2% ft 21 i—-- 2 Sy y xb+eoyn =b+ ego into Ay + By dj + Cy ve + Dy 3% t DE + Ey man &c.; in which the affumed quantities A, B, C, D, E, &c. are arbitrary and conftant, to be determined afterwards as occafion may require. Then, by taking the fluxions of this fictitious equation, we fhall have TA oca: ayy Me Bock OM = m+I x ny x è + cy) into Ay + By Cyt, ge EOE GMT into sAjf + + x By" 4 pon x Cy F247", &c. Then dividing all by 4 + cy ", and fetting the terms in order, it will be aj? = Ay? + TFR x BB tO + tan x bCyy + fo. a tcAjy 1” I +7+Fnx cB; nia de. + m+i xncAyy t"7! + m+i xneByy 1971, &c. Here the term ayy" might be tranfpofed tothe other fide of the equation by which the whole will be equal to nothing, and therefore the co-eflicients of each term will be equal to nothing, by which we fhould have as many equations as there are arbitrary quantities A, B, C, D, &c. by which they will be deter- mined. Or, making the firft terins on bach fide equal, it will be IDA == 2, ‘ot A= 5 . Then?+a x 0B + A+ m+1 X aA = 0, and fubftituting the value of A, itis 2B + 72B + — me ne sia a aor Bee eee i+ 2 ae X = ar Again, t+2n x DC +i+a x B+ a+ 1 x #B = o, or — B + | - 2cB pt ee and blind yvalucef Bei will be C= Deiana bxi+t2a Cac SECT. II, ANAKDYTIOAL: INSTEINUTIONS, 165 C — t+ mn -+-+n X ¢t + mna+2n X ace oca . And thus from one to another, till we t+ 2 xX t42n x 153 have the values of as many as we pleafe of the feveral affumed conftants; and thefe values, fubftituted in the fictitious equation, will fupply us with the aforefaid canon. If the exponents m, #, ¢, of the propofed formula fhall be fuch, that the canon or infinite feries will break off, or that any term fhall become = o, (in which cafe all the others that follow will alfo be = 0,) the feries becomes finite and terminated, or we fhall have the algebraical integral of the propofed differential formula. But it is neceffary that the feries fhould fir& break off in the numerator, or that the numerator fhould become equal to nothing before the denominator. For, if the denominator be equal to nothing firft, that term and all that follow after will be equal to infinite. Now, that the feries fhould è vani t ; break off in the numerator, it is neceffary that — —- — m fhould be equal to ° \ fome integer affirmative number. But if the exponents ¢, m, 7, of the propofed formula Mhould be fuch, that the feries never breaks off; then the expreffion of the formula fhould be changed into another equivalent to it. Thus, for example, the formula ayy Xx b +0)” thould be changed into this other, We Mey cn Am, which is equivalent to the firft, and it fhould be tried whether or not this will anfwer our expectation. If not, the formula will not be algebraically inte- grable, at leaft not by this canon. If the formula were ayy" x b= 9), | then all the terms of the canon would be pofitive. : acy bx + na : . me ao : . Let it be —— =, that is, ax * x 6-+wxl*; it will bee —~1 — — 2, n= 1, #% = 2,621; whence the quantity £ + mn + 3n will be equal to nothing, and confequently the fourth term = 0, and the others of the feries. x? i that follow. Therefore we fhall have PEA IR E Sabin * Kb Ws dene. 2a°x * 2 as — j 205 - am 3 oe 3 La ae PN a O XxX oO+ x? pars da 52 24a5 by — 16054? , 2 use eS OG. te AL di 10603? 166 ANALYTICAL INSTITUTIONS, BOOK III Let it be e het Ses PSR ER i IN aa + gy and therefore the fecond term of the feries will be = o. Hence ° — J da ee re ‘WV aa + yy — 4a é ay SET. The Rules of the foregoing Scttions applied to the Refification of Curve-lines, the Quadrature of Curvilinear Spaces, the Complanation of Curve Superficies, and the Cubature of their Solids, 75. To fhow the ufe of the foregoing Rules of the Integral Calculus, by applying it to the quadrature of fpaces, to the rectification of curves, to the complanation or quadrature of fuperficies, and to the cubature of folids; let there be any curve ADH referred to an axis AB, with the ordinates parallel to each other, and at right angles to the axis. Draw CH parallel to the ordinate BD, and infinitely near to it, and alfo DE parallel to BC; the mixtilinear figure BDHC will be the fluxion, the differential, or the element of the {pace : ABD; and becaufe the fpace DEH is nothing A M B in refpe& of the rectangle BDEC, we may take that rectangle for the element of the faid {pace ABD. Therefore the fum of all thefe infinitefimal rectangles BDEC will be the {pace comprehended by the curve AD, and by the co-ordinates AB and BD. Wherefore, making AB = «4, BD = y, it will be BC = x, EH = and the re&angle BDEC = yx will be the formula for fuch fpaces. There- fore, in this formula, inftead of y, if we fubftitute it’s value given by x, and by the conftant quantities of the equation of the curve; or, inftead of x, it’s value 4 % given TATE. | ANALYTICAL INSTITUTIONS, 169 given by y and y, and the conftants, and then integrate the formula, this integral will be the required fpace ABD, Ar Other expreffions or formule may be had for the elements of fpaces, by means of fectors, or of trapezia, which, on certain occafions, are fometimes more convenient than rectangles; we fhall hereafter fee the ufe and manner of them in fome examples. 76. For, if the curve be referred to a focus, that is, to a fixed point, fuppofe to M, from whence all the ordinates proceed ; drawing MH infinitely near to the ordinate MD, the infiniteGmal fpace MHD will be the element of the fpace AMD. Then with centre M and radius MD, drawing the infinitely little arch DK, the little fpace DKH will be nothing in refpeét of the fpace MODK ; and alfo, becaufe the little arch DK may be affumed for the tangent in D, or in K, it thence follows that the fpace MDK fhall be the element of the fpace AMD. i Wherefore, calling MD =y, KD = 2, it will be 4yz for the general formula of the fpaces, in curves referred to a focus. And in this formula, inftead of y, or of 2, if the refpective values be fubftituted from the equation. of the curve, the integral will be the {pace required AMD, Hie? 77. But if the curve fhall be referred to | a diameter, fo that the ordinates fhall not be at right angles to their abfciffes; drawing ' HG perpendicular to AG, the produ& of HG, or of FG into BC, will be the little parallelogram BCED, and confequently the element of the area ABD. Therefore the angle DBG being given, and confequently A tere the ratio of the whole fine to the right fine, which, for example, may be that of m to z; Fig. 103. making, as ufual, AB = x, BD =y, then will HG or FG be = “., and the parallelogram BCED will be oe » a general formula for this (pace. 78. It is plain, that the fum of all the infinitefimal portions DH of the curve will form the curve itfelf, and therefore that DH will be it’s element. Making, therefore, AB = x, (Fig. 102.) BD = y, and thence BC = x, - EH = y; in fuch curves as are refertedto an axis, that is, with the co-ordinates at right angles, it will be DH = Yxw + VV, a general for- mula for the rectification of thefe curves. 79. As CI 168 ANALYTICAL INSTITUTIONS. BOOK III. 79. As to fuch curves as are referred to a focus, making allo MD = 9, KD = 2, we fhall have, in like manner, “yy + 22 for a general formula. 80. But as to the curves with the co-ordi- nates at oblique angles, the given angle being HCG, the ratio of the whole fine to the fine of the complement is giv en, which fuppofe is that of # to €; when ace it will be CG = =, ree Ba Bio 2, and therefore lt a DE 2049 — DH = of xs na ee alert. Fig. 103. 81. Now in each of thefe formula, inftead of y, or x, or 3, fubftituting their re efpective values given by the other variable, and their differentials from the equation of the curve, and then making the integrations, we fhall have the length of the curve required. i 82. Let the plane AHC be conceived to Fig. 102. :1_T. move about the right line AC, the curve AH aa will defcribe a fuperficies, while the plane Arye fo AHC defcribes a folid. But the infinitefimal vA zone, which will be the element of the fu- / perficies defcribed by the curve AH. And fi Log the infinitefimal plane DBCH will defcribe a À M BG. folid alfo infinitefimal, which will be the element of the folid defcribed by the plane AHC. Now, as to curves referred to an axis with the ordinates at right angles; let the ratio of the radius to the circumference of a circle be that of 7 a VY : portion DH will defcribe an infinitefimal to ¢; the circumference defcribed with radius BD = y will be =, and therefore TV xx + yy will be the expreffion of the infinitefimal zone, and confequently the general formula for the fuperficies. © 83. Alfo, a will be the area of the circle defcribed with radius BD = y, and therefore + we will be the expreflion of the infinitely little cylinder defcribed by the reGangle BCED. Now this does not differ from the folid generated by the plane BCHD, but by an infinitefimal quantity of the fecond order; therefore the general formula for thefe folids will be — 04. But ‘SECT, Mt. ANALYTICAL INSTITUTIONS, 164 84. But as tothe cafe of Fig. 103; that is, when the co-ordinates make a given oblique angle to each other; the radius of the citcle, on which the litele zone and the little cylinder infift, it is not CH = yy but indeed GH: = eset Re "7 5 as likewife the element DH, which forms the zone, is not Wuxx-+ yy, of e QEXY 2; and the height of the little cylinder is not BC = x, mt but a/ sa +97 + but FD: x + da, Therefore the formula for the fuperficies, in this cafe, 2exy will be fais + 59 + 0] 7 85. The product of the circle with radius GH into the height FD, that is, ia age 2, is the element of the folid generated by the plane AGH. 2110773 Therefore, from this fubtra@ing the element of the folid generated by the CNNY triangle HCG, that is, so X =, what remains will be the element of the cunyy* ——, the general for- 27108 i folid generated by the plane ABD, and therefore will be mula for thefe folids, 86. As to the curves referred to a focus, becaufe of the variable angle DMB, (Fig. 102.) and confequently becaufe we cannot have the value of BD or CH, the radius of the circle, which muft neceffarily enter the formula of the quadra» ture of the fuperficies, and the cubature of the folid ; it will be neceflary, from the equation referred to the focus, to derive the equation of the fame curve referred to an axis, and then we aré to proceed in the manner before fpecified ; | -obferving that, in the cubature, it will be neceflary to fubtra& from the integral the cone generated by the triangle MHC, to have the folid generated by the plane AMD. | 287. From the differential equation of a curve to the focus, to obtain the equation of the fame curve to an axis, the manner is this following. Let the curve ADH (Fig. 102.) be confidered, at the fame time, both as related to the focus M, and alfo to the axis AMB. It is certain that the fquare «of HD, the element of the curve, is equal as well to the two fquares DK, KH, as to the two others DE, EH ; and moreover, chat the fquare of MD is equal to the two fquares MB, BD. Making MB = x, BD = y, MD = g, and the little arch DK = %, we fhall have 2% + wy = xXx + jy, and xx + 99 pari Vor, ILL L ~ Now 170 ANALYTICAL INSTITUTIONS, BOOK ITE, Now the equation of the curve to the focus is expreffed, in general, by the formula pz = «, in which p is a known funtion or power of z; and it will be 23 + pp3Z = xx + Hy. And putting, inftead of y, it’s value arifing from the equation xx + yy = 22, that is, y = ————, we fhall find 22+ppz22= ( DA e XX xx + ————, which may be reduced to this following, ppzz x zz — xw 4 DR n VA T= QZXX — 2x3%3 + wees; and extracting the fquare-root, it will be pz = It is neceflary to clear again the foregoing equation, by freeing it from a mixture of unknown quantities, by making x = 9 and therefore x = I sia, By the help of this affumed fubfidiary equation, make x and it’s ” . i . 5 a | l i fun&ions to vanifh, and we fhall have US ca i «oh Sa AR equation, if CH the value of p given by 3 fhall be fuch, that the quantity oa may be reduced to the differential of a circular arch by due fubftitutions ; and that, making the neceflary integrations, the two circular arches fhall be to each other as number to number ; then the curve fhall be algebriaical, and we fhall find it’s equation to the axis by a formula, after the manner of Certefius. In every other cafe the eurve will be tranfcendental, EXAMPLE, . : ; ZZ A Let the equation of a curve referred to a focus be ————— = % We “=——= ; and in the equation FE pus ‘ball have, in this cafe; p = > ; 7 sa CO 208 — BB Jum GG fubfututine the value of P, IL will be ie re ia A POET Make MCC me QUZ nn LZ Nb — gg | b+ zt, then 25 + 22 + 22 = #1, and 06 — lt 2 — 26m me 223 me q wherefore, making the fubftitution, it will be _——— = 7==-. - Alig + bb — tt V hh = qq For SECT. III, ANALYTICAL INSTITUTIONS. 171 For a particular cafe, let it be ce + 25 = #2, on which fuppofition it will be f = g, thatis, b+ 2 = 97> Sl Therefore bz + zx = dx, and, ine flead of 2, fubftituting it’s value, the equation of the curve will be 2W xx + yy wx - yy x da | a | % 88. The affigned canon alfo teaches us the manner of pafling from the differential equation of a curve to the axis to that of the focus, in the way following. EXAMPLE I, Fig, 104, a Let it be propofed to find the equation te Q a focus in a circle, taking the focus in a point of the circumference A, Make iAH sd, AG i +, AC = xs = bx. Refume the formula = = % ba È s Where is taken g = . Be- i ® N bh — qq iii | caufe, by the local equation of the circle, it A. G HO is bx = zz, it will be g = 2, < Then mak. ing g to vanifh, by fubftituting it’s value z, fewilbe LE me LBL ign y ee es % Cede weit P N bh = xz pz = &, if, inftead of p, we fhould fubftitute it’s value now found, it will be ZZ 77= = #, an equation of the circle to the focus, which is taken in A, i a 56 ; + Therefore, in the formula a point of the circumference. q e SS ST darei E: 0A M-PAI:E. IL . 89. Let it be propofed to find the equation of a conic fection, referred to it’s umbilicus M, that is, to it’s focus (Fig. 102.) | | Z2 Make 372 | ANALYTICAL INSTITUTIONS, ‘BOOK ITT. Make MB = x, BD =~ y; the general equation, which comprehends all the — ni PSN . ’ Cx (RI? agio eee . fections of a cone, will be this; a + —- = xx + yy; to the parabola with 1 Ù + : . si abb the parameter 22, when c = 4; to the ellipfis with tranfverfe axis = Fo aid woe SE ; ; ; 2abb if 3 be greater than c; to the hyperbola with tranfverfe axis = es with CC ci conjugate axis = ===, diftance of the vertex from the focus = wr sf i A CC pesto PA rte b be lefs than c. If ¢ = 0, it will be to the circle with diameter = 24. Put ; CX i “nt | oy e RPM Rote a < Po Re And befides, bx = 29; then czq È Gee ahh . i] . abchz n a+ > = 2%. or + — PF — =% And taking the fluxions, + =é sd) bb 3 era C ED ; : semi ECE i? LObb zabbbb aarbhb __ &bbb nabbbb ud: Do OS — and bb — qq = bb — —— an q | CE COB = COBDS . q 7 oC + CC]B aabbhh G + abchz az | pes one edera at e RS ud elt N hb — gq cav hhize — bbhbhzw + 2abbrbx — aavbbh % + abb WV bbeczz — bbbbza + 2abbbbz > aabbbb therefore p = defcribed with centre A, and, radius 7, rri rta inftead of ¢ and 7, fubftituting their refpective values, we {hall have — j = eND6b — 25 was found to be = f. Then, in the canonical equation g = Yz — == I put it, — g,- becaufe, as AC = x increafes, the arch OQ= g; V hh — 2% . I | ; | a Tate : ri ru x ; will diminifh. But ¢ = —; wherefore PAR pet RISE 1S, eae % z N ph — Zi, fe pa = %, which is the fame equation as that found at $ 88. gt. The particular formule, which are found in the cafe of curves having their co-ordinates at oblique angles, are not lefs ufeful, becaufe fuch equations may always be changed into others, which have their co-ordinates at right angles ; and after that we may make ufe of the ordinary formula. To {how this, make HG = p, (Fig. 103.) AG —=.9; then it is p= —, i #78 q=* + —-, naming, as before, AB = «, BD = y, and the ratio of the whole fine to the right fine that of m to e. ‘Therefore it will be y = m Mese Pe Maat e I sealed i | Es tag anda zq-— =q — Rd Wherefore, inftead of x and y, fubfti- 7 702 ‘74 Q _ tuung,, The quadra- ture of curvi» linear fpacese 176 ANALYTICAL INSTITUTIONS, BOOK III, tuting, in the propofed equation, thefe values given by p and g, we fhall have the equation of the curve with the ordinates at right angles to each other. But it will often happen that the primitive equation will be fimple; and yet, by transforming it, it may become fufficiently compound. Alfo, though the variables are feparate in the propofed equation, they may not be fo in the transformed equation; and what may increafe the difficulty, they cannot be feparated by the ordinary rules of Divifion, Extraction of Roots, &c. How- ever, in many particular cafes, perhaps it may not be amifs to try each method, that we may make choice of that which, in the given cafe, fhall be mot convenient, But now it will be time to proceed to Examples, in which, it is always underltood, except when warning is given to the contrary, that the co-ordinates are at right angles to each other, Be rn wg T= AO SMC i PO SDT IST LAS AER LRAT EA FE AG: Pe , 92. Let ABC be an Apollonien parabola, with the equation av = yy, any abfcifs AD = «, it’s ordinate DB = y, and the fpace ADB is to be fquared. Therefore it will be y = Wax; and this value, being fubftituted, inftead of y, in the general formula for {paces yx, it will be x/ ax ; and | by integration, it will be 3xW ax + 6. The quan- A DUE tity è is the ufual conftant, which, in the integration, ought to be added, and which now ought to be determined. In the point A, that is, when x = o, the {pace is nothing, and therefore the integral 3xw ax + 4, which exprefles this fpace, ought alfo to be nothing. Therefore, making x = o, it will be 30 x Va Xo+b=0, that is, è = 0; which is as much as to fay that, in this cafe, no conftant quantity is to be joined to the integral. Therefore the fpace ABD = 2xW/ax. But “ax = y. Whence ABD = jay, that is, 1s equal to two third parts of the rectangle of the abfcifs into the ordinate. Fig. 106. Now, if we fhould require the fpace comprehended by an affigned and determinate abfcifs and ordinate, for example, when it is x = 24; as, by the equation of the curve, it 1s in this cafe y = V 22%, this {pace will be = 44072. If the ab{cifles of the parabola fhould not begin at the vertex A, but at fome given point D; making, for example, AD = a, any line DE = «, the para- meter = f, the equation will be af + fa = yy, and y = oa + fe. Subfti- tuting SECT. ille ANALYTICAL INSTITUTIONS. t77 tuting this value in the formula yx, it will be x/2f + /x, and by integrating, + Xatx x vaf+fx + 6 will be equal to the fpace DECB. But, to determine the conftant quantity 4, it muft be confidered, that at the point D, where x = o, the fpace will alfo be = 0; fo that, in the integral, making x = 0, it will be 34/7 af + 6 = o, and therefore the conftant 6 = — 2av af. So that, to have the integral complete, inftead of adding 4, we mu fubtrad 3aV af, and therefore. the {pace required will be DECB = 2 x «+x x Saf + fi — 344 of. | Let AE = g, and let x begin at E towards A, and take any line ED = x; the equation will be af — fx = yy, and y = “af — fa. Whence yx = xx af — fx, and by integration, it will be — 2X @— 4 X./af — fa + bd. But when x = o; the {pace alfo = o. Therefore, in the integral, making x = o, it will become — jaWaf + 5 = 0, or è = jaVaf. Therefore the {pace EDBC = jaWV af — ia «Saf — fr. It may be obferved, that, in general, the parabolical fpace AEC = 2AE x EC; wherefore the fpace ADB = 2AD x DB; fo that the {pace DECB will be = $AE x EC —4AD x DB; which agrees with the calculus in both cafes, when the origin of x is in the point D towards E, and in the point E towards D. | I take the general equation to all parabolas, of what degree foever, mn i ax = y's whence it will be y = 4” x”, and therefore the formula ez mo n Eden - Ch Sia LÀ Li me a ° ° ° ; a * a’ x' x3 and, by integration, the fpace will be = — ee + 3. But, taking x = o, it is found that 2 = 0; fo that there is no conftant quantity to be annexed to it, but the integral before found is complete. Now, putting # ™m . n : ae NR ans TO rx inftead of a” x" , it will be —2 t+ or: = to the fpace required, Vow II. | Aa EX \ 179 ANALYTICAL INSTITUTIONS © BOOK III, EXAMPLE 1, 93. Let the curve be y = Xx + 4; therefore it will be yx = xN/x + 4; I Kx+axxtal” +4 But, ° . ° n and, by integration, the {pace will be — co making «x = 0, itwillbe è — — 5 . - X ar/a, Therefore the complete Mi roy Pec x si ETT citi 42 2 << I | 7 4 I integral or fpace required = BANE IE IA: Fig, 107. i . 94. Let FED be the hyperbola between the afymptotes, and make AB = x, BE = Pa and. the equation is xy = aa. Then y = —,andi È aax o . therefore yx = —; and, by integration, the ! | hie fpace will be = 2/x + 5, taking the logarithm: | NE from the logarithmic curve with fubtangent | ; ed = @ But, putting x = o, the logarithm of | ped | | is an infinite negative quantity, and therefore: A | Be E. the fpace is infinite which 1s contained -by the curve EF continued i infinitum, by the afym- ptote, and by the co-ordinaies AB, BE. sa a cs. >» . A 3 Let there be a hyperboloid of this equation a= xyy; then y =~, and there» wv ; : ae è è $ . fore yx = x/—— ; and, by integration, the fpace will be = 2/7d°x + d Now, putting # = o, it is 6 = 0; therefore no conftant quantity need be added to complete the integral. So that the fpace ABEF, infinitely produced upwards, will be the finite quantity 24/4%x, or from the equation of the curve a: Rea va 3 f . F 1 i i 3 crea 2 e A eed a a Let there be a byperboloid of this equation, 4? = «xy; then y= —7, and a3% . , . i # . yx = 73 and, by integration, the fpace will be = — — +. But, putting a; SECT. TIT. ANALYTICAL INSTITUTIONS, 179 x = oy it will be — , an infinite quantity, and therefore è 1s infinite. Where- fore, to have the integral complete, an infinite quantity ought to be added to it, and therefore the {pace itfelf is infinite. ; + Let the equation be 4°" = x" y”, which is to all hyperboloids in general ; min Moe min 7 m m . | [ ma 7 x cs then fraterna , and therefore /yx = —_— +4. Now, ifm=1, x . aa i . x 2 = 1, that is, xy = 44, we fhould have fyx = Seite è, an infinite quan- tity ; whence the {pace will be infinite, as was feen before. 9 Sr, RE dg at sy a oe RE Pet INA RA Bat, putting x = ©, it will be alfo è = 0; therefore the complete integral, or the {pace required, will be = 240°%x = 2%y, by the equation of the curve; which is therefore finite, though infinitely produced upwards towards F. ag Satie Boe Sprea that is, a? = «xy, It will be /yx = — L + 5. . But; making x = o, è will be infinite; fo that an infinite quantity is to be added to the integral, and the {pace itfelf will be infinite. If # = 1, m = 3, that is, at = 93; it will be /yx = anata B. But, making x = o, it will be 5 = o, and therefore the integral is complete. That is, the fpace will be = &datxx = xy, a finite quantity, however infinitely produced upwards, | mara, ts DI ; It willbe fyx = — — + bi Buty’. making x = o, è will be infinite, and therefore the fpace is infinite. Ha 1, m= 4, that ts; o-= wyt; cit will bé yx = 4%/ae +b. But making x = 0; 1 will. be 6 =.0% fo thatthe integral is complete, and the whole {pace = 4 {/ 45x? = 4xy, a finite quantity. ò If n = 4, m = 1, thatis, a5 = xy; it will be S 9x _ -G + 5. Now making x = o, 2 will be infinite, and therefore the {pace is infinite. In the fame manner we might proceed to other cafes, as far as we pleafe. Now let us take the abfciffes from the point B, to find the fpace BCDE. Make AB = 3) BC = x, CD =; and let it be the fame /pollonian hyper- bola, whofe equation is dy + xy = aa. Then it will be y = ee and Aa2 3 therefore 180 ANALE TICAL TS CTT UTI OMe, BOOK 314, a af i cune, 14% "a ù ne è lid Ta 3 <2 thevetore: e = PA, Then, by integration, /yx = ald +4 + f, taking the logarithm from the logarithmic with fubtangent = 4. But, to determine the conftant quantity f, making «x = 0, it ought to be f= + ald; fo that the complete integral or fpace BCDE will be 4/5 + x — ald. | If we take # negative = BA = + è, then 4/8 Pu: x is equal to 4 multi- plied into the logarithm of a. But the logarithm of o is’an infinite negative quantity ; : fo thee in this cafe, the fpace is negative, that is, towards M, and alfo infinite, as has been feen above ; and therefore the fpace between the Apolionian hyperbola and it’s afymptotes is infinite, being ini produced both ways. Let u be the a hyperboloid whofe equation is dyy + «yy = a’. It will eee a3 Cad SU as : è I : be > = Vigne whence yx = x# ae and by integration,’ {yx = 24 ab + ax + f. But, making ax = 0, it will be f= — 20054; fo that the complete integral or fpace EBCD will be = 2/42 + aîx — 2” ab; and taking % infinite, the fpace EBCD, infinitely produced towards C, will be infinite pa Taking x negative = BA = — b, the integral will be — 24 a°d, fo that the fpace will n negative; that is, it will be FEBAM, and will È Anite, however infinitely produced towards M ; as is alfo feen before. Let it be the hyperboloid of this equation 5 + x)" x y= 4°. i Ma be as as topa whence yx = CES: And, by integrating, /yx = =" ad é Ù Now, putting « = o, it will be f= >> and therefore the complete integral, or the {pace EBCD, will be — — at Taking x infinite, the term as b+ duced towards C. Let x be negative = BA = — 8; the integral will be a3 a i O towards M will be infinite. By proceeding in this manner, we may find that the fpace between the Apollonian hyperbola and it’s afymptotes, produced both ways infinitely, will be infinite ; between the firft cubical hyperboloid and it’s. afymptotes, it will be finite towards M, and infinite towards C; between the fecond cubical byperboloid and it’s afymptotes, it will be infinite towards M, and finite towards C ; between the firft hyperboloid. of the. fourth kind and it’s. | afymptotes,, wil be. = 0; fo that the fpace will be finite, though infinitely pro» TAR haat . | But — — is infinite and negative, and therefore the fpace a SECT. 112s ANALYTICAL INSTITUTIONS | 1831 afymptotes, it will be finite towards M, and infinite towards C ; between the fecond hyperboloid and it’s Os it will be finite ware ‘at and infinite towards M. And fo on. Now, to have recourfe to infinite feries: I take the expreffion of the fpace. aax BCDE, of the aforefaid Apollonian hyperbola, that is, go, reduced a ‘ 7 ax asic RATA ARO. * into a feries, will be = — — WA + arr & And, by integra». ax ala a*x3 ax tion, as &e.: which feries, infinitely continued, will. bb gn a ? be accurately equal to the fpace BCDE. And if it were fummable, it: would give us the {pace required in finite terms, that is, algebraically: and this-would be the true quadrature of the hyperbola,” But as this is not fummable, the. more terms we take of it, beginning with the so the nearer: approach. we. fhall make to the juft value of this fpace.. Now I take the abfcifs BT on the negative fide, and the equation of the curve will be 2y — 4y = aa, and. therefore yx = E 5 anes reducing to a feries, it will be s ci e ie e tegration, /yx = “= il ae = * De + ae &c. which is equal to the fpace BTPE. Taking BT = BA, the fpace FEBAM, infinitely produced towards M, will: be = aa + 144 + jaa + 144 + taa, &c.; the value of which feries being infinite,, the {pace it denotes will be infinite alfo. EXAMPLE Ivy, Figs. 108, | 95. Let OC’ be an equilateral hyperbola. "Parga the afymptotes AS, AB, and make AB Bee: Bei Lai the me. chanical curve BEF be conceived to. be de- fcribed, fuch, that the rectangle of AB into any ordinate IE may be equal to the corre-. {ponding hyperbolical fpace BCOI.. The indeterminate {pace SABEF is required. Make the ordinate IE = 2. It: has- been. — found already, that the fpace BCOI.is-equal: | [E 182 ANALYTICAL INSTITUTIONS, | BOOK 11, 5 3 at xs Li to the feries ax + 12° + o + 7a + py &c. making a and J equal. Then, | ; “x xs ae by the property of the curve, it will be 2 = « + ere + PIT, &c. and e O 22.5 34 4} . e ‘therefore 2x = wx + — CS Sars + = , &c. And finally, by integration, the & 6 fpace BIE will be = — + 2 + Lal + al + E .x = a = BA, as to the whole fpace SABEF infinitely produced, it will be == gaa T 44a + 3:44 + 344 + aa, &c. which feries 1s furnmable, and 18 = aa; fo that it is algebiaically quadrable, and the fpace SABEF, infinitely ‘produced, is equal to the fquare of BA. | : &c. Now, taking 204 (PERE a a SE XA NORD Exc V. 96. Let ATC be a hyperbola, it’s tranfverfe axis AD = 2a, the parameter = f,-E.B = «BC =, and therefore the equation xx — aa = “2, and let the fpace ABC be required. It will be Lnereioie wr SELE, and the formula will be yx = xv DELLI, Now, if we proceed to integration, we o ay LIM Mi, Bi fhould find, after the ufual manner, that the integral is partly algebraical, and partly logarithmical; fo that the fpace ABC of the hyperbola depends on the defcription of the logarithmic curve. If we would have the fpace ACHE; making MT infinitely near to BC, it’s element will be the infinitefimal {pace ITCH; and therefore the formula will be xy, in which, inftead of x, fubftiruting it’s value given by y from the equation, + p È 4 2ay aap ‘ . i x i it will be xy +ECO, the integral of which, in the fame manner, de- pends upon the logarithmic curve. And, as well in the formula yx of the firft fpace, as in.xy of the fecond, if, inftead of x in that, or of y in this, we fhould fubftitute their refpeétive values given from the equation ; we fhould likewile find integrals of the fame nature. SECT. IITs ANALYTICAL INSTITUTIONS, 0 ee Now, to return to infinite feries. I take the formula of the {pace ACHEA, i . a . 2 var . i that is, x). Then xy = barman and, for greater facility, making 242), (for the conftants make no alteration in the method,) that is, fuppofing the hyperbola to be equilateral, it will be xy =7Wy + aa; and, reducing the ‘adi ei i fa e ee edd) 5 radical to an infinite feries, it will be xy = ity aoe Senden ape were marae ts ay uae . POLE N der g3} ys &c. And by integration, /xy, or the fpace ACHEA, = ay + < — rum 2 54° + 9 x 16a% QX 12847? And fubtra&ing this feries from the rectangle xy, we fhould have the {pace ABC. &c. a feries, the fummation of which is unknown.. From the centre E let the lines ET, EC, be drawn infinitely near, and let AKP* be a tangent at the vertex. With centre E let the little circular arches KQ. TR, be drawn. It willbe AK = = a La Sei ET = Var + yy, EK = “+” And, becaufe of Similar tiangles PRG: KEM, or TEM, it will. be KQ = Mr I And,, becaufe of fimilar fetors EKQ; ETR, it will be xM gue be yy : TR =— rt i and. therefore it will be 2ET x TR = LT A the. XK IY ubi i element of the fector ETA. And, inftead of y and y, fubftituting their values given from the equation of the curve y = 4 xx — aa, (fuppofing the byperbola to be equilateral,) it will be aax ——; and by integration, / sae i dg that ZY NK — da: i Mib 2V KX — aa ERE TA RETTE O n is, the feGor ETA, will be equal to — 14/x — xx — aa in the logarithmic: with fubtangent = a; which {pace is therefore exprefled, by a negative quan» tity,. becaufe it is affumed on the negative fide. > s a È ; e x ; 2 41° By reducing the formula into a feries, we thall find ——“ij— — “* ri aN Ag— AG: 2%. 443 era + 1645 + 3247 Ce 25639 9 &c. Now, to Integrate the firlt term of the feries,. there would’ be occafion,. firft,. to reduce it to an Infinite feries. ‘Therefore it would be better to do it more expeditioufly after the following manner. Make EM = MEI RES then KP =. Make KE = p; AE = a, the tranfverle femiaxis, and the | gl femie. 164. ‘ANALYTICAL INSTITUTIONS; BOOK III. n fy E ; | : ew ax DERE | femi-conjugate = 2, Therefore it will be KQ = ua a i i yf oe WB : uae . > anc theretore SET Pa se —. But, by the equation of the curve, tia LV — aa; and by fimilar triangles EAK, EMT, it will be y = ——— } 2. Therefore zx = 24/4 — aa, and xx = i and confequently the 2000 : i 7 2: formula will be n s which, reduced to a feries, will be — + — + x is : ; it i oie de Mm a , &c..; and by integration, / i , that is, the fpace IT) . _. 4% a33 azs agli, ce 08° a ETA, will be = e + 60% DA 1004 La 1459 is 1833 ° &c. E AMPIE VI Fig. 1104 97. Let ABD be a circle defcribed with diameter AD = 2, and let the area of any half fegment AHE be required. Make AE = x, EH = y; the equation will be y = Vax — xx, and therefore yx = xYax — xx. Here it would be to no. purpofe to free the formula from its radical, or to try any other methods, in order to change it into fome other formula, which may admit of an algebraical integration, or by means of the logarithms. For this would be an ufelefs trouble, becaufe we fhould ftill be brought to a formula of quadrature or rectification of the circle; as has been obferved at § 37. And therefore we fhall thus proceed by way of infinite feries. en, GeO a Fa ° . ° ° . apo netti Mit CAS Si Refolving the formula into a feries, it will be xWax— xx = a2x2% — x. 3 di fer | xx ra xs Tr © 7» &c. $= And by integration, /yx, or the fpace AEH = Now de; SECT. III. —ANALYTICAL INSTITUTIONS. 185 ‘ Now make the radius CA = a, and let CE = x, EH = y, and the equa tion will be y = aa — xx. Therefore yx = xWa4 — xx; and reducing ee ol oti ax Sata | . this t I = 4X me om : - o a feri es, yx x me vi oe 28a? &c. And by inte x3 NOE. xf ee fyx, that ts, na fpace CEHB = ax — — cena A NARA SSE wee 6a 4043 1124%.. 549 115247? oe. And making x = a, in refpe& of the whole quadrant, it will be aa — 2a¢ — 2544 — 71540 — 15744, &c. the quadruple of which feries will be the _ area of the whole circle. Now, by means of a fector. Make CA = 4, AQ_= *, and drawing CK infinitely near to CQ; it will be QK = x, CQ = Yaa + «x; and with centre C defcribing the infinitefimal arch QS ; becaufe of fimilar triangles KSQ» QAG, it will be QS = T= and therefore MN = ae Whence a3% e” 2Xaakbax the little feGor CMN, the element of the fector CAM, will be = 5 ; h zn aac. arxrx a3x4x a3x9i which, reduced into a feries, will be = — — ntl I tie = | ies i 2a* 2a* . Soe ce &e. And by a it will be 2°, or the fe&tor CMA = 22 — si a x fel + 17 ——, &c.; and fuppofing the arch AM to ‘be half the quadrant, that is, taking x = 4, the feries is n — = + = — va 8c.; and the double of this, or 24: 144 + 1424 — iaa, &c. will be the quadrant ABC. | Inftead of taking the radius CA = a, if I had taken it — sis gpsxigg) he quadrant would have been ABC = > — sets | oo 3 5 — , &c.; and actually fubtracting every negative term from the REI term before it, [and multiplying the refult by 4,] it would be si + x = mm + Ton , &c. [= the area of the whole circle ;] which is the fame feries as 1s inferted by Mr. Leiduitz in the Leipfic A&s, for the year 1682. | Vox, II. | Bb ! EX- #86. ANALY TLOALeUNETITVUTIONA, BOOK. III Oe a Gas Ie Lad, a Fis. its | rp 98. Let BED be an ellipfis, the tranf. / verfe femiaxis AB = a, the femi-conjugate Da Le AC = 4, AE=, EH =y; whence the : SORIA ea equation will be — X aa — xx = yy, and kk PT therefore yx = a9 N aa— xx, the element of the area AEHC. But xW aa — xx is the formula for fquaring the circle BOD, the diameter of which is the tranfverfe axis of the ellipfis; fo that the quadrature of the ellipfis will depend on that of the circle. And becaufe f= ir — xx = EHCA, and /x/aa — «x = EMOA, any {pace of the ellipfis to the correfpondent fpace of the circle on the diameter DB, will be as è to a, that is, as the conjugate femiaxis to the tranfverfe femi- axis; and confequently the whole ellipfis to the whole circle will be in the fame ratio. But, as circles are to each other as the fquares of their diameters or radii, 1f we take a circle the radius of which is = “ad, that is, a mean pro- portional between the two femiaxes of the ellipfis BCD, this circle will be to the circle BOD as ad. aa :: 3.4. Butthe area of the ellipfis BCD is to the fame circle BOD, in this very ratio. Therefore the area of the ellipfis will be equal to the area of the circle, the radius of which is a mean proportional bes tween the two femiaxes of the ellipfis. Now, by the help of feries. The formula SV aa — «x, being reduced : . De de x xt n 54% to a ferles, will be = pan INTO 4 — ca — 843 Prinses pa &c. And by integration, /— aa — xx, or area ACHE, = dx — — — jo = Pes bx? ; Phin ~~ > , &c. And making « = a, the area ACB, or a fourth part of the ellipfis, will be = @d into I — 1 egli me a, KC. In the fame ellipfis, taking any arch DS, let DP bea tangent in D, AI=%, IS = y; and through the point S drawing AP infinitely near AK, which cuts the ellipfis in T. With centre A let the little arches of a circle KQ TR, be | defcribed,. SECT, TILL ANALYTICAL INSTITUTIONS. By defcribed. Then it will be AS = Vxx + yy = AT, DP = @, AK=AP SNC ae i as E Ripe 10% PR being a negative difference. And by the fimilitude of the triangles PQK, PAD, it will be KQ = aes i : AV xe È YY And by the fimilitude of the fectors ATR, AKQ, it will be TR = ae v her ‘ . Ke + yy vt das Di and therefore +TR x AT, that is, » will be the formula for the fpace ACT. This will be finally Sinton , by fubftituting, inftead of y and 2VAA — XX y, their values given from the equation of the curve. But [atm is the reification of the circle, as was feen at § 37, and 4A = XH as will be here feen alfo. Therefore the quadrature of elliptical fectors will depend on the rectification or quadrature of the circle. It would fignify nothing to take pains to free the formula from it’s radical, becaufe, notwithitanding this, | we fhould ftill fall upon a formula, which would depend on the fame circle, Now, by the means of feries, we fhould find it to be — me — + 2V aa = x 2 bxrxe ahxtic che 35haBx | È . ned a o RTP MET DR OE &c. And by integration, the fpace ATC = bx bx3 ghas sha? 35 dx? pg ‘ fa ì I iis sata rante I RT to the whole fpace ADC, a fourth part of the entire elliptical fpace, it will be igiene ab gab’ | sab 35ab : es mains es 224 ©. 2304? oe If we would free the formula from the radical vinculum, making the fubfti. tution of Waa — «x = 4 + —, it would be changed into this other, ii : aa 5% : ; ti 12 bz? z bzts which, being reduced into a feries, would be found to be 4%, — : a 6. 85 a é hed - 5 27 So le Y &es" And by® integration; 32 TW Ra yn pee ee af ne | 3a sat Jaî © gaî &c.; and making «x = a, in which cale it is allo 2 = a, it will be ad — tab + lab — jab + jab, &c. in refpect of a quadrant of the ellipfis. Bb 2 | | And 138 ANALYTICAL INSTITUTIONS. BOOK Ill. And if we fuppofe 4 = 4, the ellipfis becomes a circle with radius = 4, and the feries will be as at § 97, which will exprefs the quadrant. And there. fore, from hence it may alfo be feen, that the area of the ellipfis is to the area of the circle, the diameter of which is equal to the tranfverfe axis of the ellipfis, as the conjugate axis is to the tranfverfe axis of the fame ellipfis, (dl s EXAMPLE VIII, Pi. its, PAR eee 99. Let NAM be a cycloid, it’s ge. afi nerating circle ARH, and make AH =a, aS ee Be ey BE oR y, DF = y. The equation will be 7 = do N ax n XK IN A mm 22 Pat — But the little fpace QEFP is the element of the fpace AEQ, and therefore FP x PQ, that is, “= = XV ax — xx will be it’s formula. But /xWVax — xx is the circular fegment ASB; therefore the cycloidal fpace AEQ will be equal to the correfpondent circular fpace ASB, and the whole fpace AMK will be equal to the femicircle. But the rectangle AHMK is quadruple of the femicircle, becaufe it is the produ& of the femiperiphery into the diameter. ‘Therefore the fpace AMH will be triple of the femicircle, and therefore the whole cycloidal fpace will be triple ot the generating circle. If we would have the fpace AFC immediately; as the little fpace FCBE, that is, yx, is its element, and from the equation of the curve we have y = tala — x VE ; let the Lomogeneum comparationis be reduced into a feries, fir multi plying the numerator and denominator by V x; whence it would be mme ch 3 2 3. s. RE Re sii sx x%x È i 2 e ---?-——;, &c.; and therefore, by integration, i 2a* 8a* 16a* . ; 3 s z xaf/ax — 48 A È a a f I —— —y = 207K? — _ — —-x, &c. Whence yx = 20 SECT, I11. “ANALYTICAL INSTITUTIONS. 189 Li, i ne ati MRO. I | s . 20°%*% + ——— e o mm &c. And laftly, by integration, /yx = 34° 20a* 56a™ 1,3 : 5 2 ABE = ET PP Ei, ke. 3 15a 704% 252a* EXAMPLE IX. Fig, 113. ! A 100. Let ADR be the conchoid, CB = | li ADI ee es” DI IO {pace ADGB be required. Make CG=z, which will always be given by « and y of the propofed curve, as is plain enough, Let CE be infinitely near to CD, and with centre C, intervals CG, CD, let the two little arches GI, DF, be defcribed. It will be HI = 2, and the trapezium FDGI will be the element of the fpace required. By the fimHar triangles HIG, BGC, it will be GI = ——=—; and by the fimilar fe&ors Ree va — aa CGI, CDF, it will be DF = S222. But the trapezium FDGI = i 3% BSE — Ga ‘ A AI] nr 202 = 252 2° DF +GI x 7GD = fe al: see di ere Sea that is, ; 22V xz — da aM zx — da [al +22 — da — ala + 1a x arch of a circle of which the radius = and the tangent = 4/22 — aa,] (taking the logarithm in the logarithmic with fubtangent = 4,) will be equal to the {pace required. Alfo, the whole fpace may be had of the fame conchoid, and likewife the parts, by confidering the curve in relation: to it’s axis, In the fame Figure make AB = DG = BC = a, BM = x, MD = 9; and from the point G let there be drawn GO perpendicular to the ordinate MD ; it will be DO = “aa — xx, becaufe of the right angle GOD; and by the fimilitude of the triangles CBG, GOD, it will be BG = gist teas = MO, Therefore MD Peon, ara 4 190 ANALYTICAL INSTITUTION S&S BOOK III. = Vaa — xx + a adi = 50 = y. Whence yx, that is, the element of the Ma i egli: De Pi. DA Sparano fpace, will be x4/ aa — xx + Fal EE, The fluent of the firft term de- PA pends on the quadrature of the circle, and of the fecond on that of the | hyperbola. | | man SE EE CT LT cei EXAMPLE X, Fig. 114. 101. Let AMI be the cifloid of Diocles, the equation of which is yy = -_. Therefore, fubftituting the value of y given by the equation, the formula will be is a dg | ture of the circle. To have the relation of the whole fpace of the ciffoid to that of the generating circle, it muft be confidered, that, the equation being yy = the integral of which depends on the quadra- IN Bt oe it will be alfo YY X dx — XX = x, and there- fore yW ax — xx = xx. This fuppofed, by differencing the propofed equation ayy — xyy = x3, there arifes 2ayy — 2477 — yyx = 3xxx, that is, 2y x a— KX — yx = SE .. And, becaule xx = yVax — xx, therefore. 2) X dfn SS gilar — ve But y x a — x is the element of the fpace AMQB, and yx is the element of the fpace AMP ; then, by integrating, as to the whole fpace, it 1s /y MER LOG Se Jyx. Then, in this circumftance, it will be 2/y x a—x —fye = fy x a—~x, and therefore ae ee = gfkxf ax — 8% ; and becaufe, in the cafe of the total {pace of the ciffoid, SXV ax — xx is the area of the femicircle ABN ; thence the fpace of the ciffoid, infinitely produced, will be triple of the generating circle. SECT. III, ANALYTICAL INSTITUTIONS, | igi EXAMPLE XI. 102. Let HBD be the logarithmic to the afymptote MQ; and let AB = a = fub- Bocchi, RF 40 AK eee and ‘the equation > = x. Then the formula will be yx =_ay, and by integration, /yx = ay + 66, But, fuppofing y = 4, it will be bb = — aa; fo that the integral complete, or fpace AKHB = ay — aa. Taking any other ordinate MN = gz, it will be alfo AMNB = az — aa, fo that MKHN = ay — az. Let there be an ordinate EF lefs than AB, and equal to y, AE = . è sink ay . 3 — x; in the fame manner, the equation will be ee = x, becaufe, x being negative, it’s difference will be negative alfo. But the abfcifs « increafing, the ordinate y decreafes, and therefore y muft be negative. Now, becaufe the element of the fpace will alfo be negative, this element will be — yx, that is, — ay; and by integrating, — ay + 25. But when y= 2, it will be 55 = aa; therefore the complete integral, that is, the fpace AEFB, will be = 42 — ay, And making y = o, that is, when it is infinitely produced towards Q; it will be = aa. And-confequently the fame fpace, infinitely produced towards Q, but which begins from any ordinate EF = y, will be = ay. Pn er ze | 3 EXAMPLE XII. 103. Let the curve ABF be the traéirix, the primary property of which is this, that the tangent BP, at any point B, is always equal to a conftant right line given. Make any abfcifs ED = x, the ordinate DB = y, the arch of the curve AB = 4, and the > given right line = 4. Becaufe, as the abfcifs ED increafes, the ordinate DB diminifhes, it’s element will be nega- tive, 192 ANALYTICAL INSTITUTIONS, BOOK IIIf. tive, that is, — y. Whence, from the property of the curve, we fhall have È ‘td ° è ° ° N i I the cquation — Sr = a; and, inftead of %, putting it’s value xx + yy, © © e — iV aa oma BJ ° ° AN ' | e 9 itis x = tee ar ne This being done, in the formula for areas yx, inftead of x, putting it’s value given by the equation of the curve, we fhall have — jV aa — yy for the element of any fpace ABDE. But, fuppofing the fir& of the ordinates AE = a, and with radius EA defcribing the quadrant AQM, and drawing BQ parallel to MH ; becaufe DB = EC = y, and, by the pro- perty of the circle, CQ_= / aa = yy, the element of the circular fpace CQA will allo be — y/aa— yy. Whence’ the fpace CQA will be equal to the {pace ABDE; and fo of others. And confequently the fpace infinitely pro- duced, comprehended by the ¢rad?rix ABF, by the afymptote EH, and by the right line AE, will be equal to the quadrant AME. BEX A M6Piboti XIU, 104. Let ACB be a fpiral, and AB = a the radius of the circle BMD, the periphery of which . = 4, any arch BD = x, AC = 9; the equation will be dy = ax. Drawing AE infinitely near to AD, it will be ED = x; and with centre A defcribe the infinitefimal arch CH. Becaufe of fimilar feQors ACH, ADE, it will be CH = =, and therefore the fector ACH, the element of the fpace ANCA, willbe = = . But, by the i gf Ga ax o equation of the curve, it 1s y = —-3 therefore that element will be = = and by integration, and omitting the conftant quantity as fuperfluous, the fpace ACN will be ae and making « = 2, in. refpect of the whole fpace ANB, which will be = Late | | Let ) SECT, TIT. ANALYTICAL INSTITUTIONS. — 198 _Let the equation be general to infinite fpirals a’ = 279%; then it will be 2n 2 eect te Rare \ È ex" x : : Jy = ——, and the formula of the {pace will be ——-, and by integration, emer bm | ab™ 2n-+m ui sù | e wi me) E Mira — s and making x =d, the whole fpace will be gior ape 40 H- 2m X om It is eafy to perceive, that the fpace ABMDCNA, terminated by the radius AB, the circular arch BMD, and the portion of the fpiral ANC, will be TAM a5 becaufe it is equal to the feftor ABMDA, diminifhed by the {pace 2 ACN. But if we would have it by means of the differential formula, it is enough to obferve, that it’s element will be the infinitefimal trapezium ECHD, which is known to be = DE + CH x iCD, that is; x + a x — = aax — yyx - And, inftead of yy, putting it’s value > given by the equation, 3 Race ee usi ca : a axs ae , it will be = — —-3 and by integration, —~ — =, omitting the fuperfluous 2bb ” conftant quantity. EXAMPLE XIV. 105. Let ABM be the parabola, whofe equa» tion is 4% = yy, and make AC = «, CB = y, and let the ratio of the whole fine to the right fine of the angle BCD be that of 4 to d; to the fine of the complement be that of 4 to f; then st -will be Bid =. and CD i a CH = x, then CH x DB = CHMB, the element of the {pace ACB, and therefore the. Fig. 118. B formula will be — - And, inftead of y, put- ting it’s value given from the equation, that is, a/ ax, it will be ave and by integration, abe fan. 2bay __ 2 | ews or 222 = SAC X BD. Vou, IL 9; Ce E ig. ANALYTICAL INSTITUTIONS. BOOK If EXAMPLE. AV. 106. Let ACM be a parabola referred to: the focus B, the equation of which will be az ‘ . | . = Wednakine. BC ==, CD =a, V20% — aa an infinitely little arch of a circle, and the parameter = 24. ‘Then the infinitefimal fedtor BMC, or BDC, will be the element of the fpace ABC, and therefore 12%, or QZZ "==": will be the formula; the integral of which will be found to be ZN 24% — da | x 484/70 — da + mm. Now, taking 2 = BA = 4a, in which cafe the fpace ought to be nothing, it will be mm = o, and therefore the complete integral, that is, the fpace ABC, 1s -— V 24% — 04% And in fa&, from the point C letting fall CQ_perpendicular to AQ; the fpace BCA is equal to the fpace QCA leffened by the triangle BQC. But, making BQ = #, QC = 7, it will be QCA — QCB = 3 x a Fr xy — xy = “as xy. Therefore:BCA = ee x y But, by the pro- perty of the parabola, BC = AQ + AB = x. + a, that is, 2 = « + 4, and. yz Saa+ 20% = V 202 — da. Therefore, thefe values being fubftituted: inftead of x and y, we fhall find BCA = DI NR hen TE 24% mea, ass 6 above. EXAMPLE. XVI. Fig. 120. | | | 107. If the fourth part AC of the-periphery of a circle be conceived to be ftretched out ‘Into a right line (ae), and taking any portion: | (ae) equal to the arch AE, let there be raifed. | the perpendicular (e4) equal to the right fine 2: SECT. TIT, ANALYTICAL. INSTITUTIONS, -195 DE; the curve (45) which paffes through all the points (4) fo determined, is called the line of right fines. Producing (ac) till it be equal to the femicircum. ference of the circle, the curve will have another branch beyond (cf), fimilar and equal to the firft. — . | Let the radius be = r, any arch AE = # = (ae), the correfponding fine DE = y = (ed); becaufe the fluxion or differential of the arch, exprefled by , which means of the fine, is found to be —°—, we fhall have x = —Z rm yy. | COTTO is the equation of our curve. Therefore the formula yx, by fubftituting the ry NT — yy putting y = o, itism=rr. Therefore the complete integral is rr — rrr —yy = fpace (ade); and making y = r, it will be rr = to the whole {pace (a/c). Whence, making TH the fquare of the radius, and producing the fine DE to M, the {pace (ade) will be equal to the re€tangle DH, and the whole fpace (atc) equal to the fquare TH. 7 value of x, will be ; and by integration, — 74/rr — yy +”. But, 108. The Examples now produced may fuffice to fhow the ufe of the method. Tt only remams te.obferve, that often the equations of the curves, the areas of which are to be fquared, (and this is alfo to -be underftood in refpe& to rectifi- cations, ‘quadratures of fuperficies, and cubatures,) may be fuch, that they have not the variable quantities feparate, nor can they be feparated by divifion only, and confequently are not reducible to the formulas required. Such would be the curve, whofe equation is x3 + y? = axy, for example. | In thefe cafes there is occafion to take the advantage of fome proper fubfti- tution, by means of which the equation may be transformed into another, in which the variable quantities are feparate, or at leaft are feparable. But it «cannot be determined, in general, what thofe fubftitutions ought to be. There is need of practice, and perhaps many trials, to know when this may be fuc- cefsfully performed. Ligh ists 7 ° 3 3 —e A, E n dI. - As to the propofed equation x? + y} = axy, make y = a and making as pt e . ° 7 a3 6 a 3 e a LS mm 6 the fubftitution, the equation will be x? + = se = , that isa? == dr. si | i . "nta — 6252 | i Then, by differencing, «x = 1ETT-Z, Wherefore, taking the formula — Fa for fpaces, which is yx, becaufe, by fubftitution, it is y = —, this formula 2B 3% — O25R0—Cw È eS —., it will be 9 = will be ==; and fubftituting, inftead of xxx, it’s value now found, A4aaz% — 623% aa* Cie 2 aor Ue a ; and by integration, /yx = $2% 196 ANALYTICAL INSTITUTIONS BOOK FIR. rà 4 . ; . . AX + . . pasto oe and, inftead of zz, reftoring it’s value Spa it will be finally fyx = EXAMPLE. XVII. 109. Let the curve be aox’y? — x° = a®y?; whofe area is required. Put y = —, and the equation will be transformed into this other, 4°2 — °% 3 % al caz — ge , Bee ——_—_———; and therefore x = ——————==—- Roe 33 X daz- ar. — a°, from whence we have x = az X aaz — a3\3 aa X aaz — a3)z i vice and y = er ae yee Fence we fhall have the ele- Fai it ake as fy ate 245% Rita, ment or tne area. yr — 784 A: KX GAZ e BD com per ra s and thererore; pole” ' È - — a 2a . i pia by integration, /yx = Tr + oe And, inftead of 2, reftoring it’s. value. a®y4- 2a5y3 | ==, the area will be — —G a * È; oe È To this purpofe may be feen the Method of Mr. Craig, in his Book De Calculo. Fluentium. | EXAMPLE XVIII. The reife, BIO. Det the Apollonian parabola be given to be rectified; that is, to find a: cation of right line equal to any arch of the fame parabola, the equation of which is. aa - ax = yy. It’s fluxion will be ax = ayy, and xx = . Now the formula: for rectification is /xx + yy; fo that, fubftituting here, inftead of xx, it’s value given from the fluxional equation, it will be “xx + yy = i aan. * : : ILE = Ly 49) + aa, the element of the Apollonian parabola. ON SECT. 11%, | ANALYTICAL INSTITUTIONS. | 107 ax. = yy. Proceeding to the integration; by making the fublltution of V 49) + 4a = 29 + 2, in order to take away the radical, we fhall find it to be az az BV 2/9 + ada = go ee i aa the integral of which we may fee is partly algebriical, and partly logarithmical ; and therefore the rectification of the parabola depends on the quadrature of the hyperbola; which truth may be | difcovered after this. other manner. Let ADE be an equilateral hyperbola, with femiaxis = 4, BC = » from the centre, CD = 27, the equation of which will be XX = aa = ayy. Drawing GE infinitely near to HD, then HGED will be the element of the fpace ADIIB. But we: know HGED to be 2) AY + a2 + aa, which is the fame formula as that for the re&ifi- cation of the parabola,. excepting the conftant denominator 24. Ther efore, &c. È uf C By the help of infinite feries. I take the above-written formula for the rectification of the parabola, that is, Ly 4yy +- aa, which, being reduced to» a feries, will be y + see yy: | 45% _ 1095 a Ls tag» No. And, by integration; 25 ni 7 ay? $ f LVS ay +06 499 +-o0o Zito #7 ! ~~ met re E —- a » &c. will be any. arch: whatever. | In the general formula SX bY Ws inftead of una in the place of x,. it’s value given by y from the equation of the curve; if we fhould fubfticute, lia +aa: Hf ARH aK ; / 4an 2x ‘in the place of y, it’s value given by x, it would be which: 1s not indeed. more manageable than the other. If the parabola was not that of Apollonius, but the fecond cubic, the equation of which is axx = 3; by taking the enni it would be xx = _ » and: therefore the formula AX + JJ xXx + jy = JV zi » the integral. of which is a 2 gay + 444 + m. Bat, putting y = o, it will be m= — 44; therefore the complete integral, or the length of the arch, will be. gay + 444 © 2794 N90) + 400 mm MA 198 ANALYTICAL INSTITUTIONS, “BOOK IIT. Fig. 122. In the Apollonian parabola ADM, if it {hall be AC = #a, and taking any line CK = y, . the parameter = 34; itwill be AK = ga + y, KM =v. ‘Whence the element of the area MKCD will be jw S22", which is A C K | the fame with the element of the length of the fecond cubical parabola, except the conftant quantity 4. And therefore the rectification of this, and the quadrature of that, «is the fame thing. Whence, becaufe the quadrature of that may be found algebraically, this is alfo algebraically rectifiable. And hence, in general, if the expreffion of the element of any given curve, divided by the difference of the unknown quantity, be put for the ordinate, and the unknown quantity be put for the abicifs ; a new curve will thence arife, the quadrature of which will give the rectification of the given curve. EXAMPLE XIX. Fig, 123. | rr. Let AEM be a circle, it’s diameter Pe eee en es oat will be BE Sy Sy UE : OR et e eV as mn ee Phen yo — È; ee Iq ee ay a ee ee Ae, And therefore the ele- eC om ment of the curve FH = Vxx + yy = , and reducing it to a feries, it will be RO regia I Of gi rex I n 2 Shel xa x N Dea Ley X4- RE A + IE | 2 X2a* 2X2X 4a? 2X2xX4x 6a” 2XK2X4XOX Ea” ae 3 5 Lev by integration, it will be @?x? + - + SR + 3 — + a5-38" Hak te. BO xAKOK 74 9. E OR OSLER TORK X ax — INT, &c. Or, becaufeitis xx = ERO A ci ~, that is, by fub- 2X4X6X8 X ga* Seam on Fa ftituting yy inftead of ax — xx, xx = =. then putting this value, in- ftead SECT. III: ANALYTICAL INSTITUTIONS. | 199 e i 2N aa agri {lead of xx in the general formula, it will be xx + yy = 2 4 6y4y. + which, being reduced to a feries, will be found to be = y + =— si Bone: + 195 » &c. And by integration, it will be finally the arch FA = y aye Li 20y7 152 dale | 3a? + 5a a 4a° + ‘gat 3 &c, But if the radius were made = a, the feries would be y + eer 5A 395 gt be td > + a , &C, A AX 544 2% 40% b'x 74 a.% 46% 94° Laftly, if it were DB = «, the radius (3 oe eee at would be y = Vaa— xx, ; and, reducing to a and y= E e et +gy = N Ga = xn Fase feries, it will be ora + we 2 X = si x 4x 64° Di 2X4X 6.x 848? &c. Whence the arch EF = # + —— eva? = + 2x4 x 544 ch 2X4XKOX 748 am 1054? 2°% 4:X Gx 8 x 94°? &c.. EXAMPLE XX. 112. Let ADC be an ellipfis with: tranfverfe femiaxis BA = 4, and conjus- gate femiaxis BD = 4, BE = x, EO=y; abe È A ry Tees ca oe and yy = the equation will be GA — K%, C and therefore y= bbxwxà bbaxzk = _ ==» and the general formula Mixx + YY vy = et XX + sn aa X da — Kw | artrosi xo at — atx? + b2x3 oda = 63: | Inftead ‘400 ANALYTICAL INSTITUTIONS. | il lille Inftead of fubftituting the value of y given by x from the equation, if we | * PARA. ——————66m if aay — biyy + ae fhould fubftitute the value of. x, it would be Vxx + yy = ere : ted 2 But both of the expreffions fo found would want one of the conditions of $ 39, without which it may be feen, that thefe formulas cannot be freed from radica! figns, and fo prepared for integration. Then to proceed to feries, I take one KM at — atx + ba of the two formulas, for inftance , which alfo may be thus | | av aa — xx FR bbax . . Dr i a= -exprefled, tal Le eet and this being reduced to a feries, will be = — 44x% I1bbuxk tig STR è “jesi n — Le — — &c. And | ESISTE CCIAA mn) AIR SENESI DRS] VETRO DITEMI SION romasic ree PT rs RI 3 ee te 3 a. ‘ Y da XK adam xx ax aa—xx)\* a°-X aa — xx a x aa — sx. again, reducing every term of this into a feries, beginning at the fecond, it will x + de ci ES ae BS ; be N 1 + bbxx ens wie 1 bbwxx at x aa — 14 ‘ SA Wn og DI & Soe aie Se mn Co — ESE into D+ + +, &e. ia na, o x + de sO oe = 30 Kc, And by BETS the arch DO will be f ial I + a se # +2 into È Z+5 ary ke. è BIO. aS 241 pe SE trees into — + 4554+; &c. Sat Sat ae pat. pare ? i, aig AA ae i Ponies tlic ees ERE i ; | “È 1648 into 7a cn 6 ga? oe Liat? 3 &c. 8 x Ant a anto o ® 12548 ga® 11910? Ges And laftly, reducing.the homogeneous terms into the fame denomination, we fhall find DO = b? x3 Me “4.a2b2 oy de sa 446 — 4a?h4 +. £6 6a* 4048 tiga 640°b? — 430954 + 240255 — 55°.‘ 6 E Ok. TELO Cuba ac 6 Now, a nt SECT. IIT. ANALYTICAL INSTITUTIONS 20%. Now, if we fhould fuppofe @ = 4, in which cafe the ellipfis would become a circle, we fhall have the arch DO = « + = e sE + Gc ee Be, 404% 11209 | gx 12848° jift as was found before, at § 111. È i xY at — 8 + Ba Bb? x% After another manner, thus. In the formula avaa ome XX , if we make RL OO ee xo a4 = CCAX | PER TELI REE Pa GRETA TUE SNEED ERT bb — aa = — CC, fo that it may be , the two ‘radicals Heinen re- xf at — tx aa ctx? av aa owe AR ca? Vai 6x een 20% 8a® 16a® 128414 7” 24 843 1645 12807? divifion of the numerator by the denominator, after a very long calculation we fhall find another feries, which, being integrated, and the value of cc reftored in it’s place, will.give us the fame feries as above, which exprefles the value of the arch DO. aneeD | folved into feries, it will be SE . —— into 4° — a &c. ;s and actually making the 4 merci - 3 7 > - sno) E XAMPELAE- XXI big. Ad | 1 113. Let BD be an biyperbile with tranf- Baten verfe iemiaxis AB = a, conjugate femiaxis AB sé, CD = 9, AC = yy the equation will be xv — 44 = to, Then, by taking “A a "© the fluxions, ii De yc se — i , bV bb + yy pia _ 5 Vibytagyt Th = jaf 1 int Me RE LL erefore this being reduced ae a feries, after cee of the ways before made ufe of for ary3 405° +a* . 634 ~ 4068 cP a the ellipfis, we fhall find it’s integral, or the arch BD = y + 8a254 + 4040? + ari 64099 + 484454 + 240°? + 50° 1120"? x QX 128019 feries as that for the ca: excepting the figns, and the change of the con- {tants 4, 2. y?, &c. which is the fame CRE) Vea e 8. ka 202 ANALYTICAL INSTITUTIONS, BOOK IIle EXAMPLE KXXIL Pa Fig.119. A Qp rg 114 Let it be the cycloid of Ex- OR meee RA RMR anrcemerwrInvawisac ample VIII. of Quadratures, the equation of which we know to be y = XV ——|; Xv : therefore the formula will be “xx + yy a ae as M è : Ì Hg E A — , and therefore, by integra- tion, it will be the arch EA = 24/ax, or the double of the chord AS of the correfponding circular arch AS. And putting x = 4, AM will be the double of the diameter of the generating circle, and therefore the whole cycloid will be quadruple, EXAMPLE XXIII. 115. Let ABF be the sraGrix, whofe equation is (§ 103.) — a = 4. There- fore 4 = — = s and, by integration, any arch AB = = — ly +2, in the logarithmic curve with fubtangent = 4. But, making « = o, it Is y = 4, and. iy thereiore. 4. == 0, and’ the complete integral will be 4 = == /y. Therefore, if the logarithmic AKS be defcribed through the point A, with the fubtangent AE, to the afymptote MH; taking any point B in the ¢raéfrix, and drawing to the logarithmic BK. parallel to the afymptote, and letting fall the perpendicular KN, the intercepted line NE will be equal to the arch AB. EX. A BECT, TIT. 'ANALYTICAL INSTITUTIONS. 209 EXAMPLE XXIV. Fig. 117. 116. Let ACB be the fpiral of Archimedes of ad § 104. the radius of the circle = 4, the circum- ference se 2; .the arch BMD: — #4; and AC = ». Let AE be infinitely near to AD, and therefore DE = x. Wath centre A let the arch CH be defcribed; then it will be CH = —, and Orr. =). Therefore CO, the element of i curve, is equal | Vsyii tag a to - But the equation of the curve is ax = by, and therefore xx = 2 whence, making the fubftitution, it will be CO ra | si La a* + Tu: The integral of this, after a long calculation, which, to aoe being tedious, I fhall omit, will be found to depend on the logarithms, or, which is the fame, on the quadrature of the hyperbola. Now, by infinite feries. Firft, I make a* = bbmm; whence the formula will | be this, = mm +- yy, which, being reduced to a feries, will be 2 into 2 NE a È te | ta i: i .° CI m + —z, +” © 1g: &c.; and therefore, by integration, the Sì ay bys Oe aN arch AL os te 2, oa + Lente 9 x 1284m1? &c. And making = mila ab . a35 asb (i sani y = a, the whole curve ACB = —+or- arts 40m3 11275 ‘9 x 120m? ? &c. Then, inftead of m, reftoring it’s value iran 5 1¢ will be ACB =a + b bb b4 ~ #8 gb? Bec 6a 40a3 lives Oe late 00 Ddz If 204 ANALYTICAL INSTITUTIONS, BOOK IIS, Fig. 126. | ee If the curve ABC were the logarithmic [°D fpiral, whofe equation is ay = 6x; making RB = y, and the infinitely little arch BD = %; putting, in the general formula N xx + yy, the value of x given from the bb | : equation, it will be i Ga and by in- tegration, the curve AB = —V aa + dé. Let the curve ABC be the hyperbolical fpiral, in which the fubtangent is always conftant; and therefore, Ie@ting the fame names as above, the equation will be yx = ay. Therefore it will be Wxx + vy + yy = 2 faa +9) sa) the integral of which formula, ‘freed from the radical fign, will A found to depend on the logarithmic. By means a feries we fhall find faa +9) aa 4 yy = y into — a 2 — E cÙ | Ta — Si s &c. But if we dui proceed to the Lain n firft term cannot be integrated, but by the help of another infinite feries. Wherefore, the fum of the faid feries being integrated, all but the firft term, together with the integral of the feries expreffing that firft term, will form a feries which will be the value of the curve propofed. SSS a a! EXAMPLE ‘“AXV., ‘117. Let HBD be the ainsi. AB the fubtangent = 4, AK = x, KH = ” Di and the equation = = Xx. The value of Fig. 115 x being fubftituted in the general formula, tegral aca on the fame logarithmic. I fhall forbear to apply infinite feries, becaufe their ufe may be fufficiently feen in the former Examples. E X- it will be SV da + Ws of which the in- » SECT. II, “ANALYTICAL INSTITUTIONS. . 205 EXAMPLE XXVI. 318, Let the curve be the Apol/onian parabola, with it’s co-ordinates at any oblique angle, and whofe equation is ax = yy. This being differenced, and - fubfticuted in the general formula for rectifications, when the ordinates are at oblique angles; that is, in the formula Mi; XX = yy + o, inftead of x, it’s value given by y being fubftituted, we fhall have Lang! yy + o + 144; the in- tegral of which will be partly algebriical, and will depend partly on the qua- drature of the hyperbola. : | | EXAMPLE XXVII. 119. Let the equation be ey, which is to infinite parabolas, and to infinite hyperbolas between the afymptotes. By differencing, it will be x "x = “y, and eo, ah yy; whence “xx + yy, or the element of the curve, will be a7 Fauno + 1. Proceeding to the integration, I fhall have recourfe to the method of § 61, and fhall exhibit the formula in the following manner, DI IM è ; ) x ey .* = ° ® 4 x CI —~. The canonical formula of the faid article, or ——-, is ptt 2 +1 Ta af Bi as algebraically integrable when Anat is an integer affirmative number; and if it be an integer negative number, it will be reduced to known fimple quadra- \ tures. Now, by comparing this formula raphe: with the canonical, we 2f —_ 2 se oe +1 have 2 = o, 2f — 2 = 7, and 2 = 1. By which it will be neceffary that 1-2f42 ; | i ie he ae eg 3 tt? (hall be an integer, which I call-4 Then Aen waa is, 261 — 2 2% — 2 3 — 28 2t— 2 infinite curves. 3 + 2h = h, and confequently ro the determining exponent of the Let “206 CORMWALYPIGCAL- INS$PITUTION’'S. BOOK Ilfi Let 4 be a pofitive integer, beginning from o. Now, if b =.0, it will be foe ts {52 LT will Pe ee a, ll be FS 2) “ec Lot d be any one of the feries of natural numbers, 0, 1, 2, 3, 4, 5, &c. the innu- merable values of the exponent ¢-will-be-expreffed by the following progreffion, b= 3, ia dati tay ti, e, the law.of which feries.is manifell; and in.all thefe cafes the parabolical curves will be algebraically rectifiable ;. the fir of which is the fecond cubical parabola, Let be equal to an integer negative number; and, firft, make b = — o, in which cafe the fame cubical parabola arifes, becaufe — o and + o are the fame thing. Make 4 = —.1, and the exponent ¢ becoming = 4, it is con- fequently infinite. Make 6 = — 2, then = I. Make 6 = — 3, then i = 2, Ando od Lberetore auc infinite ae of the exponent # will be expreffed by this > ES a) Fol alae 0 a 22 > parabolical curves thence arifing will be rectifiable by means of known quadratures. The firft curve which prefents itfelf is the conic parabola, the rectification of which requires the quadrature of the hyperbola, § 110. The other cafe, in which the general formula of § 61 is either re&ifiable : eos x algebraically, or by means of known. quadratures, is when 4 e — —I1— — ti 778 is an integer. That is, by fubftituting the particular fpecies of this example, gi ka 1 Ap a ae ROSE, Sa = 4, and therefore I 25 i, the determining exponent of the infinite curves. Let 4 be a pofitive intaeet. number, beginning at 0; we fhall have ae fol. lowing progreflion, # =: 4, & 3, 3) 113, &G. Let bea negative integer, and, firft, let 6 = — o. Then the fame exponent Z = } returns upon us, becaufe — O Is equivalent to-----03-Let b = — 1, the exponent ¢ becomes equal to the fraction 2, and confequently is nothing...Luet p.es, d Siusi &c. and we fhall have this following progreflion, ¢ = 4, sex ste Sr SIME cre The fraction which gives the value of the determining exponent 7, is the fame in both cafes, only that in the fecond it is the reciprocal of the firft; fo that the progreffions ought to come out reciprocal, as in effect they ‘do. Therefore the curves determined by means of each formula are the fame, but with reciprocal exponents, that is, they are referred to two different axes. As for example, the two exponents t and £ belong to the 4pollonian pares ' ‘ UO) Jo which offers itfelf in two mariners, x? =y, that is, x =yy, and likewife yr =y, or xx = y; both local equations to the parabolical trilineum. Wherefore SECT. III. ANALYTICAL INSTITUTIONS, 207 . Wherefore thefe curves, which are included in the foregoing progreffions, are either algebraically integrable, or do not require quadratures beyond the circle or hyperbola.. But the other curves, infinite in number, require quadra- tures of a higher order. | It appears from our progreffions, that the value of the exponent ¢ is never negative. Hence no hyperbola admits of a rectification, either algebraical, or depending on the forementioned fimple quadratures, - EXAMPLE XXVIII. 120. Let ACGKA be an ere& cone, AB = a, BC = 8;0f cubatures. let AD..=-x -be any portion of the axis AB; it will be DE =y = =. and therefore, fubftituting this value inftead Oa i chbaxx 40 will be ——, and by of y in the general formula, me bbeS integration, —— » in refpe@& to any portion taken from the vertex; omitting the conftant quantity, which here is need- lefs. And making x» = a, the whole cone ACGKA will be bb | DI ui Dna = — x GOOSE 3; and it’s double, 3r . or Bcaxn — 6x3 6r > will be the whole (phere. And becaufe the a the Relght of “iù is Set to the diameter of the bafe, or 24, 1s di; the cylinder circumfcribed will be to the {phere infcribed, as = is to =, or as 3to 2. And confequently the half cylinder will be to the hemifphere in the fame ratio. But the cone alfo, whofe height is equal to the radius of the bafe, (or equal to @, the radius of the fphere,) 1s.= di; r therefore the hemifphere will be to the cone infcribed as 2 to 1 8 | Furthermore, SECT. lle ANAEYTUCALY INSELRPUT TONS: 209 Furthermore, as it is known that a is the radius of the bafe of an equi- lateral cone infcribed in a fphere, the radius of which is = 45 and the height of 5 . 3 { 3 the fame being = 3? | the cone will be = 22, and the {phere will be = 1 2 48r 3r and therefore the {phere to the cone as 4 to .°,, or as 32 tog. In like manner may be demonftrated as many Theorems of Archimedes as we pleafe, which are of a like nature. 3 Hence the manner is plain, of obtaining any fector of the fphere, which is generated (for example) by the fector of the circle BEDM. For to the fegment of the {phere generated by the figure AED, which we know to be = i cx", mutt be added the cone generated by the triangle EBA, and which ee, will be the ° £ Seer dci E is found to be = ~~ X 244 — xx X 4 — x, and the fum, DI fector required. LA AMPIE ANA, 122, Let there be a parabola of any order, È P we { ° - AO whofe equation 1s 4° E Re which, being i K converted about the axis AM, generates a para- mM I I | bolical conoid. Then it will be y= a 7 x”, 277} — 2 2 M and yy = a ™ x” ; and therefore, fubftitut- ing. this value, the general formula will be 2.774 = 2 2 27%) --2 mM+-2 ca B xx | : È Mion aay OF ' | | ; and, by integrating, , will be the folid content 2r ar X m+2 Pe ll 24m of the indefinite conoid. Or elfe, becaufe «” = = =, and therefore x m ne a Mm == —2_, by fubftituting this value in the integral now found, it will be . 2R~2 ; a mn mexyy ar X m+2 ‘ Vor, II. Ee Make 210 ANALYTICAL INSTITUTIONS, BOOK Il], Make m = 2, that is, let it be the Apollonian parabola; the conoid will be % ur? that is, the produ& of the bafe into half the height ; and, by confe- quence, the faid conoid will be half a cylinder of the fame height, and of the fame bafe. If we would have the folid content of the difh, or of the folid generated by the figure ACD, converted about the axis AB; from the cylinder defcribed by the rectangle ABCD, which we know to be = = » we muft fubtra& the pa- 222%, the remainder, 29 X m+2 rx difh. And making m = 2, inrefped of the Apollonian parabola, the difh will pe 2 ar? rabolical conoid 222, will be the content of the 242 which is half the cylinder, juft as it ought to be, the conoid being alfo half of the fame cylinder. Let the figure move about the ordinate MO, and make AM = 4, MO=f, AB = a, BC = y, CK = 4—4%,KO=f—y.. The circle, with radius CK, | will be = — x 5— x), and therefore the product of this circle into y will be the differential of KM; that is, — X bby — abxy + xe) will be the element of the folid generated by the figure MACK. Therefore, by integrating, and, inftead of x, putting it’s value given by y, it will be — x ayer Tiss i 2 su bby — ———-— + s equal to the indefinite folid, Or, UR gir whe gig ee i int ; i. 1 af 2bay KEY | putting x in the place of SI Va + a IL cc I a Now, putting x = 4, y = f, in refpe& to the whole folid generated by the figure i spinta Rag BES . ammbbf 3 | A. it will be — Ugo hi Arie ne el ACOM, i or X bbf ea TEX: And if we would have the parabola to be that of Apollonius, that is, if m = 2, ie a. It is eafy to perceive, that, in the Apollonian parabola, a cylinder on the fame | bafe, and of the height of the faid folid, fhall be to the folid as 15 to 8; and that the folid generated by the figure OAP fhall be = DI, pia 2 Let SECT. Ill. ANALYTICAL NSE Ste PER eS ort Let the figure move about the right line AP, and let it be, as pio, AB=x, BC =y; then — will be a circle with radius DC, and - will be the element .of the folid generated by the figure ACD. And, inftead of «, c putting it’s value given by y, and then tegrating, it will be "Tra N22. +I » “ar . x nd ROME pe, , that is, eo i n, ‘eda to the indefinite folid. A chbf 2r X 2m + 1 281 Xa making x = d, y = f, it will be » in refpe& to the whole folid, generated by the figure AOP. But the cylinder on the fame bafe and altitude is = CL; therefore the fold generated by the figure AMO is = — —X on But ftill, in another manner, we may obtain the folid dprieragia by the figure AOM, revolving about the axis AP. ‘Make AM = 4, MO.=f A cide with radius DC will be = — , and the circle with radius DK will be equal to zd sE heref&re — X db — xx will be the annulus defcribed by the line CK, and di x 26 — xx will be the element of the folid generated by the figure CKMA; and, inftead of x, putting it’s value given by y, it will be pani A ‘3 Mito NRE, and by integration, — x by — naar. 2Mt1. X a” Laftly, making y = ce in refpe& of the whole folid, generated by the figure 2m-|1 272 « "= bf 6. Mm —— 2 ¢ s a x buy ane a? AMOA, it will be — x bof — ar -. But, when y = f, becaufe 2M 2: 2mM+1i Xda ULI 277 . fn , and db = L_, Therefore, in a i of the parabola, it willbe x = 3 = a c the integral, fubftituting the value given by è, the folid will be x bf LR Get A as above 2% +1Tt ar Nogey ps S a00VC. oD I) EX. 212 ANALYTICAL INSTITUTIONS, BOOK TIT. EXAMPLE XXXI. Fig. 1240 123. Let ADC be an ellipfis, AB= a, | BD=s, AE=x, EO=3; and therefore tlie bb IA equation is aes X 2ax — xx = yy. There- fore, in the general formula, fubftituting the value oa given from the equation, it will be = oe x 2axx — x4X ; and by ine tegration, it will be a X axx — La » equal to Pe indefinite folid generated by the figure AEO, turning about the axis AC. Making x = a, it will be i, half of the iss and putting x = 2a, it will be 2 » the whole fpheroid, And, becaufe the cone of the fame altitude AC, and of a bafe the radius of which is the conjugate femiaxis BD, is = =, and the cylinder is = = a the fpheroid will be two third parts of the cylinder, and double to the cone. EXAMPLE XXXII. fig, 121, 124. Let AD be an hyperbola, which is PE CR | converted about BC, and let it’s tranfverfe femiaxis be BA = 4a, the centre B, and it's parameter = 4, AC = x, CD =, and the equation is ax + xx X vg PD. a Subftituting the value of y in the general chà formula, it will be > x ax + xx; and by integration, it will be — X jaxm + 4x3, equal to the indefinite hyperbolical conoid, generated by the ei ADC. Make SECT. Ill ANALYTICAL INSTITUTIONS 213 Make BC = «, and the ret as above. The equation will be È x liceali eine È bi crm i di xx — aq = yy, and therefore the formula will be — X xv — {aa, and by in- tegration, = X tx? —taax + f. I add the conftant quantity f, which, in this cafe, will be fomething. To determine what, it muft be obferved that in the point A, when x = 12, the folid ought to be nothing. Wherefore, inftead N + è ch ——————————_ l 3 I . can + Saf E of x, putting 74 in the integral, it ought to be f + pn ee gts Soe. ch caab Therefore the complete integral will be LE and therefore f = ——~ 24r Ta — tae + 150). Let the hyperbola be converted about the conjugate femiaxis HB, and make the tranfverfe femiaxis AB =, the conjugate’ femiaxis = 2, BC = x, CD = y. The circle with radius HD will be — De, and therefore es will be the element of the folid generated by the plane or figure BHDA. And, inftead of xx, fubftituting it’s value given from the equation of the curve, we cy aayy + aabh : € aay? i | fhall have — x =7—; and by integration, — X i. + aay; and 2caah making y= A, the folid will be = as SR cnr etici ANP EET NA TASER ONY gh EXAMPLE’ XXXII. 125. Let KHF be an hyperbola between the afymptotes; AD = a, DE = 4, AP = «, PH = y, and the equation xy = ab. Let the curve revolve about the afymptote AB. Then the circle with radius QH will be = 22, and therefore ; 27 e will be the element of the folid gene- rated by the igure AQHFMA, infinitely produced towards M. And, inftead of «, putting it’s value given from the equation, | it 214. ANALYTICAL INSTITUTIONS BOOK IIT, le caabby . . caabb i : it will be nies and by integration, f — oie Now, to determine f; it may be obferved, that, when it is y = ©, the folid ought to be nothing, and there- caabb È fore, f = = a an infinite quantity, and therefore the complete integral witl ae caabdb — +0; fo that the folid is of an infinite value. 4 Inftead of fubflituting in the formula the value given by y from the equa- tion, in the place of xx, if we fhould fubftitute the value of y; it would be ae , and by integration, — —_ + f. But the folid cannot be nothing ene] , except when « is infinite, and then the conftant quantity f to be added ought to be infinite, and therefore the folid will be infinite. To have the folid generated by the plane or figure BAPHK infinitely pro- duced towards B, it will be enough to confider, that as — is the periphery of the circle whofe radius is QH = x, then — will be the fuperficies of the cy- linder, generated by the plane AQHP, and confequently 29%. will be the folid r content of the hollow cylinder, generated by the infinitely little rectangle IPHO. Therefore the fum of all thefe, or / cxyx s will be the folid required. r cabs i s a finite Therefore, inftead of y, putting it’s value de , the integral will be Bini r quantity, although the folid be of an infinite altitude. cabx Lae si La In the expreffion —— of the folid, inflead of 45 putting it’s value xy, given from the equation ; it will be — rectangle APHQ. Therefore the hyperbolical folid will be double to this cylinder. And therefore the folid generated by the figure BQHK, infinitely produced, will be equal to the cylinder which ferves it for a bafe. Therefore, » But 2 is the cylinder generated by the taking x = a, and confequently y = 4, this cylinder will be = cea , which is equal to the folid erected upon it, E X- SECT. IIL ANALYTICAL INSTITUTIONS, 215 EXAMPLE XXXIV. Fig, 13%. i 126. Let HCD be the logarithmic curve; it's-fubtangent-GA.-=>-a, AB = «, BD = y, and it’s equation x = a Let it be con- verted about the afymptote EB. In the ge- neral formula, inftead of x, putting it’s value ey. 2r given from the equation, it will be ; and L MA A by integration, it will be re +/. But when: itis y = AC = a, the folid will be = o. Therefore it muft be f = — “Pa and the complete integral, that is, the folid ea by the indefinite plane. ABDC, will be = — Let the abfcifs AE be negative, and therefore = — x; and it’s fluxion alfo. will be negative, or — x. And becaufe, as the abfcifs increafes, the ordinate _ will diminifh, therefore the fluxion of EH will alfo be negative, or — y; fo that the equation of the curve will be ftill the fame, x = a - But, becaufe x. is negative, the general formula will be negative alfo, or — n. : Spiro therefore, the value of x, it will be — i » and by an ~— a + fe But when the folid is nothing, it will be y= as. therefore St, and. the: 4r complete integral will be , equal to the folid generated by the plane: cas — cayy 4r ACHE, Putting y = o, that is, fuppofing the folid to be infinitely: produced: towards M, the integral will be = = s and then the folid itfelf, infinitely prow. duced,. will be = = - But the folid generated by the plane ACHE we have- Ca? — cay feen to be ail 4r 3. then the folid-infinitely produced, generated by the. plane. LEMH, is. | Now,. 216 ‘ANALYTICAL INSTITUTIONS, BOOK III. Now, becaufe the cylinder, the radius of whofe bafe is AC = a, and it’s : v 3 ‘ E eee e : ; - ‘ height alfo = 4, is ——; the folid of the logarithmic curve, infinitely produced towards M, on the bafe with radius AC = a, will be to the faid cylinder, in the ratio of + to 1, or as 1 to 2. FELPE RG which, by revolving about the right line AB, defcribes atolid. ncvinkesAPd +, PM — 34, ABi= a, and the Fig.114,. I {i 127. Let the curve AMI be the ciffoid of Diocles, | equation will be py fera Lada Therefore, the value of «yy being fubftituted, the general formula of folids will B : ) A cut x . . cx? Cax* e , and by integration, — — — rr 2r X a—x O Or 47 CAUAX cada area eR mann È e X la—x + f. But, making «= o, the “ wf @ . ° ‘ x - N : folid ought to be nothing, and therefore f — la, TRE I . caaa,, __ caaa e a _ (Cad . complete integral — /a ca ee ui 5 Sr 18 equal to the folid generated by the figure APM. And making « = 2, the whole ; E casa caaa 1Ica3 . 7 x A folid will be = ale — —-/o — ——. But the logarithm of o is an infinite quantity and negative, which, multiplied into — 2°, makes an affirmative quantity ; fo that the intire folid will be infinite; It is to be obferved, that the miner logarithms are to be taken from the logarithmic curve, the fubtangent of which = a. | Batti i ; : i le: oxic (BY ae cati case i y the help of infinite feries, it will be a li i ea | co s &c.; and by integration, the folid generated by the plane APM will & 5 6 7 be = = ca — = &c And making x = a, in refpe& of Bar 10ra* 12r03 I4qra4? ent ha È ‘ A caz . pa Ric MES doh; it will be sy into 7 +4 +1 +1, &c. the total value of which feries is infinite, 4 E X- + SECT. 13. - | ANALYTICAL INSTITUTIONS 217 EXAMPLE XXXVI. ‘Fig, 116. iS | 728. Let the trefrix ABF be con , : x verted about the afymptote EH. In L a | the general formula one » fubftituting ° : 2r + Mps B the value' of x given from the equation FISICA, RUDI i Fe gara a = eo, $ 103, we fhall M N. N ae pom — cyjVaa — yy ar have — And by integra» “tion; it will be = x 4a— yy)? +, peal to the folid generated by the figure -AEDB, omitting the addition of a conftant, which is here unneceffary. cas Wherefore, making y = o, the folid infinitely produced will be = 7: But the folid content of the fphere whofe radius is AE = a, (§ 121.) will be = 2ca3 raat and therefore the folid eee produced will be a fourth part of that iphere. ; EXAMPLE XXXVII, £1951 32. 129. Let QBMCPT be a cylinder, | from which, by a plane through the. diameter BC, and in the direction AP, a portion or ungula, BMCPB, is cut off; the folid content of this is required. . Make BC = QM = 24, MP = QT = è, AD = #, and DH being drawn, Mall be an ordinate in the circle = 4/7 aa «= xx.. From the point H let the right line HO be drawn parallel to MP Aa DE QI, which fhall be in the fuperficies of the cylinder. Then from D to the point O let the right line DO be drawn, which fhall be in the plane BOPC. Then we fhall have formed in the folidity of the wngu/a the triangle DHO, VoL. II, Fr - which è 218 ANALYTICAL INSTITUTIONS, BOOK IIT, which is fimilar to the triangle AMP, and therefore it willbe HO = ana, | But the aggregate of all thefe triangles, TGS is Jolt phe, the folidity required of half the wugula, and therefore it will be = = f= X aa—xx; and by integra- t10n, 2 wee = . And making x = a, the aforefaid h half ungula will be SR na 100b, and the ie cc Paaba Fig. 133. | | ¥n another manner, and more ge» | nerally, thus. Let DACHEG be half of a cylinder, which, through the dia- meter CD, is cut by a plane m the | direction DE, whence arifes the ungula DBCEAD, the folidity: of which is required. Make BA = a, AE =2, BQ = #, QM = y; it will be QK = =, and therefore the rectangle PONM 2bxy ; : 5 vc 2, And this being drawn into 2byxx *, or a "i will be the element of the folidity of the ungzla. Let the curve AG be a femicircle ; Mien.» = A — xx, and the formula s and by integration, — — LX GALA + mn. Now, by ui x = o, the conftant, m, will be ent to be = daa, and therefore wil the integral of the folid complete will be 3544 — È pa aa—xx\* : and mak- ing x = a, in refpect of the whole wngula, it will be 3222, as before.. Let the curve DAC be one of the parabolas ad infinitum, and it’s equation 3, Sa Subftituting the value of y, the formula wall be i x vin x) bg being integrated according to § 29, and a conftant being joined,. and m+. 2bm*a m making x = 4; it will give for the folidity of the whole amtptiIXmatdki | 2 angula. And taking m = 2, or the ; : È: 3 qa cre — pale PL spe e« To proceed to the integrations or quadrfatures, I fhall' mil ufe of the. method explained at $ 61, and applied to the aforefaid Example XXVII:. But, firft,. it is.to be obferved, that ¢, being the periphery of the circle whofe radius i is 7, the integral p= = will give us the furface of. the conoid: But if e. reprefents any right line whatever, we fhall have the meafure of the furface. of | the angula, when a:cylindroid.is erected upon the. bafe CAB, which is cut by, FI plane. 224 AN'ALYPICAL INSTITUTIONS. BOOK 111, di paffing through the axis AB, and with the e fubje& bafe CAB forms an angle, of which the right fine is to that of the complement, as c is to r. Then the ungular fuperficies is to that of the round folid, as a given right line is to the circumference c. Operating, therefore, as explained above, : at § 61, that our formula may be algebraically integrable, or reducible to known quadratures, we fhall find z b , that it mul be ¢ = st, "3 A o) 0 I Ze Te, where 4 denotes any integer number, pofitive or negative. Bas 2h beep to pofitive and then negative, will give us thefe two progreffions : making è» any integer number, fir@t ‘The firt condition, or # = oe Lad E oe SMI BRE mah. a da SE CO SIRO AE eee a. e e Iii, TA SITI ga SLC RIE Ao 1979 #3 79 wo Tis SC. so i AER The fecond condition, or ¢ = i, making 4 any integer number,. firft pofitive and then negative, will give us thefe other two progreffions : Hei ei ee IN Sy oa ey la To this I fhall fubjoin a few fhort obfervations, I. As the two progrelfions, the fir& and the third, contain the exponents of all thofe parabolas, Which, by circulating about the axis, generate conoids, the fuperficies of which are analytically quadrable, fuppofing only the redification of the circular periphery ; and confequently all the wgule above defcribed, of a given altitude, admit an algebraical quadrature: ‘So, in the cafes of the fecond and fourth progrefiions, Tomething more is intended, as they require the NATO of the hyperbola. II. It is obfervable that, the firft feries being compared with the fecond, and the third with the fourth, the exponents are reciprocal, and belong to the . fame curve. This Mows that, as the parabolical area may be converted, either about the axis AB, or about the axis AD, and in each cafe may produce very different fuperficies ; ; if, in the firft cafe, it generates a fuperficies that is abfo- lutely quadrable, at leaft confidered in the ungula ; in the fecond cafe, on the ‘contrary, the values being reciprocal, the above-faid fuperficies will arife, which are only hypothetically quadrable. For example, the conoid formed from the fir& cubical parabola being turned about AD, furnifhes us with the furface of an ungula which is algebraically quadrable, and alfo that of the round folid, pro- _vided we have a right line equal to the circumference. But if it be converted about the axis AB, then quadratures are required. ‘The fame thing obtains in the oe cubical parabola, and quite the contrary in that of Apollonius. 9 Il Com- suor. tit, ANALYTICAL INSTITUTIONS, 223 III, Comparing thefe feries with thofe of $ 119, we may difcover, that among thefe there is no parabola of the firft or fecond feries, that is re&ifiable either analytically, or by the means of known quadratures; on the contrary, thofe of the third and fourth are all rectifiable, and comprehend all that are contained in the progreflions of $ 119. IV. Among the hyperbolas, the common one only between the afymptotes admits a fuperficies reducible to the quadrature of the faid hyperbola; becaufe | fio other negative exponent appears in the progreffions, except — 1, V. The exponents which are not found in the faid feries are thefe, ¢ = 4, 5, 6, &c. 2, 3, &c. for which higher quadratures are required, to meafure the conoidal furfaces thence arifing. E WARNPE RO XEAE ce Ea set Fig, 124. 137. Let ADC be an ellipfis, which D is converted about the axis AC, and make AB = 4, BD-= è, AE = x, EO04y; and the equation is <= = 24% — xX. Therefore, by differencing, it will be Th : ; è = + atid therefore: xa bb x a—« aryy yy “ . RO aayy . ® n O sewemees RE ees, Tes? and, inftead of — 20x + xx, putting it’s value ig given by = e og ee a yy 33 e ° 5 the equation, it will -bé xx" °° See Then fubftituting this value in bb x bb — yy ) a. , | cyjN b* + aay — bby a the general formula, we fhall have Sree ree eater * and, for brevity-fake, rbV bb — yy 3 making aa — 4b = ff, fuppofing 4 to be greater than 4, or that the axis about which the ellipfis circulates to be the greater axis (for, if @ were lefs than b, we oy Vb* + Thy » which, for rbV bb— yy ought to make 44 — 66 = — ff), the formula will be aeafons already mentioned in their place, may be freed from radicals; and the integral of which, by means of the canon of § 56, we fhall find to depend on the quadrature of the circle. But if 4 thall be lefs than 2, or the axis about which the ellipfis turns be the leffer axis, the fuperficies of the fpheroid will xen. I; Gg depend 226 ANALYTICAL INSTITUTIONS BOOK I1Y, depend on both the quadratures, that of the circle and that of the hyperbola. Wherefore the fuperficies of the urg4l2, in the firft cafe, is equal to a portion of the.elliptic fpace, which is eafily determined by means of the perpendicular to the curve, But, in the fecond cafe, thefe perpendiculars wiil give us an hyperbolical fpace equal to the fame fuperficies of the ungala. That this may be plainly feen, let ACF be the curve on which a cylindroid 1s fuppofed to be erected, which is to be cut by a plane which pafles tbrough the axis AB, and forms with the fubjacent piane CAB half a right angle. It is evident that, making a the element of the curve, fy# will be the fuper- Fig, 136. ficies of the lower wxgula, and ST will be the fuperficies of the conoid, generated by the con- verfion of the figure CAB about the axis AB; and therefore the fuperficies of the wxgw/a will be to that of the conoid, as radius to the circum- ference of the circle. Now let the two ordinates BC, DF, be infinitely near, and drawing the perpendicular FG at the point F, let it be put in DH, and reprefent the ordi- nate of a new curve MIH drawn by the method prefcribed. I fay that the area MABI is equal to the fuperficies of the wxgula, which has for it’s bafe the arch AC. The two triangles FCE, GFD, are fimilar; then it will be FC .CE :: GE «Hh DD... beretoreb DX FO GE xX CE = DH x DB. « But FD x FC (yu) is the element of the fuperficies of the ung#/2, and HD Xx DB is the element of the area IMAB, Then, thefe elements being equal, their integrals will be equal alfo; that is, the aforefaid areas. This being premifed, let the figure ACB be a fourth part of the ellipfis, the equation of which is aayy = = 24% — xx, Then the perpendicular will be FG = CO ESR LOS ECPM STEN NOLI SOE SONNETS LOR MEA TOR IRA ID O ERD Fi AR NEI Tes RS SCE EET RR SR MEI b 3 : , pe a zan ——aanx + bbxx — 2abbe + aubb, Then, making the ordinate a : RA Oe nina. ae den a P b aaa aamaa . bless Sil wall bess: = —,/ NN om 20% X 5 om a” + ab’, an equation to the curve MIH, which will be another ellipfis when 4 is greater than 4, or if AB be the greater axis of the ellipfis ACB; and on the contrary, an hyperbola, when a ts lefs than 4, that is, when AB is the leffer axis. Laftly, in the middle cafe, or when the ellipfis degenerates into a circle, we know already, that the faid furface of the xugula is quadrable, as being equal. to a rectangle. 9 EX. secT. 111, ANALYTICAL INSTITUTIONS, avy EXAMPLE XLUI, Fig. 125. LAI 138. Let BD be an hyperbola, which cir-. D culates.about the tranfverfe.axis BA. Let A be it’s centre, BA = a, the conjugate femi- i Ve ey A el Me GI D rn) A AC. | ; | «equation will be xx — a2 = 2, and there» A B C sl | bas © b AT TEN: ; fore y =ev XK == GA, and y= anv xx — aa Therefore the general formula, when the fubftitutions are. made, -will be DI pr pronao AB yh +23 om ati : chà Nine Slan RT — Ga xX CM Taree -, that is, — Aaaxnv + boxe — ats Ge kata da aay | È è è a cor x Odg 6 | : A or, making ca + db. = ff, it will be Lia Vx — — , the integral of which, ria when it is freed from it’ S radical fi fign, we fhall find, in like manner, to depend en the quadrature of Pi hyperbola. : È I MILA ve eh EXAMPLE XLIV. > # . Pi & Y/ hey, po è i a seh ES 139. Let. MD. be- an. siii hyper-. bola, between it’s afymptotes, and let it turn ‘about the afymptote-AC,: of which the (equation is ay + xy = aa; making. AB= 4, BC =", and CD’= Dax: Then’ it will be dg 4 E) 2 GEE i. nt” ; Bi di orgia DE; ane ee ie Bs ey x sa . Therefore, making the fubftitution, the general formula will be OS: + at Put /y* + at = 2, and therefore y* = ZS meg, y = =. Make thefe fubftitutions, and we fhall have the formula ga. transformed 228 ANALYTICAL INSTITUTIONS, BOOK II. transformed into this other, —2°—, which is free from radical figns; the 2r X 22 — at | integral of which depends partly on the logarithms, as is eafy to perceive. Therefore the fuperficies required, defcribed by our hyperbola, will allo depend on the quadrature of the hyperbola. EXAMPLE XLV; Fig, 126, + 140. Let ABF be the folid generated È by the traétrix, as in Example XXXVI, § 128, of which the fuperficies is re- quired. In the general formula = (where 4 reprefents the element of the curve,) inftead of 4, fubftituting it’s i GE E 2 D Pd value — Di obtained from the equation of the curve, we fhall have = 2, and by integration, — = +2. But when the fuperficies is nothing, we have y = a; therefore the conftant = om , and therefore the complete integral is <= on 2. > equal to the fur- face of the folid generated by the figure AEDB. And making y = o, then ha be equal to the furface of the folid infinitely produced. But the area a of the circle, whofe radius is / 244, was found to be = 2; then the furface of the folid, infinitely produced, is equal to the area of the circle, whofe radiu is equal to the diagonal of the fquare of AE. | EX. SECT. III, ANALYTICAL INSTITUTIONS, 229 EXAMPLE XLVI, -14t. Let CNEODAC be the ungula whofe fuperficies is required. Impof- ing the n names as at § 129, it will be QK = — = MN.. But My, the element of he curve, IS /XX + JJ» and therefore it will be = A XX + Is equal to the infinitefimal quadrilineum MimN, the element of the fuperficies of half che ungula, Let the curve DAC be a femicircle; in this cafe it will be VXxXx + yy ax A bx ————— , and therefore the formula is —_ N gaan | | Yaa — x (according to § 31), it will be — 4Va4— xx + f. But, making a = 0, it will be f = ad; therefore the complete integral will be found to be ab — bV aa — xx. And making x = a, in refpe& of the whole fuperficies of the half wagu/a, that fuperficies will be = ad. [| . And by integration Let the curve DAC be the parabola of the equation yy = @ — x; it will be VK + JI AK ies , and therefore the formula js su, i sp A 0 the integral of which depends on the quadrature of the hyperbola; fo that the. fuperficies of the amgula will depend on the fame quadrature, 9 230 ANALYTICAL INSTITUTIONS: BOOK TIT, EXAMPLE XLVII. Fig. 103. Age - = , and finally, = 2"xiz 4 wz" *s, which is the differential required, È | 151. Let it be required to difference the exponential quantity of the fecond p ee ay | degree, 2° . Make 2° = 7, and therefore it will be xP le = I And, by differencing, the differential of el x le +? x È will be = (A But, I by the foregoing article, we know the differential of x? to be x? phe + px? xi. MO hy eee ds ; cr al and therefore it will be af ply +.pafT'x x /z+— = —. Bae fee Pa xP p. E pal: fiala Therefore it will be ? = 2” wx? pixla + 2° pal ele +2 = ‘x? & for the differential required. 7 | In the fame manner, we may proceed to exponential quantities of any other degrees. | | 152. Likewife, in the fame manner, we may have the differentials of quan- tities, which are the products of exponential quantities; as, for example, of x? y°. For the differential of this will be the product of x? into the differential y°, together with the produ& of y” into the differential of xP. Butit has been fhown how to find the differentials of x” and y°. Therefore, &c. 153. From the order in which logarithmic differentials proceed, we may derive rules for the integration of logarithmic differential formulas. And, firft, thofe canons which ferve for the integration of common differential quantities, will alfo ferve for logarithmical differentials which are like to them; becaufe Vou, II. Hh | thefe i: : E eso oo Maga tec INSTITUTIONS, BOOK III, thefe are divided alfo by the risa and the integrals of thefe will be the fame as the integrals of thofe, putting only in thefe, inftead of the variable or ivs power, the logarithm or power of the logarithm of the fame variable ; dividing the whole. by the fubtangent of the logarithmic. ° == I DI . 4 ° » am ‘Thus, becaufe the integral of mx” ‘x is x”, alfo the integral of mi”~"x In the fame manner, becaufe /aT'x = lv; fo likewife (77x x i , or + f= will be Jv; fuppofing the fubtangent = 1. And, becaufe {yy /aa + yy = = X aa + yy)*; it will be allo flyaa ply x 2 = 1 x aa + Pe. Let ml” “ln x n be given È be integrated. Make év = y; then = Sy. And making the fubftitution, it will be m/”7%y x a But we know the 77) = I integral of my” se to be y°, and therefore the integral of m/ SEDE 2 will di Bef ge Ba 2 = iw, and therefore = Ie, and: 2” y = 1” ix. Therefore fai” "he x È = 1" le, | and ; | | Let it be wml” "x" x —. Make x” = y, and therefore x = —2—. And making the fubftitutions, it will be wml”~"y x die that is, ul” ~"y — 1 mx xa . J MI it x mm Sum x X—-=EK. Let it be mml”~ ‘1% x = Make /x.= y; then on = y, and /"x = He 1 7/5 y°. Making the fubftitution, it will be wml/”~*y Kia + But the integral of this is /%y”. Therefore, reftoring the value, it will Lo Sum" 1% xX de Tic a | ae Tod. DO x Re the integral of which is / 7 Then reftoring the value of y, it will be I 54: To this I {hall add a general rule for the integration of the formula Stia # nytt a] ‘oer y n "4, and fay, in general, it “Di be /y° lPIXY = = e = SIE ae ani TI, | n acini mA 24 VE de RA &c. And thus m+ 1) m+ 1) the feries may be continual in infinitum, by obferving the law of it’s -progreffion, which is manifeft of itfelf. Hence, if the exponent z fhall be a pofitive integer number, it is eafy to obferve, that the feries will break off of-itfelf, and confequently the integral of the propofed formula will be given in a finite number of terms. For example, make n = 2; then it will be x — 2 = o, and therefore the co-efficient of the fourth term will be nothing, and of all that follow, becaufe every one is multiplied by 2 — 2. So, if 7 = 3, the feries will break off at the fifth term; and fo of others. | | Make 7 = 2, m= 1; then the formula tobe integrated will be yi*y x y. Therefore the fourth term, and all the fubfequent terms, will be nothing. j2 Tt herefore the integral will be 2 Lad glen “i! an + ee i g 4 3 Now, if it were m = — 1, the feries would be of no ufe, becaufe it would be m + 1 — o, which makes every term infinite. But, in this cafe, there would be no need of a feries, becaufe we know already how to integrate fuch formulas, by what has been faid before. It remains to give the demonftration of this rule. To do which, make ly = z, and therefore 2 = cone fl Then making the fubftitution, it will be m+-I tI 7 ~I . PY Pz But y"2"y = "zy + 3 Sap re A nXn—I m+ +I 1-2: aXN—I om n—2 4. | LAZ ce air Zz a54 2" “ay, &c. And fo on in infinitum- miei 2 | m +1)” J J? aft 3 becaufe, in this manner, every term, except the firft, will be deftroyed by that. immediately following, becaufe it is 2 = oe Now, becaufe fuch an infinite - -feries is integrable, by taking the terms two by two; for the integral of the fir& m+n RFI a—J 2 ~~, of the third and fourth is — £2 cnn OE: mi 1 m+1\* 3 SEE PPE M41 t—z2 . 6 FEE the fifth and fixth is = mai ; and fo of the rel: in this. Hh2 and fecond term is integral, , 236 ANALYTICAL INSTITUTIONS, BOOK III. integral, inftead of =, reftoring it’s value Jy, we fhall find it to be at la& Pigi Net a ang Eh fer ty tbe AY Saks if Cou Bic wane ae &c, as before. 155. The artifice of finding the aforefaid feries is this. We may conceive ‘ Main, ie, the integral of y J yy to be ea , as it really would be, if 7” y were not m+ a variable quantity; but, fuppofing the fubtangent = a, the differential of M ne I » e 2 e . l f e e this integral is yl yy + 5a... This is found greater than the pro- nm He IT. ty - l Ù e n pofed formula by 2 22, fo that the integral afumed is greater than it £ nyal"7'yj mi ought to be fubtra&ed from the fuppofed integral. ought to be, by the integral o , and therefore the integral of this ny al di My nye 1a) Ha ty iS > Aa ae SI m +1)" i And here again I conceive that the integral of i Î | 2M$1I,7% ; : 7 whence the integral of the propofed formula will be —— _ . mt. I MITI ,t—mI MII »t— TI Mm sN—Y ny al . ; I al ; = +1}? = But, by differencing = a *, we fhall have eects A | miti 7 mi MAXI Zap M n)tt—m2 I Ht mT. n XI Xy ail ‘ ny al SME + TTI - 2.4 » Theréfore the integral of RAM ee Dt Mt i) ; È Mm sil ————b ny al 5 . . ina - “ONE, but is greater than it ought to be by the integral of == x +1 m+ 1] M o)N—2 . | | Mn 9 al” ‘yy. Therefore too much is fubtracted, and this integral is to be — m+-1 — 2 n X Met X y 4 at)" added, which again I imagine to be ———, So that the ua m4-1)3 I Mtl in n MPI )Z--} eee lip ny al integral of the propofed formula will be oi 1} AXHATI mel ,,n-2 al + Sar? 4, &c. And thus USO in the fame manner, the feries may be continued ix infinitum. 156. We SEC). - ANALYTICAL INSTITUTIONS 237 156. We may alfo have the integrals of logarithmic differential formule by the help of feries, which. fhall not contain logarithmic quantities, but only common quantities; which feries, therefore, will never break off, but are always infinite. — i Let x/x X x be propofed to be integrated. Make x = z + 4; then, by fubftitution, it will be z+ 4 x /z+a4 xs But, by $ 70, itis/2+4 = PA 22 23 z4 a 20% È 343 q4a4 ? &c. Suppofing the fubtangent: = .1,° Then, by actually multiplying, we fhall have z+axX/z+ax2= zs + 2° 23% a A z4% zià 28% 23% z43, PAPI . ; i 233 petit ag 24 Za* 443 5a* 2a 23% VizAz 253% ‘ a i x : z* 23 z4 aes vendi &c.; and, by integration, it will be ao z5 25 n —— : ea RE 4° agen o a MLSS g È + 60aì . 12044’ SIE J ni x ee ee So, if the formula were x lx x x, that is, z+al” x J/z+aXzZ, we mutt multiply the feries expreffing the logarithm into the power & + a)". And moreover, if the logarithm alfo were raifed to a power, as x /"« x x, that is, z+al\ x/l"z+a x &, there would be occafion, befides, to raife the infinite feries, expreffing the logarithm, to the power x, and to do the reft, as above, Rear 157. Differential formulas, or equations affected by logarithmic quantities, very often admit of integrations which are geometrical, and which depend on quadratures of curvilinear fpaces, which may eafily be defcribed, fuppofing the logarithmic curve to be given. Here are fome examples felected out of the more fimple ones. } Fig. 138. | Let the equation be y/y = x, and in the °— logarithmic defcribed let CD = y; and takin the fubtangent for unity, we fhall have AC = HD = J. Whence the infinitefimal rectangle DG, «of which the bafe is GH = FE = y, will be = yy. But this rectangle is the element of . the increafing area BDH, and therefore the fum or integral /y/y is equal to the faid area. In fact, the area itfelf is equal to the rectangle AD, fubtracting the logarithmic fpace ABDC, But. | this 238 ANALYTICAL INSTITUTIONS. BOOK III, this fpace, as is known, is meafured by the rectangle AB x CD = y. There- fore the area BDH = /yly = yh — y; as may be found by the way of analy fis. ; | I fhall confider another formula, y/*y = x. The firft member is no other than the folid generated by the fluxion HG, multiplied into the fquare of the ordinate GF; which folid is analogous to the element of the conoid, generated by the area BDH, revolving about the axis BG. Therefore the integral /y/*y = yl*y — 275 + 2y is to the faid conoid in a given ratio, _ More generally, let us have y/”y. Raifing the ordinate HD to the power m, (the index m being either an affirmative or negative number, either whole or broken, it will fuffice that the ordinate HM may be made equal to the dignity HD", and that through the point M, and infinite others to be determined in the fame manner, the curve BMN may pafs ; in order that the area BMH = {MH x jy may be equal to, or analogous to, the integral /y/”y. The difficulty will not be greater, even though the logarithms of logarithms fhould alfo be found in our expreffions. Let there be propofed yily = x. Whereas AC is the logarithm of CD; if, in the logiftic, the new ordinate IL, equal to the abfcifs AC, fhould be adapted; AI will be the logarithm of IL, and confequently the logarithm of the logarithm of CD. Let the right line IL be prolonged, fo as to cut HD, parallel and equal to AC, in the point K; ‘ through which and infinite others, determined in the fame manner, let a new curve pafs, drawn relatively to the logiftic. I fay, that the quadrature of the {pace belonging to this curve will give us the integral of the formula yiy = x. After another manner. I take the fluxion of the quantity yy, that 1s, VD + }-. and adding the term a to both fides of our expreffion, we fhall have ylly + a =x + Li and by integration, yy = * + Se There- fore, to the abfcifs AH annexing the correfponding ordinate in the reciprocal ratio of HD = Jy, a curve will be produced, the quadrature of which will exprefs the integral "ira And this will be enough to fhow how the method proceeds. 158. I fhall now go on to the integration of differential formule, which contain exponential quantities ; and let us integrate «°x. Put x = 1 + ¥, ° . . . ° . I ° (taking unity for any conftant quantity,) then it will be wx = 1+) Wy, 4 | This aN di ae es SECT. IV. ANALYTICAL INSTITUTIONS. 239 This fuppofed, make alfo 1 a gg — 1 +z, and then it will be 1 +y X Ji +y =/1+4u. Now let the two logarithms be converted into feries, by § 70; and making an a&ual multiplication of the firft feries by i eh we AEN AVE Ye ae alee + yt — oy, &c, DU n Du + 14 — ty + 145, &c. Then make a fictitious equation, fuppofing it to be « = ny + Ay? + By? + Cy* + DyS, &c. (where A, B, C, D, &c. are quantities to be deter- ‘mined by the Ree ) Therefore uu = 5’ cf 2Ay3} + A*yt + 2ABy’, &c, | + 2By* + 2Cy5 PST QA + JAC, Re. + 3By° WET ARM ey Lt Sys Whence 4 — I° + 143 — tut + 14°, &c. = tA + B+ O + Dy’, &c, — a — Ay? —tA*y* — ABy | Pit By4 — Cys + 9° + Ayt + Ay’ SVT I +12 1a), Ke. + By ve 1y4 egli. Ay’ | Ho Now, by. comparing homologous terms, we fhall find the values of the affumed quantities::to. be A m.2, Ba C iDan bet, putting thefe values in the places of the capitals, we fhall have 1 + 4 = pay arty ty + + yt + 59%, &e. Whence 14)! = y+wtxry + 195) + 13%), &c.; and laftly, by integration, LT xy =y + ty + ty eo +19 + 1), &c. 150. We may find the integral of the formula xx thus, in another manner, Make x° = 1 + y, then xlx = 71 -+-y. Reduce /1 + y to a feries, and it will be e Fishy = ye ty? ty? — 194 + +95, &c, This his fuppofed, make Yor /1+Y +APIPy+BPr+y + Ch 1+y+ DF 1+, &c. (where A, B, C, D, &c. are quantities to be determined,) and it will be 240 ANALYTICAL INSTITUTIONS, BOOK III, go =14+y + 2APigy + Ai +y + 2ABP i+ y, &e + 2Bei+y + 2CKhi+y y= | Bity + 3Ahr+y + 3A ly + 3Bisi+y y= #T1+y + 4Abart+y of cs | Mo | PTLD Therefore y — ty? + tyS — ryt + 1pS, od neem pyi= Jity tAPi+y + BPr+y + Chi + y + Dr +y, &e, —iPT+y APT Fy — 1A 14) —ABLI +9 — Biipy— Charpy = 2 di 3 A * SR I ù sa 2 E = Sie eae Se a ct NO ae I here Gt E Re Og aaa ra sio ae +32#1+9 + Al*1+y + ASI4y0 | 1 «sr Mi — 1/4*14y — Al1+y 3 + 1/51+7 : Now, by the comparifon of homologous terms, we fhall find A = 1, B=i, > Ca ap D = Hm &e.3 whence a-f 09-1 peli by PARI +y q + IB1+y + ayltity + rlol!i +y, &c. But /1+y = «le, and ; 14y= ” ; therefore, making the fubflitutions, and multiplying by «, it ) will be «xe = x + sola + Iata + 243%2% + tal + 100°x05x, &c.; e and integrating, by the known rules above delivered, it will be fx x = x +. ala — 40 + gela — GIÙ A 7 A la — Falla + Ft Fig, 139. ” 160. Now, to add fomething concerning the con- A ftru&ionof curves expreffed by logarithmic and ex- ponential equations. Firft, let it be required to Se. ee eae = ee one SUE hag lig aah «Loi er RE MN a a IT EE eR se ire Si defcribe the curve of the equation x = te . rst a BD (Fig. 139.) be the logarithmic, in which we are to take the logarithms of the propofed equation, whofe fubtangent (for example) is = 4 = AB. This 9 | fuppofed, - SECTOIVE, ANALYTICAL MAU T.ON SoMS., pee 241 Fig, 140. sp taking y = a = AB, the logarithm of y will be = o; and therefore ¥. = 0: Making, then, MN = y = a (Fig. 140), N will be a point in the curve, Taking y lefs than AB, 2 will be a negative quantity, and there- number 2 is the index.of the root of a negative quantity ; Taking I greater than AB, fuppofe = CD, it will be La AC =. But, by the CE equation, it is ar. bey. 3s ly. x, ora. Valy 3: ly. x; and therefore, making WP SS we, are mee PH equal to the fourth pro- portional of AB, a mean proportional between AB and AC, and the faid AC; which fourth proportional will be = x, and H will be a point in the curve. After this manner we may find as many points as we pleafe, and fo defcribe the curve, which will go on ad infinitum, as is eafy to perceive. | | . To have the fubtangent of the given curve, I take the radar formula ; = of the fubtangent, find the difference of the equation of the curve, whichis x = + 3/% zy X ne. Making the tai in the place of CsiWe tall have the fubtangent = 3/%y xa = 5 = rae sax, Alfo, our curve will have a contrary flexure; to find which I take the fecond fluxion of the given equation, fuppofing x confani, and I find È rit. + 3a Fond RNA AS eg n: asty Dy xi + tatti, — gdts;it, = 0; and iene j= II 3 = © 3 rt | 2. Ty me 3a H 7 20 LEA os a But, by the method of contrary flexures, it bas to Bayly be j = o. Therefore it will be 3a2yylzy — ja? yy 17 4y = 0; thati is; Py php a tal *y = o, or ly = ta. Therefore the point of contrary flexure will be there, where lE isti If the curve propofed to be defcribed were x/x = y, refolving the equation into an analogy, it will be 1 ./x :: x.y, which may be conftruîted in a like manner. | Von. ‘i : a if. fore Izy will be an imaginary quantity, becaufe the even. whence x will be imaginary whenever y is lefs than a. | agg + <<. 1 ANAbrTIICAL: INSCITUTIONA.. - seri - If the curve were x°%x = y, or xix = y, or wh = y, or, more generally, xix = y, fuppofing # to denote any power of x, whether integer or fraction ;. this equation being likewife refolved into an analogy, 1. de 32-4". y, and taking in the logarithmic any line CD = #, whence AC = /x; the multiple of AC, according to the number 2, if it be an integer, the fubmultiple, if a. fraction, will give the correfponding ordinate in. the logarithmic itfelf, which fhall be «*, by the property of the logarithmic, : Fig. 139+ ar a If the curve fhould contain quantities that are logarithms of logarithms, fuch as x°*/x = y, we fhould eafily have in the logarithmic the line expreffed by //x, by taking any line CD = « (Fig. 139.), whence it is AC = /x; and then putting AC for an ordinate. in (ae). For Aa would be the logarithm of (4e),, that is, x; as has before been taken. notice of at. ‘§ 157. 161. Let it be required to conftru& the exponential curve of the equation: x — y. Now, taking the logarithms, it will be x/x = /y; and defcribing the. logarithmic curve PAB, (Fig. 141.) with: the fubtangent AD = 1, and taking any. Tine CB: = DE’: xj it will be ' DO Then, becaufe the equation may be re- | folved. into the analogy,. 1%. #2: de. ly; ia 7 SA ah _ the fourth proportional to AD, BC, and: DC, which fuppofe is DM, will be /y; fo that MN = y. Therefore, if it be. made EF = MN, it will be DE = x, EF = y, and F will be a point in the: © curve to be defcribed. È È paia Do: The curve will cut the afymptote HM in H, making DH = DA. For, putting # = o, it will be /y = o, that is, y = DA. Making, therefore, AG = DH, G will be a point in the curve, - From the point H drawing HP, an ordinate to the logarithmic, and drawing POR parallel to HD, then OR will be the leaft ordinate, y, to the curve. For, taking the difference of the equation, it will be + + xiv = —., that is; Pe: + pala VA But, by the method de maximis et minimis, it mult "be #40; therefore yx = yxle —=.0, and therefore —./¥ = 1.22 HD: DA, 8: Becaufe. PEECT.IN,. ANALYTICAL. INSTITUTIONS. © hele ahaa ‘ pi o Becaufe 25 is the general formula for the fubtangent, and having x = from the given equation of the curve, by fubfticuting this: value in 4 XIH4/x the formula, the fubtangent belonging to any point of the. curve will be = : ie ; and for the point G, in refped of which itis AD, and. Me quently /y = 0, the fubtangent will De = =i AD, Rasa is the fubtangent of the logaritemic, : . As to the area, take the general formula ye 5 “but y x x", in the ‘equation of the curve. Therefore, fubftituting | the: ‘value of :y in ‘the formula, it will become x x, and. therefore Sa" Os the indefinite: ‘area xl HOFEADH; which, being integrated according to $ 159s willbe = ¥ » a — 1a + - ss = be oe a ae O za , &e. And mye Ù = 9 27 24: #2 ny AD= 1;itwillbe/x=o, and therefore the area Len Lol = ae ote a, no ; I 4 Tg nine &c. 3 3 that Is, mee arc a + È n° be o ax o - 162. Let «7 = a be the baustion. of the curve. | T beh’ ie è 314; hand | therefore it may be conftructed by means of the logarithmic. By taking the | fluxion of the equation, we fhall have > son + ja = ae > making the fubtangent of the logarithmic = 1. And therefore it will be x ES — LE; ; and therefore | : the ibiangent — — xa 163. Let it bew = a’; (hier why Ly, aiar may be Lattea as —ufual. Taking. the fluxion, it. wilk be xa = Ja ; and the fubtangent ala 1+ CD Ay Bie : al: Ra Here, iron RE ee > it will be yey or the eleriieat of the area, = ni, QRL — xx Ala ‘and amegiatine; by $ I 54, it 1S = area. at Other queftions | may be {till propofed, relating to sat equations; Lo Cas, for example, i in exponential equations compofed of only known AAU, wo but with variable exponents, to find thofe exponents. So, let it be e = ab Ra i the, value. of the unknown exponent, x, is required, a, è, ¢, being given, i 2 I Becaufe 244 ANALYTICAL INSTITUTIONS. — BOOK 111. Vv nazio Becaufe = DEI a be x —/a =x — 1/8, and therefore xle — lb = fa — ih, Whence x = £2. | | le — 1b i 165. Another queftion fhall be this. To find fuch a number x, as that it may be x = a, and alfo xt? = 4, Now, by the firtt condition, we fhall sis ie = la, and therefore x = a Oki ian By the fecond condi- we re tion, we thal have x +p Pix": eroi itrwill be a = lg Bow ae SE or /x= ei, ci | sr, “Then it will ‘be & si da, that is, x/a + pla = «lb, or x = pla 7 lp 14%; Java; ‘ee : Li; or elle —_ = ——*, thats: i — ram This fuppofed, I fhall | propofe ta may tele to refolve the following Problem. Ke A veffel being given of a known capacity, full of any liquor, fuppofe wine, out of which is drawn a draught of a given quantity, and then the veffel is filled up with water. Of this mixture of wine and water another draught is drawn equal to the former, and the veflel is again filled up with water. Again, of this mixed liquor another fuch draught is drawn out; and the fame operation is continually repeated in the fame manner. It is demanded how many fuch draughts may be drawn out, or how many times the operation muft be repeated, that a DeL quantity of wine may be left in the veffel. Let the capacity of the veffel be = a, and the quantity of each draught = 4, Therefore, at the firft draught, will be drawn fuch a quantity of wine as will be expreffed by 2; and as much water will be poured in again; whence, after the firft draught, will be left in the vene the quantity of wine = 4 — d. At the fecond draught will be drawn out the quantity 4 of the mixture; fo that, to have the quantity of pure wine contained in it, we muft make this analogy sas the capacity of the veflel (a) is to the quantity of the draught (2), ab = bb fo is the wine which is in the veffel (4 — 3) to a fourth proportional ath A which will be the quantity of pure wine which is drawn out at the fecond draught. Then there remains in the veffel the ayant of pure wine, — — 2 bb \2 pie È th at Is, io. Therefore, for the third draught, making alfo this analogy ; sas ste capacity. of the veffel (2) is to the quantity of a draught (2), fo is the wine in the -weffel,, o dra RE e ae St RO RIE N CR E NES CEDERE A n DITA È è SECT. Iv. ANALYTICAL INSTITUTIONS. — 245 st toa ‘artbii 7 = x Le, This will be the quantity of pure De: VENE, ‘wine, aich was drawn out at the third diane s ; fo that there will remain in va PA, i LI 3: a " e di pra ® And a È - thus, after the fourth draught, there will be left in the vetlel the quantity of “er the vetlel the quantity of pure wine, —, | pure wine, £ 3 and, In general, after | a number of draughts denoted by 2, | zia gi there will da left i in the vete the quantity of | pure wine = — —- ——-. Therefore, È if we would know. how: man y dritto muft be taken, fo “the hee fhould >... remain in the veffel a given quantity of pure wine, cdi ai for example, — part of the whole ; ; we muft make the equation = = = Se 3 which, be- | | Ù. caufe n is an unknown pie will be an exponential quantity. Wherefore, the equation being reduced to the lager thas it will be / ci ‘= Da, that roi ala = la Im + si or ‘nla =—m+ nla, and there- ve, Im fore n= er ; fo that it will be eafy from hence to ‘find. the number z, ‘ee RAI A — (A. : at rg «by the help of a Table of Logarithms. | f END OF THE THIRD BOOK. he | BOOK Ivo THE INVERSE METHOD OF TANGENTS. TA S when any curve is. given, ne manner of finding it’s ben i fubtangent, perpendicular, or any line of that kind, is called-the Direct Method of Tangents ;. fo, when the tangent, fubtangent, perpendicular, . ‘or any fuch line is given,—or when the rectification or area is given, to find | the curve to which fuch properties belong, Is: led the Inverle Method of | “T'angents... | = In the fecond and third Boa are and the Sonera: differential Ceprano of the tangent, or other lines analogous to it; as alfo, of rectifications and areas, . Therefore, by comparing the given. property of the tangent,. rectification, &c.. ~ with the refpective expreffion or general differential formula, there will arife a - . differential equation of the firft degree, or of a fuperior: degree, which, being | integrated, either algebriically, or reduced to known quadratures, will give the “curve required, to which belongs the given property. For example, let the. -eurve DE required of which the fubtangent. is double to the abfcifs, Calling the. » abfcifs x, and the ordinate Ms the formula of the fubtangent i is a ; and there- fore the equation will be Co —— oe Again, let us feek the curve, the area of which: ~ + « È. * " * ® do 248 ee LY TT CALS eT Te TION6, ye *ROOR.IV. + which muft be equal to two third parts of the rectangle of the co-ordinates ; the element of the area is yx, and therefore it ought to be /3x = 2xy, or i 24) . | ui Sx TE Spade ge would find the curve whofe property it is, that any. arch taken from the vertex hall be equal to the refpe&ive fubnormal; the expreffion of the arch is p/n + wa and that of the fubnormal IS <3 fo that we (hall have f/ax + jy = È, and therefore Vir +9) = PEE (taking x for conftant,) which is a differential equation of the fecond degree. 2. The equations which arife by proceeding after this manner, will always. have (as is eafy to perceive,) the indeterminates and differentials intermixed and blended with each other, fo that at prefent they cannot be managed, in order to proceed to their integration, fo as to obtain the curves required; and much more if they contain differentials of the fecond, third, and higher degrees. For, in the third Section aforegoing, the differential formula have always been fuppofed to be compounded of one indeterminate only, with it’s difference or fluxion. Therefore other expedients are neceflary, to try to reduce fuch equations to integration, or quadratures, which is called the Conftruction of Differential Equations, of the firft, fecond, &c. Degrees. And, as to the conftru&ion of thofe of the firft degree, we may proceed two ways; one is, to pafs immediately to integrations or quadratures, without any previous feparation of the indeterminates and their differentials; the other is, firft to feparate the indeterminates, and fo to make the equations fit for integration or quae drature. : I {hall proceed to fhow feveral particular methods for both the ways, by which we may attain our purpofe in moft equations. But very often we fhall meet with others, which will be found fo ftubborn, as not to fubmit to any methods hitherto difcovered, or which have not the univerfality that is neceflary. : | | SECTI. SECT. I. ANALYTICAL INSTITUTIONS. 249 LI Dai Se Se f Of the Conftruction of Differential Equations of the Firft Degree, without: any previous Separation of the Indeterminates. 3. The moft fimple formule which have the two variables mixed together, are xy thefe two, ni + yx, and — cd a The integral of the firft is xy, and of the (DERE —-, as is manifeft. To thefe, therefore, we fhould endeavour to reduce: J the more compounded, and that by the ufual'helps of the common Analyticks, by adding, fubtra&ing, multiplying, dividing, &c. by any quantities. that will make for the purpofe, which will be different according to different cales. We {hall here fee fomething of the practice.. Let it be yx = xx — xy. By tranfpofing the laft term, it will be yx + xy Pee and therefore, by integration, xy = tux. + db. Let the equation be xy + 2xyxy = atxx — xxyyxx 3 then tranfpofing che laft term, and dividing: | by ww, it is xy? + 2ayxy yx? = d£ +5 and . extracting the {quare-root, . w+ ye = di: s and by integration, xy = alx + 3, in the orarie with fubtangent — = a. Let the equation be yx = yy + y’y + xy, that is, yr — xy = yy + yy» The firft member would be integrable if it were divided by yy; therefore I divide the equation, and it will.be eav = = yy + y, and, .by in» tegration, it.is si —iy+yt 4. Let the equation be y° y = myx + xy. If there was not here'therco=- efficient m, the matter would be eafy, becaufe the integral of the fecond member VoL. II. K.k would 250 ANALYTICAL INSTITUTIONS. BOOK IV. would be sy. The operation would not fucceed any better, by tranfpofing the member xy to the other fide, or by writing yy — xy = myx; yet I obferve, that the dina of de 1s ae + ie % ys different from that pro- pated: mys + xy, only in this, that it is multiplied by ae ~ ", Therefore, to make the quantity myx + xy become integrable, it will be fufficient to multiply it by ta terr and, to preferve the equality, to multiply alfo the correfponding ‘member of the equation y"Y; therefore it will be y si sf a pz sa + da | A I ee ony re Bie y, and, by integration, / y ur” ‘ y mmxym +. Let the equation be the fame, but with a different co-efficient ‘in -each of ‘the two laft terms ; that is, let it be yy = myx + rey. The fecond member vi "7 nt nity ‘ye Therefore the Lomogeneum comparationis would be integrable, if it 98 è were multiplied by y #0 ". Therefore I multiply the whole equation, and it n cme "DZ i eee, awill become y phe y my x + negre af y, and the integral will be r + “nied cm >. = 173 Sy) = mye +6. 5. The differential of x°y is iy si myx ‘x. This fuppofed, let the equa- ‘tion be yy bss Pay + yt De If the lat term had # for it’s co-efficient, the integral of the fecond member of the equation would be xy. I obferve, therefore, that the differential of x*y* 1s mary 'y + nyt x ; therefore, mul. tiplying the equation by my", there will arife my tt, = ny + nyx"73%, which is found to be integrable, it’s integral being poy ty a N°y® td, But if the laft term, inftead of the -co-efficient #, had any other, or, in general, if both the laft terms were affected by different co-efficients ; or if the Ù equation is not integrable; yet I obferve, that the differential of mxy™ is mym + SECT, Te ANALYTICAL INSTITUTIONS, 25% equation were y"y = exty + eyx"~'x; I obferve, that the differential of Cn cn Cn eee ————— I Coceani d & e H «so T Ù ° ® — an a qa £ y + ey°x x. Therefore multiply the equation by Cn ns ifs Cn Fa Cn ae cn | yee , that it may be y y eat ia y + ey © x’ 'x, which is cn r+t er _ integrable, and it’s integral is fy = € y= — xy +. Here make r = 1, ¢ = ie gem 1,e= 1, that is, the ea ‘yy = oxy + yx; the integral will'be 1y*= xy Make ¢ =.2, e = 3,.2=t, r= 1, that is, the equation will be yy = 2xY + 3yx, and the integral will be: yi tt 144 ‘the equation y°Y = 2x3 + 2yx*x ; and the integral will be 13° = 2493, È sE Sy rr ae, BAN STE. DAI LMR I, CT 9; OB cn. If the equation were expreffed thus, a) x = ce) + gati x, it iss ; CH eafy to fee, that it would be integrable. For, multiplying by ye *, it would ee ee I be xx = ex’y y bey? 71). But the integral of the fecond member: . e. mai» is-known to be eS St. 6. Now let the equation be yy = ali, If. it: were: not: for: the co-. efficient 2, the integral of. the fecond member would be =, But it: will be: to no purpofe to tranfpofe:to the other fide the term yx, and to write it yy — SO a Ri cl But I obferve that the differential of 2 is ol lla 207) fo that: AS XA: Fa dn: if the RIGpoo equation be multiplied by.y, that it may be yo uni pe, Su it will be integrable, and 1vs integral will be /y © == + è.. But, more. generally, let there be any co-efficient 7, and therefore the equation is yy =. 3, ie ° | $ ne bi bere ci n° Che ua 2, I obferve that the differential of — is a; therefore, if-it: Kkza be. 252 ANALYTICAL INSTITUTIONS, BOOK IV. Nm IT, us be multiplied by y”~", fo that the equation may be Gon ey = II î ° ° ° 9° a tf-+t—t ARA yn it will be integrable, and it’s integral will be /y = = VA Thus, let both the laft terms have different co-efficients, and let the n equation be y = È. I obferve, that the differential of ‘22 is EE TAR Lee 7 : i] wean e ° e ° —— a J aay 2 7°27%, therefore, if the equation be multiplied by ym ©, fo that | 7 n | O do oa ann epee , m ae get nA at it will be int it” it may be y Ve er ; it will be integrable, and it’s n integral will be /y" i, “y is cela me integ sara L de nxy — myx : — — ,. xy mm myx + 7 ° If the equation were y aM KOS mne s it would alfo be integrable. mo | Li, L ° e ° pr ng È o o ° fix — og For, multiplying it by y= ©‘, it will be xx = SO —2 TB Bat VI mn ane the integral of the fecond member is known to be — 3 therefore, &c. Let the denominator xx be wanting in the aforefaid equations, and let the equation be yy = ny — yx. To integrate the fecond part of the equation, there would be occafion to multiply it by yar and to divide it by «x. But rp 1. as this muft be done alfo in refpe& to the firft part, it would be 22 HL XK ? which cannot by any means be integrated. Therefore let the ae of the equation be changed, and it will be —y"y = yx — my. I obferve that the ° Meo J. « Ù È x ‘a ny cp NA ® 2 e ° differential of E TTT + Therefore, if the equation be multiplied J é ì i | _ af $2—1. by y°75, and then divided by 3°”, fo that it may be —— deh = RTOS 2% 7 SECT. I. ANALYTICAL INSTITUTIONS, © 253) "x ny 73; a ° È _ y * MS "3 DOTE 2, it will be integrable, and the integral is / ar ae J x gr e b. Let the equation have both the laft terms with a co-efficient, and let it be yy =*nxy — myx. Let the figns be changed, and it will’ be — yy = myx = I * - My x — nxy m y Ra. 2% my m mumy mm — nxy. I obferve that the differential of 2% n Therefore, if the equation be multiplied by y~ © ", and divided by mmy , Lo) n n ta 5 mi nn lj fo that it may be —= Bip iL ed ite Will be inte’ I 2% an : mmy m mmy n | i + - = 1. È | | 1 Lg Nt grable, and the integral is / ——- = eee b. mmy ma my m 7. Let the equation be yy =) — ment, Change the figns, and it will be — yy = ny x — xy. I obferve that the differential of = is Nano È è n è uva n =~“, Therefore, dividing the equation by yy, it will become pee? nye Nem 47; : , 9 = oa which will be integrable, and it’s integral is But if the co-efficient x had been wanting, and the equation were y'y = ay — yx” ‘x; change the figns, and it will be — yy = yx” "x — xy, It may N” ows N Heel, ny x de ce de 4 y be obferved, that the differential of — is DA 3 multiplying thé equation by zy”, and dividing it by y°”, it will become = » Therefore, 254 ANALYTICAL INSTITUTIONS, BOOK IV. Y-7 == I + fl Ue] . N Nam. TI Le no which will be integrable, and it’s — = 5 Z | grable, and it’s 5 J | id r-+n—I. x? integral is /— —_ — = gni d J But if, inftead of the co-efficient 7, there fhould be another of a different nature; or if both the laft terms were affected by a different co-efficient, as if x ° N è FP = ] . ° ° the equation were y") = cx y — eyx © x change the figns, and it will be — y) = eye ee ca). I obferve that the differential of —_ is ne one nc eye MO toa — duale. ‘a 5 SRS NT ee ae ey oe ae ae Therefore, multiplying the equation by: eey € ee Se are Pipes, sedi; ny € , and dividing it by eeye , it will be — arn Artes ne ne eey e A ae ee ge de | eo = , which will be integrable, and it’s integral will be cey e a nC 27 uc ety e ey € ; ne x But if the equation were thus exprefied, y e N° CA — eg; mC without changing the figns, I obferve, that the differential of SL is n xv ne ne i en Era — ney © aly Can : Le al 5 therefore, multiplying the equation by my * x NC ne sa I an ne oe mey e i: DE ~ ney VI era x and dividing it by x, we thall have —— = ——_—__________——— which pe Pes ne | na de “ye will be integrable ; for it’s integral is /—— = op; 2 Pete xf 8. I have ere nae ee a a ae, ae AE NRE AT er e Tab So ae Erre tin nia SECT. I, ANALYTICAL INSTITUTIONS, 255 8. I have already faid, in the foregoing Book, § 17, that as often as the numerator of a fraction, compofed of only one variable and conftants, is the exact differential of the denominator, or proportional to that differential; the integral of fuch a formula is the logarithm of the denominator, or in a given proportion to that loga- rithm. This alfo obtains when the formula contains two variables, intermixed with each other and with their differentials. Therefore the integral of i seth OR eats after any manner, being given by x or by y,) will be ix+-y =2+b6. The integral of = = = WME DE I Me ee ehe Imtepral © of 4247 = x will be alex — yy = z+%. The integral of i o ere 0 2ay — 2yy = 2 will be //xy — yy = x & 4. And, in general, the integral of N Meo] o BI 7 cli A ici BN ee Li Se An PA will be ley me hi bit: y x 5 Da +. HW Nn Me 7 rx ay — yt And fo of any other equation whatever, which fhall have the condition affigned. g. Wherefore many equations, though they have not the neceffary condition, yet may eafily be made to acquire it, with the affiftance of fome calculation. Thus, the equation a = —.y, has not the required condition in the fir& member; but it will have it if it be divided by y. Then it will be we a _ — ; and therefore, by integration, /xy = / Wo eee Let the equation be axy + 2ayx = xyy. I divide it by axy, and it will be 3) Da 27% tu ) ks xy — in the fecond term of the firft member; therefore I fubtra& the quantity 22 xy — » This would be integrable if it were not for the co-efficient 2 from each member, and it will be SEIT ie EI ay a) = mal — and therefore, by integration, /xy = - — lx + bb. Let the equation be yxx = x°yy + yy x vy — yy. I divide it by y, and it. will be axiom x*y + 9° X Vy — yy, that is, xx + py = ary + yy xX ye And dividing again by wx + yy, it will be Se = yWy. And therefore, by integration, /V/#x + yy = ZF +4. 5 10. From 256 | ANALYTICAL INSTITUTIONS, BOOK IV. 10. From $ 31, 32, of the faid Book III, we may gather, that any formula compofed of one variable only, if it be the produ& of any complicate quantities raifed to a pofitive or negative power, integer or fraction, into the exact differ- ential, or into a proportional of the differential of the terms of the quantity ; it will always be integrable. And the integral will be the fame quantity, the exponent of which will be that as at firft, but increafed by unity, and multiplied into the fame exponent fo increafed, but taken inverfely: Or, which is the fame thing, divided by it; or elfe this integral fhall be proportional to it. Never» thelefs the rule obtains when the differential formule are likewife compofed of two variables and their differentials promifcuoufly, provided: they have the condition required. Thus, the integral of x+y x ax + y = è, (where è is any how given byxor by y,) will be 2 x # + = z+, The integral of tx + iy x V/x +9 = z will be 1 X 3 x x + y)* = z+ 4, that is, + x x+Ppt = + 4. The integral of "AR MEZ dI BI Re è AP tab De Mpg + Ep oe 2Vp3q + PP Zt % The integral of xy + yx + ay X bx xy + gp) m = & will be i 2 o li ave ° i ° ° n xy + yy n = 2 + d. The integral of 2124 Tal = 2, will be bx at) m Sace a =2z-+); And foof infinite others of the like kind. 1 X But fome equations of this kind will firft have need of fome preparation. Let the equation be wxx + ayy + yyx = 2, (where % is any how given by #,) I multiply it by x, and it will be «3x + a°yy + sy’x = #3, or HX X xx + yy + xx x gy = wz, which has not yet the neceflary condition. But it would have it if yy were alfo multiplied into LA ; therefore I add to each member the term y?y, and it will be xx x xx xe + 99 + yy x xx + yy = «3 + 39, that is, XX + yy X «x + IY = «3 RZ which is capable of 1 oes ec and it’s integral is + X x* + y*) = tyt*t bh + [ure But it is not always eafy to perceive, what quantities are to be added or fubtracted, or what other alterations muft be made in the equations, that they may be brought under the foregoing method ; efpecially when the equations are fomerhing compounded. In this way, to arrive at a folution is rather the work of SECT, II, ANALYTICAL INSTITUTIONS. 257 of chance than of art. In fuch cafes, therefore, we muft have recourfe to the Methods of Separation of the Indeterminates, which Mall now follow. o COGI Pie eno) € fe Of the Conftruftion of Differential Equations, by a Separation of the Indeterminates. az. The Separation of the Indeterminates in fome equations, although but few, may be performed by the firft operations only of the common Algebra. Such would be the equation x°x* + xyxy = a@’y*, in which I obferve, that the fir’ member is a formula of an affected quadratick, which would be made a complete fquare if the term 2 were added to it. Therefore I add this quan- tity on each fide, and the equation will be usi + xyxy + DIV) = aayy + 130)y. And extracting the root, it will be wx + tyy = pV ty + aa, in which the variables are feparated, and therefore, by integration, Ixx + tyy = SIV aa + *yy + 5. The integral of the fecond member depends on the quadrature of the hyperbola. 12. But moft frequently it will be convenient to make ufe of fubftitutions. Let the equation be aax = xxy + 2xyy + yyy. Make x + y = 2, affuming % as a new indeterminate ; and therefore x + y = 3, and xx + 2xy + yy = zz. -Then making the fubftitutions, it will be aaz — aay = zzy, that is, aaz aa + 22° of the firft member depends on the re&ification of the circle. = $, an equation in which the variables are feparate. The integration Let the equation be x) + yx x at — ceyy = —_ Mi i ————— . ; ; | 24 d ie WV we yy X Vax + yy Here I obferve in the firft member, that the integral of xy + yx is xy, and that the fquare of this integral is found exaétly in the quantity 4/44 — xy; there- fore, if I put xy = 2, in the firft member the variables will be feparated, and it will be za/4*— zz. I obferve further, that, in the fecond member, the xx by 2 integral of xx + yy is , and that the quantities in the denominator are like to this integral. Therefore, by the fubftitution xx + yy = 2p, the inde- terminates of the fecond member will alfo be feparated, and the equation will —— _ 3 be ee di, Top x a/2p ° | Vor. II Ll Let 258 ANALYTICAL INsTTrrv Tro na, BOOK IV. 247 — ays the integral of xy — yx will be had, if we divide by ww, and it will be =, Let the equation be = 2, (where % is any how given by + or ¥5) By mm yx ; 2x) — 2yà Let us fuppofe, then, — = 2, and therefore P= = £, and SS x“ a NA a aN — £, and axy — aye = “2. Making, therefore, the fubftitutions, it a> and 2a) : - will be —Z__ = &, and dividing the numerator and denominator of a x xx Wy + SY 4 the firlt member by wx, it will be _— P____=& But it was put “ CS Ia Nona Ol JI pp è sol ti 20) nie a a), and — =» th 2 it will be —1— ———7 = 3. And, be- È 7 and me ~~ 3 therefore it wi dla , be- caufe the integral of this equation is algebraical, I will go on to the integration, 20] Make, therefore, a — p = q, and it will be — Ea = 2, and by integration, 2a ! * 1X — A = ee ao But g = a—p, and p = =; therefore itis g = ———. o MISE . ° 2 Now, reftoring this value, it will be — = + è = 2, which is the curve be. longing to the differential equation propofed. If, inftead of making 4 — p = q, 1 had made p — a = gq, another integral would have been found, but differing from this only in the figns. 13. The above equation gives me an occafion of making an ufeful obferva- tion; which is, that fometimes curves do not only change their nature by taking their integrals, either fimply or with the addition of conftants, which has been already obferved from the firft original of infinitefimal quantitics; but fome- times alfo prefent us. with fuch formule, as admit of integrations which are really different, and fupply us with curves of various kinds, even without the addition of any conftant quantity; which is a matter deferving confideration, By means of the fuppofition = DE a , the equation 24) — 29% fently integrated, and the integration is found to be = = = 2%, omitting «the conftant. Now I make the fuppofition of — = È , and attempt the inte- J = 3 is pres eration. It will be, therefore, Lena 29 L, and thence 24) — 2yx = tee See ° @ ee Ad by fubftitution ghe equation will be x j — 2ap 3 , ’ ° But da ==; therefore ego gia 1: = 2 And makingp=@=%1t SECT, Ile ANALYTICAL INSTITUTIONS, | 259 249 will be — 5 x; and, by integration, È = z. Now, reftoring the va- 2y. is different from the firft, - lues, it is = 3, the integral of the propofed differential equation, which - Another integral of the .propofed formula, different from the two firft, is n) : Li 00 RX = YN + a) — YY — xk — yi | ; ) «tI x For, by differencing, ia eee ee re 48° 5, we vien; ve scree 2 le pe | «+ 2) — 2x .= 2, and ftriking out the terms that deftroy one another, it is gi nati which is the equation at firfl propofed. Make & = y, and the propofed equation is ae =). II e ufe of the fecond integral found above, there arifes the equation > + Mtg) eine therefore 2 + y = x, which is a locus to a triangle. Then, if I make ufe of v+y 2H se the firft, and of the third integral, by putting curve will be of the fecond degree. | wv eee OF. In general, let it be tina =yy. The fir® and the third integration being performed, the curve thence arifing will afcend to a degree denoted by m + 2, if m bea pofitive number. But, making ufe of the fecond, the curve will ftop one degree fhort. 14. But, however, the method of fubftitutions is neverthelefs univerfal, the greatelt difficulty of which is, that it is often very hard to know what fubfti- tutions ought to be made, that we may not work by chance, and beftow much labour unfuccefsfully. However, we fhall proceed with the greateft fecuritv in - all fuch equations, in which the fum of the exponents of the variable quantities is the fame in every term, and the feparation of the indeterminates will alwavs fucceed. It matters not that thefe equations are affected by radicals, or by fra&ions or by feries, and that the co-efficients and figns are of any kind. The fubflitution to be made in all thefe equations will be, by putting one of the variables equal to the product of the other into a new variable, fo that, if the equation be given by « and y, we muft make « = i 5 OF CL da = » (where by the denomi- nator @ is underftood any conftant quantity at pleafure,) and therefore Ja az + za ; and, making the fubftitutions, we fhall arrive at another equation, which will always be divifible by as high a power of the indeterminate x, as was the fum.of the exponents of x and y in every term of the propofed equa- - Lio tion, 260 ARAVMTICA TVS Or PTT PON. BOOK IV. tion. Wherefore, making the divifion, the letter x will not exceed the fir power, and will always be multiplied by <; whence the equation will be fo reduced, that on one fide there will be #—, and on the other fide 2, with only x - the functions of 2; and thus the variables will be feparated. For, calling A all thofe terms which are multiplied into y, and B thofe which are multiplied into x, the-equation will be Ay = Bx, and A and B will be given promif- cuoufly by x and y. Now, becaufe the dimenfions of the letter y, together ‘ with the dimenfions of the letter x, in every term make the fame number; if, inft uu of y, we put 2, it will follow from thence, that in every term of the quantities A, B, the letter x will have the fame dimenfion which, at firft, x and y had together, Whence, if this dimenfion be called N, the equation will be divifible by x?, there wi remaining 2, 4, Y, x. Let it be fuppofed, that after the fubftitution of = ui and after the divifion by «”, that which remains in the quantity A may be called C, and that which remains in the quantity B may be called D; the equation will be Cy = Dx, and C and D az + will be given by 2 and by conftants. But y == si ; therefore the equa- tion will be = Dx, that is, Dax — Czx = Cwz, and therefore n ma Gea And thus the indeterminates, with their differentials, will be a U = feparated, and the equation will be conftructible, at leaft by quadratures, WR Caz + Czx a Re È . ° It is indifferent whether you put y = —, or x = =~; for, in either of the. two ways, the indeterminates will always be feparated. But fometimes one fubftitution will give a more fimple equation, and of fewer terms, than the other, and the conftru&ion will be more eafy and elegant.. Wherefore it will not be amifs to try them both, and, at laft, to make choice of that which. fucceeds beft.. yy ANI Pio E 1 : - È A Ripe Let the equation be «xy = yyx ++ xyx. Make y = ——, and therefore y =- pie E Sa Sarge 03 x we MAZZA ZAK rd (RL Making the fubflitutions, it will be Ei dala a: da : a And reducing to a common denominator, and dividing by xx, it will be ANS + ABN = BARK 4- azx 3. that is, ave = 22%, or me E X- SECT. II. CC ANALVTICAL INSTITUTIONS, 261 - TRS AMPIE. TT ° : . 3 a & ° . x Let the equation be gxy = yyx + 40. Putting y = a. it will be 3+ zx | x tot Oe È 43% o za ax » + And, making the fubftitutions, it will be 3 = — + «°x. And, reducing to a common denominator, and dividing by «x, it will be ax3 + azx = 22% + 44%, that is, 22K — azx + aax == 4x3, and there- * az wii 15 x ra i Mo: A) IE Wa ar aie Now, making another fubftitution, « = =~, it will “i ; ) CE AA 35; a oe Sat 1 be x == a » and therefore ine or VP ED + IPP EY and, di- viding by yy,. it is appy = aayp + aapy + yppp + p°Y, that is, appy — aapy aap + ppp | sal) = aayp + yppp; and therefore em Sat E 4A PL LEI Tuet the equation be IV xe yy yx Make y = =, and i i= pt 5 sui and, making the fubftitutions, it will be ST x Vs E dann 2 SO is, «€ + BKK X Waa + zz = azxx, and, dividing by x, it will be xZV aa + 22% + Waa + ZB = 42%, or xB aa + 2% = azX — 2XV ga + 22 n I i pane hemi LIETTA n — Pane Cile ae! made Cfr 2, I fhould have had. AZ — 2V aa -- 2% this. equation, a — acest Tey . N32. + pp — p: 15. But fometimes the differentials themfelves, x and y, afcend to higher dimenfions, the condition mentioned before being, however, in the equations. In which cafes, the fubftitution of = , inftead of y, being made as before, not. meddling with y at prefent, will make every term of.the equation divifible by. neat | the 262 ANALYTICAL INSTITUTIONS. _ -* Book/i1v. the fame power of x, and there will remain in the equation only z, x, and y, with the conftants given or afflumed, but not x. Now, becaufe, inftead of y, we muff put sc Mt AIS by which the letter » will again be introduced; make 4 ND A : i ~ ota. at = 7, and, inftead-of y, write =e , and the equation will have only “| 3, 7, x, with conftant quantities given or affumed, but no longer x. Now, if we make 4. ti x ./, and if, inftead of 7, we put every where —-, we fhall have an equation free from differential quantities, in which will be only a, 2, and conftants, for an algebraical curve. By means of this curve, we may find the real values of uw. Let there be, therefore, A,B,C, &c. fo.that it may be x — Apa — B, a ="C, ec. and A, By &c. will be given only by 2, and by 1 e e . at © ar È XS e conftants, and it will be x = T3%=: &c.; and therefore ¢ = — will i HE ue x Da a, x Dex > Zo eX Sp &c. ; whence, laftly, ee e. &c.; and the logarithms of x will be dire&ly proportional to the fpaces comprehended by the curves, of which, the abfciffes being 2, the ordinates will be reciprocally proportional to the values of the quantity w before found. And the curves fatisfying the purpofe will be fo many, as are the real values (different from each other) of the letter 4; ftill obferving, that the adding of a conftant quantity in the integration of the equations asa Se &c. may again diver- fify the curves that fatisfy the demand, and will often double their number, Then x will be equal to the area of that curve, which has z for it’s abfcifs, and I : » &c. for it’s ordinate; that is, it will be equal to the integral of. >| a |» uy —-, Fo, &c. Wherefore, taking z at pleafure, the logarithm of x will be given, and confequently the correfponding ordinate x in the logarithmic will be ‘ given alfo. Then, x being given, by means of the equation y = = will y be | given alfo, that is, both the co-ordinates of the differential equation propofed, or of the curve required. Then, in reference to the different values which will be given to 3, fo will be the different points alfo, which will be found in the fame curve required. | I Mall apply the rule to an example. Let the equation be xxyy + xyxy = HZ awxa. Make, therefore, y = —, and, putting this value in the equation, inftead of y, we fhall have ax*y? + x°2xy = ax’x?, and dividing by xx, it will be cy? + zxy = ax’, Here we fee, that x and it’s functions entirely difappear, | there SECT. IT, ANALYTICAL INSTITUTIONS ; 263° there remaining only x, x, y, with their functions. But, becaufe, by fubfti- zX + KS n ) tuting, inftead of y, it’s value. . we (hall again introduce x into the i pra è Bi zx + ab MTA equation; make — = È, and therefore: y ma , and the equation will atx + 2azxt + aaté uit + azxt be —T —_——_T_ _— kT--.- a bri — aaxx; in which only enter z, x, 7, with their functions. Again, fuppofing E sides = axx, that is, 222XX + 30zxi + cali i = £, and making the fubftitution, we fhall arrive at an expreffion which a. is purely algebraical, 222 + 32% + uu = aa, fo that we fhall have the value of # given algebraically by z and conftant quantitie, Bur / = “==, whence — =—, In which equation, w being given by z, the variables will be feparated. Therefore the curve being defcribed, of which the abfciffes are z, and the ordinates reciprocally proportional to.the values of 4; we fhall have x, and thence y, by making the fubftitution of = a 16. Now, from this and other examples, it will fucceed alfo, without making ufe of this method, that they may eafily be reduced by the method of $ 14. And, indeed, if to each of the members of the aforefaid equation, xxjy + nang. = xxxx, there be added the fquare tyyxx, it will be xxpy + xyag + Lyyxne = xxx + 1yyxx, and extracting the root, xy + 1yx = KS xx + yy 3 where row it is reduced to the aforefaid general method of $ 14. Or elfe, tranfpofing the term xyxy, and adding the fquare ty, it will be xx%y + tyyjj = xXx — xyx3 + 1yyjj ; and, extracting the root, it is IV kx + yy = ax — Iyjg now reduced to the fame method.. . 17. Equations which contain differentials mixed together, and ‘raifed to ‘any power,. may not only be conftruéted in the cafe confidered at $ rs, which fuppofes the fum of the exponents of the variables to be equal in every term; but, in general, in what manner foever thofe equations are, provided one of the two indeterminates, x. or y, be abfent. This is done by making x = 2, € stot wanting, I ote = i if y be wanting ; z being a new indeterminate, and a any conftant quantity. For, by fuch a fub{litution in the propofed equation, of = inftead of x, it is plain that another will arife, which will be divifible by the power of 3; fo that it will be compofed of finite quantities only, and 3 | i therefore 264 ANALYTICAL INSTITUTIONS BOOK IV. therefore will have 2 given by y and conftants only, and the relation of y to z will be expreffed by an equation, or an algebraical curve. Therefore, in the equation x = —, inftead of ¥, putting the value that will be derived from fuch algebraical equation, we fhall have the variables feparated. Paso RE * 4 ~ Let the equation be yy*x = ax* + 2ax%y* + ay*, Make x = — ; and, making the fubftitutions, inftead of x and it’s powers, we fhall have the equation 4 424 ° 4 , Se F . 5 4 232 e gl and, @jvidine my oy it will be L — 4. = a a3 a a a3 a 23 aa Tat BRL aax = — Lar se 23 — — , a 00 roveto (CA ore and y CECO = Therefore 35 As . e ® e ® Ratei wee Jo 227 ——. If we go on to the integration, it will be a a % ae ee . k i + 3 x x= de. + = — /z, taking the logarithm from the logarithmic with the 4a a fubtangent = a. Whence we have the values of the two co-ordinates x and y of the propofed differential equation, by means of two curves, which have z for a common indeterminate, Now, as to the conftruction, we may proceed thus. Fig. 142. Taking the abfciffes in the axis QE, defcribe the curve DAH of the equation a =~ + 22 + =, and the curve RIK of the equation x = = 443 Then EH = y, and EK = x, will be the co-ordinates of the propofed differential. curve; by the conftrudction of which, mak- ing CM parallel to EK, then KM is pro duced to N, whence it will always be MN = EH; and the curve NBN will be that required. EE SECT. II. ANALYTICAL INSTITUTIONS, 265 -EXAMPLE IL Let the equation be y°x* + aayyx* = ay’, Make x = 2. ; and, making the fubftitutions, we fhall have 2° 4 22% — 455 And, dividing by J, it will be 2% + 4°24y = a'. Therefore z will be given only by y and con- ftants, and therefore, in the equation x = 2, the variables are feparated. Fit. b At. Now,, to have the curve of the propofed Papi: ee differential equation ; to the axis CE let there - be defcribed the curve IK of the equation: | 29° + dz =a’, it beng CM = y, and | MK = z In KM, produced, take MN a fh equal to the area CMKI, divided by a. Then. È a willit be MN = / 2 = », and the point N will be in the curve. 18. The method of $ 14 may be rendered ftill more general, by tranf- forming the equations which have not the condition required, of the fum of the exponents. being equal, into others which fhall have thofe fums equal, and confequently thall come under the rule of that article. This may be done two ways. One will be, to make ufe of convenient fubftitutions, for which there. can be no rule, and it muft be by examples alone that this artifice can be acquired. ‘The other is, by changing the exponents of the propofed formula. er equation, that it may be determined, at. leaft, in. what cafes, and with what fubftitutions it may fucceed, to transform the equation into one equivalent to. it, in which the condition required may be found. Thus, though the fepa- ration of the variables cannot be univerfally performed, yet infinite cafes may: be affigned, in which that feparation will be effected. 3 Vou. IL Mn |, RX. 266 ANALYTICAL INSTITUTIONS. BOOK IV. EXAMPLE I _ Now, as to the firft’manner. Let the equation be xy a4%x + az? = 223, which has not the neceflary condition. Make z3 = ayy, and, taking the fluxions, zz% = sayy. Therefore, making the fubftitutions, xV Gann + aayy = 34yy; an expreflion that may be managed by the method of $ 14. We may alfo have our defire, by putting Waaxx + az? = au, and therefore aaxx + az? = aauu, and, by differencing, 2aanx + 3azzz = 2aauir, that is, QU — AHL 233 = Jai — Zaxx 5 and, making the fubftitutions, it is wx = ere Br a a em: tte ean ei eo an EXAMPLE IL Let the equation be x3x + ae = y. Make ya +, = 2, and there- aty fote a 4+ y = 22, and y = 22%. And, by fubftitution, «3% + 2xx3 = 22%. But this ftill requires a little further reduction. Therefore make xx = 4, or x* = uu, and 4w°x = 244; whence, thefe values being fubftituted, it will be finally tum + 243 = 222, &c. 19. I fhall go on to the fecond manner of altering the exponents, and therefore I fhall take a general equation of three terms, ay «x + by’xfx + cx" y = ©; in which the figns may be as we pleafe, either pofitive or negative, If it werex +#2g+p=7 + s, it would be the cafe of $ 14. But, fuppofing fuch an equality fhould not be found between the fums of the t . Arai SR Ca ig n nt exponents; make y = 2, whence y= tz 3,y =, yf = 27, = 2%, and making the neceffary fubftitutions in the propofed equation, it will be “ gt : f-4- fo J - Ue" ae Kx + bt xb x +. tex 2° = "So. But, by the condition of the afore- faid § 14, it is neceffary that it fhould be w+ m = g? + pat + # — 1. From the firft equation, therefore, ut + m = gt + p, we mutt derive the value of the afflumed exponent ¢ = - aS which, being fubftituted in SECT. Ile ANALYTICAL INSTITUTION & 267 in the fecond, gt + p =r st + ti— I, or Sg a tmpe—r + I, will give s — g +1x pemz=zp—r Fama gq; which is the cone dition that the exponents of the propofed equation ought to have. To verify which, it will always be reducible by the rule of $ 14; and the fubftitution i ad to be made will be y = 27-97. Inftead of making y = 2°, if I had made « = 2°, I fhould have found the fame condition to be verified in the exponents, but it would have been ¢ = = and therefore the fubftitution to be made is x = 227”. 2 — p- It may happen, that the fubftitution of y =. 2 #-4 may become impoffible, that is, when p = m, or m= q. But it may be obferved, that, in thefe cafes, the indeterminates are feparable without need of reduction. In the canonical equation ay°x”x + bylafx + cry) = o, if, befides the fuppofition of y = 2’, we fhall alfo make x = #°; making all the fubftitu. gi wp+w— wy Sbaffo f WMApPW- ; Li ‘th ab 6 (tu 3% iz=0.. tions, we fhall Gnd AWS U Tn + bw2? u By the comparifon of the exponents of the firft and fecond terms, we fhould have tt + wm + W— Lt gt + wp + w—'"t, that is, #— wx 0, n=g From the comparifon of thofe of the fecond and third, we fhall have wr + st +#T—i=q94 up fw 1,0 XI5TQFIiZwxper +1. And, inftead of 7, putting it’s value, w x p—m X Seg Pi = WX. n= 9 Xxp—r + 1, which is the condition the exponents of the propofed equation ought to have. But the letter w vanifhes out of the condition ; therefore the fecond fubftitution of « = u™ is altogether fuperfluous ; whence: it may be inferred, that all the formule, in general, cannot be reduced to the rule of § 14, but only fuch, in which the condition p-—mxs—g+Ft AT GX pr + 1 may be verified. The fame thing is to be concluded. of others, when compounded of a greater number of ee which I {hall now nai to treat of, 20. As the number of terms increafes beyond three, fo, in like manner, the. number of conditions. increafes, which the exponents of the equation mult have; M m.2 iD: 263 ANALYTICAL INSTITUTIONS, BOOK 1Vy in order to be reducible by the method of § 14 . Iwill take this canonical equation of four terms, ax y x + bie? gx + ex yy +diyy = o. Putting ATA a F CATE aa r 3 ì è sE t è gt CI. CAPRE E AI and making the fubftitutions, it is az" xy + dz! xx Sit fom I o + ten’ % z + dix ¢ tu-+-t<—I. - z zo. Therefore it ought to be nf + m = gt + p. Whence we may derive the value of the afumed exponent ¢ = a Alfo, it ought to ber +.sf +f —1 = gt +), or st— gt +i = p—r ; I find az *xx + dz #°x = — 1exz 723 which is now reduced to the cafe of the faid article. 21. But, SECT, Ife ANALYTICAL INSTITUTIONS 269 21. But, without: applying particular equations to canonical ones, perhaps it may be more commodious to manage them by this method only, ELAMELE: L A ar ee snag i | hae Let the equation be aymx ° x — bay. y = ex*yy. Make xa =z, x x II tara Fg et! I 4 iz —'%; making the fubftitutions, it will be tay &— bey Ny = ce yy. But it ought to be + + H# 4+- f— 1 = gf — 1, whence I obtain E4 ; ® ® Li = Li LÌ I tnt * = é = 2, which, being put inftead of ¢, gives me this equation 2ay7z3% — by) = eztyy, which is juft the cafe of § 14. Therefore the fubftitution tosbe made, 32s EXAMPLE IL : ° Li 4 id 3 e | 2 + f Ù 4 | me Î + Let the equation be x°% + 97% + xtyy = yy. Put y me, y = te "3, ; gh dae . . L'. sh. 3 o I - iti and, making the fubftitutions, it will be x2x + 27% + int Ts — oH "Se But it ought to be + = 4#, whence I have # = i; which value, being put A : LARasa : I. 1, 3_—}: SN, oe , inftead of #, gives me the equation «2x «+ zx + jx*2 72 = 3223, which as juft the cafe of § 14. Therefore the fubftitution to be made is gia zi. ees, EXAMPLE Ill. Let the equation be ay°x°x + bx + eyax + dxtyy = 0. Put y = 2, a A ia; making the fubftitutions, it will be az°x°% + Ox + co’ mx | sdxtz*—"% = 0. Now it ought to be 22 +2=# + I, whence ¢=—r1; ) and, 270 ANALYTICAL INSTITUTIONS, BOOK Iv. ® 5 ® 7. Ske + bx ip Ses BB zt ax? and, putting this inftead of ¢, gives me the equation Da = o, which is the cafe of $ 14. The fubftitution to be made is y = — : 22. The method of $ 14 being thus made more general, I fhall proceed to another, which is alfo general in it’s kind. This comprehends all thofe equa- tions, in which neither the indeterminates, nor their differentials, exceed the firft dimenfion. : Wherefore let the general differential equation, which includes all poffible cafes wherein the variables and their fluxions do not afcend beyond one dimen- fion, be axx + 2yy + cyx + gxy + fx + by = o. The co-efficients a, 4, c, &c. may be pofitive, or negative, or nothing, as the circumftances of the particular equation may require, which is propofed to be conftruéted. As to this equation, I obferve, in the firft place, that, if it fhall be ¢ = g, both of them being pofitive, or both negative, the equation may be integrated. For then it will be £ e x yx + xy = — axx — dyy — fx — hy, and, by inte- gration, ch cxy = — 14% = thy — fi — by. But, it not being ¢ = g, I make x = p + A, y = q + B, where p and g are two new indeterminates, and A and B are arbitrary conftants, to be determined as the fequel may require. It will be then x =p, y= 9, xx = pf + Ap, yy = gi + Bg. Thefe values being fubftituted in the principalequation propofed, there will arife this following. app + aAp + bgj + bBG + cop+gpj=o. + cBp- + gAg po #12 + dj In this equation, 1f the fecond and fourth terms be made to vanifh, this will be the cafe of § 14 and we fhall know how to feparate the indeterminates, But the fecond term will vanifh, if it be made eA + cB + $= 0, and the. fourth, if it be 2B + gA -+- 6= 0. Whence, from thefe two equations, the values of the aflumed quantities A and B will be determined, fo as that the new equation will be a cafe of the aforefaid $ 14. Then it will be A = — cB — f _ — sA- 5 : i bf. eh _. ab fe ai . And, together with this will alfo vanifh the fifth and fixth, without any condition arifing from thence. That the third fhould vanifh, it is neceffary that 24B + 242A + aabAA = o. . di 7 e e 4 4 é 4 And fubftituting the value of A, itis 2B — ai + si = api, DB Sy A b bb 33 Therefore the fubftitution will be y = — > +p + a and the equation 3 bi thence arifing will be dun = — aabp — oe 24. The method of this article confifts, firft, in difpofing the propofed equation in fuch manner, as that the fluxions may continue accompanied with their indeterminates refpectively, and that a halt-feparation (as I may fo fay) may be made, by throwing into the common multipliers, or divifors, fuch quantities as hinder the operation. ‘Then taking the integrals of the differential thus prepared, compounded of two variables, it muft be made equal to one affumed variable, and, by means of an auxiliary equation, it muft give a new form to the principal equation. Laftly, taking obfervation by that which fucceeds, the operation muft be repeated till the defired feparation is come pleted, or till we fee the formula eludes all our endeavours. This method has this advantage above the others, that in trying thefe fubfutu- tions, at the fame time it informs us, which will be fuccefsful and which ufelefs. But it muft be obferved, that there are fome equations which will not admit of . 8 the SECT. IT. »* ANALYTICAL INSTITUTIONS. 295 the artifice of the prefent method, unlefs they are firft prepared according to art. The whole will be better underftood by the following Examples, lettiere n —-] EXAMPLE I. ae 3 4 gii eee Let this equation be propofed, at: Malt a = 2, in which 2 ax ty X Vax + yy — #39) ~ {tands for any funtion of « or y whatever. I fet afide the denominator, which is an affeAion common to the two terms which compofe the firft part of the equation, and the bare differential x°y + y*x will remain. I divide x by #°, and y by 9°, and then it will be xy = yx = pty? x J+ =. Fence the propofed equation will take this new form, — Lia: ee x fx VA | xx yy x Veco yy — #99 4 PO = %. Having obtained this half-feparation, in which the fluxions ¥, y, appear combined fimply with the functions of their variables x°, y?, and the other terms conftitute, as it were, a foreign quantity, which has the appearance of a e ® È ; e ‘ È E 3 multiplier; I make => + oo a —4, and then, by integration, = + 3 ° ° ° , “= p. Now, finding the value, fuppofe of x, which will be-* = BE al N 2yy | | N 2yyp — a3 and fubftituting this inftead of «, and — ae inftead of os + —- in the sar ae equation, it will be — J a = = & Wherefore, &c, 2D — i It may be recollected, that, taking a quantity at pleafure any how given by 9, a o el be eee oar pie 3 ni as p aa oe + Li Fiat Ru ge vy MV scr + yy in an inftant, we may perceive infinite fubftitutions, which will promote the defired feparation of the variables, All the other poffible ones will be ufelefs, and will leave the variables as much blended and intermixed as before. ; by which, Moreover, let it be obferved, that it often happens with the fubftitutions here explained, that in one member of the equation there may remain fome function of one of the variables x or y; in which cafe, if 2 were given by the variable whofe function remains, one fimple divifion would anfwer the purpofe. % Nnz EX. 276 ANALYTICAL ‘INSTITUTIONS, , BOOK IV. EXAMPLE II i . 2. Wy 4} x 6 » ° . . ° i è Let the equation be 22% +2" = &, in which & is any how given by y. a+t+ae+y To reduce this equation to the method, I take the integral of the numerator of the fraction, that is, yy + wy, and make it equal to p. Now, making x and x to vanifh out of the equation, by fubftituting their values, I fhall have a new equation = 2, which is reduced to the following, yp —pz = eyz. a+ 4 And this, being prepared according to the method, will be found to be P See je # = 4% È make ri — ‘= i and therefore Lp —/> = lq. I make alfo fox = ulm, where /m is fome conftant logarithm. Thea it will be /p — /g = ulm. And going on from logarithmic quantities to expo- nentials, it will be sa = m°. Therefore, in the reduced equation, making the fubftitutions of = inftead of "A — o , and of mq inftead of p, it will : eu 7 tig 0 i | 3 be mg = az, that is, g= — ; In which the variables are feparated, becaufe mM both ® and m” are given by y. PERSA NERE. ATE 2ex% + xyy + pi _ xx PI st + ata © Nas Egy formula, it will be beft to reduce it. I obferve that the fecond member is integrable, and it’s integral is V xe yy ($ 10). Wherefore I make Y/xx + yy = z, and making y to vanifh, finding that it’s powers afcend' to the fquare, and putting zz — xx inftead of yy, and 2% — xx inftead of yy, we fhall have ante + es — 23% + 27% — xr azz wer KH + dii" AZZ < Gt Let the equation be . Before we attempt this the equation =, that is, = %; which, being, DECTOII. © LANALWTICALI INSTITIMTIONA, 277 being prepared as ufual, will de ore renee Ko az ipa. OP ake se + = p, and, by integration, wz = p; and, making « to vanifh, we fhall have rw = ae ane finally, P ALA è at pp+ at mw” EXAMPLE Jv. Let it be the laft equation of the foregoing article, — app —/p = aA + g X DV + fa +6 XJ + GA + e X yp, which I undertook to conftruct, This equation being prepared according to the method, and, for brevity, making aA + ¢=%fA+b=m aA+c=%, it will be reduced to this, app + fi _ FE TBA EG bp be da perdi. see dida ; bag TRS IST fi ara ra and, by integration, 4y + — lp + — = /g. And therefore y = q tre azien E n o i oe i op + — __Î te BEA And eliminating y, we fball'have nua era n? that is, — presa HS ee: i xp+° =% EXAMPLE V. Let the equation be this already prepared, 9° X wx + by em ex 9° — XY yp M—2 | which I write in this manner, ——— x ax +99 = s In order to make ye 29 the fecond member integrable. In this I make ufe of a double fubflitution,. and therefore I put xx + yy = pp, and, by integration, xx + yy = pp. I put je — 49 = g, and by integration, — = g. Making the fubftitutions, we alfo - = J » IM {hall have 2 = X pp = g. But yy = pp — xx, and xx = gqyy, fo that-it will: 278 ANALYTTICAL INSTITUTIONS, BOOK IVe | 4 de dad gtr? will be yy = pp — gayy, that is, yy = gig and yo = wag? and | 1499) 2 VA ae gue it — 2 n oe = e Wherefore, fubftituting thefe values of y and #, we i+ na 1 — 7 «= 2 fhall have p” it is x e" in -— È 2x + ca — cp Le ui erre = fe. © = f/%; and, making ¢ = = ‘fà, and reducing to a common denominator, it is 24 we make the conftant c = o, we fhall have ay = f%, which is another integral of the propofed formula — 2, it will be — different from the firft. Laftly, putting € = — 1, a third integral will arife, pa ee Fog 25, The SECT. Il. ANALYTICAL INSTITUTIONS 279 25. The method I now undertake to explain, although much limited and confined, is yet of great ufe in fome particular cafes. By tbis the variables may be feparated in the canontcal equation ay = ypx + dygx, in which the quantities ), 9, are to be underftood as any how given by x. The quantities d, b, are conftant; the figns may be pofitive or negative at pleafure, and the exponent # may be integer, fraction, pofitive, negative, or even nothing. In this equation, then, make y = gu, where z and w are two new variables; and, by taking the fluxions, it wili be y = 244 uz; and, by fubftituting, inftead of y, y, and y”, their values 24 + uz, zu, and uz", we (hall have the equation az + auz = uzpx + bz"w"9%, in which, if two terms fhall vanifh,. the indeterminates will be feparated.. To do which, let us feign an equation between the two terms a43 = 29%, then — = px, and, by integration,. alz = fpx ; and, proceeding from logarithms to exponential quantities, it is z — ml? or 2 = m*, fuppofing Jw = 1. This laft equation fhows us the value of z, and informs us, thar, to reduce the equation propofed to two terms only, and to caufe the other two to deftroy each other, inftead of y= Vy, Spe fei we ought to put y = wm “, that is, a —=mt,oh—l = 2, And, by differencing, = = — = px, and therefore ay = ypx +=, Therefore,. in the canonical equation ay = ypx + 2y"g%, inftead of y, I fubflitute it’s, U% value now found, and it will be yar + — = ypx + by"gx, that is, 24 — by*gx, and therefore si = by” "92%. But y = 24, andy * = hae whence, finally, it will be > = ba "9X3. in which equation the variables will be feparated,. becaufe 2 is fuppofed given by x. When we came to the equation. alz = fpx, it is plain, that if p given by « is fuch, that the integral /px depends on the quadrature of the hyperbola, that is, on the logarithms, and the: quantity 4 is any number whatfoever, the relation of 2 to x will be algebriical,. and in all other cafes tranfcendental. | And here it may be obferved, that, in order to have a given equation come: under the cafe of the canonical formula, it is neceflary that the following con- ditions fhould take place. Firft, that the fluxion y may be alone, or, at leafy, multiplied by a conftant, on one fide of the equation, Then, that, on the other | fide, the firft term may contain the fluxion x, multiplied by any fundiion of » exprefled by p, and by the indeterminate J. Then, that, in the other term, the quantity gx given by x may be multiplied by a power of y. In a word,. | . making: 280 ANALYTICAL INSTITUTION $.. BOOK IV. making the divifion by y, it is required, that, on one fide of the equation, there may remain the logarithmical fluxion ae and, on the other fide, the firft term may be free from the indeterminate y, and the fecond multiplied by the dignity y° 7". If any one of thefe requifites be wanting, this method cannot take place; as we fhould not have them in the following equations, ay = yypx + dbytgx, and ay = ypow + ayy +39) X 9%. But fome formule are vary eafily reduced to the canon, by a little preparation only. For example, take this equation ay = ypx + 2y7x + yygx. Confider that the quantity px + LE multiplied by y, and that the binomial p + 6g is given n x, fo that in it’s place may be fubftituted the quantity r, alike given by x; the expreffion then will be changed into the following, ay = yrx + gyyqx, in which the method here explained will take place. And this will be iufficient to fhow the way of operation in all like cafes. EA NMPI E L Let the equation be ay = aS a yy Make y = 24, and therefore ay = azu + aug. And, making the neceffary fubftitutions, we fhall have azz + auz x ; : az ; : È de DS + zenux. Let 243 = = ve , that is, — = i ; and integrating, it x % will be @/z = fix, and therefore : = xf, If the conftants a, f, fhall be rational numbers, whole or fracted, affirmative ‘or negative, 2 will be given algebraically by x. For example, make 4 = 1, fuzx Ff = 2, fo that it may be x = xx. Then eliminating the terms cuz, ; x there will remain the two, az4 = zzuux. But zx = xx, therefore it will be ai ett te pira: —— = xx, an equation in which the variables are feparated. In proceeding to the integration, it will be — — se € 2 ua ee ie ae % which is the algebraical equation concealed under the propofed differential, nera “E Axx . — = —, and therefore — Pa + co x's that is, gog — 34% == N'Y; E X- (EE a ee ee ee a oe ee SECT. Il, ANALYTICAL INSTITUTIONS: 281 EXAMPLE Ik Let the equation be 9 = St +. > ** | Make, as above, y = 24, and xx — aa — aa y = 24.4 #;. then, making the Doo we fhall have 24 + «3 = ALUX 233% AZUX + —-. And, fuppofing az = se hat 3 pi Le ot pnt Nile is, % | 8 = aa’ KA = Ga ax f pra ee. 4 ; z3u3à 7 zi 2 = m #72, we fhall have the equation z# =. ‘which the variables are feparated, z being given by x. But it may be ob- ferved, that the quantity Pai may be reduced to a logarithmic fluxion, by 9 x a n x a % ry e 2 ® . 1 making x = Di —; wherefore, making the due fubftitutions, it will be cdi ii a = Ww % aXag=- a =. hence — = —., and therefore zz = 2 =. ——__—.- And, putting this value, inftead of zz, in the final equation, we fhall have “ _ QX% — aax ee us © x4+ax3° x ¢ os _ ataxa " ax Without making the fubftitution of «x = ———, the ey ys may be reduced to a logarithmical fluxion, by means of § 21, Book III; and we dhould have —"—a — —= =) + and conféequently va — ad 2Xaxta 2Xax*4 og FAAMPLE-= HE Let the equation bey 2 — 22 + ax Make y = 24, y = xt + ud; UNH therefore, fubftituting, it will TE zi + US = D+ zx. Suppofing x UB" pa x a UZ — 3 t — a d, by integration, z — 3 we fhall have e e i «n Yo ‘ ° Mm n - e “ Mae È + a the equation 274 = 2 u x, thatis, — = 2 x, or — = —î, | 4 “i” mk Vou. Il. Oo E X. 282 ANALYTICAL INSTITUTIONS BOOK IV, EXAMPLE IV. Sometimes a two-fold operation is neceflary.; as in certain equations which have more than three terms. Wherefore, let the equation be xy + yx = ax + #4, and let 4 be any how given in the terms of y. I difpofe the equation ou ot ~ . in the following manner, au + «4 — wr = yx, OF — + —— Aia xi È; A Make « = pg, and x = pd + gf; then, making the aisi. it will be va + on i = pi + de If any one would reduce the formula by one oO ti | Ni i eee PRES dr o ; t 1 et Cal BU pod EA ‘ b 7 peration only, he muft put ri 5 pi, that is, s ; - y which we find g given by y. But the operation will be performed more neatly in the following manner. Make — e — = pj, then — 2 = 2, and, by integration, = ua baking, othe the other terms of the equation va + a = gp, and, inftead of g, fubftituting it’s value ee it will be ap Ly, this ig 4 È eda a wi, then 4 im + mm, and making the fubftitution, it will be 4 + 2 = mn + nm. Sup- pofe = a = mn, that is, = 2 =. Therefore x ni be given by y, and in e e È mM the remaining equation, after the terms ne » Ma; have been chasatasogi that is, in the equation # = mm, the variables will be feparated, and it will be Li = mes 26, Still, after another manner, the variables may be feparated in the cano- nical equation y = pyx + gytx. Make px = i 4 * : . Imu X 2 I-nX pz P | : x . A a bs ns ; ; Making the fubftitutions, it will be y = eile cee Bhat a ) I—n X% I X px pz + yr , OI — 2X pr = pys + gy"B; and therefore, dividing by py, 1-72 X px cee ee di a Laftly, dividing by 22, it will be SECT. IL ANALYTICAL INSTITUTIONS, 283 I—nw MX zy #, ed ae q% | 9° i — gi i EN © vid aie Te one i i ione vie "i 14at iS : Toso: and, by integration, — Jj 9 , KX = zi And, becaufe p and gare fuppofed to be given by «; and z alfo, by the fubftitution of px = , is given by x; the variables will 1-4a4X3 be feparated, at leaft tranfcendentally, Refuming, therefore, the equation of the firft example, ay = 2° + yx, pe f a ’ i TAO . I is ¥ = : +=, it will be p = 7H n = 2 So that, fub- od eb ‘ftituting thefe values in the final equation y'~” = z/£, it will be — se J “f=, and the fubftitution px = will be Di tin And, I-nX%z % making f = 2, a = 1, we fhall have = = — —, that is, z = —. And % KA I I e e pe I eS eT RATE EON, therefore — = ——f/ — xxx. And, by integrat Fan Ge TI, 7 sf y unteprahony pisano + Pa PILL ICI that is, 36 — e DI = x, as before. And fo we may proceed with the other Examples, ENAMPLEEL O V, Let the equation be ax*yy — dxtyy = ayyxix — by yn? x + 4°x — 4°, which, x Si — Sx divided by ax*y — dx*y, will be found to be y = a= A a > Which sa cafe of the canonical equation. Therefore it will be p = —, qa= E densi ; i î ab X x* s= —1. And, by fubftitution, px = —— : x % pera will be Via pee whence z = «x. Then, putting thefe values in the final canonical equation, y!7* = AI © & Axx —b x ATX x3 z se » we fhall have yy = x I= s In which the Varani are fe. parated. 002 27. If 284 ANALYTICAL INSTITUTIONS, BOOK IV, . ° fiems T o ° Ù ° 27. If the canonical equation were y 7 $ = px + gy"x, where p and g, in a like manner, are any how given by #; the indeterminates may be feparated by making gx = =, and x = oe + For, making the fubfitutions, it°will e ® bi 71 «e JT. HN è» . nel. _ PX yz : nzy RA ALTRE + 03: Deg er gn ae ri + =, that is, TT rg and, dividing by 2, AE ee Ne a eae on Pie ays” oe pe a dg Map E SUE Narada mae ae 18," ome an equation in which the variables are feparated. For an example, let the equation be 2a*xyy = aayyx + 2bx8x, that is, DES barak UE, dae e I a sona i eae ra agita i E sac It will be z È da, ——» and therefore we fhall 2bx3z ; x CIOÈ e. have 2 = (25, But gg = —— =, ande = 2. Therefore ic will be & aAZZ 2% 29 yy 2bxx È : Wok bas i i pe = =f) and, by integration, Si gen + ¢; an algebraical curve. Alfo, the general formula y°7!y = px + gy*x might be conftructed, and confequently the particular example, by means of the method at § 24. 28. Before I finith this Se&ion, I fhall add one obfervation, that. fometimes the indeterminates are involved and mingled with differential quantities, when it may be allowed to modify the co- efficients ; and this fucceeds efpecially when the exponents are formed of the co-efficients; and thus making a kind of circuit in the reduction. This artifice chiefly takes place in Phyfico-mathe- matical Problems, in which magnitudes of very different kinds mingling together, we are more,at liberty to make ufe of fuch conftant quantities, as beft ferve the prefent purpofe. For an example, I fhall propofe to myfelf this equation, #°% + dy + X — = yy, which, being prepared according to the method of § 24, will be bey mi.) = I | DA AN came | SAX F at Make, then, me se “pe and we (hall x G have the value of y = DK and yy = ppa + Thefe values, conveniently fub- ftituted, will give the equation x°x + 2epa7'x = #°pò; and, dividing by 2 : ‘ — 20 > w—Com] » ° Thar ; ‘ a, itwillbe Tx + depx Tx = pp. Here it.is plain, that, an equality being given between the exponents of the indeterminate x, that is, between m—2¢ and — c— 1, the variables will be feparate, the bomogeneum compa; rationis 1p being only to be divided by the binomial 1 + bcp. Now, putting | M, 26 6 * & # SECT. III. ANALYTICAL INSTITUTIONS. © 286 m—2¢ = —c—1, it follows m + 1 =; fo that, expounding the con- ftant c by m + 1, we Mall have our defire. If ¢ reprefents unity, which we are at liberty to fuppofe, it will be m= 0; and if.¢ = 2, it will be m= 1. And fo we may go on. The artifice here explained may be applied to all other equations of a like n» Yr. kind ; for example, to this following, xk a sE cl — di). For, putting f =7 — I, or —7-= I, the formula will be thence abbreviated by making ufe of the logarithms. gi a i © 8 Of the Conftruttion of more Limited Equations, by the Help of various Subftitutions. & / Serer are AD 29. In the equation x7 + ay") X p = xy — yx x 9g, the indeterminates are always feparable; where p and g are promifcuoufly given by y and « after any manner; algebraically, when, in every term of the quantity p, the Sum of the exponents of x and y is the fame, and thus likewife in every term of the quantity g; but it is not required that the fum fhould be the fame in p and Ge | al ol I The fubftitutions.to be made are y = f#z2+1, and ax = t x a? F az2\et1, Thefe being fubftituted, refpectively, inftead of x, x, y, , and making the neceflary operations, after a very long calculation we fhall come to this equation, aì pazzin bi Now, becaufe it is known, that, in every term of », the fum of the expo» nents of x and y is equal, as alfo in every term of g; making in them the fubftitutions of the values given by ¢ and 2; in every term of p, # will have the fame power, as alfo in every term of g a fame power; that is to fay, that the homogeneum comparationis will be multiplied by a pofitive or negative power of ¢, or the firft member will be multiplied or divided by that power, and therefore the variables will be feparated. y P E As, 286 ANALYTICAL INSTITUTIONS. BOOK IV. As, for example, let the equation be «x + ay x Vy = xy — yx x 4; n my i jal a it will bea — 1,p0p=Wy, qg = 4, and therefore — = SE CORRE ; é V a3 — ax X NY a x z a But y = /z; therefore it will be — = ME ADI vi V 43% — azs In the fame equation the indeterminates may be feparated, when alfo the i cm n) x . è ° . . am 7] . — 1 e exponent 7 is negative; that is, when the equation is this, xx + ey yXp 2 = Ky — yx X g; and the fubftitutions are y = #z1-%, and a =f x i+” ; n= 2 = gene ptt d ge e ° e wane 7] ome 2, — 7 a = azz\i-—2, Thefe will give the equation ¢ Dm — AL ; a} = azzii—a the fame as that above, only with the figns of x changed. And though the | . ta Lr We equation were alfo thus expreffed, y"x + axty x ayn = XY JR X G5 it follows that this alfo is. conftructible by the fame fubftitutions. —n-1=cC 30. Let the equation be more general, xx + ay ot y Bai ny + ox X g. The variables will always be feparated by making the fubftitutions of NI “a+ r DE Sarti, anda =f x a acz » where s and r are numbers affumed at pleafure ; fuppofing, however, this condition, that the quantities P» 7, are given algebraically, and in fuch a manner, that, in every term of the quantity p, the exponent of y, taken as often as the number ¢ denotes, may exceed, or be exceeded by, the exponent of x in the fame excefs; and fo in every term of the quantity g; but it is no matter that the excefs in p fhall be the fame as in g. Thus, for example, if c = 3, it may be p = dy*x* + fy"5, s c È Misia, 63 0 n 3 * &c.; and it may be g = gy?a? — byx2, &c. It is eafy to perceive, that it cannot be c=o. | Making the due fubftitutions, inftead of x and y, in the propofed equation, ye I xa at SX LL — 1-05 we fhall have this following, — = fp 16 ao) per Pe ALE] de gra n È atac cisti For SECT. III, ANALYTICAL INSTITUTIONS, 287 For example, let it be ax + ay *y x = xy + yx Xx. Makes =1, r= 2; itwillbex=1,¢=1,p= = q = x; and, making the fubfti- È È * | —3: x x xy. tutions in the laft equation found above, we fhall have — # °¢ = =. . i | se ae But, by the fubftitutions made, x = #7! x @ + az*)*, and y = tz. Theres AE OE È; ; —2 3 - fore xy = 2 Xx a@-az ~'*. Whence we fhall have Se See | —af—c—f g1. But let the equation be ftill more general, w~ + ay ¢ YXRXp = fey + ox x q, which comprehends, as particular cafes, the two canonical equations of the foregoing articles; that is, that of § 30, when it is f= 1; and that of § 29, when itis f= 1, ande = — 1. § A The indeterminates are feparated by means of the fubftitutions y=#/z/%X2+1» s a RE isis | ik we ace oc (@tt and x =f Xxa+ ; the condition concerning the quantities p and g being fuch, thar, in thefe, the exponent of y being multiplied by 4, may exceed, or be exceeded by, the exponent of x multiplied by f, by the fame excefs in each term. The fame quantities p, g, may alfo be fractions, or mixed with fractions, and rational or irrational integers, whatever they may be. And the indeterminates will always be feparable in the equations, provided that and gq are given by x and y in fuch a manner, that, the affigned fubftitutions dana made, fuch quantities may arife in their place, that they may be the produ& of two, one of which fhail contain z, and not 7, the other ¢ and hot a. i The faid fubflitutions being made, we fhall have this formula, pal Ade di — fom fin —se r ve Jn+f = sa g SI cf è. 141 — ——/ z = cr» % e | I gpl x at ee € TI if. EX. + 298 ANALYTICAL INSTITUTIONS, BOOK IV. EXAMPLE IL Let the equation be «xx + ayy x y = — 3x4 + yx x ax. Let it be, as q i DI SY Ty DED wee Lf, el be fi ee IR rp = i and, making the fubftitutions in the lat formula found above, we fhall have mt ax ee RITA 1-3 ee pare —t 32 = =. Buty Kt te eat xa pati; I —2|3 a = 74% _ ” t 21% therefore it will be — — = -—— :_ as was to be found. EXAMPLE” IL L rat : - she Let the equation be x + ay) x ayx + yes = 2%) + 3JX X JI myn, Letts f= hy f= bs it will be ¢ = Gn Sues 2 Th a es Pp = aytx + yyaT, g= yix — yxx. And, making the fubftitutions, it will be 5 Da I r Ie 2 n = e ee -| 3 -— = ne = — gi _ 33 0% X atiaz 3 — ja 3z xX atiaz 5 4, “Tuoi —s Ù x : ; 2 2 Sg oe 2 Pai me PI, ia de = gre az Xat3az 313 +27 x atiaz 309 in which the variables are feparated, as was required. 32. In the equations (1) pxy'y = pytx + GX, (2) pi) = — pyre + 9% (3) apay = bpyrà + gx (4) ap” j = —bpy'à + ge, where p and q are any how given by x; the indeterminates may be feparated, by putting, as to the firft, y = «2; as to the fecond, y = —; as to the third, b b Ri a HAZ; as to the fourth, yun az, * This equation evidently admits of a fimpler form. EDITOR. Py sECT, ITI. | ANMALYTPICNE INSTITUTIONS, — _ 289 As, for example, let the equation be abbuyyy — 2x yyy = bxty — 3bby?x + 3xxy°x, which I write thus, 25 — «x x 260999 = bxtx + bb — wn X — 394%, This being referred to the laft of the four canonical equations, it will be PE Db n eRe eye ee ea Pherciore’ wer mull pue 3 TL Z . wry a TO ZB z3 . ; \ y=<>,J = = — Y= YW = 7. And, making the fubftitu- em | x tions, we fhall have 2bhx Se GRA = xt + ghb — 3xx xX x 234 eR ae —————— die ELE REN FRA RAR "> —; that is, 225 — 20% x wx2z3z — 12% = bat x + 3bb — 3%%x x n° i ik — 23x; and, making the ufual multiplications, it will be 204x223 — 24%z23 ] 5 Ae da bea x Li È e es s ao 43 €; that By coz, 2hha — 2x3 33. Let the equation be axy + dyx + cyte Tae 4 fe” y7%5 o. In this the indeterminates may be feparated, in general, by putting x = «°7'2°7, — ’ x n £ “andy = 2 "3; for, making the neceffary operations, we fhall come to the Me Mod I Mh ffl 7 | I - equation I = 7 x as + fu Zo poe 1 MOS +b Ck pa \ —1I. MA Mt. ua ie, Z = He 1 X — D24 "U — (24 u, thatis, — = wa Ron La ET NAM b I Mi Me N41 1-m X a+ fu +1 XxX b+ cu As, for example, let the equation be ax) — ByX = cyyxn = fuxyy. Then awh De» =", mw =, 1 heretore Ji pue ms oe =, and y = —, that is, au . AVÙ -- AUY x = ote And theretare’ ae = era, Whence, making the due fubflicu- i atuy ay — QU) caayuu — caauuy aauuy : fons, we thal have —— — #-x. ee egg sep that is, atuy + abeuy + aacuuy + faauuy = abyu + aacyui, and therefore — = x 5 | i abi + aacun au + abu + aacuu + aufuu . ; yee gly 34. Let the equation be Sater) oY STS generally, POLE. to ~ dba + ay’ x i | ba tayo! = x : x x filer: 1 bo ee ee i = y. The indeterminates will be feparated by putting dx + ayn” = 24” Vor. II. Eb Whence * See the Note at the bottom of the preceding page. 290 ANALYTICAL INSTITUTIONS. BOOK Iv. I— 7 x | ra spugna, feet i Ix aright cage eee n | Whence y == ————, and therefore y = ~———-_____ x a” a” Lis seed = grupo Peck ee e fer 1 . E SLITTARE Hof Ln e DX x= —I 1 fur i—r L | PANE ea — bx . mi « ne s putting thefe values of y and ~ in the propofed a" % I bit 7 nO ae Pd abt)" . general equation ; and dividing by ————; , it will be a” I I ter il, > tereko ——- fune, —— Ke x z Zbtiarxze x+i-r xX — bx = eas DERE | ere OS a ati ae a UX ) «ne. ZI ta aa, I °° È Il. I. Ri e a e A — DIR ENZO Mmm NOK 3. and therefore ——_ __——— Mn” fe owt X mg 13 + mr—mt x bz — mnb If you fhould have terms with negative figns, you muft proceed after the fame manner, and in the final equation there would be no other difference, but that of the figns themfelves.. Ue i ye — bec? + af du Ul Mit i pr pn ur oe n y; the indeterminates would be feparated by the fame 35. Alfo, taking a more univerfal equation, as fubftitution. | EXAMPLE IT. Let the equation Ras by, Make 4/dbxx — ay = wz, and V bbax — aby _ bbax — zznx +. __ 2bbax — 22Zx% — Wee bibi therefore y = a ee ee And, making the SECT. II. ANALYTICAL INSTITUTIONS, 291 aa bbxx — SzKK abixi — 2bzzuio — abxxxx È x the fubftitutions, x os DECRETI = > that is, a20bxx a3 a — aazzKn = ben — 2b23e% — abenzed, or 2bexzzz% = 2bFenk — 2b23 0% _ 2bzzx An x Cod a ; = PR LD TL Oe < AAZBX aabbxx ; and therefore TOTI joe oe è x eae ea x EXAMPLE II, xy : DEMO, SETTORE pee jo ae Let cap aeamicass be eee Make 7 — bbx* + axyy = 3 2xx, and therefore y = VEC ob amd dA oi pe ee aa Where- Wai + bba3 a = fore, making the fubftitutions, we fhall have — n ab BAK as nz + 3zaeuee + 3bbaxa Vee AIR, i Adi 3 Paste res ma E TORE » that is, bzzmwx + D'uma = n°223 + 12°x08 + 2bbzwnx, or bzzuxx + Bxax me izianx — bb = #°2%3 5; and therefore — — ov ZZZ baz — 323 — 2bbz + bi" 36. By the fame fubftitution as above, .the indeterminates in this equation alfo may be feparated. : Var lun tnit=thdfer | eo ae Soa a. aioe week d poe bina ui f Ur, ml f a ean cx n Xx. Make dx + ay” = xs, it will ba + ay x | eee I-m t—r : gni feti. m è ta then Sy. sai = alte {f-ful . — f Eom gu J da fhall have X into 292 ANALYTICAL INSTITUTIONS BOOK IV. EI 1 n 1777 VU A} 9 a : X. Wherefore, dividing the $ uti. TA gi a , inftead of x and multiplying the whole by @ # 2; and Writing # n xa Zl a , which is the fame; and, uniting the dimenfions of the letter x, we (hall find the equation to be divifible by I- 22 tu -—~in-+-t—ru-+-nr—r ukb—-n—tmn—rutinr x n) > and that being divided accordingly, it will be m & MDa ss, t+ I 2 - O) ber 6 U . = \ ee © È ® ° CO) o + — bx into z*—b ” = ca a zx. And laftly, dividing i UtI—2 I— m7 XZ m.& ra again by 2° — b' ” , it willbe EM ==! y omy 4S" x be + ubi 1 —— Ym SÌ eae i Pa can 2% Xx" — b , that is, — = Ie. z mn z A—uU—I Li , at ES ai n ceci ili Li n CEE mnca no KB" — di + mr — mt X 2" + mt — mr Xb E. X A MPL &E..- yy ARE Let the equation be = —. Put dbxex — aaxy — abxy bbax — aaxy — abay DDA mm LZ n 242% bbx — xxx ve : ryan ag 3 and y = pg see oe e Making, | di A NR. Isca bbx — xe — 2x2% kak therefore, the fubftitutions, it will be n aS rr ee And, inftead of dbx — 22x}3, writing x? x 25 — 2zz\3, and multiplying the whole equation by aa + ab)* x ax, we fhall. have #° x 40 — zz) x — wz, and therefore y = SECT, 111, ANALYTICAL INSTITUTIONS, © 293 z13% bOX% — BZN — 2023 = a4 + ablt x o" And, diyiding by #3 X 20-—22)?, it will be dx — 22% — axed = aa + av\* x bb— zz) Xx ine that is, Cc bbs — 22% + aa+ab)* x bb = 223)? x — - = 22% And therefore Ea i 22% racco Fane (SORE EER EE PROTEOME _——@@n6uW@uieulsii@ SON AEGEAN: r a__t_—t___t_i -— 4 sare Db oo es ees bb — 23) 9 x aa + abl A 37. The fame fubftitution will ferve, in like manner, for a more general bi TAR x Par Ul — n =lmn-rutbtiordntv : x equation, oi = x n “x. Alfo, it will ferve bi + ay x | f; . yo 1; — ife 1- mi | for the equation “at 3 x,. by. making, bee eri ay’ . È pa QUIS te si hat tiene | bit cx” + ay’x ” = x; which; if m = 1, will be a particular cafe of § 27; and if it be ¢ = o, will be a particular cafe of $ 36. Moreover, we - zt r n 7\e ° Me 1 : : o o bw. hy x. Ms may alfo conftrué the equation == TSE > t r n T\m ax +bx + cy x when it is cb = 2k; making ufe of the fame fubftitution, an + be + cy yg _ ptm—r—i+ed—mt.. = fa X's mt one ze | | Now, if it fhould be alfo è = 0, è = 0, the equation will be a. particular. cafe of the firft equation of this article... . Thefe e ions. e tru&ed; ———Z = gy 2% } 38. Thefe equations may be ae ed, ide ay ang MI. { Pa) + = gx" 'x, by putting; for the firft, cy? + fa)" = 2, and for bt cy” + fx”! the fecond, cy” + peek = z And, as for the firft, it. will be then Ny See (2 x. oy pont 1-% x — fa” eit atti RO API ‘“; and. therefor 2 » and J ra Nae apr xX —_3 Z — fx; and. therefore, . ¢ c i 1-% making the fubftitutions, we fhall have az. # % = nubegx + muegza + auf, that. ANALYTICAL INSTITUTIONS. BOOK IV. 294 1-% 3 ae gn I e asd A | To iii topi Var I CAS beth fecond. ave thall hay = that 1s, nubco + nucgs + auf sl : ca "E 92 casini Gn FT} at a = | ple yand therefore y XK ii fe Stili Ze taf” "xs £L {2 4 uz : :c Tesi Lu az u z own JT EA begnu + cgnuz + mafu and, making the fubflitutions, « aly ———- = g9x%, where 7 b+ a" +p" and g are any how given by # and conftants ; if it be g = _, the indetermi- Likewife, if we take a more general equation, = 22 “Por a oem L ten I-—% 7: le a A di [rn © e Xx 37% 3 — Ps and, Cose and pes E nates may be feparated, by putting, in like manner, ey” + p ì SET IDG aie gie a è making the fubftitutions, the equation will be wbegugx + neguzqx + aup = i I--% ye "ale : ae I =? az * %. But if we fuppofe p= 7% then it will be nbcgu + ncguz + au EXAMPLE L " è à he ASS RRR TR a I 34 | ~ * Let the equation be 45) = 62°x — 3bbxVcy + dx, or a ghi, , 2b— of cy + be s a bi ù i ad making ZA om bx Seema Make cy + dx = 2, it will be y = ; 2a3zz — asbà sa 366%, or 2422 = 683ex — Zbbezx + a°bx, the fubftitutions, —-——— 2.432% : nd NN and therefore Ghic — 3bbca + a®b E X- SECT. IPT, ANALYTICAL INSTITUTIONS, 295 SOA REI Mea PINCO os = aax — 2hbxx. Make b + Mv 3 + cax — bxx 34a2x — 6bzxx 3 and, dividing by 4 + 34 + 3z, it will be gazzz a+ 35 + 3% = aax — 2bxX ; that is, ZAZZ3 = AX — 200x% + 3aab% = GAX n 2bxX.. 39. The equation, or canonical formula, ax°x + cyyx*x = +, has not it’s indeterminates feparable in general, whatever the exponent m may be; yet they are feparable in an infinite number of cafes ; that 1s, the exponent m sita receive infinite values, in which the defired feparation will fucceed. To GEE which I make ufe of a method not unlike to that of § 23. Make y = Ax? + x; where the quantity A, and the exponents p, r, are arbitrary conftants, to be determined as exigence may degne and # in a new indeterminate gine Therefore it will be y = = PANI i pe rte di xi, and yw = AAx’? of aAx?*"s 4 tt". Wherefore, fubftituting thefe. values in the propofed formula, they will give this following, ex°x +. cA An? tx + 2cAtxft"*+"% + ett te = pAx erty + rte "x sà a. Let us fuppofe cAA = pA, 2p+a=pT—1r = ach ; that is, p =. — i I —n—i, A=——,r=—2n — 2. By thefe, in the laft formula,, will vanifh the fecond, third, aft, and fixth terms, and it will be reduced to gU=A 2172; ax x + tte” at . That is, (dividing by “ore > ax'tant2, — MZ » + ditx ns) (DI. an x fp ctx = = 7, making m + 22 4- 2=K, and 7 —- 2 = X, I refume 4 296 ANALYTICAL INSTITUTIONS. - BOOK IV. ay è ‘ mM ‘ ; Jai È I refume the propofed equation ax X + cyyx"® = .y, which, putting y = } . ° ° 8 - Mi « e ° ° o ° —, is transformed into this other, ezzx x + cx = — 2; in which is put, & È asrabove,. 2 <= Bx? + x 4, where B, 4, ©, are conftants, to be determined as, before, and 4 is a new indeterminate quantity. Therefore it will be 2 = @gByt le + vue Tx + wu, zz = BBw? + 2Bxl* "x + aux. And thefe values being fubftituted, we fhall have aBBu te + 2aBusl tt” 2 + au tg 4 ci = — gBal i — vue — di. Now, if we fuppofe iBB = — Bg, 2g +m = 9-1, —v = 20B; that issg+ta=— 1, B= LA ~, V = — 2m — 2; with thefe in this laft formula will vanifh the firft, fecond, fifth, and fixth terms, and it will be reduced to auux 3" Tix + nN + ; —2m—2- : aes ees — 2,772 — 2 24-14-22 - Rie So RTS TT ig int dis, (dividing by x Peo ge cn jmm? + È 6: 0108; ° e e AUUK x 2 — 4, or (G) cx x + auux x = — 4; making 2m@+n+ 2 = è, and —Mm—- 2 w. Now, in the ‘propofed equation, the indeterminates are feparable when Mm =n. Wherefore, alfo, in the formule marked D, G, the indeterminates will be feparable, when it is m+ 27 + 2 = — n — 2,20# +n +2 = — m — 2; becaufe m obtains two values, that is, m = — 3% — 4, m = "74; which being fubftituted, the feparation of the indeterminates will fucceed. For then,.in the propofed equation, the indeterminates will be — FT — feparated when it is m = Se ; alfo, they will be feparated in the formula ra: MEO fae Sla | DP; G, when itis K = — — a, becaufe there are other two } step de Seo values of m, that is, m = ae m = he By the fame way of argumentation, we may have infinite other values of m; _ —]Jn— 12 — sn — 12 — oz — 16 — mn — 16 asma LET, m = PELI n= ID, me I —, &c.; 1) 7 Ci 9 : zhtixtTan—- 4h TOR . È | n gener fio sc f and, in general, a , taking 6 any integer pofitive number, beginning from unity, Putting any of thefe values in the propofed equation, wre Mall have the indeterminates feparable. It SECT. IT, ANALYTICAL INSTITUTIONS. 20% It may be added, that the indeterminates will allo be feparable in the pro- pofed equation, when the exponent w is fuch, that, by the method of $ 19, it may be reduced to a cafe of $ 14. This would be the place to make ufe of two Differtations of the very learned Mr. Euler, inferted in the Memoirs of the Academy of Peterfourg, Tom. VI. But, becaufe of the fubtile manner in which chat author proceeds, I fhould be obliged to exceed thofe limits which I had fixed to myfelf, intending only a plain and fimple Inftitution. I {hall therefore leave the curious reader to feek. them in the book itfelf. | PROBLEM LI, 40. To find the curve, the fubtangent of which is equal to the fquare of the: ordinate, divided by a conftant quantity. Making the abfcifs equal to x, the ordinate equal to y, the fubtangent is. always x, which therefore ought to be equal to =. Therefore we fhall: , Or ax = yy, and, by integration, ax = Zyy, or. have the equation a == 24% = yy, which is the Apollonian parabola, — If the fubtangent ought to be equal to twice the abfcifs, we fhould have the: yo tis oe F equation So ae and therefore — = goa and, by integration, 3/x + 1/4 = ly, (where the aanieat t/q is added, to fulfil the law of homogeneity,) that is, IV ax = = ly; and, returning from ine logarithms, fav = y, or ax rs which is alfo the fame parabola. If the fubnormal is to be conftant, it will be #3 = iy CAR les de gee. and, by integration, tyy = ax, or = 248, which is again the fame pa- sali dai the fubtangent be triple of the abfcifs; it will be a = 9, "È = cca x I, and, by integration, /W44x = Jy, or aay = 9’, which is the firft cubical parabola, Vox. II, % Q4q Let 298 ANALYTICAL INSTITUTIONS. BOOK lv. Let the fubtangent be a multiple of the abfcifs, according to any number m ; È L x SL nt Li Prata it will be Pai SS ew, tat 18, —_ = -— , and, by integration, / T's = ly, a 772 «o J în . . or @ x =", a curve of the parabolic kind. Ci YX 2a” | x Gs }; e e Let the fubtangent be SESIA. thew the equation Is =—- = —— , that +4 ada . ” ee a . ax A xx nali i 2 : 5 Pi 1S, 2YX + yxx = 2axy + xxy, Or oa And, by integration, it will be /y = 1/24x + «x, and therefore xx + 24x = yy, an equation to the hyperbola. 24xYy — ane yx e 7 : 20X)) — 2x3 ; then the equation will be 2— = 225 + 3x4 pane oes ay + 340% sd that is, ayyx + gyanx = 2axyy — 3x3. According to what has been already delivered at $ 18, I endeavour to reduce this equation to a cafe of $ 14. Let the fubtangent be i 23% Therefore I make y = ——; and, making the fubftitutions, it will a 3 be 2% + g2zxxx = 4233 — 6x32%, where now it is reduced to the faid cafe. Wherefore the indeterminates will be feparated, if we put z = a sa ae; and, making the fubftitutions, it will be —— =~ ses + ser ee X aa ue xp = 3 = ye *P Tbe a 2 , that is, gaapx — 3p°x = 4xppp = baaxp, and therefore — Gi ca ci ae ; and, by integration, /x = me n . And, reftoring the value MISA PR pt saapp of Di that 1S, af 2 3 it will be x = Siam that 1S, finally, ay Ran SR ANA | Tn. The two fubftitutions made of y = 2, and &. z, in order to feparate the indeterminates, plainly fhow us that it would have been fufficient if, at firft, we had made but one of them, or y = e : But we might have obtained our defire fomething more expeditioufly, by writing the equation thus: 3yxxx + 345 = 2axyy — ayyx; which, divided by xx, will be 39% + 3x) = ui ; and, by integration, 3xy = 2 , that is, tay = xx, the Apollonian parabola, when we omit the conftant m, Let SECT. III, ANALYTICAE INSTITUTIONS: 209 : x3 X 4x3 sn (1% Y Let the fubtangent be a i. x ——=; the equation will be = tai Tr ay 34x — ay x that is, 4x3) — axyy = 3xxyx — ayyx, which | write in another manner, thus: 403) — ayxax = axyy — ayyx. I obferve that the fecond member would be integrable, if it were divided by xxy; I divide, therefore, the whole equation, - + 4%) — Bye 0%) — ays ; , ; whence it is 2 TUT — 2 723. 1 fuppofe the integral of this fecond y xa ¥ a ‘a CORTA °- ° ° member — = 3; and, making y to vanilh out of the equation, it will be a = %; that is, petti = 2%, which may be conftructed by the x method of § 14, or elfe prepared according to the method of § 24, it will be x X Ciara. Therefore F make ne “ae and,, by integration, dz4x = la*p, or 2% = a*p; and therefore, making x to vanifh out of the final: £ , ata p ad a 3 Sud : equation, we fhall have, laftly, ni X 7 3, that. is, 44 = 2*z, and, by integration, a*p = 325; in which, reftoring the value of p, then that of 2, it. will.be xx = ay, which is the 4pollonian parabola. nego ata x1 s. the equation will be I aa risi. at lata a+ latex. yx s 5° ax +alat+x i A ee ~—, thatis, — = —_ . In order to proceed to the integration, I J ata xX late , make 4 + x x /a+ « = 2, and therefore 2 = x x la + x + ax; (fup= pofing the logarithmic with the fubtangent = 4.) Thefe values being fubfti- ae we 2 PE : GERE: ai tuted in the equation, it will be paro and integrating, itis.y = 2, that atexlatx Let the fubtangent be is, y= a@+a x Za+x, a tranfcendent curve, but which is eafily defcribed,. fuppofing the logarithmic.. PROBLEM II. 41. To find the curve, the area of which is equal to two third parts of the rectangle of the co-ordinates. The formula for the area is yx, and therefore we fhall have fà = LES de te Ps. Rava 4 pie : ee. ae ; i whence yr = 44 + 4yx, that is, yo = 24Y, or se a and, by integra» Q4q2 tion, g00 ANALYTICAL INSTITUTIONS BOOK IV, ‘tion, as before, it is /V/ ax = ly, ax = yy. The curve is the fame Apollonian parabola. - Let the area be equal to the fourth power of the ordinate, divided by a - . e = at ® i s a Avs @ i 6 7 y imm C ] ” & AI conftant fquare; then it will be /yx = ae that Is, yx = ——, 00 aax = 459) s and, by integration, 3aav = y5, the firft cubic parabola. Let the area be equal to the power denoted by m of the ordinate, divided 27 == T, 1 wt ® “ ry e e 771 ) by a conftant; it will be /yx = 2, Waigo — è » a curve of a a the parabolic or hyperbolic kind, according as # — 1 fhall be pofitive or negative. PROBLEM III. 42. In infinite number of parabolas being given, of any the fame kind; to find what that curve is, which cuts them all at right angles. Let the equation of the curve required be ?- is = y°, which, (2 being confidered as arbitrary, and fufceptible of infinite values,) exprefles infinite parabolas ; and (confidering mand in the fame manner,) expreffes any kind of parabolas. And, firft, let them all belong to the fame axis AB, (Fig. 144.) with vertex A, and different only in their para- meters. Let AC be one of thefe infinite parabolas, in which AB = x, BC = Ye From any point C let the tangent CT be drawn, and the normal CF. It is wv known already, that it will be BT = —. Let DC be the curve required ; and, becaufe this ought to cut the parabola perpendicularly in the point C, in an infinitefimal portion it muft coincide with the normal CP in the point C, Therefore CT, the tangent of the parabola AC, will be likewife perpendicular to the curve DC in the point C, and confequently, at the fame ume, BT will be both a fubtangent to the parabola, and a fubnormal of the curve required, DC. What is faid of the parabola AC agrees with any other of the fame kind, Therefore the problem confifts in finding, of what kind is the curve DC, SECT. III ANALYTICAL INSTITUTIONS, gor DC, whofe fubnormal is = — . Now the general expreffion of the fubnormal 1S a; which, in this cafe, ought to be taken negative, becaufe, in the curve DC, as AB, or x, increafes, at the fame time BC, or y, decreafes ; and there- fore the differential equation will be — = — d- ; and, feparating the vari- i MX il ° 3 s TARR Pe nyy pin ables, DPR ip and, by integration, "n Lyy + aa, or —— = 2naa — xx, which is an equation to the ellipfis. And, becaufe the parameter p does not at all enter here, the folution will be general for the infinite parabolas that may be thus defcribed. If the exponent # of the equation p” “x” = y” is fuppofed to be negative, fo that the equation may be x°y” = peat, in which now it is pofitive; it will belong to infinite hyperbolas of the fame kind between the afymptotes, the fubtangents of which are — —, and the fubnormal of the curve DC ought alfo to be equal to thefe. Then it will be — = =— 2, or x = We 2naa nyy ° ° MIXX ri And, by integration, = = 30) + aa, oF xx — = =~, an equation to ne the hyperbola, Fig. 145% x hi If the infinite parabolas AC, QC, &c. È Ne i — 7. N of the equation p””z” = y”, fhall have all the fame parameter, but each a different vertex in the fame axis; that is to fay, if one of them be conceived to move always upon the axis parallel to itfelf; from a fixed point A (Fig. 145.) making any” abfcifs AB = #, and taking any curve QC, whofe abfcifs is QB = 2, and ordi- nate BC = y; then will allo — D_ be ‘ xe the fubnormal of the curve DC required, and therefore equal to the fub- B Db P tangent BT of the parabola QC. Whence the equation — LA — = ca gia but, m o by the equation of the parabola we have z = Di , and therefore — 2 — ue IAA ZII pn 302 ANALYTICAL INSTITUTIONS BOOK IV, m OR CS n MY n . apeek Rk SD ae a | - ; we that 15 Da oD Fe ny > and,.-bys inteptation, x = UP a | Mm {Zn 778 NUD n Va , the equation of the curve required, DC. m X 27% — im If the parabolas are the pollonian, that is, m = 2, # = 1; the integrated equation would not be of ufe in this cafe; for, making the fubllitutions of the values of m and x, we fhould have « =: — È . But, taking the differential equation, it would be x = — Ip x ee an equation to the logarithmic. Therefore the curve which cuts the infinite Apollonian parabolas at right angles will be the logarithmic MCN, the fubtangent of which is equal to half the parameter of the parabola. Let the parabolas be the firft cubics, that is, 2 = 3, #.= 1; it will be 97° 3 ‘= — 3 or xy = 3p, and the curve DC will be the hyperbola between it’s afymptotes. Let the parabolas be the fecond cubics, that is, # = 3, # = 2; it will be # = — $V py, or xx = 3°y, and the curve DC will be the common parabola. Taking other values for m and z, we fhall have other curves. If the parabolas AC, QC, &c. befides having a different vertex on the fame axis, fhould have their parameter variable, that is, equal in each to the refpective diftances of the vertex from the fixed point E; taking any one of them, QC, make EB = « the abfcifs of the curve required DC, the ordinate BG =, LO =ip' > parameter; tit will be QB = x — , and the equation. of the infinite parabolas 9°" X 4 — pl" = y", and the fubtangent BT = = n x % = p, and therefore the equation — = =f — Xat p. If the parabolas be Apollonian, that is, m = 2, n= 1, it will be p= te + 4/ixx — 333 whence, making the fubftitutions in the equation — 4 — _ Xx — p, it will be — = = x F2V1xx = yy, which may be reduced to a feparation of the indeterminates by the method of. $ 14; and then we may go on to the integral, which will be algebràical. If SECT. III, ANALYTICAL INSTITUTIONS. ” 303 Fig. 146. N A the infinite parabolas AC, QC, &c. of the equation p” "a" = y° fhall have the fame conftant parameter, the axes parallel, and the vertices variable in the perpendicular to the axes; that is to fay, if one of them be moved in fuch a manner, as that every one of it’s points may defcribe perpendiculars to the axes: Taking any one of them, EC, (Fig. 146.) 4mdecalino “ANE > EB =, BC =, MC = #; and, drawing to the parabola EC the tangent CT, produced to V, then MV will be the fubnormal of the curve DC required. Now, becaufe it e Be ne, it will be MV = 3 whence we fhould have the equation Men ud MZA wie +, a ig | ‘ —— ee ——: and, inftead of y, fubftitutine it’s value 9 # 2” . given ny ua 9 DE) 2A Dp hess by i Me mM « - MBN Dir: xa ; the equation p 2" = y,, it will be, finally, eo: forata it that is, P 2/1 = 72 MLZ A Z x WW m — — aw iat aids By intedration, of = ito. NE eta Leti ha A 5 3 rte the ip m 2am 3 n X 2mM—n Xp n equation of the curve required, DC. Let the parabolas be the Apollonian, that is, m = 2, n= 1; it will be 2 or 2px" = 2*; and therefore the curve DC will be the fecond Kk = cani af | | cubic parabola, of which the /atus recfum will be to that of the parabola AC as g to 16. It 1s to be obferved, that, in this cafe, the pofition of the curve DC will not be that marked in Fig, 146, but will have it’s vertex in A, cutting the inferior part of the Apollonian parabola at right angles; that is, meeting the convexity, as in Fig. 147. ich Another kind being pitched upon for the parabolas AC, alfo the curve DC will be a parabola of another kind, | PRO. 304 ANALYTICAL INSTITUTIONS, BOOK IV, PROBLEM IV. 43. Upon the right line AD let the right line AC infift at half a right angle; the equa- tion of the curve AB is required, the property of which is, that the ordinate BD may have to the fubtangent DF, the ratio of a conftant line a, to BC. | Make AD = x, DB = y; it will be CB = yo x. Whence, by the condition of the pro. blem, we fhall have y. "a :34.Y=— x; and therefore the equation ax = yy — xy. Now, to feparate the indeterminates, I make ufe of the method of $ 23. Wherefore, putting x = Ay + p + B, and x = Ay + p; and, making the fubftitutions, it will be aAy + ap = yy — Ayy — py — By. Now, in this equation, the indeterminates will be feparated, if the firf€ and fecond terms of the bomogeneum comparationis be made to vanifh ; that is, if A = 1, and B remains arbitrary, which, for brevity-fake, I will make B= o. Therefore the fubftitutions to be made will be x = y +p, x = Y + p, and the equation will be ap = — ay — py, that is, ny = — y, a tranfcendent curve, and which depends on the logarithmic, PRO BEE BMY. _ 44. To find the curve, the area of which is axy + ba y; where tle abfcifs is x, and the ordinate y, as ufual. Therefore it ought to be /yX = axy + dx°y°; and therefore yi = Axy + ayx + ch'a + ebiy'); or, making a — 1 = mM, it is myx + any + chy li 4 ebx®y’— “y = o. To feparate the indeterminates in this equation, we may make ufe of the method of $ 33, putting «x = uo ef" and TSE wee whence ° Caml Cu=2 + —1 é=2- dei we Pen 147278, and ) = 1-27, Now, making 3 SECT. III, ANALYTICAL INSTITUTIONS 305 making the fubftitutions, we fhould uu an equation much compounded, ‘and which would require a very long calculation. To come, then, to the point with brevity; refuming the equation /yx = bay + axy, put «°° = g, whence the equation will be /yx = bg + axy, and therefore yx = dj + axy + ayx.. This fuppofed, I make ufe of the method of § 24, in the form of which I write the equation thus, axy X — x ni — oa = dj; then I put - ix i - * Sa va and then integrating, it will be = == =; =. (ARI 3) = p. Wherefore, making the neceflary fubftitu- tions, we fhall have the equation = "i —— = dj. Now, to exprefs the quantity af by the affumed quantities p, g, we muft confider, that x y = = 4% that is, y wine ge I—@ » xz e ‘ Bans Cc, Macon gras. But we have allo = y; therefore Cui mc Pal Ne ? we @€— ae + ae ; I AO e Or: = ops ‘and, luftly, 8 mt gtr Se Ki gsr Ee Then male € 7 ° DI ® ry | € = . VICE ATA È ing this fubftitution inftead of x°, we fhall have the equation ap*—4+4 5 DUE = ZAC ee b 7 i 3 Ae - porn eee . » ° oasi i — j that is, ag—@t46 p = bgt 64% Gs and, by integration, q eé—ae+ae ae ac e—ae-+ap—I. a Ger C—aetac. mm be — bae + bac fi per rro aa | ae = ac x p gg gary rupi x q a 8; which is the equation of the curve required, It is plain that this curve will be algebraical, at leaft when the quantities a, c, e, fhall be rational; and, on the contrary, it will be tranfcendental when one of thefe fhall be irrational. I fay at leaft, becaufe,. making: a, c, e, ra- t.— e. tional, the curve, however, will be tranfcendental if e = €; or if a = ; ; ; — & or if ¢ = 1, and at the fametime @ = 1; or a = 0, and allo e= 1 And in feveral other cafes, which it is not neceffary to enumerate. Vor, Il | Re rd Ce 306 ONAL Y TI CAIL- A NSTIT.U TAI BOOK IV. Of the Redu&ion of Fluxional Equations, of the Second Degree, ee. 45. WHEN the differential equations of the fecond degree are fuch, that the rules here explained for integrations may be adapted to them, as well in cafes of feparate variables, as in thofe that are mixed; nothing elfe remains to be done, but to apply the faid rules, and thus, by means of integration, to reduce them to firft differentials ; therefore there is no need to add any thing further about this matter. If, after the formule thus reduced to the firft degree, the indeterminates will not then be feparable, as is often the cafe, nor fhall be in any wife conftru&ible; it is not the method that is in fault, by which the fecond differences are refolved, but rather that by which the firft differences are managed. Therefore we ought to employ our induftry about the redu&ion of the differentio-differential equations, that, by the rules already taught, they may be made fit for integration, which may be attempted feveral ways. 46. One way will be, to make ufe of the common expedients of vulgar Algebra, by tranfpofing the terms, by multiplying or dividing them by fome quantity, and fuch like. But, firft, before any other thing, it is neceilary to recollect, or to know, if, from paffing from firft to fecond fluxions, there be any fluxion that was taken for conftant, and what it was. And befides, that as, in the integration of firft differences to finite quantities, there is always added fome conftant quantity ; fo, likewife, in the integrations of fecond to firft - differences, fome conftant quantities fhould be added. This fuppofed, det us proceed to fome Examples. | E X- ELI a SECT. IV. “ANALYTICAL INSTITUTIONS. 307 EXAMPLE 1. 3 i : by __ 20y% + asj . PA ge TL ego Let this equation be propofed, rs A: in whichx = Y/xXx ge is the element of.a curve, and is fuppofed conftant. I write it thus, ei da 20yX + AXY. | aa As î is conftant, the firft member will be integrable, even though it fhould be multiplied or divided by any funétion-of y; and I obferve, that the fecond would be fo alfo, if it were divided by 2Wy. Therefore I divide the whole fi ; È b M è» è 2 so * è i ; ‘ 1 4 equation by 2Vy, and it will be 22% = 2 22, and, by integration, it ac / 29 Der i tan. Les ROC MER : will be — = 4XWy + aiva, which equation is now reduced to firft : mt+i x 20" | | | fluxions. In the integration I have added % for this reafon, becaufe it is conftant; and I have multiplied it by aa, to preferve the law of homogeneity. EXAMPLE I. n va — yy pre Let the equation be-f = » in. which yx. is taken for a conftant. I BLY — 2955 ; * e È A A IPR 33 Jyxa? and, by integration, becaufe of yx being conftant, it will be SS = I II e PI Ù = gas multiply it by 2y, and it will be aff = + RYN. Rre es EX i 308 ANALYTICAL INSTITUTIONS BOOK IY. / EXAMPLE III. Let the equation be f = ——* SITA , in which let x be conftant, and % the element of a curve, that is, V xx + yy = # Therefore, becaufe x is conftant, it will be DB = #4; and therefore, fubftituting the ve of in the equation, ayy ui — ayyiu JUÙ — yuu ; and, multiplying by 2y, itis afy = it will be fi 5a Poe A pat 2yy wit cin 2yy@i , ia . oe PICCIONI Zz a that si fy = 3 and, by integration, 2f/y = pre + NXX Sax Again, after another manner. Inftead of 4, putting it’s value in the equa» tion, it will be f = SHe te: and, multiplying by 2yy, it is afy = 2yyxà nie ayy3 — 24995 ; - aye + 2993 — 299} i . : ey ens that is, 2fy = oe ripe ; and, by integration, aa sie xx es) Si iva n + nix, __—_—___—__m—mÉ__tt_ttt_euc@cmuu[@—ttetiòrt i EXAMPLE 1V. Let the equation be ax = Heals, in which Jet x be conftant. Multis plying by x, and dividing by «, it will be = = yy + yy; and, by integration, becaufe x is conftant, it is axl + Ax = yy. Now, if we fhould make the affumed conftant A = a, we fhould have axle + 6X = yy 3 and, proceeding to integration, axlx = 20). EX. SE sera ES SECT, 1%, ANALYTICAL INSTITUTIONS, 309 EXAMPLE Vi: Let the equation be f = a in which # is the little arch 9%) | or element of a curve, 7 is given by x and y, and no firft fluxion ts yet taken for conftant. I divide it by y3x3, and multiply it by 2, and it will be af QX) Ul + 2yuux — 2yxun _— afyti aycxynu È DYYUÙUAI e 2yy ica | — = ee, or eS; and, by 343 pratyit yee plat 4 ‘ vii | uit integration, 2/ D dl ot eel er ee E ne + hag But it may truly be faid to be a thing impoffible, to make ufe of this method in fuch equations, in which the quantities are intricate and compounded, when we do not know the integrations pretty nearly before-hand, which we are to make. Wherefore I fhall go on to other methods. 47. In the folution of problems, when we are to proceed from fir& to fecond fluxions, it may be much more convenient not to affume any fluxion for conftant, though we are at liberty to do it: that we may be able the better, when the formula is under our infpection, to determine that to be fuch conftant, by which the expreffion may be much abbreviated, and moft readily integrable. The Examples will beft make this method to be underftood. | EXAMPLE I. ; sh a ae poi, Jor po * eo ove L i a È Let the equation be f = eee 22., which may arife without having taken any fluxion for conftant. To fhorten this formula, I confider, what may be that fluxion which, taken for conftant, will deftroy two terms of the bomogeneum comparationis, and leave only two in the equation; and I find’ there may be two, ‘that is, xy and —. Therefore make xy = c, and taking the difference, it is «) + xy = o. Then multiplying by x, it is xx} + xx == ©, by which means the fecond and fourth terms of the homogeneum difappear | me : e è ° i )3 es x out of the principal equation, fo that we fhall have f = ra . But, as it is 4 wy + 310 ANAL VTEC Ade I Wee LIO RI, BOOK IVe ey + xy = o, itwill bey = — i whence, by fubftirution, f= — re Lei f= — IE 22353? ° agi uao E es But xy = €, and D + k& Z6cx 3 and, daily. Ja = Lane and, by integration, therefore f = A Petes YY bak ua I) + ax OS Aa e ieee vel (FX 2 mm HOF, came to the Sf. eA pe i appli When I ca | rye 53 ome AVA ; 4 * 3 equation f = cere , we might more briefly have gone on to the integration, ° © e ‘ . a Se : sa re : by multiplying by x, and difpofing it thus, fx = 243 aay? where, becaufe a e È Ser < I de i xy 1s conftant, it will be /fx = — — — —- +7, as before. 4x% 4Avxyy Now let us make conftant the quantity ear Such a fuppofition giving ul —l = o, and alfo — xyx + xxy = o, takes away the fecond and third è ° ° ? 4 si Li ESSE: as + ax terms from the principal equation, and changes it into this, f= nt d, ai ia ra 3b whey ; HEE multiplying by x, it 1s fx = agree the integral of which, (becaufe of si i i i Lor — conftant,) will be found to be ffx = — — — x ie LA ce» dle Axx 440)) ft, as above. 48. But, to know nearly what fluxion may be taken for conftant, it may be obferved, if, in the propofed equation, there be two, three, or more terms, which, being multiplied or divided by a quantity which is common to them, they may be reduced to be integrable; then making the integration, their integral may be taken as conftant, and fo proceed in the manner fpecified. If not always, yet fometimes, at leaft, we fhall fucceed in our attempt. DS 4A) — apx + ay I refume the equation f = ror , and obferve, that the two terms x%) + xx, being divided by x, will become xy + xj, which is an integrable quantity, and that it’s integral is xy. See, then, upon what account we may take this quantity for conftant. In like manner, I obferve, that the — KB HK two terms x*y — xyx,, if they be divided by — xxy, will give us ———— 3 an integrable quantity, the integral of which is =; therefore the fluxion = might alfo be taken as conftant. | For SECT.IV. ANALYTICAL INSTITUTIONS, ZII For example, let the formula ay x xj —yX = 3Y%° — yxy? — xxy° be propofed, in which the variable 2 is any how given by y. I difpofe it thus, MAI. + yyzy? = pay + gyx* — xxy?, and obferve, that, if the bomogeneum comparationis be divided by yyy, it wil o dr x — ay ‘ i I he integral of big | which is ibis Therefore I take - for conftant, and make «È zee Sane Rares At vX ee me XI ° ‘ thence dn = o. Whence the propofed equation will become xi + yy3y° = 0, thatis,z=— ye and, by integration, becaufe of — y conftant, it will be 2 = # + % 49. In an equation of the fecond degree, when either of the two indetermi- nates are wanting with all it’s functions, and only it’s firft or fecond differences enter in the formula, any how compounded and raifed to any dignity; the integration, or reduction to firft fluxions, will always be in our power, by help of a fubftitution, This will be, to make the firft fluxion, which is flowing or indeterminate, equal to a new variable multiplied into a conftant afumed fluxion, or which may be affamed at pleafure, in cafe that no other be ap- pointed conftant. For example, in a given equation, let x, at firft, be fup- pofed variable, and y conftant; make x = py, and taking the fluxions, on the fuppofition of y being conftant, it will be % = py. Making this fubfti- tution inftead of x, and the equation being managed by fubftituung the values taken from the equation x = py, it will always be reduced to firft fluxions, Or, perhaps, it may be more convenient to make the firft fluxion of the variable, which 1s wanting in the equation, equal to a new indeterminate, multiplied into the firft fluxion of the other. Making the neceflary fubfti- tutions, and having a due regard to the fluxion which, at firft, was taken for. conftant, we fhall have the propofed equation reduced to firft fluxions. EXAMPLE I, uy >] in which # is fuppofed conftant. Make, therefore, x = pi, and by differ. Let us take again the equation of the firft example of § 46, a = ey Cc - encing, * = pw, Then, fubftituting this value, we fhall have di = € 312 ANALYTICAL INSTITUTIONS. BOOK Iv, EIA hae is DE 100 and therefore BS — 2ayp ha Fr ti. , ur te ae ereror i — 2ayp + apy, | beta which equation, divided by 2/y, is integrable, and the Ae is —~— me x ac” de py te; a E = PVI +g. But p = —, therefore ——— = aXyy+gh m+ x20” . | SRE I SRN I A REMERON ESS, a EXAMPLE IL Let the equation Be Syyxx = — un, where f is given by y, is the element of a-curve, and yx is the fluxion taken for conftant. Therefore I make 7 = pyx, and, by differencing, it is % = ypx; and therefore, making the fubfti- tutions, it is RL ig = — y°ppx°, that is, fy = — pp. Whence, by integra» ILA ee + JV ‘ tion, 2ffy = — pp + 2m But pp = — > Wherefore, making the fubftitutions and the redu&ion, we fhall have x = ————+—______ I vamp 1 =2yffj Now I reduce the fame equation by means of the other fubftitution mene tioned before. Make, therefore, x = pù, and & = p% + pi, whence % rca x — pu Ux See, era Making the fubflitutions, the equation will be Ayppyin = But the fluxion yx is affumed as Sallie whence we fhall have yx + yx = 0, that is, x = — 2, ork=— o, And, fubftituting this value again in in the equation, it will be fppyyy = L + È. This fuppofed, we may go 7 sE on, and make di + > = -L, whence py = g, and therefore big = = or fy = an tear th integration, {fy = ae +m But qq = ppyy = ee Gy ME yw tien a aK Therefore i I fy | "a A ; . e ” whence we may derive, as above, x = A a E E X- un “Lar SECT, IV, ANALYTICAL INSTITUTIONS, 313 Be AMP bak I refume the equation of Example III, $ 46, fyxx = xx + yy — yy, in which x is conftant ; and make y = px, and therefore y = px. Making the fubftitutions, it will be fy?xx = xx + “gf — ypx 3 and, making x to vanifh È 3:97 4 già | by it’s value = 2, we fhall have n Fei w — Li; that is, fy3yy = yy + ppyy — YpPyp. And, dividing by yy, it will be fy = ig. And, by dara SS) =— DD — = ee m. And, da of p, fubfti- tuting it’s value = —, itis f{f7 = — a — as + m, that 1s, 2/fy = vie + yy È mairet tene (e a, Leone + 2m; and therefore x = VA I 50. If, in the propofed equation, no fluxion has been taken for conftant, one may be taken at pleafure, and the operation may be performed, as is done at § 48. As, for example, the equation of Example V, § 46, being given, in which no fluxion is affumed as conftant, that is, Dx = xyuit 4 yuuX — gti, (putting yx inftead of 7,) if X be made conftant, it will expunge the term guuX, and the equation will become fy*yx* = y4 — yi. Now, to reduce it, we muft put # = px, whence % = px. Thefe values being fubftituted, we fhall have /y*yxx = ppyxx — yppxx, that is, 719 = ppy — JOP i 3 which equa- tion, in order to proceed to integration, I write thus, /y’ i= = ppy x — MI sui Pp T herefore, integrating by the method of 9. a4 aforegoing, {fy = — i Pom: and, ae ae the value of PS) a + m. If % be taken as conftant, the term yx%% will be expunged, and the equation will be fyiya? = xyi + yi, and therefore we muft put x = pu, x = pu. Thefe yesh being fubftituted, we fhall have fyy x p°% = py + we that ne D +9 } . . . stai ii “RION Dagro bi, oe ee a then, by integration, i will be {fy = >» + m; and reftoring the value of p, it will be /f9 = = ann ae Vou. II, Sf 51. To uu 314 ANALYTICAL INSTITUTIONS, BOOK IV. 51. To affume at pleafure any fluxion as conftant, in equations wherein there is none already fo taken, may make fome equations fubje& to the method of $ 49, which are not fo already, becaufe of having both the indeter- minates finite quantities. And this by afluming fuch a fluxion for conftant, as may make all the terms to vanifh, in which is found one of the finite indeter- minates, thofe only remaining which include the other. For example, let the equation be 4? — xyy = yxx + 2xyy, in which no fluxion is taken as conftant, If we make x conftant, the firft term of the bomogeneum comparationis will vanifh ; and if we make conftant, the laft term will vanifh ; and, in either cafe, there remains only one of the indeterminates, Therefore, appointing x to be conftant, the equation will be x7 — ayy = axyy, eo Li x « ° e ° © 3 Put 7 = ag ,y= £* , and making the fubftitutions, it will be x? da di | oy aa Qxppxx x app ; : SEs EE eee ® N ” Fo ee then, by integration, it will be dx = —laa— pp + Im, and therefore x = , that is, aax — ppX = axpp, or 772 ° Ga = Bp A And, inftead A si : ay i È . maxx of p, reftoring it’s value -, it will be x = —,thatisa=_—_—,, 4 x rea mete BAX mm aay | EP, OY MX” = aarx? — aay", | 52. But when the taking at pleafure a fluxion for conftant, does not fucceed- in eliminating one of the two finite indeterminates, or if the conftant fluxion be already fixed, fo that both the indeterminates remain in the equation ; there is no general merhad as yet difcovered, how to proceed further. ‘The methods here explained may fometimes have their ufe, as alfo the ufual expedients of common Algebra, fuch as multiplication, divifion, &c. As, for example, in the equation xxyyy = xx — xx, which, being divided by xx, will be yyy = “a, and therefore is integrable, (fuppofing y to be conftant,) and the integral is tyyy = — + my. Sometimes a fubftitution may make the propofed equation within the reach of the method of § 49. And, indeed, the equation xv = D + + ypyy, which is not fubje& to the canon of the aforefaid article, will however be fo, if we make yy = 3; whence it will be x”%X = £ + 38. 53. Wherefore, in cafe that in the equation there fhould be already a conftant fluxion, it may be of good ufe to change the propofed equation into another equivalent SECT. IV, ANALYTICAL INSTITUTIONS, 315 equivalent to it, in which no fluxion is conftant. To do which, let there be a general equation y = px, where p is a quantity any how given by x and y, and let x be conftant. By taking the difference, it will be f = Pica Baba ag DIS i then, by differencing, without making any conftant fluxion, it will Dop ini Wherefore, the value of p being fubftituted in the equation XX 3 = px, we fhall have y = aS. So that, in any propofed equation in dii X is conftant, inftead of y, if we put it’s value, so, it will be changed into another that is equivalent to it, in which there is no conftant fluxion. But, becaufe often other more compound fluxions may be affumed as conftant, or have been at firft aifumed, it may be of ufe to render this method more univerfal. Let us take this general equation y = mpx, where p is likewife given, in: any manner, by x and y, and m is any function whatever of x or of y, or of both together. Let mx be conftant; then, by differencing, it will be j = mxb. Bot 7 = =~ ; and bydifferencing, without afuming anyconftant, it is p= miy — mi) — MYX ° . ° ° ° I oi. ST 3. gem azar Wherefore, fubftituting this value in the equation ¥ = mx}; inftead of p, we fhall have y = equation, in which mx is conftant, if, inftead of ¥, we put it’s value now found, it will be changed into another which is equivalent, in which no flaxion is conftant. wey ni e % DI i mY n ie Wherefore in any propofed. mx After this manner equations being made complete, that is, fuch as may have- no conftant fluxion, in proceeding to the reduction, we fhall be at liberty to take: that for conftant, by the affiftance of did we may beft attain our purpofe... EXAMPLEE"L" Let it be propofed to reduce this equation, xxy= y# = n) + da in Fac: x is conftant. Therefore, inftead of j, putting it’s value. = ——, (for Sf2 in 316 ANALYTICAL INSTITUTIONS. BOOK IV, in this cafe m = 1, and m = 0,) it will be xxy — 9° = ax} — ayx + way — x}X, in which no fluxion is conftant. Whence, making y conftant, it will be found to be xx + ax + ax = py; and, by integration, xx + ax = yi, which is an equation to the hyperbola. i,’ [carie] EXAMPLE I. xyy + av DA DAX n XXX è ; } wy Ty +9% Ade = in which the fluxion Liye aa + Xx 9X is aflumed as conftant. To transform it into another, in which there is no conftant fluxion, becaufe in this cafe it is m = y, the value of $ to be fubfti- Let the equation be — . yx) — yy) — yYX : . A ; tuted will be a and therefore the equation is — > — LO VI), — NEI) — AKIVX AX — XXÀ : i ; O a SES ca SIE et be reduce this, making xy a conftant yey aa + xx 1 x Ù by bs 6 a? ® ee xy fluxion, in confequence of which it will be xf + xy = o, that is, —y = =; i ° e ° ° ° ) ° e ay x X ni then making the fubftitution, itis — 2 —HX +e +1 +5 = 0°, y y Ps aataa * oN ew — aax ; : >. _ 7,44 + xx : that is, — 7 = ad and, by integration, — /x = /—— == Jy, Here I fubtra& ivy, becaufe it is a conftant quantity. And, taking away the ° I. i e | ° ° logarithms, — = a, that is, «°9 = 4°% +. EXAMPLE III. Let the equation be — = ~ da = 2, and ya a conftant fluxion. Therefore, inftead of ¥, I put it’s correfponding value, , and it Jay — YF — WH % i cu > = LTD, in which there is no conftant fluxion. Wherefore, taking 5 conftant, it will be «x = xXx + yy. Which equation is the cafe of § 49, and therefore it’s reduction is known. gECT. IV. ANALYTICAL INSTITUTIONS. ZIE 54. The method explained in the foregoing Seétion, at § 24, may be alfo of ufe in differentio-differential equations, by proceeding nearly in the manner there purfued. Here is the practice in fome Examples. EXAMPLE I, | ne 3 es I refume the formula of the fir& Example of this Section, I a te n u Cc ty ~in which 4= Sixx + yy is affumed conftant. It will be oe = 29X + xy. AC o by; I prepare it after the following manner, -—-+— x x = , where I ob- 29 ac' x 2y ferve, that the two quantities under the vinculum are integrable, by means of the logarithms. Therefore I make = + = = 2, and therefore ix + /Vy. = /p + lu; (Ladd Ju, becaufe of « conftant,) that is, xy = pà. Where- fore, in the propofed equation, inftead of — + a , fubftituting it’s value. 772 «n ] è « è ‘ ae pu aula, . du _ by gu SLI 4, and, inftead of x, 1t’s value +, it will be — = —-, OF = E: ’ 9 9 NZ; : sip = 3 Pp ttt — 2. pot a oe ; and, by integration, b-p = y But p = 2 and a » DY 5 , za sen si 2.49 2ac mt i X 2ac ie oe —__*_.., as in the Example quoted above. | mt X 246 therefore, laftly, bi + xy = EXAMPLE: IL Za er td oe ‘ È È È | Mar ty = =, in which 9x — xy is conftant, Let the equation be — o DA The fecond fluxion x, divided by the conftant xy — xy, will give us an e e e e. co integrable quantity, and therefore I write the equation thus, Toga 9 318 ANALYTICAL INSTITUTIONS, BOOK IV» x xX ye — xy xx + yy rr grano yx — xy is fammable when it is divided by yy; therefore I prepare the —xe __ ayy CD Ge yy x Vine + yy - But I obferve, that, in the fecond member, the quantity equation according to this method, and it will be E x TRE Make aoa = p, and, by integration, — = ». Whence, I | 7; ny ayy yx — ay xx + yy X V cx yy making the fubftitution, we (hall have , from whence we can expunge x or y, by means of the equation a =p. Expunge x from the fecond member, by putting it’s value py in it’s place, and we fhall have vai i» ie RO » in which x is a conftant fluxion. In refpect of this, the method will be of no ufe, for we fhall have p + m = o, and confequently every one of the terms of the general differential equation of the firft degree, except the laft, will be infinite. But, in this cafe, the reduétion is eafy, without any further artifice. 1 write the equation thus, «j = ayyx. Now the integral of the firft member is xy — JX, that of the fecond is 74yyx. Therefore the equation is x) —- yx = tayyx sb dx. | | ss 56, The SECT. IV, ANALYTICAL INSTITUTIONS, 321 56. The fecond canon comprehends all thofe equations, in which the fum of the exponents of the indeterminates, and of their differentials, is the fame in every term. Suppofing x and y che two indeterminates, and x to be conftant, thefe are reduced to the cafe of $ 49, by putting x = c,and y=ce“;c being full the number, the logarithm of which is unity, and a, ¢, are new aderi minates. To fhow the method, I fhall take the equation ax my DOSE Py? 7? {= bey "7 'x5y°7? = dî, which, though it be but of one dimenfion only, and of three terms only, yet the method is general notwithftanding, and will ferve for any number of terms and dimenfions, if the conditions be obferved. Therefore I make x = c°, y = ct; it will be x = e“; and, becaufe is conftant, we Mall have oa + cuit = o, that is, Z = — #4. It will be alfo J = <¢? 4 e" and Pa e NI ou + ti + ti, Buti = — uu; there fore j = c° X 7 + 245. Wherefore, thefe values being fubftituted in the propofed equation, it will be at7 "7a? x Fe GP + be" TH? x t+ am? =? + 200. And, becaufe in this the indeterminate z is wanting, we may proceed by the method of § 49. Make 4 = 27; it will be «= zi + 2f. But 2 m= — me = — ze}; therefore 7 = — £_ — zii. Wherefore, fubftituting thefe values, we fhall 5 have at” 2??? & Foe? + ot TA KEE el? — —_ EL % me ai, Ore 3 MIR) eb ep Cee eee a En + 27, a differential equation of the firft degree. From hence it may be feen, that the propoled equation do have been reduced at the peas by putting a = hie and y = e, For example, let the equation be xxy — yxx = yyy. To bring this to the è ‘ . ° — 2 .. —1 .. os È . canonical FT I write it thus, xy “xy—y xx = ¥. Then it will be Sigs IO kei me emia DA Haro, 1, 9g = 2. Wherefore, thefe values Sai fubiticured in the differential canonical equation, here before found, we fhall have the equation reduced, tt XL 265 — LU ‘z2i = AE zi; D zi + ceti az zzi Of —— sm —— TI die; ti é You. Il. Ts | If , that is, 220 — 2ztti = — US 322 ANALYTICAL INSTITUTIONS BOOK IV, » i ; et tii — È z If we proceed on to the integration, it will be —— = -—, and therefore, by integrating, # + — = —— + f, (where f is a conftant to complete y integrating, — = - ; p the integral.) that is, #2 + 2 = — / + ftz. But, by the fubftitutions, rer. oo ana EY. ST Pee ae __ Y 2 _ 29% <= 3, KSC, y=, it will be ST N oe a , and therefore 2 = EPL wherefore, fubftituting the values of ¢ and z, we fhall have Da ‘omg pi 57. The third canon comprehends all thofe equations, in which one of the two variables, whatever it may be, together with it’s differentials, always makes in every term the fame number of dimenfions. But we muft here diftinguith two cafes. One is, when the differential of that variable is conftant, which forms the fame number of dimenfions. The other cafe is, when the differential of the other is conftant. +7 «| 2 42 Ox x Saale = x” j, in which the fum of the exponents of x and * is the fame in every term. P and Q are any functions of y, and x is conftant. As to the firft cafe, let the canonical equation be Pr”, ‘ : u RT ! To reduce this equation, make x = e, where alfo ¢ isa number, the logarithm . . . O . ° . . de of which is unity, and # is a new variable. Therefore it will be x = e°4; and differencing again, making x conftant, it will be c°4 + ¢ wi = 0, that is, 4 = — i. Thefe values being fubftituted in the equation, we fhall have py"*? + Qiay”t37? ale ge which, becaufe it does not contain 4, will be under the canon of § 49. Therefore I put # = zy, and it will be 4 = ay + 2¥; but é=—-% 2 — 2°)°; therefore we Mall have 2j + zy = — 227y; and thence j = 239) — 9 % & equation before found, it will be py *? ty Qany™ 4 = —z mal m+ i Va Wy" t's; and, dividing by y”*7, it will be Py + Qu) = — 2845) Wherefore, thefe values of % and ¥ being fubftituted in the | miri .m2+2 di n] — x '%, an equation of the firft degree. Therefore we might at fir& have made x = ¢/”, and thus have reduced the equation at one ftroke. 3 | For ‘tins 4 ni q i d Il ; È i Dt E AO IRE I O ANTE rea P PA N : hi i ' k Ti SECT. IV. ANALYTICAL INSTITUTIONS. 323 For example, let the equation be 2axxy + axxy = 2XXyY + ax), in oe which let x be conftant. Put x = SP, and therefore x = 376%; and x = e Si X zy? + zy + yz. But x is conftant, and therefore 2299 + zi + 9% = be Whence: ee ae, Now, the values of x and x being fubfli» | tuted in the equation, we fhall have 242°)" 4 azyy = 22)? + 2y}; and, E pg + 27 X fubftituting the value of ji, it is 242%" + azy X a am )2 — CY i A di rei se ; . ° ° ° 22, that is, dividing by jy, 42°) — azd = — 2%, or ay = ZE — 22 pe i 4 a I ha . And, by integration, ay = — — + +. Laftly, reftoring the P n Ù Pi zy < value of z, which is given from the fuppofition made of x = 9, that is, z= > we fhall have the equation reduced, ayxx = xxyy — axxy. 55. As to the fecond cafe, let the canonical equation be P eer + Li mil I metro i) ‘ $ Qu "gay o8ti = 47%. in which let y be conftant, and P, Q»: any Laica: of y. & x ® ° os ss po 0 Put, as above, x = e“, and therefore x = c°%, Xx =e’ + chum. Make the i Ù e . ° ° . I E ce fubftitutions in the canonical equation, and we fhall have py”* at Qing” pt me ye tt — un” "a, which, becaufe it does not involve wz, is fubje& to the canon of § 49. Therefore I put « = zy; and, as y is conftant, it will be # = zy; and then making the fubftitutions, we fhall have pyr! + Quy tt = tit 4 2°7'"z; and, dividing by 9”, it will be Pj. + Q2r; sta getty + 2's) an equation of the firft degree; which might have been Sy reduced at once, by putting, as above, x = e°”. For an example, let the equation be 2xy = ax — yx, in which let y be 3) x zy + ay + yz. But y is fuppofed conftant, and therefore 7 = o, and conftant. Therefore, putting # = e°*, thence x = zy x of? shel des ae ‘thence % =” x x 229Y iy + zy. Wherefore, making the fubftitutions in the propofed equation, we fhall have 2zyy = azz) + 49) — 2%YYY Ys and, dividing by y, it will be 22y = azzy + a3 — zzyy — y%, which is a differ- ential equation of the firft degree, Ttz To 324 | ANALYTICAL INSTITUTIONS, BOOK IV, To go on to the integration, I divide the equation by 4% — yz, whence it is 2 CPN i: % 2) & ie FRA i eg DI aga Set tgs A = zy. And now, if you pleafe, making . ‘ A — I ag ufe of the method in $ 24, by integrating, we fhall have ee + #; and, laftly, by reftoring the value of z = HI we fhall have the equa- tion reduced, yx + x) = ax, where the conftant # is neglected, which was introduced in the integration. This example has ferved to fhow the application of the method; for other- wife fo ae operations would have been unneceflary. Indeed, the equation itfelf, 2xx = ax — yx, might have been redu ced in an inftant, by only tranf- pofing the term yX, and writing it thus: 2xy + yX = eX; for, as y is conftant, the integral of the fir member is yx + x, as plainly appears. 59. To what has at already faid, concerning differentio-differential equa- tions, in which no firft fluxion was taken for conftant; another metiiod may be added which is more univerfal, and Ei will lerve for all iuch as are compre- -m+iI hended under this canonical formula, z MIL dj ="; in which 2 is any how given by the fun&ions of x ber Ye To reduce this, appoint the fluxion a for conftant, where g is any how. given by the functions of # and y. Then make È = p. Now, becaufe = is conftant, it will be, by differencing, gx — xg = 0, that is, X = =; or, inftead of = s writing it’s value p, it will be x = 49. Befides, make y = up, and taking the fecond fluxions, fuppofing p conftant, as being equal to È, which is conftant, it will be j = #9. Therefore, in the canonical equation, fubftituting the values thus determined, inftead of x, x, y, and j, we fhall m-<$<- 1.741 have the equation i ia sel” and, dividing by % yeti; Vi i m+. pi it will be 27! q "4 + È — = "i, or q 4 ne ana: And, PA 241 ett by integration, cara + gf = art a and therefore 4 = DO : MmHPIXEZ SECT. IV, ANALYTICAL INSTITUTIONS 325 nt me xe? a Bot ge so. se Then 2 me x P x “ i I i / mt oe se © ives : ui g + Mt 18 » an equation reduced to firft fluxions. 60. Concerning this laft equation we are to obferve, that, if the quantity 2 be given by x and y in fuch manner, that to the quantity g fuch a value may be affigned, alfo given by w and y, that the indeterminates may be feparable in the equation, and therefore that it may be conftrudtible, either algebraically, or, at leaft, by quadratures, we may have the curve, on which the differentio-. differential equation depends. And, becaufe the values are many which may be affigned to g, the curves may be many alfo, and every value of g will fupply us with a different curve, either tranfcendent or algebraical, which will fatisfy L 2aayx)y + aaxj3 xy o . ° ° e . CX applying this to the canonical equation, it will bem = 1, z = — j therefore h a L bh : b Ayre a Te N the queftion. Let the equation be —; = aayy. Now, the reduced equation is 2 DE mi X99 +2 2, I take q=&3 it will be = st VT + 2¢¢, that is, re = xxW xx + 2g; the integral of which plainly depends on the quadrature of the hyperbola, and the curve will be tranfcendent. | 61. In paffing from firft to fecond fluxions, either we affume no fluxion for conftant, or we affume fuch an one as is moft eligible, as faid before. Where~ fore, in finding the integrals of formule of the fecond degree, becaufe we know what fluxion had been fo taken, we know alfo how to proceed, and the rules for it have been explained, But there are an infinite number of problems, which require fecond fluxions, without our knowing what conftants are involved in the formule thence arifing. It often happens, that we cannot arrive at the analytical expreffion without the afliftance of the conftants; and likewife, it fucceeds fometimes, that the equation . may be refolved without recurring to the conftants. Thefe two cafes, therefore, ought to be examined, and we fhould feek for fome criterion, to diftinguifh one from the other. And, becaule examples will perform this better than any thing elfe, I fhall take this following. It is required to find fuch a curve, that it’s abfcifs, raifed to any dignity, may be directly as the fecond difference of the ordinate, and reciprocally as the fecond difference of the fame abfcils. Therefore we fhall have this analogy, 7 % 326 ANALYTICAL INSTITUTIONS BOOK IV. ‘ La sta. And confequently 2x°& = aj. In this equation I find the fecond differences both of the abfcifs and of the ordinate; but I cannot know what conftant was aflumed,.or whether any conflant was affumed or no; fo that I cannot know what courfe I am to purfue. I fay, in the cafe of this equation, that no poffible curve will fatisfy the Problem, fince we pafs from firft to fecond fluxions, without the affifiance of conftants. On the contrary, the conftants being determined, we may find curves that will felfil the conditions of the Problem, but they are infinite in number, and different in their nature, as varying by the change of the arbitrary conftant which is aflumed. To diftinguifh one fpecies from another of thefe equations, we may make ufe of the method, or canon, which will arife from the following Examples, and which will ferve in all fuch cafes, wherein the Integral Calculus does not for- fake us, | EXAMPLE LI [ Let this equation, STA = xy" i = yj, be propofed. I fay, this is oné of thofe formule to which we may attain, without taking any quantity by way of a conftant. Let the variable z be any how given by « and y. The demonftration will be made general, as far as that can be done, by taking the fluxion = as conftant, in which g is a function of « and y, any how combined. Wherefore I put a = p; and, becaufe the firt member of this equation is conftant, the fecond p will be fo too. And, as it is x — qb; if we pafs to fecond fluxions, it will be x = gp. Now make y = wp; and, taking the fecond fluxions, on the fuppofition of > being conftant, we fhall have y = vp. Wheretore, fubftituting, in the principal equation, the values thus determined, there will arife the equation mel ma ..m+I qh ep I PE ig m4 a pop 4-) —*_— =a up ; and, dividing by p7*t, an equae D tion will arife which is free from the unknown quantity fp, and from it’s miti. Raum ety più 4 «ff Wy Taking the fluent, therefore, by fun&ions, that is, SECT. IV, ANALYTICAL INSTITUTIONS. 327. by the rules before explained, not omitting to add the conftant g, it will MAI pert be rap Pi 3 acco which equation gives us « = 2 X Mm-l K&B I én +1 lm + I q + gm +g . And, becaufe y = ap = = , making the neceflary fubftitutions, we fha'l have the equation reduced to it’s fimpleft ftate, that is, | a SST EI » zx pr mb I Viene di Tet I: e St eel From the foregoing manner of operation, we may deduce the following Coroliaries. oe I. The quantity z being determined, if the laft equation can be conftructed, even by quadrarures, fo that 1 may but be executed, it is plain that infinite curves will agree to our formula, which will change their nature by changing the affumed conftant fluxion a And every value of the quantity g will £upply us with a new local equation, either algebraical or tranfcendental, II. Although, if the value of the fymbol 9g be altered, different curves will arife; yet it is certain, that, 1f we make the additional conftant g = o, we {hall always have the equation y = 2x. In which cafe, it matters not what fluxion = is taken for conftant; becaufe, the given quantity g vanifhing, the variable g alfo vanilhes. III. Here, then, is a token by which it may be known, that we fhall arrive at our primary equation, without affuming any fluxion as conftant, and that, in fuch a fuppofition, it’s integral is 2X = y. For, recalling to our view the + -m+1 Im. % .m.. . . > expreffion 2" 'x"% +1 xy — yj = o, and again differencing the integral 2x = Y; without affuming any conflant; thence we fhall have ex + zx = 5; if, by means of thefe two lalt equations, we fhould make to vanith out of the principal formula, firft y, then x, with their functions, we thall find +im.. m: mb1 mim. moi 24 £ ee: EE eg nt TE MS tI salti oR and yy ay %z MI Psa —J ey Ze Op IV. The 328 ANALYTICAL INSTITUTIONS, BOOK IV, IV. The primary formula being managed as above, and the equation being mb i found reduced to the firft degree, that is, y = 2 xg rae aa oo we fhould pafs on to the integrations, which fometimes will be out of our power, according to the various values of the exponent m of the fraction z given by x and by y, and of the quantity ot which is taken for conftant. However the reft may proceed, the aforefaid aes being determined in infinite particular cafes, the local equation of the curve is alio difcovered in finite terms; when we proceed to the firft, and thence to fecond difi:rences, keeping fill the conftant La which our principal formula will prefent us with. But, changing the conftant, different formule will be found. I can affure nothing further, but this is very manifeft, by turning back again the fteps of the Analyfis, iis fame thing happens by taking the firft fluxion 2 for conftant. For, making the operation according to the method, (which i fhall omit for the fake of I i i i : a : ; mt t brevity,) we Mould arrive at the reduced equation x = £L— — ZL xmg + £ dae >» vf in which it may be obferved, in like manner, that, making £ = o, it concludes by reftoring the equation x = > , expreffed by fir& differences. VI. Affuming fome limitations that are more fimple, that is, # = I, x Son we, and gt wy 1 we make'ule of the conflant a in Cor. IV, the | I ‘ x im ICE i i ; formula y = = X 9 ve + gm a £! tT will be changed into this following, 4 = ax xx + 2g, which admits of analytical integration, Now, making ufe of the expreffion contained in Corol. V, that is, x = weg na X mg + x arifing from the afumed conftant ae and keeping {till the limitations of NAM IL = v./%2 is not integrable without the help of the logarithms, and coe ME) gives us none but tranfcendent curves. iS 12 sty, and =) Merestelults theexpreflion = y, which Therefore \ SECT, IV, ANALYTICAL INSTITUTIONS. 329 Therefore it is plain that we may arrive at the differential formula of the (me wt 2... x .mMmb I «ae SE A i , fecond order, 2° bali + — xy i, = y , without taking any conftant ; in which cafe the integral 2% = y will take place; or, fixing for conftant the fluxions = x ae for example-fake, and then the fame integrations will be made as before, that were found in thefe {uppofitions. EXAMPLE IL Let us take the equation xx = ¥ + yy. I fay, we cannot arrive at it, without taking fome conftant, except in one cafe, in which it is m = — 1 To fhow this plainly, I fhall manage the formula in the manner following. Firft, I take x for conftant, and thence X =o. Then — — = y, and by integrating, / du = 98507" x = cl. Make e’ = 2, it will be y/ = /z, and therefore y = 2. and, inftead of y, fubftituting this value, we fhall have # =. But c”? = 3, therefore x = %, and x = z = ¢’; and therefore = y, an equation to the logarithmic. Secondly, I propofe to inveftigate how it may fucceed on the fuppofition of another conftant, y for example, whence ¥ = o. I make x = sy + oy where s is a new variable, and ¢ a given quantity. I go on to fecond differ- ences, and it will be « = 5y ; and, making the fubftitution, it is x”; 4 = JJ, or 773 è ite fc © n ZAR ‘o : . Ge l'ala: awl ayer a ° . Si “west But y = Tm then si + ci=x «3 and integrating (omitting g met fasta to add a conftant), tss + cs = ———, or S++ ¢= cranio MO MORO iii: Mi 1 +1 ; ; iii . s$=5FexXj=}j ara + 065 therefore i LU | 20 PF Logi + cc | * See § 46. EDITOR, Vor. IL | Uu , I proceed 339 | - ANALYTICAL INSTITUTIONS BOOK IY, I proceed to inquire if poffibly the logarithmic curve may be concealed under the laft formula, which being found above, in the hypothefis of x being conftant, it may likewife have place in the other fuppofition of y being —m| 1 n o n . 2 conftant. Making ¢ = o, it is neceffary that the equation Ve. rn peg fhould be verified, or elfe 2x7”?! = =m +1 x an And, that the equa- tion may be found, the fame quantity — m +- 1, both in the co-efficient and the exponent, ought to be = 2; for this to obtain, it follows, that it muft be NI = = I. ° VA = oe . e è Therefore, in the formula x x = Y + yy, by limiting the value of the exponent to m = — I, we come to a differential equation of the fecond degree, without affuming a conftant, the integral of which is the logarithmic expreffion — = y. In any other cafe we could not obtain the forefaid expreffion, without fixing upon fome infinitefimal quantity of the fir& order as a conflant. EXAMPLE III It remains that we fhould propofe a differential equation of the other clafs, at which we cannot arrive without affuming a, conftant. I refume the problem: To conftru& a curve, in which any dignity whatever of the abfcifs may be in a dire& ratio of the fecond fluxion of the ordinate, compounded with the inverfe ratio of the fecond fluxion of the abfcifs, "The equation is dx"% = aj. Make x = gf, 3 = up; and perform the operations, as in the firftt Example. Taking the fecond fluxions, we fhall have x = Dj, j = wp; and, fubftituting thefe values, it will be 2x°g = au; and By integration, fox" 9 au tg. Buty = “p= ae then'ap = = /bu"4 = sul. Making g = 0, in this cafe, whatever be the value of the fymbol 9, 3 it gives us a different curve, if alfo we do not put the exponent m = o, by which the hypothefis will be deftroyed, and the problem changed. The fame thing may be faid if we make conftant the fraction la. and from 8 hence ER Pol ee Se ne SECT. IV. ANALYTICAL INSTITUTIONS: 33I hence we may conclude, that it is not poffible a differential equation of the firft degree, without the benefit of a conftant, fhall reftore our formule, when it is differenced again; for, if it were fo, it would be manifefted in any affumpuon of a conftant ; and alfo, the analyfis evidences the contrary. _ PROBLEM I. 62. The radius of curvature being given, any how expreffed by the ordinate of a curve, to find the curve itfelf. As, when the curve is given, to find it’s radius of curvature, it is called the Direct Method, or Problem of the Radin of Curvature, of which we have treated already ; fo, when the radius of curvature is given, to find what curve it is to which it belongs, is called the Inverfe Problem of the Radii of Curva- ture. Wherefore, let the radius of curvature be = 7, and be any how given by y, the ordinate of the curve; and we may take any one of the formule for the radii of curvature, which we pleafe ; but, firft, for the curves referred to a 3 ; Pass uy —, in which x is conftant, and & is the element focus; as, for example, os # bina of the curve. Then we fhall have the equation r = —*——-; or elle, it being | LIS om yy $$ = Xx yy, itis si = yy ufe of x MRO, 3 I $$ = NK + Vs becaufe of x conftant, and r = Soe + To reduce this equation, I make ufe of the method of $ 49; and therefore I make i = px, whence s = px. Then, making the fubftitutions in the equation, it will be 7 = » or elfe ca an me ae and then, by inte- gration, becaufe 7 is given by y, it will. be 7 = DB ue 0s, citi sir tees pa. r terso x ieee? At? ; therefore the curve will be —"— = / + 5, an equation reduced to firft fluxions, becaufe, r being given by y, the integral fe. may always be had, at leaft tranfcendentally. Uue : Another 332 ANALYTICAL INSTITUTIONS. BOOK IV. Another way. I write the equation, r = os 53 a sa = , in this manner, — = 5° — IX}. — yay proceed the ordinates BE of the curve required AEC, I draw BF perpendicular to EB, terminated at the radius of curvature EQ; and, making BF = p, EF = gq, by the known ne of the normal and fubnormal, it will be g = a, peg x da i a a And, by taking the fluxions, on the fuppofition of x being conftant, it will be ¥ = “ine . And, mak- ing the fubftitutions in the principal equation, it will be sl = 57 — pa + a Pan fcr ae therefore . = qqX —yyp + pyy. And, becaufe it Is ge "a , it will be a = 9qV + ppy — ypd. But, becaufe of the right angle EBF, it is pp = 99 — yy, and pp = gi — yy. Wherefore, making the fubfitution, we fhall have au = 29) — y¢3 And, multiplying by y, and dividing by gg, it willbe = = Epi, and, by integration, it is ye +hb=Z., But ua therefore » + d ee LE ion q iv * Sr Vik + yy It may be done thus more fimply, by avoiding fecond fluxions. Taking the infinitely little arch EC, let the chord CED be produced, to which let BD be perpendicular, Now, it we make BD = Ps by what has been faid at $ 115, Se&.V, BI, QE = r= È » and therefore wh = p; and by integra- tion, becaufe r is given by y, it is Pest du = sforp = J TE x +53 Vik typ by the place now quoted. rs + db pra +) ‘ na actual integration, (omitting the conftant 4 for greater fimplicity,) xe a ——_ th r 2 "fa Saar 2.2 2% + Ve Vas te erefore 2°X* + 0° ax + 83°, that is, dy = ax, which is the logarithmic fpiral of Example V, $ 128, Book II. Inftead Let it be y= IV ua + bb + 56; then it will be f 9 Then, from the point B, (Fig. 149.) from whence See ne Ig È e det io SECT. IV. ANALYTICAL INSTITUTIONS, 333 / = QE, EH; that is, y.p 3: = » % and therefore x = Da, or = 5 and by integration, ST + b= ip. Make z= y, then fe +b5=/ yx Vex +jj then by/xx + yy = mx, and therefore by = &V mm — bb, which is the logarithmic fpiral; and, when 6 = 4, m = aa + bb is the fame as the above-cited. and by integrating, /y = Ip + i= * that is, y= a oa a = 63. For curves referred to an axis, the formula of the radius of curvature is $s | ba Bae putting x conftant ; and therefore the equation will be 7 = — 5 I put y = gx, whence ¥ = gx; and, making the fubftitutions, it is r = Sv rie ly cae ca rn iX 1+99È Tet 2" ; and, inftcad of %, putting it’s value 2-, it will be r = atta ili sa g — 47 as . that is, 2- = = a - And, by integration, [= +h = ———. But 1+99)% r Vi + 99 J J x = —; therefore /— + 6 = ————-; q + 3 Fi r We re yy Let r = te; then it will bef" +b = —=—. And, 2aa 4yv + aa 7 Viù yy : ; eee «da 2 by actual integration, omi'ting the conftant 4, it is nf tr sah ha vy + aa Vik + jj that is, 2yy = ax; and by integration, yy = ax, which is the parabola ot the firft Example, $ 122, Sect. V, Book IL Inftead of the radius, let the co-radius be given, which make = 2, the formula of which (fuppofing & to be conftant,) is “re. Then = maf and making y = gx, ¥ = gx, and making the fubftitutions of thefe values of ma ° ° 4 oe 4 r" : ; © sh o J and x“, it will be cali post A that 1S, 2 DL — di 3 1499" tion, Sf _ +52 Yi + gg. Whence, if 2, or the co-radius, be in fuch And, by integra» manner given by y, as that / a be a logarithmic expreffion, we fhall have a * This equation, as well as the fubfequent work, would have been clearer and fimpler, if m had been put for the conflant number of which the logaritom is è, ÀADITOR, differential Mii: 334 ANALYTICAL INSTITUTIONS, BOOK IV. differential equation of the firft degree expreffed after the ufual manner; in any other cafe, it will be exprefled by logarithmic quantities. p 3 3 Let it be z = ET, we fhall have the equation /—— ua ei + ge | ae 4y3 + aay — JV 1 + gg. And, by actual integration, (omitting the conftant 4,) it is [aoe ee hy , and therefore —2.— = ——. And, fubftituting N yy +4aa WI +97 Ig AI the value of g, it is 2yy = ax, and, by integration, it is yy = ax, the fame parabola as before. 64. In the fecond place, let the radius,: or co-radius, of curvature be any how given by the abfcifs x ; it is plain that, in chis cafe, we cannot make ufe of the fame reductions we did in the firft, becaufe we cannot have the fluents f--—, or =; if r and 3 are’given by 4, r % x Taking, therefore, the formula of the radius of curvature, in which x is ; CIO sa 3 3 Ù conftant, that is, + È for curves referred to an axis, (for, in thofe referred tf to a focus, the radius, or co-radius, cannot be given by the abfcifs,) it will be FAT NE * e e vr = £24+222., and therefore, in the fame manner as before, I put Weston i = ge, fy = ggx4; and, making the fubftituti eva | whence j = 9x, yy = qqgxx s and, making the fubftitutions, r = “27 74'*3, I — AKG : ; z —g ° : x — 9 is ok . Rit peor and, by integration, /— + b = ——~., which is an equation reduced to firft fluxions ; becaufe 7, being given by x, the fluent i may always be had, at leaft tranfcendentally. And, fubftituting the value r PO x — y My sf—-+s=- —=—. of g, itis f—— + Vea Let it be r = 24/442 — 2ax; then it will be f——— +5 = ; 2V 44a — 24% i =, And, by actual integration, omitting the conftant 4, it will be 4 aa tes i sl N 4aa N: 20% — 7 i 7 ti IT ——=-. And, by fquaring, and reducing to a common i ci i i denominator, it 1s 444X%X = 24XXX — 2axyy = O, that is, y= XV ——, an equation to the cycloid of § 131, Sed. V, B. II. Inftead SECT, IV. ANALYTICAL INSTITUTIONS. 335 xk bh yy. I meio id And putting, in like manner, y = 9X, itis j = 9x, y nas. qIXX ; and making Inftead of the radius, let the co-radius be given; then 2 = + 99XX the pes Re inftead of y and yy, it will be 2 = i that 1s, a = i, and, by integration, f+ E Bole Seger a the Zomogencum comparationis is the arch of a circle ; dite fot; if the co-radius . But the integral of fhall be given in fuch manner, as that Sf — is alfo the arch of a circle, and thefe arches fhall fo correfpond, as to be to each other as number to number, we fhall have the equation reduced to firft fluxions, and exprefled in common quantities. cae. 2 ut the OV 2ax — “x ni + 99 integral of the firt member is the arch of a circle, the tangent of which is SO cnx — AX a Let z = 24/244 — xx; then it will be / ; and of the fecond, is the arch of a circle, the ori of which is g. Then it will be SESTO se — q= Ti therefore y = tga — x 2 an equation to the fame cycloid. PROBLEM II, 65. The radius of curvature being given in any manner, in a curve referred to an axis, to find the faid curve. The formula for the dalia of curvature is > » making s the element of the curve conftant; whence the equation will be r = on Call the tangent of the curve 7, and the fubtangent p. It will be oe = #, and, differencing in the hypothefis of s conflant, it will be 7 = agi that is, 7 = ee x Da Wherefore, making the fubftitutions, it will be 7 =" But, becaufe | we 336 ANALYTICAL INSTITUTIONS BOOK IV, : ; Nate” a SR ty ave have p = 2°, and ¢ = 2, it will bex = Bo = —. Then, fub- J 7 y J ° : : ts ftituting thefe values in the equation above, we fhall have r = ae . But Vit — yy i 2B a yi 2 i ci pp £ = yy; therefore r= — - pZEVI=- )); f eae - sii The firft member of this lat equation is in our power, at leaft tranfcen. dentally, becaufe r is a function of s. Then, in the fecond, the indetermi- nates will be eafily feparated, if we make g = = » by which we fhall have a . $ very fimple equation, — = VI = 99 In the formula r — Pes 2) if, inftead of ¢, we had taken it’s value Y/pp +, o img we fhould have found r = st i and, making ca = 2, we fhould alfo fu dp 3 + % have had a very fimple equation, “A The two differential quantities —— and are the expreffions of de the element of the arch of a circle. Whence, if the integral / — fhall be algebraical, or fhall depend on the logarithms, or on higher quadratures, the rectification of the curves required, and the value of the radius of curvature, will fuppofe the quadrature of the circle. But, on the contrary, each of them may be algebraical, if the integral / — agrees with a formula of the circular arch. Retainin i RE de g one of the two equations, for example the fecond, — CUTE J Then, fubftituting this value into the equation, we fhall have y = FLL 1-4zz X Vi+zz becaufe i = 2. — Vp + yy, and p = 2, it will be i = yi + Le . Now, it being i = 24/1 + 2a, we (hall have alfo if = XxX + Jy = EL, and therefore i = —. Make SECT. IV. ANALYTICAL INSTITUTIONS. 337 Make the given radius of curvature r = 1 + ss. Then the equation x Ss Nate race I +23 =} from whence we ob- TE Sn =— will be changed into this, TREE tain z = s, and therefore r = 1 + zz. Subftitute this value in the equation Bz Secs types Deir and it will be ‘meee... And, by mteoration i I+zz X ViI+za | J V1 + 2% pcs 5 a iid Omitting the conftant, it is y= VI + zz, whence z= Wyy — I. Then, becaufe I retained x = a 5 Ut will be finally x = —— » an equation of | | yout the curve required, on the affumed fuppofition of the radius of curvature. It’s conftruction depends on the quadrature of the hyperbola. sy — ys I take the formula of the radius of curvature, — = +=, in which no XS firft fluxion is conftant. I difpofe the equation thus, amb Es 86 3g | x J 5 r The integral of =. — — is 2) — li, which I make equal to 7p. Then it will be ia ~ È = Dal and “— = p, and then the equation will be — E Bais Sead BF Xs RIA otc eb 3 ahiage 5 Call Sa But p = =-, and 7 ISO + vy therefore 4° “N Consi e e ° © 5 . ‘ DOLL.) And, fubftituting this value, it will be — = —£ an equa- 7 4 Jy yy | “a 1+ az* But x = 2j, and si = xx + yy = zzji + II. Whence = a : 5 Therefore, after whatever manner we operate, the integral /— will always be brought, either to the rectification or quadrature of the circle. Let the co-radius 4 be any how given, to find the curve. Take one of the three formule before, that, for example, in which + is taken for conftant; that $ q Is, Pi at in which it is put 5 = gj. The radius will be r = : n a : ee us. x i : $ sq uo and, putting this value in the formula, we fhall have id x qeVqq = 1 But i = gj, and x = jWgg — 1. Whence, making the fubftitutions, it will be — orig - zi But 4 is given by s; therefore, &c. Here it may be obferved, that, as the integral /—— is equal to an expreffion 3 r of a circular arch; fo the other integral / — will be referred to the quadrature of the hyperbola, or to the logarithms. 66. By like artifices and expedients, or but little different from thefe, many equations, or formule, may be reduced to fecond differentials, which are exprefled by third, fourth, or higher degrees of fluxions. And, firft, the method of $ 49 may be extended, (yet within certain limitations,) to differ- ential equations of the third, fourth, &c. order. That is to fay, equations of the third order may always be reduced to the firft order, provided that either one or the other of the finite variables, x or y, is wanting in them. Thofe of the fourth order may be reduced, if, befides one or other of the two finite variables, x or y, one or other of the firft fluxions, x or 7, be wanting, together. with their refpective functions, Thofe of the fifth may be reduced, if both the finite variables, and both their firft fluxions, be wanting in them. Thole of the fixth, if, befides all this, one or other of their fecond fluxions be wanting. And fo on. Let the equation be x + xxj = x* + 94, in which & is taken for conftant. I make, as ufual, pv =, and therefore px = $, and px = j. Wherefore, making TAC ARIE n Bol SECT, IV, ANALXMIIGAL: UNS TIT OT 1 Op 8. 339 making the fubftitutions, we fha!l have xxp + x39 = x*+ + 94. But 9 = p°x*; therefore it will be p + xf = xx + ptxx, an equation reduced to the fecond order. Make further gv = p, retaining x as conftant, and therefore gx = p. Then, by fubftitution, it will be gx + px = xx + ptxx, that is, Gtp zw + pix. Batx= i therefore ¢ + p= - + ee, which equation is now reduced to firft fluxions. Let there be a fluxional equation of the fourth order, ¥ + x} — Xx} = 0, in which let x be conftant. Therefore I make px =, and thence px = 7, and px = j, and px = }. ‘Therefore, making the fubftitutions, we fhall have p + xp — xd = 0; an equation which is a cafe of the foregoing Example, and which therefore we know how to manage; and which will eafily be reduced to firft fluxions. | The method of § 49, found fome time ago by S. Count Fames Riccati, was now firft known to me; but the foregoing application, as alfo the fecond inverfe Problem concerning Radii of Curvature, I have learned of him only fince the fecond Tome of the Commentaries of the Inftitute of Bolonia is fallen into ny hands. And, indeed, fomething too late for me, becaufe I was now at the clofe of the impreffion of this my Work; nor could I take the advantage of the other learned Differtations, neither of P. Vincent Riccati, fon of the aforefaid gentleman, nor of S. Gabriel Manfredi, therein inferted. Therefore it muft fuffice that I have juft named them to the readers, that they may there find them, and be improved and inftructed by them. 67. Having fhown the aforefaid application, or improvement of the method of § 49, I fhall go on to other equations, and to other expedients. Therefore let the equation be pyjj = pxxj — apxxy — pxxy, in which p is any how given by x and y, and now the element of the curve, i, is taken for conftant. Becaufe _ $ is conftant, it will be xX = — yy; then, fubftitutin& this value inftead of «xx, it will be pyjj = pxxj + 2pYjj — pxxy, that is, ftriking out the Ra Wo ORE ani es I + pxx), or I Lula And, inftead of jj, putting it’s value — xii, it will be Le =a Lu <-. And lattly, integrating by the logarithms, /p = /y == /x —/i, i being conftant; and 3 therefore p = =~: which equation is reduced to fecond fluxions. i q x x2 7 ., et 340 ANALYTICAL INSTITUTIONS, BOOK IV. Let the equation be box — 352% — 43% = o, in which » is any how a ; ra è ee given by x and z. Let us affume the following fictitious equation, B”2"x = conftant; where m, 2, r, are unknown exponents of powers, to be determined by the procefs. Then, by taking the fluxions, we fhall have rhe xT de co ° cose co co Jo è es ° ° = e 772 I . jin T..f n] noi "Se mb” "53" = 0, which, being divided by 6 x x , will be reduced to rb2% + nbX3 + mb2xX = o. This equation being com- pared, term by term, with the principal equation propofed, we fhall have r=1,%= — 3, m = — 1; wherefore, inftead of the fictitious equation b” x" 3c" = conftant, we fhall have the true one, a = conftant, which is the integral of the propofed equation. Alfo, by the way of the logarithms, we may obtain the fame integration. I refume the equation 42% = 352% — 53% = o. I divide it by 43%; it will be n. _ 2s ~ a = ©, and by integration, /x — /z3 — Jb = toa conftant logarithm. Therefore =i is equal to a conftant quantity. ADVERTISEMENT. 68. I sHart finifh thefe Inftitutions with an Advertifement, which is this; that the ingenious Analyft muft endeavour, with all his fkill, in the folution of Problems, to avoid fecond fluxions, and much more thofe of a higher order; and this by means of various expedients, which will offer themfelves commodioufly on the fpot. Such artifices may be teen, as they are made ufe of by famous Mathematicians, in the Problems of the Elaftic Curves, the Catenaria, the Velaria, in that of Ifoperimetral Curves, and in others of this kind; the folutions of which may be feen in the Leipfic A&s, and other works of this nature : by which a learner may acquire fuch fkill and dexterity, as will be very beneficial to him, END OF THE FOURTH BOOK. AN AN A DDITFON TO THE FOREGOING ANALYTICAL INSTITUTIONS; Being a Paper of Mr. Colfon’s, containing a Specimen of the Manner in which Two or more Perfons may entertain themfelves, by propofing and anfwering curious Queftions in the Mathematicks. HE Manufcript of this little piece appears to be a firft draught, and only a part, of what Mr. Co/fon intended to draw up: yet, I perfuade myfelf, it is {ufficient to point out to the readers of it the way in which feveral perfons may amufe themfelves with propofing and anfwering Queftions of this kind. Thoie readers, who with to fee more of this, may find it in the VIth Section | of Mr. Colfon’s Comment on Sir Isaac Newrton’s Fiuxions. ‘They may alfo, with a little attention, propofe and folve, in the fame manner, any of the Queftions in thefe Volumes. ‘© A Problem is fuppofed to be managed between two perfons, the Queri# and the, Refpondent : the Data are fuch numbers or quantities as are given or fupplied by the Querift; the 4fumpta or Quefita are fuch as are aflumed or found by the Refpondent. ’ PROBLEM LI ““Querist. I give you three numbers, 4, 5, and 10; Irequire a fourth, — Responpent. I affume x to denote that fourth. Q. So that, if from the product of this into the third, the firft be fubiratted, | R, Then. 342 AN ADDITION TO THE FOREGOING ANALYTICAL INSTITUTIONS, R. Then the remainder will be denoted by iow — 4, Q. And if the remainder be divided by the frft, 104 — 4 > R. The quotient will be denoted by —— Q. The Quotient will be equal to the fecond number. . . 10% — R. Then the equation 1s e iss = 5s whence 10x — 4 = 20, and 2 log = Bay and x = ei PROBLEM II. «_-_666@GE-— <--> Sect. V. Of the Conftrudion of Loci which exceed the Second Degree. § | | Page 210. Tao ways to conftrudt the bigher loci tee a 207 221. The firft manner is by finding an indefinite number of points ibid. 222. Lhe ordinates at right angles to the abfcifs ome 208 223. An example of de eferibing the curve by points —_ ibid. 224. The fign of the axis is ambiguous in even powers —— 209 225. To find 1 where the curve cuts the axis eee See ee ibid. 226. The more points are found the better a n ibid, (227. Lo find when the curve can have an afymptote —— ibid. 223. Afymptote found by changing the equation n 210 229. Cautions to be obferved in finding afymptotes —_ Ibid. 230. To find whether the curve be convex or concave towards its axis ibid. 231. Further to determine the forms of the curves, with examples 211 232. Examples to determine when the ordinates are real stall ia 214 233. Lo de:ermine the fame when the equations are irreducible —— 218 234. which may be done by finding points ne . 219 235. An objeclion obviated sri ls SRP ta ibid. 236. Example for de cher Tg the forms of the loci from the equation 220 237. Another example for the fame purpofe —— o 221 238. Example of the curve called the Witch —— 222 239. Another example, being the Conchoid of Nichomedes ——— =. 2239 240. «Another cafe of the fame curve oe —— 1285 241, A third cafe of the fame curve PRE ii 227 242. The metbod improved of defcribing curves by points e 228 243. improved by the conic fections RR | | 231 244, e by parabolas of higher degrees see one ibid. 245. Ti he firft cubical parabola conftructed "RVORIETESITI prio 232 246. The fr n of the fourib degree confiructed civili ibid. 247. —— of the fifth degree conftrutted oe 23% 248, —— of any degree ee ne 234. 249. TANA of other fucceeding parabolas —_ ibid. 250. Reduplication of the curve produced by fquaring the equation | — 251. Conftruciion of hyperboloids RENT as ia 236 252. of higher byperboleids —— —_ 2 226 ACE Otber byperboloids confirucied ee —__ ibid. 254. Ob= 352 TNDEX.,==VOL, I BOOK I, SECT, VI. Page i Obfervation on the forms of the firft paraboloids n 237 255. of higher paraboloids and byperbolcids in ibid. i 256. Curves of feveral terms conftructed, in three cafes nn 238 257. Example of the firft cafe —— n —_ ibid. 258. Another example n a een 239 259. Another inn —_— ee ibid. 260. The co-ordinates may make any angle uu i 240 261. The fecond cafe of curves conftructed waa 241 262. The third cafe conftruéted, with a general example ne ibid. 263. To feparate the indeterminates when involved —_ ibid. 264. Example of the conftruction of thefe loci ate! mee . 242 265. Another locus conftructed seni a ibid, 266. An obfervation —— —— —— 243 ‘ 267. Conclufion of the examples —— | | ibid. Secr. VI. Of the Method De Maximis et Minimis, of the Tangents of Curves, dI of Contrary Flexure and Regreflion ; making ufe only of Common Algebra. Page Di To find the maxima and minima of quantities. by comparing the i ; equation with another which hath equal roots ena . 269. To find the fame by multiplying the equation by an arithmetical pro- È gres fe 245 © M ono. Tangents and perpendiculars to curves, how found [RARA 247 Ù 271. Exainple of this as | Pah, oe 272. How to choofe a convenient progreffion — = 249 | 273. The problem falved another way aL ibid. 274. Points of contrary flexure and regreffion, what, and bow found . 249 275. To diftinguifo contrary Pexures from regreffions, and maxima from minima imac 251 VOLUME INDEX,--VOL. If, BOOK Il. SECT. Ie VOLUME II. BOOK Il... ‘ THE ANALYSIS OF QUANTITIES INFINITELY SMALL. ns Sect. I. Of the Notion or Notation of Differentials, of feveral Orders, and the Method of calculating with the fame. ì: Variable quantities, what ca oom 2. Conftant quantities, what. Notation of each —— 3. A fluxion, or differential, what —— on 4. Fluxions, bow expreffed. A proof, from the incommenfurability of fome quantities, that there are infinitefimals of feveral orders ee 5. Fluxions of the higher orders, bow reprefented —— 6. Turorem I, with its corollaries; foowing the exiffence and fome pro- perties of infinitefimals of feveral orders —_ . Tneorem II. Other properties of infinitefimals e Vot. Il. Z.2 20. . Tueorem III. The verfed fine of an infinitefimal arch is an infinitefimal of the fecond degree; and the difference between the right fine and the tangent of that arch is an infinitefimal of the third degree —— . Coroll. 1. Lz an infinitefimal arch, the tangent, arch, chord, and right fine, may be affumed as equal — ant . Coroll. 2. If the radius of an infinitefimal angle be alfo an infinitefimal, the arch and its right fine will be infinitefimals of the fecond order, and | _ the verfed fine will be an infinitefimal of the third order — . Coroll. 3. Lnfinitefimals of the firft and fecond order in curve lines . Scuotr. If the firft fluxion of either the abjcifs, ordinate, or curve, be taken conftant, the fluxions of the other two will be variable. The fuppofition of a conftant firft fluxion fhortens and facilitates calcu- lations —— —_ — . The foregoing conclufions are not affetted by the angle of the co-ordinates . A LEMMA. What is the ratio of angles to each other ae . Turorem IV. 16. Coroll. Some properties of the involute, evolute, and radius of curvature ; ne need steed . Tnrorem V. 18. Coroll. 19. Coroll. 2. Properties of three per- pendiculars to as many points in a curve, infinitely near to each other, and of the angle at the middle point — ipa Page 1 1A ibid. THE- 3554 INDEX.e-VOL, II. BOOK II, SECT, II $ , Page 20. Turorem VI. The difference of the verfed fines of two equal infinitefimal arches of circles, the diameters of which differ only by an infinitefimal, is an sig mal of the third order | —— 13 at. Turorem Vil. 22. Coroll. Properties of infinitefimals when the curve 15 referred to a focus —— urna 14, 15 23. SCHOLIUM I A difficulty obviated Sa ee 16 24. SCHOLIUM II. Some further obfervations on infinitefimals. ‘Two tm- portant circumftances to be confidered. Caution to be obferved in the ufe of them ao —_— ane ibid. 25. Rule to find the fluxions of fimple quantities conse 17 26. when the quantities are multiplied together — — 18 dI, for finding the fluxions of fractions, with examples —— ibid. 28. —-— for finding the fluxtons of powers —_ eee 19 29. Finding of fecond fluxions, or higher orders ee 21 SecT. II. The Method of Tangents. è Page 30. Finding tangents to curves by a general formula for the fubtangent 24 31. Second fluxions have no place in finding tangents ead 25 32. Several formula for the tangent, fubtangent, normal, Sc. a ibid. 33. The angle made by the tangent and Jubtangent may be found ibid. 34. The fame things may be done when the curve is referred to a focus 26 35. Example 1. Yo find all thefe lines in the parabola __ ibid, 36. Example 2. To find the fame in parabolas of all orders te 28 37. Example 3. Yo find the fubtangent for the Alonso byperbola, and all others, between the afymptotes —— 29 38. Example 4. To find the fame for the circle 30 39. Example 5. Zo find the fubtangents in ellipfes, and isaenbiias of all orders —— mn nia Ibid. 40, To find the afymptotes —— a a sn 41. Example in a general equation cn —_ ibid. 42. Another example to find afymptotes —_ ——— 33 43. When the angle of the co-ordinates is not a right angle —_— 34 44. When the curves are not algebraical but mechanical —— — 35 45. Example 2 the cycloid — — aa 36 46. Another way more general __ n 37 47. Example, when the given curve is a circle nn —_— 38 48. Example 2. When the given curve is a parabola — 39 49. The fubtangent found from the generation of the curve n ibid. INDEX—vVOL. 11 BOOK II. SECT. III. Page 50, st. Another example of this ore nine 49, 41 52. Another way n —— ro 41 53. More generally —_ i IO ibid. 54. Obfervation when the curves become right lines a na re ibid. 55, 50. Tangents drawn to /pirals ; with examples me 42, 43 57, 58. The formula of the fubtangent more fimple; with an example 44 59. Langents drawn from the generation of the curve —— 45 60. Particular cales of this —— a 40 Or. Drawing tangents to a curve by means of another curve n ibid. 62. An example of this in the ciffoid —— mene 47 63. Lhe fame thing done more expeditioufly —— ibid. 64. Drawing tangents to a curve from its relation to another curve ame 48 605. Example in the quadratrix —— sent 49 66. Drawing tangents to a curve from its relation to two other curves ibid. 67. Example in the logarithmic fpiral —— —— 50 68. A difficulty flarted in the bufinefs of drawing tangents, when the ex- preffion of the fubtangent becomes _ . ae 69. This may be removed by multiplying by arithmetical progreffions 52 70. This method confirmed by recurring to the firft principles of fluxions 55 ni. The fame difficulty removed in the conftruétion of curves —— 56 Secr. III. Of the Maxima and Minima of Quantities. § ° Page 72. The foundation of the maxima and minima, and their formule for ordinates sean cope 58 73. Applied to curve lines wea me —— ——e 59 74. The ufe of this method —— — 60 ns. Exemplifed in the circle eee a no ibid. 76, 77. More examples —— ae 61 78. To diffinguifb a maximum from a minimum —— 62 79. Another example —— momen —— ibid. 80. A difficulty removed —— _ ibid. 81. An example _— e meee 63 $2. A difficulty folved nen eo 64. 83, 84, 85, 86. Otber examples ne eee nn 64-68 87, 88, 89. Problems to find maxima or minima | — 68-70 go. To diftinguifh between a maximum and minimum —— 70 gi, 92. Other problems sm coma —__ 71 93. A problem with a conftruction renne a 72 Ze 42 ve SECT. 356 INDEX.—VOL, II. BOOK Ile SECT. V. Secr. IV. Of Points of Contrary Flexure, and of Regreffion. Formule for points of contrary flexure, or regreffion, when the curve is referred to an axts — bt when the curve is referred to a focus —— How, by thefe formule, to find the points required — To diffinguifb contrary flexure from regreffion i Of another kind of regreffion —— —_— 100, 101, 102. Various examples me dgr 104. Examples wib conftruétions ann — Secr. V. Of Evolutes, and of the Rays of Curvature. 105. Of involutes and evolutes —— re 106. Fundamental properties of the/e curves wer aT he 107. Another property — — bea Tae 108, 109. To determine the center of curvature of the involute wae 110. The co-ordinates may make an oblique angle —— 111. /be co-radius, what, and how to find it —— 112. When the co-ordinates are at oblique angles —— 113, 114. Other ways of finding the formula of the radius of curvature 115. Formula for curves referred to a focus — 116. Thefe may become curves referred to an axis coins 117. The fame otherwife —___= ia sota 118. Otherwife for the co-radius nio er 119. Thefe curves can have but one evolute essa ara 120. corollary n — cara 121. When the radius of curvature may change from pofitive to negative 122. Example iz the common parabola n mestre 123. To find the equation of its evolute 124. Lhe evolutes of algebraical curves will be algebraical and reGifiable 125. Example i the common hyperbola, and to all “setto and byperbolas 126. in the ellipfis, or byperbola rtl Foe 127. ———— in the logarithmic curve — | RELA 128 in the logarithmic /piral ‘stona ors i 129. ———— din the byper bolic Spiral — Te 130 in all fpirals in general — —— tas in the cycloid | eee 132. Points isp regrelfion of the fecond fpecies rr Si Se eines : 5 aes ae Ma een aa È I NDE X,eeVOL, II, BOOK III. SECT. Ti BO Onde JT, OF THE INTEGRAL CALCULUS. . 99f Sect. I. The Rules of Integrations expreffed by Finite Algebraical Formule, or which are reduced to fuppofed Quadratures. § | 1. To find the fluents of fimple fluxions, when multiplied by any power of the . vartable quantity —— —— ee 2 when multiplied alfo by any conffant quantity | 2. when both multiplied and divided by any powers of the unknown | quantity —— —— —— 4. A conftant quantity fbould be added to the integral —— 5. Lo find the fluents of complicate fluxions when they can be refolved into Simple ones —_—— — — ___ 6. — - if raifed to any power —— oy » sh except when the index of the variable quantity is a negative unit 8. In this cafe, we have recourfe to logarithms | —— o. Conftruction of the logarithmic curve —— —— 10. nother defcription of the logarithmic, with confeftaries —— 11. Fluents reduced to the logarithms, or logarithmic curve —— 12. The Notation of logarithmic quantities — 13. Lhe logarithm of a negative quantity — sits 14. Lhe logarithm of powers or roots mito ti mo. of produlis or quotients — — eee 16. Thefe fluents require alfo a conftant quantity to be added a 17. Some cafes in which the fiuents of fractions may be found —— 18, 10. When the fluents of other fractions may be reduced to logarithms 20. Eluxtonary expreffions prepared by reduciion — —— 21, Complex fractions prepared by fplitting them into fimple ones —— 22 when the denominator of the formula is the produc of equal and unequal roots —— cita —— 23. Reduction by a partial divifion — 24. If the roots of the denominators cannot be found algebraically, yet they may be found geometrically — 25. Some of thefe roots may be imaginary sn ee Page 110 ibid. ibid. III ibid. 112 ibid, Ibid. ibid. 183). 114 PRE 116 ibid. 117 118 ibid, ibid. 119 120 12I ibid. 123 124. 26. Fluents 358 INDEX™VOL. 11° BOOK itt, “SECT. IL § Page 26. Fluents reduced to the arch of a circle a —— 124 27. Formula reduced partly to a circular arch, and partly to the loga- rithmic curve -—— a 124 28. Radical formula which admit of algebraic fluents — 126 29. [Reduction of a formula with a general exponent —— 127 30 if that exponent were negative —— 128 31. Other algebraic integrals found _— —_ ibid. 32 more generally, with feveral examples —— 130 33. Other formula algebraically integrable —— oo 131 34. Formule /ometimes algebraical, fometimes logarithmical —— 132 35. Certain formule freed from the radical quantity by fubftitution ibid. 36. Other examples — —— —- 19 37. Formula requiring the rectification of the circle — +96 38. Formula containing two radical quantities freed from them by fub- Stitution a are —— 136 39. Conditions requifite in formule which may be freed from radicals ibid, 40. Rational frattions, having complex denominators, refolved into others 138 41 when the numerator is multiplied by any pofitive power of the variable quantity —-— —— —— 139 42 when the denominator is multiplied by any power of the unknown . guantity ce sc aaa eigen era 140 43. <1 convertible formula ——— ibid. 44. Certain binomials refolved into their real component parts — 141 45. Otber binomials refolved —— —— 144. 46. Binomials refolved into trinomials —— ao 145 47. The integrals of thefe formule may be bad by the quadrature of the circle and hyperbola —— —— See DAN 146 48. If not otherwife, by geometrical confiructions a ibid. 49. Trinomials refolved —— —— oe ibid. 50. Lrinomial integrals of cther formule obtained by logarithms and circular arches | eres rea cre 147 51 When the index is negative, reduced to the former cafe —_ ibid. Da When the numerator is multipiied by any power of the variable quantity ae Vai tee de "sr 148 53 When the denominator is multiplied by any power of the variable quantity —— eta ——— ibid. 54. Praltions in the exponents may be removed —— 149 55. Other fractional exponents changed into integers vi ibid. 50. Another formula integrated by the circle and hyperbola — 150 57. Obfervations on this —_—— la cece ibid. 58 IVben the exponent of the multiplier 1s negative —— 151 59, 60. When the other exponent is negative, or both are fractions = 152 61. Other cafes confidered —— eed —— ibid. 62. Integration of a formula ix which the denominator is a multinomial 153 63, ——— When INDEX+eVO Li ti. BOOR-IFI. SE CT. TL, 359 Page When the numerator. is multiplied by any power of the variable quantity —— — bane 154 64. Count James Riccati’s method of integrating fractional formule, of which the denominators are multinomials § 63. Sect. II, Of the Rules of Integration, having Recourfe to Infinite Series. § Page 65. Quantities reduced to infinite feries by divifion pene: 159 66. —_T__ by the extrattion of the fquare-root — 160 67. Infinite feries found by a canon —_ adagio? ibid. 68. An infinite feries raifed to any power by the fame canon pease 161 69. The logarithmic formula integrated by a feries } — 162 70. The fame more explicitly —— ee Perec ibid. at. 4 radical formula integrated by a feries — 163 72. Approximations by thefe feries —— E ego ibid. 43. Reference to James Bernoulli for certain properties of feries ibid. | 74. A general canon for the fluents of binomial formule —— ibid. Sect. III. The Rules of the foregoing SeQions applied to the Re@ification of Curve-Lines, the Quadrature of Curvilinear Spaces, the Complanation of Curve Superficies, and the Cubature of their Solids. Page ‘A A formula fer jinding the areas of curves referred to an axis RETTE ho: 76. for curves referred to a focus 167 77. for curves referred to a diameter when the angle of the co- ordinates is oblique si 78. A formula for the recitfication of curves, the co- ordinates being at right angles ee ae 79. when the curves are referred to a focus i 168 80. — when the co-ordinates are at oblique angles — ibid. 81. In each of thefe cafes to rectify the curve mala ibid. 82. A formula for the fluxions of tke fuperficies of a round folid ibid. 83. . of the round folid itfelf a 94 of the fuperficies when the co-ordinates make a given oblique angle 169 85 for the Solid in the fame cafe ibid. 86. How to proceed when the curve ts referred to a focus ae ibid, 87. Reduction . A fubftitution when the co- ordinates make an oblique angle . Another general example . The quadrature of a mechanical curve « —— ——- of the logarithmic curve . a ——— 0f the tractrix, reduced to the circle . ——_ 0f /pirals a , ———_—— of the hyperbolic conoid 2 . Reduction of a curve from a focus to an axis pier Pe | . Reduction from an axis to a focus » — non Example i a conic fection in general — A general method of this reduction, with examples The.quadrature of the Apollonian parabola, and of all parabolas Several otber examples, fome by logarithms, fome by infinite feries of the hyperbola ———k of the circle, by feveral feries — , — —_ of the ellipfis, by feries ‘ah Live ——T__ — of the cycloid, by feries cence —_—_——.——- of the concheid, reduced to the circle and hyperbola —-—_—_—— of the ciffrid, reduced to the circle n face ‘of the parabola, when the co-ordinates form an oblique CD coniare] angle —— of the parabola referred to a focus ee of the figure of right fines —— . Quadrature of curves by means of new Jubfitutions . Another example of this . The rectification of the Apollonian parabola, and of the fecond cubical | parabola —_—_ of the arch of a circle ———_————— of the arch of an ellipfis —— ——————. of the hyperbola — Ro —T——— of the cycloid aoa em —— of the tracirix . — of the ftiral of Archimedes, and of the logarithmic Spiral —— of the logarithmic curve of the Apollonian parabola, when the co-ordinates make an obl.que angle ——— of infinite parabolas and Raa n . The cubature of the cone — — —— of the /phere — — — of parabolic conoids of any order of the {pberoid (cea eccitazione i between the afymptotes DRILLED IIR SPIRA, AE Ne , e of the conoid generated by the logarithmic curve a Page ibid, TRI tbid, 172 175 176 178 ibid. 16 159 INDE X,=—=VOL. II, BOOK III, SECT. IV. 361 § Page 127. The cubature of the folid generated by ciffoid —__ 216 128 — by the tractrix neers om ary 129 —— of feveral forts of ungulas — ibid. 130. ——~ of a Segment of the parabolie conoid nn 219 131. Obfervation vu ibid. 132. Complanation of curve furfaces ; and firft of the cone —- 220 133 when the cone is fcalene mons me ibid. 134. of the {phere — —— —— sia a Ker of the parabolic conoid —_ —— 22S 136. of various parabolical conoids, which are quadr'able, and which are not oe -—— —— | ibid. 137 of the fpberoid me n uc 225 138 of the byperboloid =. —— | — aay 139 of the equilateral byperboloid aoe oe ibid, 140. Lhe fuperficies of the folid generated by the revolution of the tractrix 228 141. The fuperficies of an ungula of a paraboloid —— 229 142. ————— of the parabolic conoid, when the co-ordinates form an oblique angle - —_— —— — cee 230 143. Obfervation — ie —— —— ibid. r@mtd(q1i(019@<@‘ll‘Q eg, el 00 E RC Ee Sect. IV. The Calculus of Logarithmic and Exponential Quantities. § | Page +44. Lixponential quantities, what —___ —_ 291 145. of Several degrees —— ibid, a To find the fluxion of a logarithmic quantity — ibid, 147 of any power of a logarithm 232 148 of any power of the logarithm of any power ibid. 149. ————— of any power of the logarithm of a logarithm ibid, 150. of an exponential quantity aes 233 15! ———— of exponentials of the fecond degree __ ibid. 152 — of produéts of exponentials od ibid, 153. To find the fluents of logarithmic differential formule — ibid. 154. The integration of a general logarithmic formula —— 235 155. The artifice of finding the preceding feries —— 236 156. Integrals of logarithmic formula found by different feries ——. 237 157. found by quadratures | —— e ibid. 158. Exponential tormula integrated by feries —_ —_ 238 The fame thing done in a different manner —— 420 160. Logarithmic and. exponential curves confirucied, their fubtangents | = found, Sc. — — — —— 240 Vou. Il. 3A "161. Con- 362 - INDEX,-VOL, Il, BOOK IV: SECT. II, $ Page 351. Conftruétion and quadrature of an exponential curve tees 242 162. Tbe fubtangent found of another —— nee 243 163. Another exponential curve confirufted, and tts area found —— rid. 164. Variable exponents found, the ro of the quantities in the equation being conftunt cone ost —— ibid. 165, 166. Two exponential problems _ ee 24.4. BOOK. IF. THE INVERSE METHOD OF TANGENTS... . ‘a Page 1. Definition and illuftration —— alare 247 2. Further explanation of this matter. Two ways of proceeding in st 246 Sect. I. Of the ConftruCtion of Differential Equations of the Firft Degree, without any previous Separation of the Indeterminates. Page 3. Reduttion and integration of differential equations aoe 249 4. Other examples more compounded —— —— ibid. 5. Other examples of reduction to integrability —_ — 250 6. More examples of this reduttion i —— 251 7. Other examples ——- n es 2:53 8. Reduétion to logarithmical forms — —_ ae 26% o. Other expreffions reduced to forms of that kind —— ibid. 10. Other more complicate examples of integration —— 256. Seer. II. Of the Conftruction of Differential Equations, by a Separation of the Indeterminates. § | | Page 11. Example of the separation of the variables seen 257 12. The reduction of differentials by fubftitution ri ibid. 13. Some INDEX—=VOUL, II. BOOK IV. SECT. IV. 363 $ | | HT + Rage 13. Some ambiguities in integrations —— saa ih 253 14. Some difficulties in the choice of fubjtitutions imme 259 15. Differentials eliminated by fubftitutions —ae pei 261 16. The, fame example otherwife reduced == = ———— mm | 263 17. The feparation of the variables ; and defeription of the curves ibid. 18. More examples of the feparation of the variables I 205 19. Lhe variables feparated by altering the exponents —— 266 29. Separation of the variables by a canonical equation mo 207 21 without the canonical equation | AR 269 22. 4 canonical equation, or method, for fome fimple cafes | 270 23. A general method of fetarating the variables — 272 24. A tentative method of doing the fame, with examples 274. 25. Another method of Jeparating the variables, of ufe in particular cafes, : with examples | 279 26. Another method of feparating them in a canonical equation eer 232 27. Another canonical equation n 284 28. A reduction by the exponents n n ibid. Sect. III. Of the Conftruction of more limited Equations, by the Help of various Subftitutions. Page 29. The feparation of the variables in a general formula by fubftitutions bi E 308" in a more general equation 286 31. in an equation fitll more general ame ‘287 32. in four other equations 288 33> 34» 35» 3% 37, 38. Examples of feparation in more complex equations 289—293 39. Other fubftitutions for separating the variables in a canonical equation 295 40. Lrom the property of the fubtangent, to find the curve 297 41. From the area given to find the curve —— —— 299 42. A problem concerning parabolas cut at right angles by a curve 300 43, 44. Two other problems —— —— ~~ 304 Seer. IV. Of the Reduétion of Fluxional Equations of the Second I, Dearest, «dro, “u: : | Page 45. Rules for the reduction of equations containing fecond fluxions —— 306 46. Examples of paffing from fecond to firft fluxions ibid. 3A 2 47. Integration § Page 47. Integration of fecond né without afuming a conftant at firft 309 43. To know what fluxion may be taken for conftant 310 49. Reduttion to firft fluxions by fubftitutions — ZII 50. When no fiuxion has been taken for conftant, one may be iis taken at pleafure —— —— 313 51. By this affumption fome equations are brought under the ui a § 49. 314 52. Other methods fuggeffed for this — — —— ibid, 53. Reduction by changing the conftant fluxion o —— ibid. 54. Example by a method before explained —— 209 55. Reduction of fecond fluxional equations by a canon o 318 56. Integrations by another canon —— 22% 57 by a third canon reina rn 232 58. Second cafe of the canonical equation nn n 323 59. Another method, more general sc a 324. 60. An obfervation — 4 da 61. Difficulties in thefe reduchons, arifing from conffants ibid. 62. A problem in the inverfe method of the radius of curvature, when the curve 1s referred to a focus GE 63. when the curve is referred to an axis ee 23 64. when the radius, or co-radius, is given by the abfcifs 334 65 the radius being referred to the axis, to Jind the curve 395 66. The foregoing methods extended to equations in which there are higher orders of fluxions ae n | 428 °, 67. The fame Subject continued een ——— 339 68. Conclufion i a a 340 AN ADDITION a —— serena 341 | ee We Fee = TELE LE @@@e@ INDEX--VOL. II, BOOK IV. SECT. IV, ERRATA, ca < RI 4 tag ee Be et ae en a me Sty ere a ee ER RAGT.A. Nore. When the letter 5 is joined to the number of any line, it is counted from the bottom of the page. VOLUME L. {n the Plan of the Lady’s Syftem of Analyticks, Page, Line. xl. 11. After the word branch, infert a comma. In the Body ori the Work. AI.” 3.0. Dele as. 125. 7. Inftead of 2aacex, read 2aacx. And in the head-lines, on the right-band pages, from p. 209 to p. 223, inftead of Sect. IV., read Sect, V. VOLUME Il. Page. Line. 9. In fig. 11, the perpendicular to AC is drawn from the point G, inftead of E. FI; The Small letter i is wanting in fig. 15. 15. 4.5. Inftead of each, read one the, 16. 9g. Infeadof EG, read EF. 24. In the head: ee, inftead of Book I, read Boox is 64. 7.5. After the letter a, infead of —, read = 113% Inftead of art. 9, read 10. N. B. All ra articles from 9 to 22 are numbered too little by 1. Vasi 366 ERRATA. Page. Line, 125. 20. Towards the end of the line, after the word radius, dele the comma ; and inftead of adding, read added to. 189. 9.b. After =, infert the letter a. É 205. 8. Inftead of x, read =. 216. 6.5. After =, inftead of a, read 1. 295. 13. Ziftcadof in, read is. aoe ye = wp) i 1%. db. lh cad 0 Lit AA read o Ro A. > I Beat Bes LOA eee La | 339. 3. Inftead of gx, read qx. si N. B. The name of the city Bologna is in a few places printed Bolonia, as it was ; found in the Tranflator’s Manufcript, but I take it to be erroneous. ‘ EDITOR. 4 | o) il dA A LETTER i | AGE ESITO ER FROM PHILALETHES CANTABRIGIENSIS. Reprinted from the Gentleman’s Magazine for November 1801, N the Gentleman’s Magazine for November lait, pages 997 and 998, is a Letter figned Philalethes Cantabrigienfis, the delign of which is fo laudable, that I gladly embrace this opportunity of contributing my mite to it by reprinting the Letter; conceiving that it cannot fail of the approbation of all the fober and difcerning part of mankind, and that, if the fuggeftions of it be duly attended to, it will prove very beneficial to thofe who are of a different character, as well as to the public in general, 3 EDITOR. Dec. 10;- 1801. ‘Mr. URBAN, | OF. 7. ‘ THE following paflage, taken from the preface to the fourth volume of the ‘ Scriptores Logarithmici,” lately publifhed by Mr. Baron Maferes, appears to be written with fo benevolent a defign, and points out 305 A LETTER PROM PHILALETHES CANTABRIGIENSIS, out to the Great obje@s fo worthy of their attention, that I wifh it were more generally known; and therefore fhall be glad to fee it in the Gentleman’s Magazine. ‘The paflage begins in the ixth page of the preface, where, {peaking of | Dr. James Wilfon’s “ Hiftorical Differtation of the Rife and Progrefs of the Modern Art of Navigation,” the Baron fays, 6 It is full of curious hiftorical matter, and has fuggefted to my mind a with that fome perfon of affluence, fond of the fubje& of navigation, and who fhould have been indebted to it, perhaps, for his rank or fortune, would caufe a collection of all the authors on that fubje&, whofe works are mentioned in this Differtation, to be made, and reprinted in a handfome manner in a fet of quarto volumes, of the fize of thefe volumes of the Scriptores Logaritbmici, under the title of Scriptores Nautici. Such collections of learned tra&s on particular fubje&s, under various titles fuited to the feveral fubje&s of which they treated, would be very convenient in the prefent ftate of {cience ; which is extended to fuch a variety of fubjects, and difperfed in fuch a number of different books, that it is very difficult and very expenfive for a perfon, fond of any particular branch of fcience, to procure himfelf all the books that relate to it. Befides the collection called Scriptores Nautici, relating to navigation, there might be a collection called Scriptores Statici, relating to the do@&rine of ftaticks, or bodies at reft that form an equilibrium, or counterpoife to each other; under which head all the books of merit that treat of the /ever, the inclined plane, and the other mechanical powers, would be comprized, and thofe that treat of the catenary curve, and of the partial immerfion and the pofitions of bodies floating in liquids of greater fpecifick gravity than themfelves, and of many other curious fubjects of the like nature. And there might be another collection called Scriptores Phoronomici, relating to the do&rine of bodies in motion; under which head would be comprized Galileo’s Mechanical Dialogues, of which the 3d and 4th contain the doctrine of the fall of heavy bodies to the earth with the law of their acceleration, and of their motion on inclined planes, 6 | and A LETTER FROM PHILALETHES CANTABRIGIENSIS, 369 and of the-motion of pendulums in circular arches, and of the motion of pro- jectiles, which (abftra€ting from the refiftance of the air,) would defcribe parabolas; and under the fame head would be comprized Mr, Huygens’s tract on the motions of perfectly elaftic bodies ftriking againft each other, and his admirable treatife De Horologio Ofcillatorio, or on the motion of a pendulum- clock, and his tract on central forces; and all Sir Haac Newton’s moft pro- found, but very difficult work, called the Principia, or Mathematical Principles of Natural Philofophy, with the feveral Commentators on it, and Herman’s Phoronomia, and Euler’s work De Motu. Another collection might relate to the finding the centres of gravity of different bodies; which is, I believe, a more fubtle and difficult fubje& than is generally fuppofed. This collection might be called Scriptores Centrobarici. And another collection might confift of all the writers on opticks, under the title of Scriptores Optici. This collection fhould _-comprize the work of Euclid, or that which has been afcribed to him, on this fubje&, and thofe of Alhazen, and Vitellio, and Roger Bacon (the learned Englifh monk), and Antonio De Dominis, and Willebrord Snell, and Des Cartes, and Huygens’s Dioptricks, and his treatife De Lumine, and other works of his on the fubje& of opticks, and James Gregory’s Optica Promota, and Dr. Barrow’s Ledtiones Optice, and Sir Ifaac Newton’s Leétiones Optice, and his Treatife of Opticks, or Experiments on Light and Colours, and Molineux’s Dioptricks, and Dr. Smith’s Compleat Syftem of Opticks, and Harris’s Opticks, and many papers in the Philofophical Tranfa&ions relating to the fame fubject. If fuch feparate collections of authors were publifhed, every perfon who was devoted to any particular branch of thefe fciences, (and no man can attend to all of them, or even to many of them, with any great profpect of becoming mafter of them,) might buy the collection which related to his particular branch at a moderate expence.” “On this occafion I beg leave to make another remark or two. ‘The importance of the art of navigation to this ifland, in times of peace as well as of war, is generally acknowledged ; yet it may be juftly doubted whether it has been encouraged here in a degree fuitable to its Vor. II. 3B importance, 379 A LETTER FROM PHILALETHES CANTABRIGIENSIS. importance, or equal to what it has received, in the laft fifty years, from other nations ; certainly riot fo as to excite equal emulation amongft men of fcience *. In fupport of this aflertion, 1 might enumerate the prizes which, from time to time, have been given by foreign academies for improvements in navigation and aftronomy, and recount the learned tracts which have been produced in confequence of that encouragement ; but I fhall at prefent wave this fubje&. ‘In all civilized nations, arts and fciences have been confidered as making a part of the education of the Great, and as being under their patronage. Amongft the men of rank in this country, in former ages, are to be found the names of Napier, Bacon, Boyle, Newton, Macclesfield, and Stanhope; men who excelled in fcience, and patronized it in others. May I then be allowed to fuggeft to the nobility and gentry who, of late, have made a confpicuous figure in Wefiminfter-Hall, and to all others of rank and fortune, who, although their names have not yet graced the columns of the London mews-fapers, are waiting their time and money in the feduétion of the wives and daughters of their friends, or in other idle and vicious amufements, that, if they would exchange thofe vicious amufements for the innocent and rational ones purfued by the men whofe names I have mentioned, and, inftead of fquandering away thoufands on courtefans, lay out a few hundreds in printing fuch Scientific traéis as the worthy baron has mentioned, and in the fupport of Genius firuggling with poverty, it would undoubtedly be much more * I am aware of the rewards which have been offered by acts of parliament for the difcovery of the longitude at fea, and not unacquainted with the manner in which 20,oo0l. has been beftowed. for A LETTER FROM PHILALETHES CANTABRIGIENSIS, 371 for their prefent honour and future fatisfaGion, as well as for the good of mankind.’ ‘ PHILALETHES CANTABRIGIENSIS. Omne animi vitium tanto confpeciius in fe Crimen habet, quanto major, qui peccat, habetur. [eee e e. —_!_!_!_!__ e e e TT Tota licet veteres exornent undique cera Atria, NOBILITAS fola eff atque unica viRTUS. Juv. tea TOA AAR CETO REI COD ESENTI I PRETI salirono n Printed by Wilks and Taylor, Chancery-lane. ws