IN MEMORIAM FLORIAN CAJORI '// rv^ '^^^^^ o ^. ^ Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/fluxionalcalculuOOjephrich THE FLUXIONAL CALCULUS. AN ELEMENTARY TREATISE, IN TWO VOLUMES, tllJSIGNED FOR THE STUDENTS OF THE UNIVERSITIES, AND FOR THOSE WHO DESIRE TO BE ACQUAINTED WITH THE PRINCIPLES OF ANALYSIS. VOL. II. COKTAINIXG THE CALCULUS OF FLUENTS, FLUXIONAL EQUATIONS, AND THE CALCULUS OF VARIATIONS. BY THOMAS JEPHSON, B.D- I LONDON ; PRINTED FOR THE AUTHOR. PUBLISHED BY BALDWIN AND CRADOCK ; AND SOLD BV THU BOOKSELLERS AT THE tTNTVKRSITIES. 1880. LONDON: PRINTED BY THOMAS DAVISON, WHTTEFR lAUS. V. '3- ADVERTISEMENT. This second volume concludes the work which I have undertaken to submit to the publick. I have been compelled to omit several subjects which formed part of my original plan. The discussion of certain curves ; a chapter on curve surfaces ; the application of the calculus to the finding of the centres of gra- vity and oscillation, and to some of the branches of natural philosophy ; these, together with other sub- jects which are usually discussed in the elementary treatises which are published in this country, were to have constituted the fourth or miscellaneous part ; but I have entirely omitted them, that the work may not swell to an inconvenient.size. It is the less necessary at the present day to introduce these sub- jects into our treatises on pure science, as there have lately appeared publications in every depart- ment of the mixed mathematics adapted to the use of students. CONTENTS OF THE SECOND VOLUME. PART SECOND. FLUENTS. Page 1 21 23 24 47 53 CHAP. I. Rational forms Cotes' and De Moivre's theorems The construction of fluents Irrational forms .... Logarithmick forms Circular forms .... Miscellaneous praxis CHAP. II. Integration by infinite series; and be- tween cei'tain values of the variable by ap- proximation. Integration by infinite series . . .72 Definite fluents by approximation . . 78 Miscellaneous praxis . . . .93 FLUXIONAL EQUATIONS. CHAP. III. The theory of arbitrary constants . 96 CHAP. IV. Fluxional equations of two variables q/' the first order and degree. Separation of the variables ..... 105 Homogeneous equations . . . .109 Equations integrated by substituting so as to destroy some of the terms . , . .116 Linear equations . . . .118 Riccati's equation . . . .123 Exact fluxions ..... 129 Euler's criterion of integrability per se . . 131 Integration of exact fluxions . . ,132 On finding the requisite factor . . . . 137 Transcendental forms . . .147 Miscellaneous praxis . . . .149 KnRaRnfvy VI CONTENTS. Page CHAP. V. Fluxional equations of a higher degree than the first . . . . .153 Clairaut's form . . . . .158 Singular solutions when the primitive is known . 162 when the primitive is unknown . 168 Praxis on singular solutions . . . 181 CHAP. VI. Fluxional equations of tzvo variables of a higher order titan the first. Fluxional equations of the second order . .185 Homogeneous equations . . . .193 Fluxional equations of the wth order . .201 Exact fluxions . . . . . 206 Integration of exact fluxions of the second order , 210 Linear equations . . . . .211 D'Alembert's process when the roots are equal . 213 Lagrange's method of the variation of the parameters 231 On changing the principal variable . . . 237 Formulae which have no independent variable . 24.5 Miscellaneous praxis ... . 250 CHAP. VII. Total fiuxional equations of more than two variables. Equations independently integrable . . 252 Homogeneous equations .... 257 Equations of a higher degree than the first . . 259 Equations of the second order . . . ib. The equation of condition that a function of any order may be an exact fluxion . . . 262 The integration of exact fluxions of any order . 269 Equations not independentli/ integrable . . 272 Equations of a higher degree than the first . . 276 Equations of a higher order than the first . . 277 Application to geometry .... 278 Simultaneous equations. (D'Alembert's method) . 279 Miscellaneous praxis . . . . 284 CHAP. VIII. Partial fiuxional equations. Equations of the first order and degree . . 288 Homogeneous and linear equations . . . 296 The integration of equations of more than three va- riables ...... 297 Lagrange's theorem for reducing partial to total equa- tions ..... 302 The theory of arbitrary constants relative to partial equations ..... 304 •Singular solutions .... 306 CONTEXTS. VU Page ' Equations of a higher degree than the first . . 306 Equations of the second order . . . 313 Equations of the nth order . . .316 Equations of the second order and of the first degree, but of more than one dimension with regard to the fluxional coefficients . . . .317 Monge's method of integration .... 318 Linear equations ..... 324 Laplace's method of integration . . . 326 Euler's method of integration . . . 328 Equations not linear (Legendre's methods) . . 331 Certain forms integrated by particular methods . 333 The construction of surfaces . . . 335 . On the determination of the arbitrary functions . 339 Miscellaneous praxis .... 345 CHAP, IX. The integration of fluxional equations hy series and hy approximation. The method of approximation .... 349 Sir I. Newton's rule . . . 350 The method of successive substitution . . 352 Continued fractions . . . . 356 Equations integrated by series . . . 359 • by approximation . . 378 by continued fractions . 379 Partial fluxional equations integrated by series . 382 An exception observed by Poisson . . . 387 Praxis . . , . . .389 CALCULUS OF VARIATIONS. PART THIRD. CHAP. I. Variations of functions . . 393 Examples . . . ' . . .411 CHAP. 11. Isoperimetrical problems . . 424 ERRATA. Page 7, 1. 6 from bottom, /or cos. ^^"" ^^ \ read cos.?^^— i^^. n n 13, 1. 2 from bottom, for — , read — . n n 2 14, last line, /or 2, read — . a^sm.- .-^1^ ,2., IT n 15, fori. 2, read — sm.(n—m) — tan.— i n ^ n , «!i 1 — xcos. — n 35, L 12,/or ^/l— ^^ ^^^'^ Vl-^^''- 52, 11. 2 and 3, insert z^-^ and x^"— 3 in the denominators. 55, 1. 3 from bottom,/or sin. a, readsin.x, andfor sin.2w— 2x, read sin."— 2x. 69, 1. 5, for — !— , read—^. 66, 1. 13,/br A, read niA; and 1. 15, /or cos.«2, reati cos.«2. 73, L 8 from bottom, /or -. read — ^ — — and divide the exponents of a in the series by q. 114, 1. 23, /or art. 22, rrarfart 24. 145, 1. 5 from bottom, /or a'^ read ar. 166, 1. 6 from bottom, for — , read — . dy dy 179, 1. 10, /or Qx% read Qx^pS. 218, 1. 19, /)r a", read a'", 223, 1. 7 from bottom,/or a:, rea \ s / ' s 1 + a;2 p p ^ T -^x ^^'1 - 9, 8^g •'• dx ~ (1+^2)2' djc^ "■ (1 +^2)2+ (^1 ^^2)3? •• 1 1 R or u = -^ f- — . X^ X ^ Next, to find the part corresponding to ^. Here, cos. 9 = 0, sin. 9 = 1, r = l, .•. jr = +V— 1; a = ^3(1 + x% or ^ = 3^^(1 + x^) + 2^7^ .-. ^—^ . . . =-- (2^'') = 2; and the equation for determining a and b becomes + V—l a + b= ±\/ — 1; ••. a = 1, b = 0; or X J. 1 <2: the required part is -; ; whence w = \-- x"^dx 4. Required to integrate —^ x^ - px""-^ + qx""-"" - &c. = du ; where x"^ - px""-^ -\- qx""-^ — &c. ={x-a) {x — h) (x—c).., and m < n. . X"* ABC Assume _|_ _____ I __^___ ^«_pj.n-i_j. 5^n-2_&c. x—a'^x—bx — c + • • • &c. ; and to find the coefficients a, b, c, • • • with- out the aid of the calculus, we have x'^ = A{x — b) (x — c)- • • + b(x — a) (x — c) • . • + c(jr — «) (jr — b) + ... CHAP. I. RATIONAL FORMS. 5 + &c. ; which obtains for all values of jt. Let x = a; then «"' = A(a — b) (a — c) - - -, or a = r— 7 r . ^ ^ ^ (a — o) {a—c) ' • • Next, let a; = b; then ft™ = B{b —a) (6 — c) • • •, or B = (6 _ a) (6 _ c) . . c" Similarly, c, which = 7 r—r tt , and all the re- ■^ {c — a) {c — 0) ' ' ' raaining coefficients may be determined. ^^ _ Adx Bdx , cdx , „ JNow du = 1 r T !-•••; whereiore X -r- a X — X — c u = aI{x — «) + ^l(x — h) + cZ(,37 — c) • • • It follows, from the original assumption, that m must be less than n ; for the resulting value of x^j viz. — A{x — b) (x — c) ' ' • + B(.r — a) (jr — c) • • • + c(a: — a) (a? — ^) • • • 4- &t3. cannot rise to more than n — \ dimen- sions, be the coefficients a, b, c • • • what they may. The cases of equal and of impossible roots may be pro- vided for as in Vol. I. Ch. II. Arts. 60 and 61. x'^dx 5. Required to integrate -^ x^ - px"-^ + g'tT"-* — &c. = duy when m is not less than n. 1 A B Here assume x^ — px^~~^ + qx""'^— &c. x—a x—b . c ^ ^ I H + &c. ; then a = a?— c *' (a — ^) («— c) • • •' 1 1 ^ "" (6 - «) (6 - c) . . .' ^ ~ (c - «) (c - 6) . . . ' ^^• and we have Ax^dx Bx'^dx cx^dx du = -\ r + -f &c. x — a x—b x—c = Adx\ 07'"-^ + ax'''-'' -h a'^x'^-^ • • • H l t x—ay + Bdx I jr"*-i + bx"^''^ + b^x""'-^ . . . -j \ c x — b J + &c. 6 FLUENTS. CHAP. I. Whence u = aI — + =- + rr • • • + aH(cc—a) > t m m—l m—2 ^ j + &c. ""(A + B+ • • + -J^-ZT (""^ + *B . . .) ^^ ?7i ^ m + a'^AHpc - «) + 6'"BZ(a7 - Z>) + &c. by substitution = a'^'" + b^x""-^ + c'a?'"-'^ + • • • + e'^'"-"+^ + - • + oTkl^x - «) + 6^B/(ar - 6) + &c. Differentiate this value of w, and equate it with the pro- posed form ; and there results — — = mA'x"'-^ + (m- 1)b'j:»'-2 . . . 0?" — px""-^ + qx""-^ - &c. ^ / + (w — 71 + 1):e.'x'"-^+ • . . a'^A 6"»B + + r + &c. x—a x—o Clearing this of fractions, we have an equation of the form x"^ = a'A'a;'»+"-* + (6'a' + c'b').^"'^"-^ + (e^A' + /'b' + ^'c')^»»+"-3 H + (k'A' + I'b' . . • + m'E.')x"' + . . . + «"*a(^ - b) (x — c) Ib'^Bix-a) (a;— c). • . + c'^c(x—a){X'-b)'" + &c. This equation is to obtain whatever be the value of x ; and consequently (Alg. 347) all the coefficients before that of ^'" are to = ; or the first 7^ — 1 terms a', b', c', &c. each = 0; and we have u - e'^'^-^^^ + . . . + a'^Alix - «) + 6"'b/(:c -b) + c'''d{x — c)-\ Cor. When m is less than n, all the coefficients a', b', c'« • • = 0; and wehayeu = a'^Al(x — a) + b'''Bl{x — b)-\-c'"cl(x'-c) + • • •, which agrees with the preceding article. If the theory of algebraical equations were perfect, it is obvious that all rational forms would be integrable by the above method ; since whatever be the numerator, the pro- CHAP. I. RATIONAL FORMS. 7 posed form may be separated into parts which shall coin- cide with either this or the preceding article. But there is no known formula by which an equation of n dimensions may be resolved into its factors; and, in consequence, par- ticular methods have been invented by Cotes, De Moivre, and Waring, for the purpose of exhil3iting the fluents of certain rational forms ; some of these we shall now proceed to give. 6. Required to resolve the trinomial a^^—^ka;'^ + 1=0 into its quadratick divisors, where h is not greater than unity, & Assume A; =cos.^; (then vol.1, ch.2, 37, cor. l.)if 2cos. — = 7i 1 1 ^ 1 . .^ X -\ , we have %cqs,Q = x^ -\ ; i. e. it • • • • x^ — S^cos. — +1=0, then shall a?^'* - 2^'' cos. (J +1 = 0; n or the roots of the former equation are two of the roots of the latter. Now, C0S.9 = cos.(27r + 9) = cos.(4ir + ^) =cos.(2(w — V^it 4- 0) (1) ; wherefore COS. — has n different n values corresponding to the same value of 6. It can- not have more; for taking the next value of cos.S, viz. .(^nif + 5), there results cos.f Stt ^ j, which = cos. — ; or the same values recur after the wth. Since cos.0 also = cos.(27r — 9 ) = cos.(4w — 6) • • • • = cos.(2(n — 1 )7r — 9) (2) ; from these there will hkewise re- suit n different values of cos. — ; but they are the same as the former taken in a reverse order ; for cos. • • • n cos fa 271- + A 9.'rr^ — COS.l 2* ) = COS. \ n /. n We obtain then n different quadraticks which are the divisors of the proposed trinomial ; and substituting suc- cessively the terms of series (2) for those of series (1) taken in a reverse order, we have x'^'' — 2.r"cos..r4-.r2) + — tan.-i -^7- I6' = sin.— . 8. Required to integrate ;; = du, when n is odd. Here l+a" = (l+a;){I-2a^ + jr^)(l-26j7-fa;2) . . . T tJ'Tf where a = cos. — , b =- cos. — , • • • (Art. 6, Cor. 4.) Whence jlj:^ " 1^ "^ 1 _2«^ + ^^ +l~26a; + ^^'*" w^'" ^ 2.12 ~2a^ 2x--2hx or ■ — = \- -[- — 1+a;" 1+a; 1 — 2aiP + :r^^l-26^ + j72 * Let w = 2m + 1 ; then since there are m terms with quadratick divisors (Art. 6, Cor. 2.), if we subtract the left side of the equation from w, th^ first term on the right from 1, and each of the remaining terms from 2, the equation will not be destroyed ; and there results n 1 2ax-'Z 9.hx-2 l-f-^" 1 + ^ l-'2a^ + ^^ 1-26^ + ^2- •" dx 2dx < ax — I bx—1 ^ ^^"^7i(l-i-x)~~l l-2ax-{-x'-'^ l-2bx-\-x^ ' ' '§ and integrating, •N where 7/1 V- ^7,, « \ . 2a' x — ai , : v u=l.{\^x)n — 1(\ ^2ax-\-x'')'\ tan.-i — r-f « = sin.-j ' n^ ^ ' n a' V n -\\-2bx-^x^) + -i^nr^~\y = sin.?-. n^ ' n b' J n In order to correct these fluents from ^ = 0, add to ^ — « .a tan.-^ — ~ the constant quantity tan.-*— ; then their sum X a' a^x ~ *^" ax-^a^ '^^^'^lT^' ^^^^^ begins from ^ ^^ X——b X = 0; and similarly tan.* ^—rp, when corrected . . . • 12 FLUENTS. CHAP. I. — tan.-^ - — r- ; which must therefore be substituted in the expression for ^i. doc Similarly, we may integrate and exhibit ~ when n is odd, and -; — — ^ when n is even. x^dx 9. Required to integrate ; -^ = du, when m < n. First, let n be even ; then 1 4- x''= (1 -2ax-\-x^) (l—2bx-\-j:^)(l — 2cx+a;^-) - • . T S'TT B'TT where a = cos. — , o = cos. — , c = cos. — • • • n n n WV. ^^' - ^^-^^ 2x-2b Whence j^^„ _ ^_2^^^^,+ ^_2j^^^^+ * ' ' To find the first term of ?^; we have the coefficient of the first term of du = -('2x — 2d) ; which • • • nx'^-^-^X x S -—^x^-"^ + —-j==----(hx-g), where ^ = — cos.f?* — m — 1) — = cos.(7» + 1) — , and £• = — cos.(^/ - 7n) — = cos.w — ; or the first term of du 2dx hx — g- . „ = — ■ -z — - — -, — - ; whose iluent n l—2ax-\-x" 2 C jo^—ah x — a") t = \h.l ^/\-2ax■^x'^—- — 7-tan.-^ — :-}- ifa'=sin.— n (^ a a' y n h ,,1 ^ „^ 2h! , x—a _ , . ,,, = /(I —2ax + x^)A tan.-^ — ;-> foi" ff=ak+a'h' ^ „^ ^ ^ 27i' ^ a!x ^ = /(J — 2«;r + jr2)H tan.~^ :; , when corrected n ^ ' n 1 - ax from j: = 0. ^. ., , , , p , 2c/^ hx—g Similarly, the second term ot du = -, — ^.^^ — ; — - CHAP. I. RATIONAL FORMS. 13 where h = cos.(m +1) — and g = cos.w — , which may be integrated as before, and thus the whole fluent may be ex- hibited. Next let n be odd. Here l4-^" = (l-f ^) (l-2«.r+.r2) (l-26^ + .r8) . . •; and consequently all the terms are the same as before, ex- cept the first. The coefficient of the first term o£du = 7; butherethe factor 1 + ;r = gives x = — 1, and consequently the co- cient = (-ly efficient = — ; and integrating, the first term 1(1 + a). x'^dx 10. Required to integrate -, = du^ when m nl^ d aJ ) 14 FLUENTS. CHAP. I. ^*7/i « ^ 2h' x—a. = Z(l-~2aa: + a:n+ — tan.-^ — r-, £or ff= ah -\- a' h' n ^ ^ n a ° Jh ^Ji! cix = 7(1 — 2aa? + ^")H tan.-^r; -, when corrected n ^ ^ n 1 —ax from J7 = 0. Similarly, the third term oi du^ • :; — r^— ^ — -. where g = cos.m — and h = cos.(m + 1) — , which may be integrated as before ; and so on. To integrate the first term, we have its coefficient = x'^ — ;^2x ; but 1 — x^ = (I — x) {I -\- x) = 0; which gives TIX J? = + 1 and X ~= — \ ; wherefore, when decomposed, it is — J -^ f- -= — ^ V ; the fluent of which is the first n t 1^^ 1 +^ J 1 1 -{-X 1 term of u ; which therefore = — I = or III — x^), n l—x n ^ ^ according as m is even or odd. Next let n be odd ; then 1 - a;" = (1 - ct) (1 — 2ax + ^2) (1 _ 2hx + x"-) . . . and consequently all the terms are the same as before, ex- dx 1 gives a? = 1 ; wherefore the first term —J^ — . = = - J_Z(l - a:). cept the first ; and its coefficient is — -— , ; but 1 — x = n '.m — 1 i_ ,»,w — m — 1 II. du = = ~ dx. l-\-x It appears from Art. 9, that any term of ?/, when 7i is even, may be represented by cos.m — /(I — 2a; COS. f- ;r") . icr . X sm. — > + 2sm.m — tan.-^ r- n ^ itr l~^cos. — n CHAP. r. RATIONAL FORMS. 15 cos.(w — m) — 1{\ — 2arcos. f- x^) n ' n ^ n . . ^ i<7r xsin.i'n' ^ ^ 71 , tit 1 — a;cos. — n where i is any odd number not greater than n. „ 1 , .i"^ 1 .. iT^ . But COS. (n--m) — = cos. (n^ — m — ) . . . n ^71 n ^ n^ = — COS. 711 — ; or the logarithmick parts destroy each other, and the term may be represented by the circular function X sm. — ^ . lit n — sm.m — tan.~' r— . n n . t^Tf 07 cos. If 7J be odd, then the whole of the first term = -^ lil -^x)^ '- li\ + x) which = 0; and consequently in either case u may be exhibited as a cir- cular function free from logarithms, by substituting in the above expression the odd numbers 1, 3, 5, . . . for i. vim, — 1 I ^n—m—\ Thus, (1.)/ ttV"''''^ 4 . It — — sm. m — tan.~^ n n 4 . .% f — sm.??2 — tan.~' 71 71 X sm. \ —X COS. — 71 07 sm. Stt 71 \ — X COS. Stt ITt 4< . iir -] sm. 7/1 — tan.~^ n 71 07 sm. 1 — 07 cos. ITT 16 FLUENTS. CHAP. I. Similarly, the following forms may be demonstrated. = COS. w — 1(1— 2/1' COS. 1- .r-) n n n cos.m^ — /{I — 2^ COS. h x^ n n ^ n cos.m — /(I— 2;rcos. h a?^) where i is the greatest odd integer not greater than n. When n is odd, i = n; and the last term becomes 2 4 . . cos.7W7r/.(l + x'Y = q= — /(I + a-) according as tti is even or odd. /y,m— 1 __ ^n— m— 1 ^ ^ sm. — 4 . 2 -^- 1 — X COS. — r- 5 flf^ 1 l + xvT+ x^ ^ _^ 3a;(l — a?^) ^•^fl.'^= -T^(l -^)-4C0S.-g-/(l -2JCC0S. -g- +'^') 2^ a7sin.-g- + |sin.-g-tan- g^ 1 — a:cos.-^* .^-sin. — TT 2 + i-sin. -pptan."^ 2 — TT 1 — .rcos.— scsin. — - TT 3 H- |sin. -5-tan.-^ -- 1 — ircos. — r- 3 — -icos.— /(I ~ 2^cos.— + jr^) . 27r + |sin.— tan-- ^ 1 — ^COS.-TT- 3 = 1^ 1 + -^h ^,H- :Uaur'f-l± ^ l-x ^M-^+a?^ 2a/3 l-jr^ CHAP. I. RATIONAL FORMS. 19 ft -|sm.^tan.--— I — /rcos.-r- 3 + icos. -^?(l + 2a7cos.— + jr^2) . It a?sin.- . TT 3 + jsm.— tan.~^ O It \-\- X COS.-r- O 11. /*:; dx — sin.— rtan.~^ •^ 1 + x* 4 a:sin. -r- J+a?% . It ^ , 4 \ — x cos. 1 ^sin.--r- . Stt 4 + sin.-^tan.-- -^ J — ^COS. — r- 4 = — =tan.^ V2 1 — ^2 1 +jr4 a: 12. /.; -6?a; = Itan.-^^i 1 + |tan.-':r . ^ , 2j? r= ytan.-^= r + |tan.-^:j "" ^ * 1 — 4.r-4-^"*' 13. Required to resolve- — •^rn i^ into fr actions *whose denominators shall he the quadraticJc divisors of the trinomial. Assume as before k — cos. 9 ; and let a, b, c, . . . . = cos. — , cos. , COS. , . . . then 1 — 2kx''-\-x^'' n n 71- = (1 - ^ax + x^) (1 - ^hx + X'') (1 ^ 2c.r + ^0 . . . . c2 20 FLUENTS. CHAP. I. and consequently ^_g^^,_^^, - = • • ^a-2x 26-2.r + T— 7Tr-7-T+ or 2nkx'^ -^.nx^'' ^ax—^x'^ 26r— ^^^ l-2kx*'^x^''~l~2ax+x''^l - 26a; + ^2 + • • • to w terms. Add 71 to the left side of this equation, and 1 to each term on the right ; and there results n{l^x^ ") l-^x^ 1—^^ l^^kx^' + x^ ~" l-2ax^x^ "^ l-^^bx + x' "^ ^^ 1—2JCX'' + x''''~n i -x^"ll -2ax +x'-'^ l-'2bx -\-x^'^ ' J 1 1 - :; x=l- V '"x^-'JV' X^J }_ / „ IV x' (f-i)' X'^~' — X ,2w— 1 ('■-ij- Let the first term be the one which is required ; then, since the factor is 1 - 2ax -[-x^ — 0, we have ^' -j ~ 2a X & 1 ^ = 2cos. — ; wherefore jr^'*"^ + ~i;r=a ~ 2cos.(2» — 1) n 1 4A;* — 4 and the required term is — . -rr — - ^ n l—2ax + x^ 1 s. r /c' = sin.O. a—h 1 2^*^ Similarly, the second term is — • .. _ ^ , g , where A = cos.(2w — 1) , and so on. CHAP. r. cotes' THEOREM. ^1 dx By means of this article, the form A _ qt » , o < ^^y be integrated and exhibited. 14. Required to integrate -i _9z. « . o^* ~ '^"j ^^^^^ h<\. T. Simpson integrates this form in the following manner. Assume k = cos.9: let a = cos. — , o = cos. , . . c = cos. — -- . . . ; then 1 ~2A;a;"+jr2« = (1 — 2a 1 /. which is integrable. x^'^^dx By the same method the form - — r- — , may be se- ssfdx parated into fractions of the form - — [^ each of which Jl ~~ liCLX ~i~ X is integrable. The student may see these forms exhibited in T. Simpson's Fluxions, vol. ii. pp. 104. We shall conclude this part of the subject with Cotes' and De Moivre's theorems, which were invented for the purpose of resolving binomials and trinomials into their quadratick divisors. 15. Cotes' Theorem. If any point s he taken in ao the radius of a circle whose centre is o, and the circumference be divided into n equal parts, ap, pq, qr, . . . then shall sp x SQ x sii . . . = Ao" — so". Also, if AP, PQ, QR, . . . be bisected in p, q, r, . . . then shall sp X sq X sr . , . =: ao" + so". FLUENTS. CHAP. I. Draw the ordinate pn ; join OP, OQ, . . . and let AO = r a = cos. Z AOP so = ^ b = COS. z Aoa &c. = &c. (Eu. 2. 13.) si>2 = po^ 4- OS"- ^ — 2so X ON = r"' -f .r^- — ^arx. Similarly , sa^ = r^ — ^brx + a;% sr2 = r^ — 2crx + 07^ . . . ; wherefore sp^ x sa* x sr^ . . . = (r^'-2arx-\-x'^)(r^—2brx + .r^) (r" — 2crx -\- a^^) , . , But it may be shown as in Art. 6, Cor. 3, that (r^ — 2«?'a7 + x'^^'ir'^ — 2broD + a;^)^ . . .= r*" — a?" ; whence sp X sa X sR . . . = r" — a" = ao" — so". Also, since ^, p, g, q, r, . . . divide the circumference into 2n equal parts, we have sp x sp x s^ . . . = Ao"-" — so"-" ; ^o^" so'^" wherefore s» X sq x sr , . . = • — ;; r- = ao" + so". ^ ^ AO^—SO" If s be taken in oa produced, x is greater than r ; and it may be shown in the same manner that sp x sci x sr . . . = so" - AG". Cor, 1. Since the radii vectores, reckoning from the last but one, are the same as those from the first, we have Ao" r^ so" = SA X sp2 X sa^ . . . If n is odd, and asm be the diameter of the circle, one of the divisions will fall on m, and we have Ao" ^ so" = SA X SM X SP- X sa^ . . . n Cor. 2. Ao" + so" = s;?- x s^^ x sr^ ... to -^ terms. 16. De Moivre's Theorem, If the divisions of the circle begin from some other radius OB than that in which s is situated, then shall sB^ X sp2 X sa^ X sr2 . . . =Ao2^' — 2ao" X so"cos.9 + so^'*, where L aob = — . n For as before, sp^ = po"- + os« - Sso x on = r^ + ^^ - 2r^cos. -^—^ ; and consequently SB- x sp"- x sa'^ X SR^ . . . CHAP. I. THE CONSTRUCTION OF FLUENTS. 23 = Ir'^ - ^rx COS. 1- ^^) (r^ — 'iLrx cos. 1- ^^) . . • = (Art. 4), 7-2" - 2ra:cos.9 + x"-" = AO^" - 2ao" x so"cos.9 + so^". 17. Required to construct the Jluents of ^__cy„ , "T With radius oa describe a semicircle apm. Let OA = 1 ; suppose fl = ON = COS. Z AOP a con- stant quantity; and sup- pose OS to represent the / variable quantity x. if "^ S O Then joining sp, if os become os and sa be drawn per- pendicular on sp, d. I OPS = the limit of Z. sp^ = — ; but stf : 85 : : NP : sp ; whence d . Z ops = the limit of sin.AP .dx . . . ^ dx NP X SS SP2 1 sm.AP - X Z.OPS. Also xdx xdx — adx adx x-'-9.ax-\-\ x"—2ax-\'lx'^ — 2ax + l xdx , a , and, inte- grating, At — ^ -y =lVx'''—2ax-\-l + z OPS. & °''^jc2_2aj7-(-l sui.AP 7 SP . = I \- COt.AP X Z OPS. OP x'^dx The form — — ^ by division becomes dx -\ — ^ — - — — —, and is therefore integrable ; and gene- x'^dx rally — — - — —■= is integrable by this construction, if w is a X tiCLX -p J. positive integer. 24 FLUENTS. CHAP. I. P - Let m be a fraction = — ; then substituting y—xi^or p xidx qif-^'^-^dy , . , . , X — y% we have — - — ^r —r = o! q — :rn > which is to be integrated as in the preceding article. The forms -j^Jl^, -^^, dividing by , dx ^adx — xdx _ 1 — 2a r + x% become f- — — r: ~:r and • • • • ' X x^—^ax + 1 dx 2ax~^dx — dx . i ^ • ii -— -\ — r-, and are therefore integrable. x^ x'^-2ax + \ ' ^ If m is a negative fraction, it may be reduced to an in- tegral form as before. De Moivre (Misc. Anal. pp. 60.) integrates the form x'^dx 18. Irrational Forms. If these can be rendered rational by substitution, or any other artifice of calculation, they are to be integrated as in the preceding articles, and their fluents may be expressed in finite terms: if they cannot be rationalized, their inte- gration, when possible, is to be effected by the method of continuation. 19- It has been shown (vol.i. ch.2. art. 46.) under what con- ditions binomial forms may be rationalized by substitution ; and it may be here observed, that the more general form f p L y^ } F [ ^""j (« + bx'^)\ {a + hx^)', [a + bx'^y, . . . {;r"-Wr, where f represents any rational function of the quantities which compose it, ma}^ be likewise integrated by the same method. For substitute a -\- bx"" = y «'•• ; then jt" = - — 7 — , and the form becomes F I [ ^ f, J ' !/'''•"> y"» • • • J-^r'""''^^; which is a rational function of^. CHAP. I. IRRATIONAL FORMS. S5 dx Thus may be rationalized by substi- (1+^)^(1+^)^ tuting (1 + ocY = «/, or j/6 = 1 + ^. This substitution enables us to effect the integration, even if the binomials which compose the function are fractional, provided that they are, as before, the same binomials. rationalized by substituting ' ^ ' ^7^ = y'"*- The form (x +a/1 + x")i ndx, where r represents a ra- tional function either of x or of ^^ 1 + x'' may be always rationalized by substituting x -{- ^\ -V x'^ — y"^* If different binomials compose the function, the form can in general be integrated only by particular artifices, of which a few examples have been given in the first volume. The case in which there are only two different binomials of the form -y/^ + ^-^j \/a} -\-bx is integrable by sub- stituting « + 6^7 = (a' + Vx)y^\ for then we have X = T, ^5 ^a + bx = - — , va' + b'x • • • = — ; or the proposed form is reducible to one Vb'y' — b which contains only one binomial surd ^/b'y^ — b ; and it may be rationalized by a second substitution. The form ; — , , „. ; , ,. , is integrable by substitu- {a\bx''Y{g-\-hxy ^ ^ tion (vol. i. ch. 2, art. 50.), if r be a positive integer, and p + ^ also an integer greater than r. ^ , . a-\-bx^' . gy—a For substitute y = r — ; then x"" = ^r—i — j • • • ^ g'-\'hx" b — % ••^ -V6-%>--^ '^''- n\b-hy) Ib^hy {b-hi/y I n " {b-hr/Y^^ ' FLUENTS. CHAP. 1. Also « + 6^» = « + ^-fc2> = ^=^'^ ; and o—hij b — hy ^ ^ b-hy b-hy' _ bg-ah {gii—df-^dy , {bg—alif^^ n ^ {b-hyy+^ "^ {b-hyY^'i which can be expressed in finite terms if r be a positive in- teger, and JO ^- q an integer greater than r *. 20. In the following forms, integrable by the method of continuation, we shall suppose the indices of x both within and without the radical to be integers ; for if they are not, they may be made integers by substitution ; thus, let the 1- p J. form be (a + bx"") x^dx\ substitute 3/^= t, and it becomes Q{a 4- by^Yifdy. Also the index of x within the radicals may be supposed positive ; for if it be negative, it may be made positive by substitution, or it may be brought out of the radical. We shall further suppose, that there is one term of the quantity within the radical into which x does not enter, as it may be reduced to this form. 21. In a binomial form ^ the i7idex of the variable with- out the radical may be increased or diminished by its index within the radical. For let it be required to reduceyxPa7"'^"c?,37 = b to the form fx^x^dx = a, where x = a + bx"". Assume p = ^p+^x''+'^ ; then dv = (jp -^ l)nbxPx''+''dx -h (v -\- l)xP+^x"dx, = [{p-i.l)nb-]-{v+l)b] xPx^'+^dx -^{v + l )axPx''dx, orp= {(p^l)n +lv + l)\bB + (v + l)aA. (a). From this equation we can find either b in terms of a, or A in terms of b. If it be required to reduceyjc^cr"-"^?^; to the formfxPx''dx ; here dA = xPx'"~'^dx, dB = x^x^^dx ; wherefore assume * The reader should consult a very useful paper in the Philo- sophical Transactions, June 4, 1816, on the Fluents of Irrational Functions by Mr. Bromhead, who by means of inverse functions integrates forms more general than those given by any preceding Avriter. CHAP. I. IRRATIONAL BINOMIALS. 27 and F = ^(p + l)n + (v \-l — n)\bB + {v-\-1 —n)aA, ={np+v-{-l)bB-^(v-hl—n)aA, (/3). Let v + l =m or v = m—l ; then (a) and {(3) become ^p+,^^_„__ ^^ _|_ np)bfyLPx"'-'dx + (m —n)aJkPx'^-''-Hx 3 ^^ ma ma " I These two formulae by which the index of a;* without the radical is increased or diminished by its index within, are Hirsch's Table I. Formulae 5 and 3. Cor, 1. Hence by successive substitutions the index o^ x without the radical may be increased or diminished by any multiple of w. Cor, 2. The form (a) fails when m = or v = — 1 ; in (a-^-bx'^ydx this case aA = ; or representmg the exponent of the radical by a fraction, dA = . •^ X Substitute y'i = a-\- bx'' ; then {a + bx"") '^ = i/^. Also • • • bx''=y^—aovl.bx''=lhf-a), .-. — =-^ ^ ^, and • • ^ ^^ X n 2/^— a dA = - - ^—- -, which is under a rational form. n y^— « Cor, 3, The form {(3) fails when m-\-7ip = or p= ; in m this case dA = (a f bx'') ''x'^'-hlx; which is rationalized by substituting a + ^JT" =xy ; when it becomes — - — - — y-^. If either of the constants a, b be supposed = 0, the form becomes a monomial ; and there is no difficulty in the integration. ^ FLUENTS. CHAP. I. 22. EXAMPLES. Ex. l./x-^a?-'"-W^ = x-p^^^-'«*" (m-\-n)a ^ T—\ wT - 7—^ Tlf^~^^~''~''~'dx. {m-\-np)o {m-\'np)nr For here a =Jx~Px~'^~^~^djr, .*. ^ + 1= — m — w... .-. p, which = x-p+'^-*'^-", = (( — ;? + 1 )/i— m — n)bfx~Px-"'-^dx — (m-^7i)aprPos-'^-''-^dx, .- x-p+^;r-"*-" {m-\-n)a ^ fsr^x-^-^dx = - . , 7 — ■ — ^7 /k-Pa:-"'-^-'dx. ^ im-\-np)b {m-\-np)o'^ Ex. 2. Required to reduce/ to the form dz Here d\ =. . or v — — n-l.ov r + 1 = — ;/, and/} = — i-; wherefore assume p = ^bz^ — a.2~"; then , 7ib dz f/p = — — n x/bz''—a.z-"~^dz = nadA . . . Ex. 3. Required to integrate :. Va + bx"" x^'-'dx 2 Let dA = — - or A = ^ //« + 6^'* ; ^/a + bx" 'no ttB = r; ac= ^« + 6jr'^ Va + bx"" First assume v = ^/a + 6a7^ j", .*. cZp = Next assume q ~ A/a-{-bx''.x'^''^ .*. 6?ci ^ ^/a + bx"" ' Snb ^ - 2p 2aA w6 x^'^-^dx -\-n Va + bx'^.x'^~^dx = ~^-dB + waJa, .-. b= htt— "ztt- ^ Va -h 6d7" .. .. 5jib + ^n x/a + bx^^.x^^'-^dx ■= -^dc + 2w«c?b, . CHAP. I. IRRATIONAL BINOMIALS. — 1 Ex. 4. Required to integrate A/a" — <^"a?'* dx = r/?/ To reduce this to f^/al" — x'^.x'^ dxy assume P =(a^ _ ^.«)^a:^ ; then dr=-'-^du + ^{aT-x'^-x'' dx na =:.—9>ndu-{'——^/a'' — x'',x'^ dx, ov u^--fVa-'-x\x'- dw ^^ To findy* a/«"— ^". X dx = dF, substitute a" = b% x^ =y ; 2 g then Jf = — a/6^ — «/'^^i/ ; and f = — cir, area, rad. = by . = ?/, = — cir. area, rad. = a^, absc. = a?*, = — cir. area, absc. = — ; wherefore a2« . x'^ (a"-^")^.r^ M = TT— cir. area, absc. = — -, pr . a* x^^dx Ex. 5. y — ^= , where w is a positive integer. a/« + ^^^^ Let cLa = — , as = - , ac = 1st, Assume p=x^cr, for t; = 0; then cZp = bdB + x^tZ.r P — aA X^JT-OA 26dB + «£?a; .-. p — 26b + aA or b 26 ~ 26 2dly, Assume q = x^^^ . ^i^gn ^^ = /^^c + x^.So^^Ja; . . = 46c^c + 3adB, .-. a = 46c + 3aB or c = -^^ — . . . 46 ^0 FLUENTS. CHAP. I. x^o;-^ Sax^'x Sa^A + 46 4M'' 4.26^* 3dly, Assume r = x^x^ ; then dR = Gbdo + 5adc ; R-.5«c x^x^ 5axJx^ 5.Sa°x^x S.Sa^A .'. D= — 7^7 — = -m n-rrr- + - 6b eb 6Ab^ 6.4.2^.3 6A.2b^' From which the law is sufficiently manifest ; viz. that x^'^dx _ if '^^"""^ ^n-l ax""''-^ ■^ 2A/(2ri-2)(27z- 4) "~fi3 • • • I " 2/2(2/Z-^)(2li-4)... * 6"' The coefficient of the last term is + or — , according as 71 is even or odd. Similarly, it may be shown that ' x-'^+Hx I- c jc^"* 2ma x''"'-'' ^ ^a->rbx^ == "" I (2;;m^1^ "" (27«+l)(2m-l)F * * * ■^ (2m + 1 )(2m - 1 ){%n - S)b^ - ' * * | This latter form may also be integrated by substitution as in vol. i. c. ii. art. 46 ; and in either case the fluent does not contain a circular arc or a logarithm. Similarly may be exhibited the formsj" Va-\-bx'^x^'^dx andj\/a + bx^ . x^"''^^dx where b may be either positive or negative. Ex.6. du = ^^^ Vct—x Assume v — x~^ Va — x ; then dv = — x~^ ^Ja — xdx x~^dx , x~^dx — \ = - fldw + 1 • -^ a—x ^a—x Since — . cannot be further reduced, for t; + 1 = 0, Va —X 1 . , x-^dx —'^dy , J substitute «/ = A/a—x\ then — = -; .:ap = —aau - ^.' and p =- «« i ^ -r-^, or CHAP. I. IRRATIONAL BINOMIALS. 31 ^ %a^ a^ - Va-00 __ .Ja — x 1 a^+\/a— :r 23. Required to exhibit fyiPx^^-^'^^-^dx in terms of j\Px^-^dx. Let f?A = KPx"'-^dx ; cZb = xp^^+"-' J.r ; dJc = xPa^'^+^/i-' j^ . • • • Jk = xPar"*+<'-^)"-V.r; and dh = x^^r'^+'^-V^. 1st, Assume p = xP+^.r'" ; then c?p=(/?+ l)wZ>xPa:"*+"-'X^-Hx, Let dA=xPx'''-^dx, dB = xPx"'-''-^dx, dc = x^x^'-^^'-^dx... dK = xPa?"'-<'^-i)"-^c?x and dh = x".* '"-'^"W^. 1st, Assume p=xP+^a7»»-" ; then dp = (p +l)nbxPx"^^dx -\- (m - n)xP+^x'^-^-'dx = (np + m)bdA + (w - n)adB, . . P -(w — /z)flB ■. A = {np + m)b S9, FLUENTS. CHAP. I. 2d]y, Assume q = x p+^x^^-^" ; then ^ Similarly, c = ^jj^-^^^^^;^ and K = — ; , rr 77- ; wherefore fx^x'^-^dx . . . . {np — (r~-l)n-^m)b ^ — xP'''^{kx'^~^ — bo:"*-' 2" + c:r"*-^" — • • • ± La^'"""''"] . . . X (»2 __ rw)aL/xPx'"~''"~^t/a:, where A = -, -, ; . . . . ^ ^ , {np+m)b' (m—n)a (m — 2n)a ~ {np—n-\-m)b ' ~ {np — 2n-\-m}b ' im — Or — V\ri)a {np — {r — \)n -\- m)b This and the preceding are Hirsch's Table I. Formulae 7 and 9. Cor. When m is a positive multiple of /?, the series will terminate, and the fluent can be expressed in finite terms; which agrees with vol. i. c. ii. art. 46. 25. EXAMPLES. _ , x^ dr Ex. 1. du Va-kbx'' 2 h^dii Substitute w^ = ^" ; then du ^ ^—^— ^ Va + hg' To integrate — — -^^ — = dQ, let '^ ^ . = ^b, and Va-^-by"^ ^/« + %2 -=^- = d^. ^/a^-bif Assume r = y \/a -|- bj^'^\ then dv ~ dy ^ a + 6j/~ . . , + — ^^— ^— = «(/a + ^Z» Jb ; or b = ^, . ^a + by^ 26 Assume o.—y'^^a + 6y ' ; then Jq = 3fl£?B + 4Mc ; or CHAP. I. IRRATIO^'AL BINOMIALS. Sti Q.—SaB __ a 3av 3«^a 2^/a-\-by^c y^ Say ^ 6a'^ ^ ,' . , — --^^ 3n n Ex. % J — =1^ m terms o\f Let flA = — , ; flB = (Zc = — . . . ^/k =— — ^ , and ^m.(m-l)(7«-2) ■"^m.(7/i-lX77?-2)...3.2.1 S (gm -l)(^77?-3)-- 5.3.1.c^ _^^ "*" ?7i.(77i — l)(772-2)...3.2l~'^'^ T* 26. /w a binomial foryn the index of the binomial may be increased or diminished by any integer. Let it be required to reduceyx^ + '';r"^jr where r is an in- teger to the ^ormJ'^Px''dx when x = a + bx'\ 1st, Let d^. = x^x^dx, ds = i^P+^x'^dx; then B =/xP ''^x'dx=faxPx"dx+fbxPx''+''dx = aA -^/bx^x'^-^^dx. 'ButJxPx''+''dx may be reduced, as in art. §1, to the form a by assuming p n= xp^*^"+»; by which we find b in terms of A. 2dly5 Let dc = x^^x'dx', then c = wb + b/xP+^x'+''dx. 38 FLUENTS. CHAP. I. Butyx*^'a7^+"J^ may be reduced to the form b by as- suming Q = xP'^^x"'^^ ; by which we find c in terms of b and A, or in terms of a alone. Similarly, d —fiP^^x^dx may be found in terms of c, b, and A, or in terms of A alone. The index of the binomial has been diminished by unity at each assumption ; but as the same equation gives a in terms of b, this method enables us to increase the index by any integer. 27. Required to exhibit J}L^^'''x'"'~^dx in terms of. . . fsPx'^-^dx. Let Ja -x^x"'-'dx^ c^B-xP+^a7'«-^Jar, r7c = xP+2^'''- V.r . . . cZk = xP+'-'o^'^'-^fZ^, di. — ^p^'x"'-'dx. 1st, dB = xdA, .-. B = aA +fbxPx'''+''-^dx, Assume p = x^'^^x"'; then dv = nb(p + l)xPx"'+''-^dx + mxP^^x'^'-^dx = (up + w 4- m)bxPx'^^^~^dx -{■ mad a ; w hereforey6xPa?"*+ "~' dx xP ^^x"^ — ma A np +?? x^^'^x'^ ' B = aA -\- np-\-n-\-m, npi-Ti'\-m jip-\-n-\-m -^P+l^m 7ip-\-n-\'7n or A — — 7 r^T — f 7 — -7T — B. {p'\-\)na (p-t-l)«a 2dly, dc = x^B, .♦. c = aB +fbxP^^x'''^''~'dx. Assume q ■= x^+^or'"; then d^ - nb(p + 2)xP^'x'''^''-'dx + mxP'^x'^'-^dx = . . . (np + 2w + m)Z>x^^+^d7'"+"-Wd; 4- madii^ xP+2^m np-\-2n^m ""'' ^ "" ~ (^2)^ "^ (p + 2)na ^* . ., ^ xP+V* np + Sn+m Similarly, c — — 7 — -^jr 1 — 7 — -75T — d . . . and ^P+rj,m np+rn-\-m K = — -; r 1 — ; t L : whereiore we have (p-\-r)fta (p-\-r)na fxPx'*'~^dx = — a;'« ^ axp^* + BxP+2 4. cxP+2 ... 4- Lx^+*- 1 . . . -f (jip •\- rn + w)L/x^+'Vr'"~^c?^; where . . . . . . 1 _np-\-n^-m np-^2n-\-7n CHAP. I. IRRATIONAL BINOMIALS. 39 _ wp + (r— 1)W+»J ~ (p^r)na Co?'. If either « or 6 is negative, and x be so assumed that X = 0, we have (yxP+'*^'"-Va:) (np-i-n-^m)(np-\-2n-\-m).„{rtp + rn-\-m) ^'^ 28. Required to exhibit fy^p-'x^-^dx in terms of ' • ' Jk^x^'^-'^dx. Let d^^^Px'^-^dx, dB^^^^x'^-^dx, dc = xP-''x'^'^dx • • • dK=x^'-^^x*^~^dx, dj. = x^^x'''-^dx ; then it may be shown by the same method that fyj>x^-^dx -x"^]^ kx? 4- BxP-^ + cx^-=* ... 4- Lx^^+i I + (/) — r 4- l)wai/xP""''a?'"-^J.r ; where A m-\-np n{p-l) = — o , ~^B> • • • n(p—r-{-9) np n(p — l) B = — «A, c = — ^^7^ — r «B, np—n-\-m np^^n-^m -ax. tip — (r— l)w + m These are Hirsch's Table I., Formulae 10 and 8. EXAMPLES. x^dx x^dx Ex. 1. Required/ ^^^^ny in terms of/ j-^. _ x^dx , ardx x'^'^'^dx _ ^ x'^+^'dx dB = rfA + — -, or A = B + /(i^^,. ^ , x'^+^'dx , ^"*+» lo reduce 77-- — -— = of, assume p= =— — - ; then • • . (1 +07^)2 ' 1±X'' dv —(m. + \)dB + nd¥y or + f = -b ; whence by substitution there results A = b ; or • • • n n r-^fL.-._^l_ fn-n-^l x'^dx ^ (1 ±xy ~ n{\ ±x'^) ~ n ^ r±?' (Misc. Analyt. p. 55.) m 40 FLUENTS. CHAP Ex.2, du {a + bx'^Ydx Substitute x"^ =7/; then -~lx = ly and — = — -, whence /& X Tiy 2 Idy du = -{a + hifY-^. To reduce this, let ^a = ^Ja + by^ - , • • • • • • £?B = (a + hiff^-y t/c = (« + hy'Y -^ ; then .... (a + hy 4^ 3 ' • ' • B = «A + ^v^a + ^y^ ydy = ak + 5^ ,c =;= ttB + hf{a + hy'^yydy = ub + ^ ~^-^ ' . *-^ = a-A + 3 + ^— . But A = tfA , .To + ^ t/x/a-\-by'^ ^/a-\-by^ — Va + by^— ^/a — s/a I. \ 1- \/« + by- ; whence b-y c or -^ = aM ^-^ + b'^x^ Similarly it may be shown thaty 7 1 ^/a-{-bx''— Va ^ 4fa + Sbx a^ b^x^ Sa^(a-\-bx'^y 29. In a binomial form the index of the binomial may be increased or diminished by any integer; and the index of the variable without the radical may be increased or diminished by any multiple of its index within. CHAP. I. IRRATIONAL BINOMIALS. 41 Let it be required to reduce Jk^ ^'\i''^^"dx = f to the form Jk^oo'dx = A, where r and t are integers. First reduce f to the form JkPx"+^"djc = b by art. 26; and then reduce b to the form a by art. 21, cor. 1 ; by which F is obtained in terms of a. Thus/xPa;"»'^'"-ltZ^=-^'"+''" |axP'^ + bxp =^. ••+LXP+' ^ + {nip + /' + v) 4- m)LfKP+''x'^'"'-^dx; where .... _ 1 _n(p+l-^v)-\-m 7i{p+^-\-v)-{-m ^ ~ (p + l)w«' ^~~~(p+2)na ^' ^~ ip + 3)na ^" L = ^'^V-"^ -±^l±!^K(art.27). {p \-7')na A\soJxPx'''-'dx=xP^'x'» { A - B^" + c^^" • • • ± L^'"+<"-^)" I + L(n(p -^ v) + p)bfxPx''''"'-^dx; where 1 np^-n + m np-{-2n-^m A = — ; B-— , ; — a; c = -^^ — -77-^ — b; ma {m \-n)a {m-\-'zn)a L ~ -7 -, 7— r — K (art. 23) ; and by means of these (m + {v-l)n)a ^ ' ^ y^pr/jomxtn-i^^ may be obtained in terms ofJyJ'x'*'~^dx. For the exhibition of this and other general forms, the reader may consult T. Simpson's Fluxions, vol. ii. pp. 46. -r^ , , \^a:^-\-x^dx a°dx dx Ex. \. du= —^ = r + ^ X^ x/a" + 37- 07 y/ «- + ^* dx To reduce = = dF, assume p = x~^ V«^ + ^'^ ; then dp = - ^^^^' + ^'^^ + ^- wliich shows that x^ X\/a"-\-x^ dx dv =— 2du 4 — ; whence X Va^-h^r'^ 1 , x/a^'+x^-a x/a-' + x'' n = 7k- I Ex. 2. /{a \-bx'']Px"'~^dx = f in terms of • • ^ • • y(a + hx'')p-^x'^^''-\lx = B. Let dA = {a f bx'^)^^x^~^dx ; then F = aA + ^b ; and to find A in terms of b, assume p = (a + hx^yx""-^ then dv = iipbdB + m{a + bx'')Px'"'~^dx = (np-i-m)bdB + tnadA, p np-{-m. p npbB .', aA — ~ OB : .*. F = , or • • • m m 111 ^ VI *» FLUENTS. . CHAP. T. y(a H- bx'^yx'^-^dx ... Similarly it may be shown that J^{a -\- hx"yx^~^dx • • - x^-^{a-{-hx''Y^^ m-n ^^ , , ^, = XT — TYT J7 — r-rx/(« + bx^'Y^^x'^-^-'dx, nb(p-{-l) 7ib(p-{-iy ^ ' dx Ex. 3. Requiredy-nj -~ between :r = 0, or = 1. dx dv dv x'^dx A = B + / TT- — -r. Assume p = ;r(l + x"^)^ ; then • • • •^ (1 -{-x-y- \ ' J ■> 2x'^dx ^ xHx '''' = '^^-(-iT^^'---./^(T+iJj-^ = -('^-^)'°'^ • • • A = B + i(A - P), .-. B = -1(a + V). x'^dx Also B = c -\-St. ^g. Assume Q=:a;(l -\-x'^)-^\ then 3b a 3 '' d , ••• c = — +^= -g-(A + p) + — ; whence 3 , , 3 1 3 cr 1 Ex. 4. Required the fluent of (1 — x-) ^ dx between 07 = 0, ^ = 1 . Let dA=(\—x''ydx\ d& = i\-x''Ydx\dc=:{\-x'^ydx\ 2m— 3 2n— 1 . . , cZk = (1 - aj"^) 2 f/o;; Jl = (1 - a;2)~2 dx ; then • • • B = A — y(l — oc'^Yx^dx, Assume p = ^7(1 — x'^Y -, then Jp = c?B - 3(1 - x'^Yx^'dx; .'./(l- ^2)^^2^ .... B — p 4b p ~ ~W ^^ ^ - "3 ■" 3"' J _5 c = B —7^(1 — x'^Yx'^dx, Assume q = ^^(l — x^Y '•>'" then '-'dx = 7- ■- - ^^-^ -^-fxPx"'-''-'dx - . . . •^ (2/?;? + m)c {2np -\- m)c -^ ^ ' f-^Px^^-^^'-^dx. {2np + m)c' Cor. By continued reduction the ^orm J\Px'''-~^dx may be reduced to two of the formyx^;r'^^ '•^*-'cZ^'and/xP^"*+<''+'>^'-^c; c = n(p + 1)(6^— 4«f) — m{^ac — b); d = ((2p + S)n + w)ic; K — (p f 1)(6" — 4ac)na. If the /orm of the expression for JxPx"'~^dx had been known, the coefficients a, b, • • • might have been found by differentiation and equating the coefficients of like terms. Trinomials may be also integrated by reducing them to binomial forms by taking away the second term ; of which examples have been given, vol. i. ch. 2. 34. If in a trinomial of the form JxPx'^~^dx, m is a posi- tive multiple of Uf the integration can be effected. h , ¥ For substitute «/ — x"" + -q"? ^^en x = cj/® + «— — = by substitution c(2/2 + y(;2). Also ^"=3/—^; whence • • • 1 / b \!L'-i x^'~^dx = — ( y ~ Q~ )" ^^' ^^^ w^ have •^ = —/(«/' + ^y\y - ^f'' ^y ; ^^^^^^h, if -^ is a posi- tive integer, can be resolved into binomials of the form y(^- + ^^Yy'dy^ and is therefore integrable. m-\-2pn-\-n n 46 FLUENTS. CHAP. I. Cor, Since du may be put under the form h (a^-2" + hx-^ + c)^.r"*+2pn-ij^^ substitute y — x-^ + — and X = «j/-+ c — — = rt(z/'^+A:*). Also x =[y — -\ "; whence w = - — /(2/' + ^0^(^2/ - g^ j " ^y '■> ' / ' or the proposed form is likewise integrable when is a negative integer. Ex r ^^ ^^ r dy _\ ydy '-^ xWa + bx-^cx^ ^a^^ t^k' a^ y'^ + k'^' , , ^ , , 4fac — b'^ where ?/ = ^^^ + ?r? and /c^^ = ; — . ^5. A quadrinomialoftheformfKPx''dx where • X = « f So;" H- cjc'^" + {T^^** m«^ be reduced to three of the formfyJ'x''''''dx^ fsPx'"'''''dx, Jxf\x'-^^''dx, Let Ja = ■sPx''dx\ dB = xP.r"+"J^ ; dc ;= x^cr^^-^^dT; • • • dE = x^a^^+^^^ar. Assume f = xp^^^**^* ; then we have dp = (p + 1)1 nbdB + 2ncdc + SnedE I + (v + l)\adA + bdB + cc?c + ^Je | or p = 7naA -f- . . . (wp + n + 7w)5B + (2np + 2^1 + ??/)cc + (3?i/? -\- 3n-{- m)eE, if V = m — 1. By means of this equation, and by eUmination, all the formulae of Table 3 may be deduced. The fourth Table contains the corresponding formulae for multinomials. See also T. Simpson's Fluxions, vol. ii. pp. 138. The difficulty of integration increases with the number of the terms within the radical. We shall conclude this part of the subject with a trinomial form which may be always rendered rational. 36. The form in which -j- is a rational function of x and A^a -h bx \ cx"^ may be rationalized. CHAP. I. LOGARITHMICK FORMS. 47 For first, let the roots of « + 6zr + cr^ = be real ; and let a -{- boj -\- ccc^ — (p — qx)(p' — q'x). Assume a + bx -{- cx'^ = {p-qxYy"-^ then {p—qx)'i/- = (P - q^)ip' - q'^'v) and 2/2 = ^^^, or .v =g^» .•• - . P^Q^ PQ^ Also Va -{■ bx -\- co;- = {p — qx)y — ■ „ y ; • • • j«y J. whence, by substitution, du becomes rationalized. Next let the roots oi a -\- bx -V cx^ = ^ be imaginary. TT T ^ 4- ^ y'^—a Here assume a -\-bx ^- cx^'-^ic x -\-yY or .r= -^ ^, .•. b^2c^y i_ i_ 2{by —c'^if- — ac'^\dy ,, = i dx- -^^ ^- ^^. Also ^/a^bx^-cx"- = c'-x+y (b - 2c^yy by — c^'vP' — cic^ = — ^-j ; which substitutions will rationalize du, b — 2c^y This latter substitution may be used even when the roots are real, provided that c is positive. For other forms still more general, the reader must con- sult the works of Euler, Lagrange, and Legendre ; the result of whose labours Lacroix has collected in his Calc. Int. torn. ii. pp. 48. du The integration of du, when -y- is a rational function of X and ^a + 6a; -J- cx"^ + ga;^ + hx"^ is made to depend dvC tV'^dx doc upon that of the three forms — , , and -r— — — -- where ^ R R {x'^-\-a)K R = v'a + /3^^ + r^*. 37. Transcendentals. These are to be integrated either by the method of In- determinate Coefficients or by that of Continuation. The same fluent may be frequently developed in two or more different series ; but that is to be taken which, under the circumstances of the case, will either terminate or converge the most rapidly. 48 FLUENTS. CHAP. I.- 38. Required the fluent ofx^a'^dx = du. Assume u = a''(AJc'' + bo:""^ + cx"~^ +•••)» then 3^, which =a'x'' = la.aHAx"" + Bar"-* + cx"-^ . . .) f dx ^ >-;••• + a'(riAx''-^ + (/2 - 1 ) Ba:"-2 _) J wherefore (Alg. 347), Ala = 1 ; b/« -f wa = 0; 1 ^ c/a + (/z — 1)b = ; . . . or A = y- ; B = — ^ ; • • • a series which will terminate when w is a positive integer. In this example it appears from its form that the co- efficients are constant quantities ; but in general it is neces- sary to consider them to be variable. To integrate this form by the method of continuation, we have a" n Jaf'.a'dx = x"" .-J j- f^'^~^ .a^'dx; and using this as a formula, we have a^ n — \ fx^-\a'dx = x^-^.- — fx^-'^.a'dx, fx^-'^.a'dx = x''-'', — fx^'-'Ka'dx, a" c 71 x"""^ n(n — l)x"~^ ^ whence u = -^-^ x"^ ^ 7— ' ' i la I la la^ 3 C(yr,fx^€^'dx . . . = Saf-^. ^-^ /-^ &c. [. 39. Required the fluent of x'^^'a^dx = du. If we suppose n to be negative in the former series, it will not terminate ; but since du = a'' . x~''dx, we have /a^ . x-''dx = - 7 TT-T-T + t/^'- ^"""^ '^^ ' • • ' ^ {n—l)x'^'^ 71 — 1*" from which CHAP. I. LOGARITHMICK FORMS. 49 a^ la , a" la.a'' whence u = This series cannot be continued beyond n terms, for the « + 1th term fails. The nth. term is , r— - „ ^ ^ f . . . . a'dx There is not any known method of integrating , ex- X cept by expanding a* by the exponential theorem, and in- tegrating each term separately. If w is a fraction, either positive or negative, both tliis and the preceding series are interminable ; but either fur- nishes an analytical value of z^ The formyZi^pc^a;, where p is any algebraick function of ^, may be integrated by the method of continuation in two series by considering it either asyp . a'^dx or di^Ja" . vdx. If p is an algebraick function of a\ the form may be changed into one which is not transcendental by substituting a*' = ^, 40. Required tJie fluent of Ix'^ . x'^-^dx = du. Let s: = Ix; and assume u = az"* + bs"*~* + cz^~^ • • • where a, b, c • • • are variable ; then du, which __ s»»^«-i j^j == z'^Ja -\- z'"-'dB + z'^-^dc • • O ^dx . ^, „ dx > X ~ X J or (c?A — x'^-^dx)^'^ + f (Zb H k"'-^ + . . . . [dc + -^ y^-'' • . . = ; wherefore (Alg. 347) • • • dK—x''-'dx — {)% (Zb+ = 0; Jc + ^^ =:0; .. . X X ^" mx^ " m{m — \)x"^ or A = — ; B = — ; c = ^ — ; • • • whence n n" ' Tf VOL. II. E 50 FLUENTS. CHAP This series will terminate if m be a positive integer, wliat- evcr be the nature of n, the value 7i — excepted ; in which case we have du = Ix'^dlx : wherefore u = r-. ' m + 1 Cor, When w is a positive integer, the last term of the . . 1.2.3...m , , , series is + ;;;-^^ — a;'' ; and consequentlyy^t j;'" . x'^—^dx, if we suppose ;r = 1, in which case Ix — 0, becomes • • • "^ :;:ti — according as m is even or odd. Since Ix =— I — , we have Ix"" = I ( / — ) according: x' -\ x) ^ as m is even or odd ; and consequently (y ( / — j ,x'^~^ax) = H ^-[— in either case. To integrate the proposed form by the method of con- tinuation, we have /^"^ . x^'-^dx = — rz"'-'^x''-'dx; from which f%^-^,x^-^dx = fz^^'-'^x^'-^dx^ mz'^~^ m{m — 1 m(m— 1)...2.1 whence u = — • } ^" 91 (^ n n- + ~ n Ex. 1. Required the fluent of v'"x''dx = du, where x^ Here i; = — Z..^^ = - 2Z^ ; .*. f?« = ( — ^yii^'x^'dx ; .^ by the article CHAP. I. LOGARITHMICK FORMS. 51 u - (- ^r.,7:j:i I ^" " i;;-^ + (ni-i)^~ w(m— 1)...2.1-| Ex. % Jixfdx between x — ^^ x — \. ... Also by the article, we have fix . xdx = ^1 ^-^ - "2 I • fix'' , x'dx= ~ ^ Ix^- -^Ix + -^ ? • ^* C , 3 , 3.2, 3.2.1 J flx\x-dx^ -^S /^3__/^.+ _^^ _ ___ ^ ;r2 07^ 07'* whence ^^=^--2^+1^-^+"' ?2 ^^ 07 r 07^ 07^ 07' ^ _ c x^ x^ 7 +^^"'11:2:3- 1:2:45+ •••!■ But (vol. i. ch. 5, 15, ex. 10, cor. l)x'' , Ix"" = when 07 = ; and /o7 = when 07 = 1 ; wherefore Similarly it may be shown that ^ , 1 w w'^ ?i^ (/r"'o7'»-i , or + WA^"-^ Vin — 1)b^"-2 — . . . ( ' y y ^ 7ih.dx whence df^ — dx = 0\ d^-\ '- = ; y (w~ l)BJtr ^ , nxdx dc + -^^ = ; . . . or A =^ ; aB — or . . . y VI -^^ ■R ■= ny; dc — — n(n — \)dx or c = — n{n — Y)x\ . . . xdx do = ?z(w — l)(n — 2) — : or D= —n(n — \)(n—2)y,., ^^ \/\-x^ ^ wherefore M = ^'(^"_w(w— l)z"-2 + w(/?,-l)(;j-2)(7J-3)z"-* . . .)) + ?/(n;s'*-'— ??(7i-l)(w-2)2'^- i + 7i(w-l)(;2-2)(w— 3)(w-4)2"-s . . .) I Similarly, if ^ = v.s~^ . w and (?m = ^Vw, it may be shown that ■■■■■I + w(/z-l)(fi-2)(w-3)(w-4)2«-5 . . .) where as before x = sin.z and y = cos.z. These forms may be also integrated by the method of continuation ; which, in general, is to be preferred, because it does not assume that the form of the series is known. Thus, to take another example, let du = z'^xdx where z = tan.~^^. Here.. = -^-/j-^^ = -^ -f.dx^^l^ .^^^^ CHAP. I. CIRCULAR FORMS. 55 '^-'^^fYV^.-^fi^^=-%-^'' + /a/1 -H a;2 + «e2 = -— zx -\-U where s — sec. . z, li 45. Required the fluent ofsm.mx sm.nxdx = du. cos.(a— B)=cos.Acos.B4-sin.Asin.B ) . cos.(a + b)=cos.acos.b— sin.Asin.B ) sin.Asin.B = icos.(A — b) — 4cos.(a + b) or • • • . sm.mx sin.nx = 4cos.(m — n)x — 4cos.(m + 7t)x; whence du = ^cos.(m—n)xdx — ^cos.{m + n)xdx ; and, integrating, ^sin.(m—n)x sm.(m-\-n)x '^ . u = 1. < • > (1.) "" I , m--n m-^n ^ Similarly it may be shown that ^m\Am—7i)x %\x\,{m-\-7i)x') ,^ ^ rcos,mxcos.nxdx=^] — -^^ — -^ ^- —Y (2.) ♦^ "^ (_ m — n m-\-n ^ ^ '' ( co^.{m-\-n)x cos.{m—n)x) fsin.macos.nxdx = — 4< ■ 1 > (o.) 46. Required the fluent of sin.w.r sin.w.r cos.pxdx = du. cos.pxdxi . ^ , .7 du = — < cos.(m — 7i)x — cos.(m + n)x > .*. II = 1. Jcos.px cos.(m — n)xdx — \Jcos.px cos.(m -\- n)xdx ^ ^ sm.{ p — 'm-\-n)x six\.(p-\-m—7i)x -% ~^i ^—m + n p^m — n \ ^ i sin.(/?— w— w)^ m\.{p-\-m-\-7i)x ^ ""^C p — m — n p + m + n S It is obvious that we can integrate by this method any form which is the product of the sines and cosines of the multiple arc of the first dimension. 47. Required the fluent of sm J" xdx and ofcos.''xdx. sm,''xdx =— sin."~"^a7. Jcos.o:; whence ysin."^J^= — sin."~^a7 cos.o; + (n — l)ycos.-a: ^n.'^~'^xdx, = — sin."-^a; cos.o: + (w — 1 \/(l —sin.x)sin.^''-'^xdx, or 'nfim.''xdx= —sin.^-^o; cos.o; + (n ^l)Jsm^~^xdx, sin."~^^ cos.a; w— 1 _,. , , 1 11 or fsm:'xdx = H f^m,''-'^xdx, by which we have 56 FLUENTS. CHAP, I. „ , sin."~%cos.a7 w — 3 . . ■ . , , sin."~^^cos.a? n — 5 ^. , , wherefore /.. , C sin."""^a: w— 1 . , . fsin-xdx = -cos.^^ ^ +^^^^— ^-sm."-='^+ • • • • (yi-l)(w-3) . ^ r/(w — ^)(w — 4) -^ In this formula substitute -^ o^ for 07, and it becomes ^ « , . \cos."~"^^ w — 1 /cos. .37aa:=sm.^-< h— pr-cos."~"*a7 + . . . . *^ I n n{n — 2) •^7 ^r. jrcos."-^^ >■ +- --——^--rr~^fcos.''~''xda;. These series will always terminate when t^ is a positive integer; for if w be odd, and = 2m -\-l, they contain m + 1 terms, of which the w + 1th coefficient will be _ (w-1)(a?-3)...4.2 "^ w(w— 2)...3.1* If n be even, and = 2m, the series will contain m + 1 terms, of which the ?w + 1th will be ; ^r- r-r — x, ' 72(72— 2)... 4.2 Cor. When tz is odd, since the fluent is expressed in terms of the sine and cosine alone, it will not exceed a certain limit when the arc is indefinitely increased ; but when 71 is even, since the arc itself enters into the expression, the fluent is infinite when the arc is infinite. // /*? cLit* 48. Required thejluent of- — s— and of —. sin. X COS* X f- — ;r- =.— /'sin."~<'^+^);r(^cos.a7 = — sin.-^^+i^^rcos.a? — (w + l)ysin.-<"+2)a;(l - sin.2a:)Jar, ^. _ dx cos.o: _ dx P dx ^^ cos.a; n dx . ^^ / sin."+=^a;"'""(72 + l)sin."+ia7~''^Tl*^ihr^' ^" diminish n by 2, and we have 'm CHAP. I. CIRCULAR FORMS. 57 _ dx COS.X n—2 ^ dx „ , . , J ~ — ^ — = ~ 7 TT~- — 7~r + 7 / ~ — ;r~r-; n-om which we have P dx _^ cos.^ w — 4 dx ^ sin."-*^- {n-5)sm.--'x^W^^ sin."-«^' wheretore - sin.'"-^^ + -sin.'"-^';^ m-{-n I m+n — 2 + 7 — r- — ^T7 — — — TTSin.'"-^.r i (7?z— l)(w— 3)(m — 5) ^ . « {m + n){m-\-7i — 2){m-\-n — 4iy If 772 is a positive integer and even, the proposed form is thus reduced to fcos.'^xdx, which has been integrated, art. 47. If m be odd, the form is reduced to/sin.^ cos.^'xdxy cos ^'^^ cc which =— fcosJ^xdcoB.x = '- :; — . If it be more convenient to reduce the index of cos.^, we have either by the same process, or by substituting in (a) — X for X, and interchanging m and ?/, .... J'&mJ^xcosJ'xdx • • • sin.'^+^^cos."-^^ w — 1 ^ . o / / X ^ -f / s\\\."'x cos.** ^xax (b), from m+n m+n which we have n-\ :COS. fsv[i,^xcosJ'xdx= — '— \cos,^~^x-\- , ^. ^ m+-n (^ m+-n — 2 (n-l)(n-3) ^ 7 ^ (m + n-2){m + n-4) $ + 7 -T7-^ »x. . .. fcos.''-^xsmrxdx; which {m + n){m+n—2){7ii+n-4iy reduces the proposed form, either io J'sm.^xdx, or to . _ sin.'"^^^ .. . -J fGO^.x^in,"^xax — — TT"» according as n is even or oaa. CHAP. I. CIRCULAR FORMS. 59 The proposed form may be also changed into one that is algebraical by substitution. For let sin. a; = z\ then cos.^ = //i — ^^ and • • • dx = :. Whence ?/=/(! - z") =^ ^"^il^, which may be rationahzed when either — ^ — or is an integer (vol. i. ch. 2. 46, and vol. ii. ch. 1. 18). 50. Required the fluent of — ^^7— . ^^\x\,'^ocdx n ' ^ / = — /sin.'"'"*^.cos. ".racos.^ •^ cos.''a; -^ sin.'^-'^cos.-^^^ar ^-1 ^.. ^_2 _„ ,, -0.1 sin."'-'^cos.~"+^.r m— 1 ^sin."'"^^ _ , ^ 1. 1 . — I / (ix (c), which IS m — n m—W^ cos.''a; ^ ^ the same as (a), n being negative, and the proposed form is T ., . ^ dx ^%\x\.xdx 1 thus reducible to/ — or / — = 7 tt -— ;— , ac- •^ cos.^'o: '^ cos."^ (w — l)cos.""^^ cording as m is even or odd. If it be more convenient to raise the index of cos. j?, we must refer to the form (b) ; which, when n is negative, becomes JvciJ^xdx _ sin.'"+^a7 n-\-\ sm.'^xdx •^ cos.''^ ~~(w — w)cos."+^a7 m — n'^ cos."+^^' sin.'"^^.37 sin."*+^^ m—n sm.'^xdx '^ cos."+2^ ~ (n + l)cos."+ia7 ~ n-rl'^ cos.'*^ ' ^" ^ ^^ diminish w by 2 ; and we have sm.^xdx sin.'^^+^iT 7n — n-\-2 smJ^xdx ^ cos."^ "(w — l)cos.^-'^~ n — l •^ cos.'*-*^ ^^^ ' -^ which we obtain %\n.'''xdx sin."'^^.r r 1 w — w+2 ^ cos.'^^ ~ w — 1 ^cos.*""^^ ""(/« — t5)cos."~^^ (m— 7«-f2)(w— w + 4) -^ ^ (7i-3)(w— 5)cos.^-^a; 5 * • . . (m — 7Z + 9.){m —7i-\- 4)(yra — w + 6) sin.^'.r^a? (w-J)(n~3)(?«-'5) -^ cos."-«^ ' 60 FLUENTS. CHAP. I. which reduces the proposed form iofsinJ^xdx if n is even ; and it odd. to / . COS.J7 sin.'^jT^^o: lo iind / , we have •^ cos.^ ^s\x\."'xdx ^sin."'''^xdx f =f (1 - cos.«a;) cos.o; cos.a? m — 2, ^s\n.'"~^xdx - . „ 1 — r /cos.:rsin.*"~^ ocos.-a: ^ Also/ T— — = 2/^-—- = ^-Tr-= -2cot.2j:. '' sin.^iT cos.^iT "^ sin.^2.r sin. go; dx Ex. 3. Jw = sin.'^iTcos.j: ^-. , £?.r cos.jcdx dx co9,.xdx Here aw = - — 1--^; — - — = \-—. — r— sin.^o; cos.o; sm.^o; cos.a: sin.-o: co&,xdx _ 'x-\-2x 1 ] 1 H : — 7 — ,\ u =■ I tan. sin.'*^' * ' * 4 sin.^ 3 sin.'^^* It appears from the preceding articles that the form ysin.'".rcos.V(icos.^ ch. 2, 64, may be integrated; as also r- ; which, ••• ® «+^tan.xf 64 FLUENTS. CHAP. I. substitutinoj y = tan.ar, becomes r wn , — ?T> a rational form. ' ^^ _. . , - ^ ^ sin.^Jz , _ cos.2;^^ 55, Required me nuents of — —7-^ — and of — —. ^ ^ c/ a-\-b&m.z ^ ft + 6cos.^ _ _ . ^ sin.2(i2 sin.z b 1 o hnd / T— : — , assume ^^ — = A H —7—: — *^a+6sin.z a + osm.z a'\-b^m.z Att + A^ sin.z + B , ^ ,, . = 7-^ :orA« + B + (A6— 1 jsm.z = ; • • • « + c>sm.z - wherefore Aa + b = and Ah — 1 = ; whence a = -7 o and B = — ; and consequently the proposed form is re- dz duced to f 7—: — , which has been already integrated. _ . ., ' n (ios.xdx . , J By a similar assumption,y 7 may be reduced to u ~i~ COS.JT r "^ . '^ a + b cos.^ The same reductions may be also effected by actual di- vision. _, _ cos.zdz - sin.Zf^z , 1 he lorms r-- — and ; are the same as • • • a + osm.z a-\-bcos.z md sin.z ^ dcos.z , i , • n ^ 7^ — and — 7 ; and consequently their fluents a + 6 sin.z a-F6cos.js ^ -^ are -T-^(a + ^ sin.z) and — -j- l{a + bcos. z). 56. Required the fluent of , , 77 ' ^>.« and of • • • {p-\-qs\i\.z)dz (« + 6sin.z)" (p -^-q COS. z)dz A sin.z ssumey («-^^cos.;s)" ~ (o+Tcos.z)"-^ -(B + ccos.z)«i.^ , ,.„ . . 1 T • V 1 + / i ^ — ; ; then diiterentiatinff and dividing by ^ («+6cos.z)"-^ . ^ ^ ^ dZ) we have CHAP. r. CIRCULAR FORMS. 65 p + qcos.z ACOS.2 (n — l)bAsin.-z {a-\-b cos.;^)^ ~ (a-{-b cos.^)"-^ {a + 6 cos. zy B + ccos.^ , . + p — TT 7T—,i or » + qcos.z = Acos.zla + cos.s:) • • (a+ocos.z)"~^ ^ ^ "^ + {n — 1)6a(1 — cos.-jc) + (b + c cos.z) (a + b cos.;^?) or = ((2 — 72)a + c)6cos.-^ + («A + bB -\- ac —q) cos.^ + (w — 1)6a + «b — p = 0; wherefore (2 — m)a + c = 0; flA + 6b + ac — g' =: and (w — 1)6a + «b — p = 0. From which we obtain c = (n — 2)a; wherefore (w — l)aA -\- bs ^ q = 0, or B = -r - (n - 1)^; wherefore (w — 1)5a + -y — {n — l)-y p = 0, or . ^ ^9~ f^P . ^ _ 1. __«_ e?'- V _ ap--bq By this assumption, the proposed form, if n is a positive • ^ . , ., 1 .(b + ccos z)dz , . , , ,. . . integer, is reducible to / r- — ; wluch by division ° -^ a \-b cos.s; -^ csf bB — ac ^ dz b b ^ a-\-bcos.z' T^ . ., . . (p + qs,\n.z)(/z By a similar assumption we may integrate — ^- — . •^ I J & (a + 6sin.<^)" rri, r ,. (i? + ^sin.;2)tZ^ Jp + qcos.z)dz The forms/ 7 , . -^ and/ '; /, - — '— separate •^ («H-/>cos.2;) *^ (a-\-bsin.zy ^ P pdz q ^m.zdz pdz ^^^'^{a\rbcos.zf^(a-{-bcos.^y'^(a + bsi^zy'' ' ' ' q cos.zdis pdz pd.^ I 7 — — y— . rz ; of which / ~ — and / 7 7—: , ^ (a + 6sin.2)"' ^ (a + 6cos.^r ^ («+6sin.2)~' are particular cases of the article, q being = ; also sin^ cos..2t/:f / ;^ — —7 r-and/3 ^ — : may be chanijed by sub- '^ («+6cos.2)« ^ {a^b 9,m.zf ^ .^ J . . . , ^ . di) _ \ 1 stitution into the iorms + / -. ' ■^ — 4- {a + byY n-\ {a + by)"-'' VOL. II. F 66 FLUENTS. CHAP. 57. Required thejluent of «""= dxi^'^zdz and of al^^ cos^zdz. far'sm,''zdz - • - = —,/a/"^sin.*^~^z • dcos.z = — a''*=sin."~*<2f C0S.2 + wA/a"*''sin."-'2Cos.;^Jz + {n— l)7a"^-sin."-2^ cos^^zdz „,. . ., , mAa'"-\sin."j2 m^^A^^ . ^ , = — a"'-sin."-^2; cos.2 + fa"^- sm.^'zdz n n '^ + {n ~\)Jar^sm.''~^z{\ — %m.''z)dz, or /a'"^ sm.^rf^ -= ««« sm."-*2 cos.z n -^ n + (?i— !-)/«"'- ^m,''-''zdz, Oj^~ sin ** — '^ ^ "1 or/a*"'' sin. "^(Zz — — r-r- — --) wja sin.2 — w cos.2 S ^(^i — ]^ H r"r~i — ;; A'"^ sin."~^^j2; ; where a ~ la. By this formula the proposed form is reduced toya"*^c?2 = , if w is even ; and if w is odd, to fa^~ sin. zdz; which by the formula, 7i bein'jSj = 1, is — i- ) wzAsin.s-cos.-s >• ' ° ' ma" -{-I I i Similarly, we may integrateya"*" cos.'* W^. 58. Required the fluent of e^'' sm.m^ cos.nzd% = du. - — ] sin.(w + n)^ + sin.(m — ii)z ' ; which reduces u to forms which have been already integrated. 59. Required thejluent of e^"" sin.'^z cos."^(Z;?. In order to integrate this form, sin. '"2; and cos.'*^? must be developed in series in terms of the sine and cosine of the multiple arc, and each term may be then integrated sepa- rately. These are all the forms which are usually given in ele- mentary treatises. But in the application of the science, fluents frequently occur which are not included in any of the precedii.'g forms, and which must be integrated by inde- pendent methods. It is also requisite that the results sliould be under a form convenient for computation ; and we shall therefore conclude this chapter with one instance out of many which might have been selected of the necessity of e^'-dz i du — —7^— < s CHAP. r. CIRCULAR FORMS. 67 adopting certain artifices of calculation in order to obtain the fluent in a convenient form. 60. Required the fluent ofl(\ + n cos.z) . dz — du. l{\ + wco«.^) =:= wcos.z — 4w^cos.% + l^i^cos.'^ — . . .; and if each term be multiplied by dz and integrated, we shall obtain a series which —u ; but in integrating the forms cos.zdz, cos.'zdz'f • • • (art. 47), the results, when added to- gether, will not be expressed after any simple law. It is manifest that since they may be expressed in terms of the powers of cos. .v, they may be arranged in a series of the form A -J- B cos.-^ -|- c cos.^z • • • ; but this being a laborious process, Euler employs the method explained, vol. i. ch. 4, art. 11. Let 1(1 -{- ncos.z) = — a + bcos.2; — ccos.2z + • • • ; n s'm.z then — 1 =— Bsm.s -f 2csm.22; — . . • L ^-n cos.^ Clearing this of fractions, and expressing cos.:; sin ^, cos.^ sin.2^, cos.z sin..'i;s • • • in terms of the sines of the multiple arcs by means of the formula cos.;^ sm.mz = -i sin. (771 f 1)^ -I- i-sin.(w — 1);^, we shall have nsm.z — nc sin.;: + 2c 2 3/ZD ~ ~2 %m.9.z ~ 3d sin.Bz + wc + ^7ZE wherefore — — b | «c -f « or c = 6d B— W 71 4C— UB 2nc 1 . ., , 8e ; and similarly, f = — Snj) &c. expresses all the coefficients in terms of a and b. Now to find these ; we have l{\ + cos.;/2;) — «COS.2 — i-w-cos.^^ -{- j-z/^cos.^ — 372 ' which • • • and expressing cos.^z. we shall find cos.^^ • • • in terms of cos.22;, cos.32:, • • /(I +COS.7J2 '-{ 2T"'"2l ^2.4.6 6 "^ - &c ,n V l.S n^ L3.5w^ 3 ^ 2T4.6 "5 "^ 2.4 \ COS.2 r ^ 68 FLUENTS. CHAP. I. __J_^ K3^ 1.3.5 7i6 *''* "^ "" Y 2 '^2A 4 "^2.4.6 6 + * * * To sum this series, differentiate both sides of the equa- tion, and multiply by n ; and we have ndjv 1 1.3 1.3.5 ^ fl?w 2 2.4 2.4.6 = (1 — w^) "~ ^ — 1 ; whence dn dn . . rfA = = — — ; and mteffratuiff, ,1-v/l-//^ A = / h cor. To find the correction, a = when n = 0; but in this case, I becomes I -pr ; and to find this value, n- ,1- Vl-n' ^ l_,/l-7i2 substitute V = I z or e" = ; wherefore (e") = : =■ TTi ^^ (p) — ^ ~^''> ^^^ ^^"' 2wx/l— ^2 ^ ^ ,2-2 v/l-w' sequently a = Z j^^ . B 1 1.3 w3 1.3.5 71^ Next to find b; ^^ have -=-.^ + ^ -g^ + g^^ -y + •• • w^^B 1 1.3 , 1.3.5 , = (i-w^)~^-l, or aB = — — — , and mte- 2v/i-w^ 2 2(1-^/1 — w^] gratmg, we have b := 1 = n n n for correction = 0. Substituting these values in the series assumed for l{\ -j- cos.wz), and putting for the sake of brevity m = , we shall have /( 1 + cos-w-^-) .... CHAP. I. CIRCULAH FORMS. 69 = - I — + 2?w n 2m^cos.^z COS.JS— TW'^COS.S^ -i ^ ... ; wherefore ,2m 2msin.2 2nfsm.9.z Sm^sin.S.^ u=—zl- + — /I + n 7i 1 4 9 . . . 2m* sin. 4s . . + cor. Cor. Supposing w = + 1 and n — — 1, in which cases m — +\ and m — — 1, we have 2cos.3js 7(1 + cos.2;) = — Z2 + 2cos.2; — cos.Ss; H ^^— — ...4- cor. o ,,, X ^ 2cos.3s /(I — COS.2;) = — 12 — 2cos.;^ — cos.2;2 ...-|-cor. But 1 + cos.« = 2COS.2-— ,and 1 — cos.« = 2sin.2— ; and by substitution and subtraction, we obtain % ^ r cos.3:s: cos.5^ cos.7^ > Ztan.-^=-2|cos.;^+-y- + — g— + —^ + ... ^. 61. MISCELLANEOUS PRAXIS. 1. du = 1 .*. u — \5f— — r^, where • • • i\-{-xy+{\ + xy ' s -f 1 %^^ — \ + X. dx ^' ^^ ~ t o , g\4 - The equations for determining u are \^Cl' -f- X ) A + p 3b+q 5c + r , B = -^rVi ^ = A o V ^"^ ^^ == ~";r-;— ; where .... 2a^ 4^2 6a^^ 1 ^• A = — tan.-i — , p = x(a^ -f x^)-' , q, = x{a^ + ^^)~% and CL Ob 3./- ^or v^«^ + ^ («2 + ^^)^, 1 4 2.4 '-^ .r2" v/l4a^2 ^i.-x ^ 2//-1 70 FLUENTS. CHAP. I. 2n-^ ^ (2n— 2)(^«— 4) "" (2w - ] ){2n -Sf' "^ {^n - 1 )(2w- 3)(27i- 5)^* ~ ^ (2n-2)(2n--4)...6.4 ^_^ ^ (2w-2)(2yi-4)...6.4.2 ^/ 1 + -r^ ^(2«-l)(2«-J3)..7.5.3 S ' 2)J— 1 5./^''*~^^^'(l-^^)"2~ between a? = 0, or = 1, = • • • {m + l)(w + 3)(?» + 5)"-m + 2n - 1 V v/T^^^/ ^ , £cdx ,, - ^ 207—1 , 07 + 1 07^-1 07+1 V 07^+07 + 1 + V3tan.~^-r ^. 9. du = k==— ... . ^/?^«. 10. /r"rf^ ^ -^^ I T - 2^ + "aT - - 5 //o-/o7 c 1 wo; 7i'^o;® ^ ri^x^Jx^' r 1 «07 ^^07- ^ ■^ "TS"^^^" 4n"^"5^ ^"" ^ "^ 1.1.2 ]'¥~'5^^ 'W '" \ j,_ . . . . ^'io; 5 coso" a dx " •^ (« + /;sin.07)- a^ _ ^^ a |- 6 sin.o^' « ' Ir'^ a -h i.sin.07' CHAP. I. CIRCULAR FORMS. 71 12. fe^sm.^xdx — 3-7: jsin.^^r + 3cos.^a;-f Ssin.o^ — 6cos.a;> 13. Jm = sin {a + 6.2?) . sin.(m + noc)dx. ^sin.{a — m-\-(h—n)iv) sm.{a + m + {b -\- n)ai) ''' '' "" 2{h-n) 2(6+^^) • 14. du = a -\- b sm.x + c cos.x .*. u = I . , where ^2 = 52 _|_ ^^ and z ~ ^sin.ar + 6-cos..r. If ,9^ < a% w is a circular form. If s^ = a^, the form is reducible to • • • p dz_ •^ « + 6sin.2;* dz . c . D . 15. f -, = A2 — Bsm.2; + -7r-sm.22 — ^srn.S^;-! — ; •^ l+wcos.;2 2 3 1 2. _. 2b where a = ■ ; b = — (a — 1);c = — — A; .... 2c 2d D = b;e= c; • • • n n 72 INTEGRATION CIIAl'. II. CHAPTER II. Integration hy Infinite Series; and between certain values of the Variable by Approximation. 1. When the known methods of integration fail, our last resource, as has been already observed, is to develope the form in an infinite series, and to integrate each term sepa- rately (vol. i. ch. 2). The kind of series in which it is de- sirable that the form should be developed depends upon the nature of the question. The main object is to obtain one which under the circumstances of the case shall converge : this is indispensable ; and in order to diminish the labour of computation, that is to be preferred which converges the quickest. The subject of this chapter is of the greatest im- portance in the science of analysis, because the forms which arise in the solution of physical problems are for the most ]:)art either irrational or trancendental ; and they are fre- quently of that complicated nature as to be integrable only by developement. There is one circumstance relating to these forms which is favourable to their integration. They are required between two given values of the variables X = a, oc — b^ called the limits of the fluent ; in which case it sometimes happens that the fluent, which in its inde- finite form is developable only in a series, may be expressed in finite terms. Instances have been already given in the former chapter. If t = «, which is the beginning of the fluent, causes it to vanish, this value of :r is called the origin of the fluent. The notation proposed to represent a fluent integrated between the limits b and a isf^, placing that limit below corresponding to which the fluent is to be subtracted. Thus, f^jr^^c/x = andfl "-=-- = ^. (Encyclopae-, dia Metropol. Art. Int. Calc. p. 832.) These are definite fluents, for the correction disappears in the process of integration. The notation y;^xcZj expresses that a either is, or is assumed to be the origin of the fluent ; CHAP. II. BY INFINITE SERIES. 73 and this is also a definite form ; thus fix'^'dx— = — , which does not contain the correction. When the fluent between its limits cannot be expressed in finite or known terms, we must endeavour to approximate to its value. So many instances of developement have been given in the first volume, that in this chapter we shall add only a few examples to show that the same fluent may be expressed in different series ; and shall then proceed to the theorems for approximating to the value of a fluent between its limits. 2. EXAMPLES. Ex. 1. Required y(« -f hx'^Yx^~^dx when it is integrable only by developement. Since ]) is not an integer, represent the exponent by V 1 — ; and let du = (a -{■ boo'^) 3.1 ^ . ^ , fsin.'^O 4sin.'2« 4.2sin.O) 5.3.1 /s.n.« = - cos.«|-^ + ^^ + ^-^^ ^ +g^gO, Now the limits are z = a,z = b;oY since ^ = « cos.^^ + 5sin."^, 2" they are ^ = 0, ^ zi — , and consequently we have y. In the problem the origin of the co-ordinates is at the centre of the sphere, r =r rad., and ^ is the vertical co- ordinate. Let T =: the time of ~ an oscillation, or the time that elapses between ;s = a, z — b ; then the result is ^ = ^^^^__ 1 1 + (L\\. , (^i-^Yv If,the corpuscle oscillate in a vertical plane as in ex. 3, r — b we have a znr^ and y^ ■=. — — and •^ /^ (t (^\fr -h\ /1.8.5VA-6V ) •^=27 2;;;r + y (^) +(!iA~6; \-^) + •••}• If the oscillation be small or ?- — b nearly, we have 3. Required to obtain the "value of a fiuent between its limits by the method of approximation. Let the proposed fluxion be du =■- xdx, where ?/ —fji\ and let it be required to find the value of u between x — a, X = b^ where b -^ a — It. By Taylor's theorem, du h (I'^u k^ f(oo + h) =fx -f - -+^ U~^ • ' -fx f X y + ^^^ j^g + -J-, j^g^,. . . . CHAP. II. BY APPROXIMATION. 79 / In this equation suppose x = a-, and represent the cor- responding values of x, -^, — , ... by Y, y', y", . . . ; then h h^ h^ we have/(a + h) ^fa V Yy + y' y;^ + y" y-^ ... or If/i be small, this series converges and an approximate value ofyjxJ^ may be obtained. But in order that h may not be of a determinate magnitude, let us suppose the in- terval h — « to be divided into ii equal parts, each = h\ then by increasing w, h may be diminished ad libitum, for h — a _ n Represent as before the values of x, t-? -t-^ • • • when X — a^ by y, y', y"' . . . ; also, when x — a f //, by y,, y\, y''^ . . . ; when x — a ■\- 2^, by Yg, Y'g, Y^'a? . • • &c. ; when x=::a -V {n—\)h =^h — h,hy Y„_„ y'^^^, y"„_i ' Also denote the corresponding values of u by u, Uj, U2» — u,j_| ; then tlie required value of ;/ ory^^xdx is u„ — u. It has been shown that U, - U,-Y.y4-Y',j--^+Y",j-g^+... U«— 1 — Yjj_i ^ + Y ,j_i 1 o "I" ^^ «-2 -j o q Whence u„ - u =:= yj Y + y, + Yj . . . + Y„_ii -1- j^|y' + y', +y',...H- y',_,[ i- ... to /z terms. By assuming n a sufficiently large nuniber, any degree of 80 DEFINITE FLUENTS CHAP. II. convergency that may be required can be given to this series. This method fails when any of the fluxional coefficients of u between the limits become infinite. 4. In the preceding article, if we suppose the fluent to re- present the area of a curve, in which case x may be called the ordinate of the fluent, its value is expressed in terms of the 1st, 2d, 3d, . . . w — 1th ordinates and of their fluxional coefficients. The value of the fluent may be also expressed in a series in terms of the 2d, 3d, ... /z — 1th, nth ordinates and of their fluxional coefficients. For by Taylor's theorem, we have, taking h negative, __ dVn h d-Vn h^ or U„ - U„_i = Yn-r - Y'„:r^ + y\ 1 M.2 ' " 1.2.3 •• ^ and U„_i-U„_2=:Y„_iy — y'^_, y-g + Y^_i ^-^T^ h . h^ .. h U; - U = Y, -p- -Y,,-7, +Y\ 1 ^1.^ • M.2.3 Whence u^ — u= y/ y^ + Yg . . . + y„ I " Jy",+ y",... + y",^ "^ 1.2.3 — ... to 72 terms. The advantage which this series possesses over the former is, that since its terms are alternately positive and negative, the values of u^ — u computed by means of it are alter- nately greater and less than the true value; and the com- puter is thus enabled to ascertain to how many places of decimals the result is true. 5. Euler has also given the following approximation. Since the first series gives a result which is too small ; and the second, if we stop at a term which is positive, a result CHAP. IT. BY APPROXIMATION. 81 which is too large, if we add the series together, the semi- sum will give a nearer approximation than either taken separately ; thus we have f\y.dx = Y $ Yi + Y2 . . . + Y„_i + i:(Y + Y„) ? + -f Y^ \ Y\ + y", . . . +- y'U + i(Y" + y\) ^ J 5 y'" — y'" I + &c. The propositions contained in this and the two former ar- ticles are geometrically illustrated in the Princ, vol. i. lem. 2, in which the first term of the first series is the sum of the cir- cumscribed parallelograms ; the first term of the second, the sum of the inscribed parallelograms ; and the first term of the third, the sum of the inscribed trapezia. It is supposed that none of the terms which compose the coefficients of A, A®, A^, . . . suffer a change of sign ; or that none of the intermediate points of the curve offer any peculiarity. Ex. Required to approximate to the value o£fle ""dx. __i dx _ll d^x --i f 1 2 ) Herex = . s^ = . '7^' d^ = ' '{^^l^^l^ for = when x = 0; and consequently a: = is the origin of the fluent. Divide the interval x = I into n equal parts, in which case, nh = I or h = — ; and we have from the series, art. 5, n _- J- -I VOL. II. G 82 * DEFINITE FLUENTS CHAP. II. 48w* ' + &c. . If we suppose n = 10, and compute the required fluent by the terms here stated, the result will be true to 6 places of decimals ; if we suppose n = 20, the result will be 20 times more accurate. (Euler, Calc. Int., vol. i. art. 327). 6. From this example the advantage of the method of approximation over the common method of dev elopement is 1 very obvious. For if we develope er^dx in a series, the terms when integrated become infinite at ^ = ; and we cannot obtain the required value of the fluent. There are on the contrary other examples which, though readily integrated by the common methods, fail to be in- tegrable by approximation in consequence of the resulting values of y, Yi, . . . Thus to find the whole fluent of dx which = — 2Vl — oc -\- cor. = 2, we have a/1 — ^ y^ = X ; or this method fails. See also art. 2, ex. 2. In these cases the form may be frequently changed by sub- stitution into one which will not fail. Thus let it be re- vdx t) quired to find the whole fluent of ^, where — is a proper fraction. (1-^) p Here ^ = x when ^ = 1 ; but substituting (1 - x)~^ I — X = y'^, the form becomes — qvif-^-^dy; which does not fail either when ^ = 0, or .r = 1, or at any intermediate value. 7. It may be observed in general ,of the method of ap- proximation by equal intervals that, ca^teris paribus, the result is the more accurate in proportion as the change of the ordinate is the less in passing from one interval to an- CHAP. II. BY APPROXIMATION. 83" other. The number of the intervals which must be taken in order to obtain a given degree of accuracy must of course depend upon the nature of the fluent : i; will be the greater in proportion as the change of the ordinate is the greater. When the change is infinite, the number of the intervals should also be infinite; and this is the reason why the method fails in all such cases. If we would calculate by this method the value of the hyperbolick area whose equa- lion IS y = -rz r- between x = 0, x = 1, we should fail, ^ (1 —wy though when n < I the area is finite (vol. i. ch. 9, 3, ex. 5) ; and the reason is, that at a; = I a very small change in the value of^ produces an infinite change in the value of «/; otherwise the area would not be finite. It is upon these considerations that Euler proposes an- other method. He substitutes only for that part of the fluent's course which fails in consequence of «/ becoming infinite. Ex. l./i 8. EXAMPLES. xdx Instead of dividing the whole value of x into intervals, each = h, let the last interval be w ; then to find the value of the fluent between 07=1,^ = 1 — w, substitute x=l — ^ ; and since the limits of ^ are o and w, the form, which becomes — _ , may be considered nearly the same ^1-. (1-2)3' J ^ as — =, which integrated between ^=o, 2=0;, is 2 / - If in this example the exponent of the binomial in the denominator had been greater than 1 , we should have had w in the denominator of the result, or the fluent = x ; which agrees with vol. i. ch. 9, 3, ex. 5. Here to find the value of the fluent during the last in- terval w, substitute x ~ a — z\ then G 2 84 DEFINITE FLUENTS CHAP. II. 0-0 du = (a — zydsi a'^dz V4«^«-6aV + 4a2'-«* 2^T / ^ fl V ^ ^ Sa"^ a^ 2^ z'' 4« ^ 32a^ {u) — a^'cy^^l— — -^ \ which process may be con- tinued to as many terms as may be necessary. Ex.3. d« = -==^^=. Here x may not be greater than a or less than h ; and the ordinates at both the limits are infinite. The ordinate also during the progress of the fluent admits of a minimum value; and in consequence the proposed form cannot be conveniently integrated by the method of approximation bv equal intervals, and we must have recourse to substitution. Euler substitutes x = ^{a -\- U) — \{a — 6)cos.^; then a — X — \{a — 6)(1 +cos.^) and ^ — 6=4(a — 6)(1— cos.^) or V(« — x){x — b) = i(a — 6)sin.^ ; whence du = vd&; by which transformation the ordinate is not infinite at either limit, unless there is some peculiarity in the function p. Generally if du = where p and q are ({a'-x)P[x—by)^' less than Sn ; then if p and q are unequal and p > q, du may be put under tlie form -, and substi- {(a-xy{x~-bYy^ (a—b\i-P±l . i_Z PriB —^ J ^njj}d& sin J « (1 — cos.^) 2" . No general rules can be given to direct the student in the choice of the transformation which may be the best suited to any proposed case : this is a kind of knowledge whicli can- not be taught in an elementary treatise ; it is to be acquired only by a diligent perusal of the best authors. CHAr. II, BETWEEN LIMITS. 85 Euler in the Calc. Int. ch. 8, has given the following theorem, which is sometimes useful. x^ — ^dx 9. Thejluent of ^^fr^'^ x =0 to x =\ is equal to thejluent of ;^ffom x ^= \ to x ^=^ ^. For substitute 1 — ^" = ?/"; then when or = 0, 2/ = 1, and when ^ = 1, ?/ = 0. Also (1 — ^") n = y^-^\ and since a:^ = (1 — «/")«, we p — n have xP-^dx = — (1 — y^y^y'^-^dy^ wherefore p — n ^ ^"-^dx ^ _ {l-f)~y''-'dy ^ y^-^dy *' n — q J yn—q J n — p ' and since the values of 3/ are to be taken the same as those of 1 1 n oc^^dx .„ x^-^dx ,^ , X, It results that/i — =/? — . (Legendre, Exercises du Calc. Int. p. S22). Cor, 1. These forms may be rendered rational when X^'' 1 For substitute z^ =-• = -, ; then 1 -f- 2"= , and 1 ^^ 1 — d;"= r ;: or a" = TT— — ; wherefore 1+2 l4-;s» Ja? d% %'^'^^dz dz X Z 1+2;'* ^(1+2'i) _ , x^^dx dx / X"- \« z^'-^-^dz But du = ^=_.(^__j) =_^, ara. tional form. z^-- according as m is odd or even, 2sin. — n _ Ix . . and the sum = + — . But in this case another logarith- mick part is to be added, viz. /(I +:i') = when^ = x , Ix = -\ , according as w is odd or even ; and n — n ^ consequently the whole sum of the logarithmick parts, as before, = 0. CHAP. II. BETWEEN LIMITS. 87 Next to find the sum of the circular parts ; the first part m' aJx = — tan.-^q =5 when or = x , n \—ax W —d 2 / 'jr\ . Ttw .;t — tan.~^ = — { cr ) sin. — . n a n\ n / n similarly, the second part = — X'Tr 1 sm. , ana so on ; and consequently the whole sum St . = — (sin.a + sin.Sa + sin.5a ..•-[- sin.Z;a) i + Mn.Aa.)l St . -(sin.a + 3sin.3a + 5sin.5a To sum the second series, we have sin.(^f l)a s = cos.a + cos Da + cos.5a • • • + cos.A;a = — ^^. ; %sin,a and consequently differentiating and dividing by da, we have, — (sin.cc + 3sin.3a + 5sin.5a- • • + A:sin.Z;:a) . • • • (^ + l)cos.(A; + l)a sin. (^4-l)a cos.a "" 2sin.a ^sin.^a k cos.{A;-l- l)c6 sin.^a 9, ' sin.a 2sin.^a ' To sum the first, let s = sin.a + sin.Sa • • • + sin.^a ; then Ssin.a .s = Ssin.asin.a + 2sin.asin.3a • • • +2sin.asin.A;a. But 2sin.asin.wa = cos.(/2— l)a— cos.(^2 + l)a; wherefore Ssin.a .s:=z 1 4-cos 2afcos.4a ••• +cos.(A; — l)a — cos.2a — cos.4a ••• — cos.(/t— l)a — cos.(A:-|-l)a — v.sJk + l)a or s= — pr-^ . ^ ' 2sin.a Hence the sum of the circular parts _ cr v.s.{k-{-l)oc T r A:cos.(/^+l)a sin./ca -i ~" n sin.a 7i'^l sin.a sin.^a y When n is even, Jc + I = n and the required sum cr v.s.mT -r r {n — \)cos.mx sin.(mT — a) ^ n^( sin.a sin.^a 3 n sin.a . m<7r wsm. — n , whether m be odd or even. 88 DEFINITE FLUENTS CHAP. II. When n is odd, k = n — 2^ and the required sum ^ V.S'irri'X-'Oi) a, the function u becomes negative. Lagrange asserts that the common principles of the calculus fail in explaining this result — *' Ainsi les principes du Calcul DifFerentiel sont en defaut dans ce cas," p. 93. of m/j CHAP. II. BETWEEN LIMITS. 91 coefficients respectively of un — _^ and -jt;—{ — '^^h ; or and m are determinate magnitudes, and h not being contained in u, -= • is a constant. These functions vanish at ^ = 0, because at that point we have Ui = u and -j ^n-i ~ j n-i ' wherefore, by the lemma, cor. 2, they have always the same ^ sign during the limits of A; and this sign is positive, because their first fluxional coefficients are positive. Again these functions may be shown to be the first fluxional coefficients of ^* L2 "■ ^ d¥^ " d^^ ~ 'doc''-' * T S (and these vanish J^-'^Ui d"-2u d^'-^u h^ h^ iath = 0; ^^^ dhF^'^d^^~d^''T~^'i:2 3 consequently, as before, they are always positive. By means of ti similar processes, it may be shown that the two following functions are always positive during the limits of h. /," C du h d'^v k^ M 1.2. ..n dx""-' 1.2" {n-\) and Qi - u - ^ . y-^ ' yJ^ ^;^i • iJJ^Yf — m 1..^.. ,n' du h d^'-^u h''- Butu,= u+^^ 1 • ^^- 1.2...(n-l) - ■ • "^ d^* 1.2. ..n + • ' • Whence by substitution the two following functions are always positive, ^^'l,2-"n~' Id? ' \.^".n^d^' ' 1.2 ..(71+1)*" S 92 DEFINITE FLUENTS CHAP. II. and 3— . t-t; — +-r-rrT • r?^ tt + "^ and consequently, m • r-^r and m • tp-pj are limits; the ^ •^' 1.2---71 1.2.-.71 one greater, and the other less than the error of the result. Cor. 1. If any two magnitudes be taken, the one greater than M and the other less than m, these a fortiori are also limits to the error of the result. Cor. 2. By a similar process the limits may be found when u is a function of two or more variables. Ex. I, V = x"". Here Ui = (jc + hy .-. -j-^ = r(x + hy-^; ^'=.(r-l)(.M)-.-^"= • • • • ■ • • r(r — 1) •••(/• — TO 4- l)(x + h)""^ ; which are all positive. Now M = r(r — 1) • • • (r — w + l){x -\- hy~'', and m = r(r — 1) ...(/• — w + l)a;'*~" .*. the required limits -'-^^^^^f^V + ^r* •- and ^ ^—^ ^^r'-^/fc" ; or the value of (x + hy lies between r— 1 r(r— l)...rr-w4-l). r-1 and x" + r;r'-iA + r - —^x''-%'^ • • • Ex. 2. u = Ix or Ui = Z(j7 + h). The first fluxional coefficient is ^, which provided that J- + 7i is not = 0, neither becomes infinite nor changes its sign. Also ^ = ± (x + hY ~' accordmg as n is odd or , .. , „ 1.2.3... (to-1) even ; whence, it w be odd, m = and CHAP. II. BETWEEN LIMITS. 93 m = — _i_h\n — ' ^^ H^ + /f) IS between k h^ h^ ^ , h h^ A« and Ix \ Y- ... ± X 2x^ n{x-\-hf- Ex. 3. Uj = sm.(j7 H- //). Here -7-^ = ±sin.(a: + A) or ± cos.(.r-j-^), according as n is of the form 4/, 4i + 2 or 4z + 1, 4^ + 3, which include all the possible cases. Now -|- 1 and — 1 are evidently limits in all the four cases (cor. 1); whence we have sin.(^ + ^), where .r + 7i is not > — , always between h . 7i^ A3 J^n .m.x + cos.a;y - sm.^— - cos.^^^^+ . .. ± j^^_. Cor. 1. If ^ = O5 we have sin.Zt always between 1.2.3"^ 1.2.3.4.5 "I^^^' Cor. 2. If 07 = -^, we have cos.A always between ' ~~ To "T 1.2 ' 1.^.3.4 - 1.2.. .?* In these series n is greater by unity than the number of the term at which the series stops. 14. MISCELLANEOUS PRAXIS. . ^ dx ,^ . 1 x' , 1.3 x^ 1.3.5 x^ 1-x (l-x)lal (l-a7)/« (I— ^)W^ 1.2.3 ^ 94 DEFINITE FLUENTS CHAP. II. °'=^+ii+Ti2-+r+T+ 1:21-3- • ■ • Za;.^^ __^r J_ 1.3 1.3.5 ^ '^••^'' TflTp" ? "^ 2.32 "^ 2.452 "^2.4.6.72 "^ ' ' 5 ft A —- - n ^^ - /-i _f^__ ^ ^' ./ 1 4. »3 —0/ , — y 2. — o ,-q' ^+^ (l.-a;3)T (l~:r3)T 3V3 . _^_ _ ^^ 2^ _ ^1 ^^ /'i_f!^ ^ rp3 1 1 5 1 rp7 f , 2t^ (2t2)2 (2t^)3 ^ - also =.-.tJi + _+L^j- + L^H.... J ^ -«/ _ £l!?H 1 L3 13.5 ^ 7"^ ^^- 2t r ~2f2 + 22T^"2H'« + *" ^ .r. dv ^ ^ 1 1 1 between a; = 0, 07 = 1 . 13. J'o — = — = the fluent of the same fluxion between ^,/x a/\ —X- CJffAP. II. BETWEEN LIMITS. 95 X — ^ , 07 = 1, where a may be any value of 07. 14. The function a!" is always between , , h ^ h' , A" and 1 +la .—-\-M,rr-7^... + /<* 1 ^ l.^*" ^ 1.2...71* 96^^^ FLUXIONAL EaUATIONS. CHAP. III. CHAPTER III. FLUXIONAL EQUATIONS. The Theory of Arbitrary Constants. 1. Def. 1. The oj-der of a fluxional equation is deter- mined by the highest fluxional coefficient which it contains. Def. 2, The degree of a fluxional equation is determined by the highest power of the coefficient which marks the order ; the equation being clear of radicals. Thusj^ ^/y" + x^ — 6 — «//? — ;r = is of the first order and second degree. Def. 3. Equations in which y and all the fluxional co- efficients of z/ =fx are of the first degree are said to be linear with respect to ?/, because the fluxional coefficients are of the same dimensions as a right line. 2. The genesis of fluxional equations. It has been shown, vol. i. ch. 4, art. 27, that the successive fluxions of M = are also equal to nothing; and conse- quently all equations which result from their combination obtain at the same time, or are simultaneous. Let w = be an equation between x and y involving n constants «, 5, c, . . . ; then differentiating on the supposition that X is the principal variable, we may by means of the re- sulting equation, uJ — 0, eliminate one of the constants as a, and obtain what is called a first derivative of the primitive w = ; if we had eliminated h oy c or , , , instead of «, we might obtain n different first derivatives. By again differentiating and eliminating, we may obtain second derivatives, in which two constants have disappeared; and the number of these will manifestly equal the number n — I of combinations taken two and two in n things or n . — - — . Alg. 230. Similarly, the number of derivatives of the rth order, or in which r constants have disappeared 71 .{n — l)....(n— r + 1 ) • "^ TS r * CHAP. III. THEORY OF ARBITRARY CONSTANTS. 97 Cor. 1. There are n derivatives of the n — 1th order, in which only one constant remains ; and one n\\\ derivative from which all the constants have disappeared. Cor. S. A derivative of the //th order has n first pri- mitives, each mvolving one arbitrary constant ; n . — - — second primitives, each involving two arbitrary constants, and so on ; and one original primitive or fluent involving n arbitrary constants. 3. If the constant disappears by differentiation solely without elimination, the equation is said to be derived imme- diately from its primitive; thus, the immediate first de- rivative of x"- — 9.ay — a^ — b = is 9.xdx — 9Mdy = 0. All such derivatives are said to be the exact fluxions of their primitives. There are primitives which do not contain one constant more than their derivatives; these must be considered as in- C07nplete fluents^ or as particular cases of a canonical equa- tion. When we retrace our steps, and from the derivative return to the primitive, it is always supposed that the latter has one constant more than the former. The value of this constant is to be determined from the conditions of the pro- blem which has given rise to the fluxional equation. Cor, It is manifest that the immediate derivatives, what- ever be their order, are all of the first degree. If then an equation is of a higher degree than the firstj we know that it is not an immediate derivative, but that it has been derived from its original primitive by differentiation combined with elimination. 4. There are other fluxional equations which may be deduced from the same primitive. This will appear in the course of the chapter. We shall not call them derivatives, but shall appropriate that term to such equations as are derived from their primitive by the elimination of a constant. 5. When it is said that an equation of the uih order can have only n first primitives, it is to be understood that these primitives are all independent equations. Garnier produces the following example, as if to show that the number of the first primitives may exceed the order of the equation *. * It does not appear from the context what Garnier's object is VOL. 11. H 98 FLUXION AL EQUATIONS. CHAP. Ill Ex. (1 - x)d^y - Mxd'y = 0. It may be shown that this equation has the following 5 first primitives, each of which contains a different constant. (1 - a;)d'y - 2dxdy + ^cdx'^ = 0. (1) x{\ — oc)d'-y — {x + \)dxdy + (y — d)dx^ = 0. (2) x\x — \)d"y + 9.xdxdy - ^{y - c")dx"- = 0. (3) (1 - xYdy^ d"dx^ = 0. (4) (1 - x^d'y- (1 -x)dxdy — {y - c''')dx^ = 0. (5) But of these only three are really independent ; for add- ing twice the equation (2) to the difference between (3) and (1), and substituting c'" = 2(c" — c' — c), there re- sults the equation (4) ; also subtracting (2) from (1), and substituting C'^' =: 2c + c', there results equation (5). 6. A derivative may be derived from its primitive in two different ways ; either by differentiating the primitive, and then eliminating the constant ; or by solving the primitive vi^ith respect to the constant, and then differentiating. It is manifest that the resulting equations, though they appear under different forms, must be essentially the same ; since the same constant has disappeared. Ex. a2 -^ay - a'^ -b =0. Differentiating, 9>x — 2ap = .*. a = ~ ; which sub- stituted, gives x^ - -^ — 6 = 0, or (^2 _ jw __ p p 9^xyp - .r- = 0. (a.) Next, solve the equation with respect to «, and there re- sults a — ~ y -t Vy~ 4- ^'^ — 6 ; and differentiating, — — p -\ ■_ or p \fii^ -^ x^ — b — yp — X ' " ^ ^hf^x'^-b ^ ^ ^^ = 0. {/3.) The equations (a) and (/3) may be shown to be the same by solving (a) with respect toy?, and reducing it. 7. The degree to which the derivative rises is equal to the number of dimensions of the constant which has been eliminated. in producing this example ; he prefaces it with " Nous ferons ici une remarque importante qui n'a pas echappe a M. Dubourguet, qui I'a presentee avec beaucoup de developpement dans son Calcul InUgral, p. 272." CHAP. III. THEORY OF ARBITRARY CONSTANTS. 99 The second method of deducing the derivative has this advantage over the first, that the resulting equation is solved with respect to p. If the primitive contain n constants, by adopting the second method we may obtain a succession of derivatives of different orders, in each of which the highest fluxional co- efficient rises to only one dimension. Cor, If we eliminate x or y from two derivatives of the same order, the resulting equation, which contains one con- stant more than either of the derivatives, shall be of the same order, but its degree shall equal the degree of the variable which is eliminated. 8. If we eliminate p by combining any two of the n first derivatives, there will result an equation involving the n constants, which is therefore the original primitive. 9. An equation which is homogeneous with respect to cc and y^ does not admit of a second derivative. For when the equation is homogeneous, and of n dimen- sions, dividing by oj", it takes the form of a function of — , and consequently the values of — may be found in known terms by the theory of equations. Let in be one of y them ; then — = we or j/ = mjr, therefore ^ = w«, a con- stant quantity ; and the equation cannot be differentiated a second time. 10. If from a proposed equation of the first order there be derived one of the second, either immediately or by means of elimination; and from this latter, there be de- duced by any process whatever a first primitive essentially dif- ferent from the proposed equation, eliminating f from the two equations of the first order, there will result the re- quired primitive involving one arbitrary constant. Generally, if from a proposed equation of the first order, there be deduced, immediately or not, a succession of equa- tions of every order as far as the rth ; and from this latter, there can be deduced one of the r — 1th order involving one arbitrary constant a ; there will be altogether r equa- tions, from which, if the fluxional coefficients be eliminated, there will result the primitive of the proposed equation involving the arbitrary constant a. H 2 100 FJ.UXIONAL EQUATIONS. CHAP. III. 11. EXAMPLES. Ex. I. 2/ + flj: + ^ = 0. There are two first derivatives, j/ — a:p-\-b = 0\ p + « = j ; The second derivative is g' = ; or the equation admits not of a second derivative. Ex. 2. .y + «a7 4- c VI 4- tt" = 0. The two first derivatives are 3/ -' xp -\- c ^/l +^2_oi p -\- a =0 j' and the second is ^ = 0. If p be eliminated from these, there results the original equation. Ex. 3. x^ -9.ajj - a'^ - h = 0. The immediate first derivative \s x — ap — {a) '^ and X eliminating a, which = — , another first derivative is ^xv x^ x^ ^ — & = 0, or {x'^-b)p^-'^xyp-x''-=^ (/3), from which may be diediucedi p \/x^ + y'^ — b—yp—x = (7). If J9 be eliminated by combining (a) with either (/3) or (7), there will result the original equation. The second derivative of this example may be deduced in four different ways. First, differentiating (a), we have 1 — aq = or a = — , p and, by substitution, a: — 0, or xq — p = is the se- cond derivative. X Next, differentiate a = — , and there results P = ^ or ^<7 — p = 0. It may also be deduced by eliminating h from (y). For, differentiating, p{x+yp) q Vx^-Vy^-b +-^^=^% ~ 2/y - jo^ - 1 = 0, in ^/x^ •{■y^ — o ——————— tfT) -\- X which, if we substitute for y/x"^ + y^ — b its value ^^- > P CHAP. III. THEORY OF ARBITRARY CONSTANTS. 101 there results ^^^^^ -\-p^'—yq—p^ — \ =0 ov ocq —p = 0. Lastly, differentiating j;^ ^ — 6 = 0, we have /? f' p^ p^ ( p ) i P P } and consequently xq — p =^ and 7/p -\- x = ; of which the latter is not a first derivative ; it is the equation which results if we eliminate a and b by combining the primitive with (a) and (y), taking the negative sign of the radical. Ex. 4. 2^j/ + «3/2 — bx'^ = 0. The two first derivatives are {xp — y){bx — «/) = > {xp-y){ X -\- ay) = Oy which may be obtained either by differentiation and elimina- tion ; or by reducing the primitive to the forms — + a -=0, — 4--^^ 6 = 0; and then diffe- y y- X x^ rentiating. The equation being homogeneous does not admit of a second derivative; or ^ = 0. (Art. 9.) It appears from the first derivatives that xp — y = is also an equation of the first order derivable from the pri- mitive. This equation may be deduced by the elimination 2<2? bx^ of a and b ; for, since \- a — = 0, therefore, dif- y y- c . . \ xp bx bx"p ^ , xpy—if , ferentiatmff — -^ ^ —0,'.b = -^-^ — — ; and y y y y x^p—xy differentiating a second time, making g' = 0, we have ^ ^ -yp+xp'' {xpy-y'^{xp -y) _ x'^p —xy {x'p— xyy x\xp—yY x^ ^ ^ The same equation will result, if we eliminate b first and then a. Ex. 5. y = 6(«2 _ ^2), The two first derivatives are ^«/ + ;?(a^ — ^r^) = ^ . yp ^- bx =^ ^y and the second derivative is xyq + xp'^ — yp — ^. 12. The Integration of Fluxiunal Equation.^, 10i2 FLUXIONAL EQUATIONS. CHAP. III. Since the first derivatives contain one constant less than the primitive ; the second derivatives, two less, and so on ; it follows, e converso, that the canonical or complete pri- mitive contains one arbitrary constant more than its de- rivative of the first order, two more than its derivative of the second order, and so on ; and that it cannot contain a greater number of constants, because more could not have disap- peared by differentiation and elimination. If from a fluxional equation of the tith order, we can obtain by any process whatever an equation u = 0, which involves n arbitrary constants, w = is called its complete primitive or Jluent. It follows also, e converso, that every fluxional equation of the second order has two primitives of the first order, each involving adifi'erent arbitrary constant; and generally, those of the 72th order have n different pri- mitives of the n— 1th order. As these two propositions admit of direct demonstrations, they shall form the subject of the following articles. 13. The complete value of y in terms of x as deduced from afluxionul equation of the nth order involves n ar- bitrary constants. Since y =fx, therefore (Maclaurin's Theor.) y =/• +f f +./" r^ +/•" 1J3 + • ■ • °f "'hich f is a constant quantity; for it is the value o^ fx when a: = 0. First, let the equation be of ihe first order, F(.r, j/, p) = 0. Here j9 is a function of {x, y) ; and therefore y.', which is the value of/? when ^ = and y =/*•> is a function ofy*. Differentiating f(,27, y, p) — 0, there resultsy*(a7, j/, p, q) = ; and supposing or = 0, we have y =f and p a function ofy! ; consequently {q) orf" is a function ofJ\ Similarly it may be shown that/.'", /.'% • ' • are all func- tions ofy*. Next let the equation be of the second order, f(^, y, p, q) = 0. Suppose X = 0, then it appears from the equation thaty." is a function of {f,f') ; and by differentiating, it may be shown as before thsLtf."',f.^^, • •• are all functions of (/.,J^') ; or the expression for y involves two constants ; and so on. This demonstration holds good even in those cases in which the developement given by Maclaurin's Theorem is only an analytical and not a numerical value of y; and the CHAP. III. THEORY OF ARBITRARY CONSTANTS. 103 reason is, that the number of the constants in y —fx can neither be increased nor diminished by any analytical process. Vid. vol. i. ch. % J23. Cor. It appears that the arbitrary constants possess this peculiar property, that when y is developed in a series by Maclaurin's Theorem, they are equal to /*., f.\ /.", • • • respectively, the equations being continued as far as the number of the constants. ] 4. To afluxional equation of the nth order there belong n different primitives of then — \th order. For when 3/ =fx (Taylor's Theorem), ^, 7X h h^ h' f{x +h)=y +p. J + q, — ^r.j-^ + .- . Suppose h ~ - x; then /(a; + h) =f(x — x) =/., for f(x — :r) is the same asfx when x = 0; and the equation becomes/. =3^-i?.-Y + g.j;2-^. j^^jg + •••(«)• For y substitute p, which is a function of x, and (a) be- X x^ comes f' = p-q'Y'^^'r2 ^^^' Similarly in (a) substitute q, r, • • • for y, and there results/." = (7 — r . Y" + • • • (7). First, let the proposed equation be of the first order, By differentiation, it will appear as in the preceding article that^, q, r, - - - are functions ofx^y; and if these values be substituted in (a), there will result an equation involving one arbitrary constant/ Next, let the equation be f{x, y, p, q) = 0, Here q, r, s, • - • are functions of x, y^ p\ and sub- stituting their values in (/3) and (y), there will result two equations of the first order, which are primitives to the proposed equation ; and so on. 15. If then from an equation of the second order there can be deduced by any process whatever two equations of the first order, each involving an arbitrary constant, these are the first primitives of the proposed equation ; and every other equation of the first order from which the proposed 104 FLUXrONAL EQUATIONS. CHAP. III. equation can be derived must be a particular case of one of these primitives. When these primitives are determined, eliminating p, there will result an equation which involves the two con- stants, and which is the canonical fluent, or original pri- mitive of the proposed equation in as general a form as possible. Generally, if from an equation of the Jiih order, there can be deduced 71 equations of the ?z — 1th order, each in- volving an arbitrary constant ; eliminating all the flux ional coeflicients, there will result the original fluent involving n constants. CHAP. IV. FLUXIONAL EQUATIONS. &C. 105 CHAPTER IV. Fluxional Equations of two Variables of the first order and degree. 1. The two principal methods of integrating a fluxional equation which contains two or more variables are (1), by separating the variables, and then integrating each term ; and (2), by finding a factor which renders the equation an exact fluxion. When the variables can be separated, the equation is said to be integrable, as it is supposed that the fluents can be ob- tained by the common rules. It may be here observed, in general, that it is usual with analysts to consider any problem as solved which has been reduced to one of a lower class. Thus, an equation of the second order is said to be integrated, if the problem can be made to depend upon one of the first order, though this latter equation may be such that we have not the means of integrating it. 2. When the fluents are transcendentals ; the algebraick relation between the variables may be sometimes found. ^ ^ dx dy Ex. 1. — =^ + ^ — = 0. Integrating, and correcting the fluents by a like quantity, we have sin.~^^ + sin.~^^ = sin.~^c; and to find the alge- braick relation between x and z/, let sin.~^jr = x, sin.~^^=Y ; then c = sin.(x f y) = sin.x cos.y + cos.x sin.Y . . . — a? v/ 1 — «/2 + ?/ ^1 — x^. Ex. 2. dtan.-^x + Jtan.-^y + dtan.-^z = 0. Integrating, tan.-^c = tanr^x -f tan.-^i/ + tan.~'2; or 3. SEPARATION OF THE VARIABLES. All equations of the form xdy + Ydx ~ 0, where x and y are functions of x and y, are integrable. 106 FLUXIONAL EaUATIONS OF TWO VARIABLES. CHAP. IV. Q/fj citJC For, dividing by xy, ~ -{■ — = ; in which the variables are separated. Ex. 1. ydx — xdy = 0. TT^- . 1. , dx dy ^ , . Dividing by .rj/, ^ = ; and integrating, X y Ix — ly =. Ic ox X := cy. Ex. % (1 + x'^)dy = 2j^dx, {J9J is y^ ' Here -^ = ^i :; and ^y^ = tan.-^^ + c. y- ^ Ex. 3. ^¥ — y^dx = V^ax — x"dy. dy dx • , «/ , ^ - Here — : = .'. sin.-^^= t;.g.-^— • . . ^b'^—y'^ ^2ax—x'^ o a , , I2x X^ b ~ = sm. / -.*. y— — \/'Zax — x% This solution is not the complete fluent of the proposed equation, since it does not contain an arbitrary constant. 4. All equations of the form s-Ydy + x'r'dx = where x' and y' are any functions of x and y, are integrable. "Ydu ^^dl! For, dividing by xy', -j- + ' = ; where the varia- bles are separated. Ex. 1. x^ydx + (3j/ + \)x'^dy = 0. Dividing by x^y, x^dx = dy .*. &c. 5. An equation is integrable, if by any artifice of cal- culation y one side of the equation may be rendered an exact fluxion, and the other a function which contains only one of the variables. „ ^ xdx-\-ydy Ex. 1. - , \ - = Ydy. Multiplying by 2, -^^^±^ = gyjy l(x' + f^) = yrdy + c. CHAP. IV. SEPARATION OF THE VARIABLES. 107 Ex. 2. '-te^ =-<%,. Dividing by y, - ^ ^ = ^ .'. lxy= l— ov xy'^ = c^. Ex. 3. adx = ydy — xdy. Subtracting ady^ a{dx — dy) -{■ (x ^ y -\- a)dy = .*. a(dx—dy) y ^ c -J^ , dy = ^^ ^^ /. — = / or X = ce «-f y — a. •^ x—y^-a a x—y-\-a Ex. 4. adx^ = ydy'^ — hdxdy. Here the variables may be separated by the solution of a h h^dif quadratick ; and we have dx^ + — dydx + —r~ _ (toy + b^^ Ex. 5. ydx = y^dy + y^-dy + xdy. __ ydx— xdy ^ -, ^ 2^" Here-^-^-^ = ydy +dyr.j=^+y. Ex. 6. aHx = (x + j/)%. dx {x+yY dx -\- dy __ (x + y)'^ -\- a"^ ^ dx + dy dy~ «* ' ' dy ~~ if- ' ' {x -\-yy + a^ dy . . (x + y) y + c =■ -f .-. integrating, tan.~^ '— = or X 4- y = a tan. . This example may be integrated by substituting 2= ji"+^. Ex. 7. {xdy + ydx)^a' — x'^y^ = ^ ——^, (x^ + yy^ The first member of the equation is of the form dz ^/a?-—z^ which is a circular area ; and the fluent of the second is 9.[x'^ + y'-Y- Ex. 8. dy — (x -^ yydx = oTdx. Substitute z = x -{- y .-. dz — dx — z'^dx = a^dx .•.••• 108 FLUXIOKAL EQUATIONS OF TWO VARIABLES. CHAP. IV. dz ^ dz Ex. 9. dz = x^dx + xi/d^ + y^dx where z — yjc. Here a;c?2 = x^dx + ^V%/ + ^^^dx or xdx{x^ + i/^) + ;r^j/ ( — , — , • • • ). ^ \x ^ X J For any term as x^y^^z'', where p -V q -\- r = n, being divided by ^" gives ^^ = r — J •( — ) which \x' x) Cor. 1. By assuming t = — , v = — , • • • an homoge- X X neous function may be transformed into one which shall contain one variable less than the original. Cor. 2. The theorem of the article does not necessarily include transcendentals ; for logarithmick and circular func- tions when expanded cease to be homogeneous. 7. If du = Adx 4- -Rdy-{-cdz-\- • • •, where a, b, c, • • • are ho7nogeneoiis functions of (x, i/, z, • • •) of 7i — 1 dimen- sions; then shall ax + By + c;s -\- • • - = nu. For, in the function u suppose that there be substituted y z=z tx, z = sx, &c. = &c. ; then it takes the form u = vx* where p = f(^, .?,.•. ). 110 FLUXIONAL EQUATIONS OF TWO VARIABLES. CHAP. IV. Let dv — pdt 4- qds + • • • ; then du, which = x''dv + nvx^'-^dx — n?x^-^dx + x''{pdt + qds + •••)• But du = kdx + -R^ldx 4- xdt) + cisdx + xd&) -f • • • ; wherefore, nvx^~^dx -\- px'^dt + qx'^ds + • • • = (a + b^ + C5 + • • • )dx + -Bxdt + ca:^* + • • • ; and, equating co- efficients, 7/PJP"~^= A -f B^ + C5 + • • • = A + — + h — X X or A^ 4- B2/ + c;s -f • • • = wp;r" = ww. ^ ' t^M du du Cor.l.nu = ^^ + ^^y+^^z + ... ^ ^ T^rl ^ ^^W ^'^ ^W Cor, 2. When 7i=0, — ^+j-«/+ -7-^ + -.- = 0. dx dy^ dz Cor. 3. (n — l)du = xdx + yd& -j- -^c + . . • For, since nu = Aa; + bj/ -|- . ' • , we have w dx dy I dx X ^ , du xdy—ydx du J y\ ,. , , . or du=-jycx —-^^-^ x d[~^y which being an exact fluxion, it follows that -j-x = 01 -— ) * and • • • dy ^\xj du = (p(^-\d{ — j; and consequently u = rf ~ j. 9. Required to investigate the relation between an homo- geneous function of given dimensions, and its fiuxional co- efficients of any order. Let w = an homogeneous function of (or, 3/, 2, • • •) of w * This is the converse of the proposition established in vol. i. ch. 1. ; viz. if M = F.r, du = <^xdx. It follows immediately from the nature of fluxions. CHAP. IV. HOMOGENEOUS EaUATIONS. Ill dimensions; for x, «/, • • •, substitute tx^ ^j/> * • • ? then, since the function is homogeneous and of n dimensions, u becomes f'u, and the equation becomes t'^u — r(^ir, ty, • • •)• In this equation, suppose t to become t -^^ h, t at the same time being made = 1 ; then (vol. i. ch. 4, 20). ^l.^Xdx^ dy'^ -^ dz^ dxdy ^ 4- &c.; + &c. \ whence, expanding (1 + JiYu and equating terms which include the same powers of the indeterminate quantity h \ we have du du du ,. d^u d^u d"u 2d-u 2d-u ^ ^ dx" dy'^^ ^ dz'^ dxdy ^ ^ dxdz &c.==&c. +^2^^- 10. Homogeneous equations of the first or dei' and degree are always ititegrahle. For, let vdx + Q,dy — be an homogeneous equation in which p and a are of n dimensions ; in it substitute ^— = z X ox y — zx\ then (art. 6) p = x'^'EZ and a = x^fi\ wherefore by substitution, the equation becomes, when divided by x^^ Fz.dx + fz.dy = ; and to eliminate dt/, we have dy = zdx + xdz, and consequently ¥zdx -\- Jk[zdx + xdz] = 0; dx fz.dz ^ 1 , . , 1 or 1- TT- = ; where the variables are separated, X FZ-\-ZJZ ^ ' and the integration can be effected. Ex. 1. y'^dx + jydx — x'^dy = 0. Substituting y = xz and dividing by x% ..... fl IT tl/S ^ z'^dx -\~ zdx — {zdx + xdz) = .*. — = — and c =xe~v, X z^ Ex. 2. xdx + ydy = nydx. Substitute y = xz, and dividing by ^, 112 FLUXIONAL EQUATIONS OF TWO VARIABLES. CHAP. IV. ax + %{xaz + %ax) = n%ax or — = , ; — - . . • • ^ ^ X 1— W2 + 2^ \ndz—%d% \-ndz . . = r — _— ; and, inteffratino^, Here n may be either > , = or < 2. 1. n > 2. Let a and — be the roots of 2" -wz -h 1 = ( Alg. 292) ; 1 «- + l , 1 then n = a + — = ; and a a l—ttz + %'^ 2(«2_l)'' 1 ^^ ^'-^ 2(a^-l)a3-l "" "V X "*" aV "" 2(«'^-l) ««/-a: •• • • • , , «'- + l ,ay—a"x 2. n = 2. 1 ^ = c — r or Z(a; — «/) H = c. 1 — ;^ ^ ^^ x—y S. n<2. Let n = Scos.fl; then I — n% i- z'- =(z - cos.9)- + sin.^fl ^ dz 1 z— COS.9 ... .'. / -: z= — — -tan. ^ — : — 7— =, adding ^ 1-71^ + ^2 sin.^ sin. 9 ' ^ 1 COS. 9 . rw • 1 -T — rtan. ^-^ — - in order to correct from ^=0, as m ch. L 8, sin.9 sin.^ ' ' ' 1 , ;^sin.9 , -: — 7tan.-*r 1: .\l.s/x^—nxy-\-y^ sin.6 1— ^cos.fi •^ -^ COS.0 ysin.^ + -T— rtan.-'-^^^ ; = c. sin.e ;r— ^cos.^ CHAP. IV. HOMOGENEOUS EQUATIONS. 11^ Ex. 3. xHy = yHx + x'^dx. Substituting y = x%, and dividing by a?% ssdx + xdz • • • =^%^dx -f dx .'. a:6?2; = iz^-z+Vjdx .*. — = -; r • • • X «* — 2 + 1 .-. /^ = -Ltan.-i?:^ .•.&€. Ex. 4. y^d[y = ^yxdx — o^^dfz/. Here it will be more convenient to substitute x — yz\ and dividing by 2/^, dy — ^z{yd?i + 2;c^^)— ;s24y = %^^2+2;22c(y — ; QX y'^ — 2x'^ = c^^ is the required equation. . {if'^-2x'^y Ex. 5. «2^f^^ = («.r — -v/^^ + y'^)dy' Substitute x = y%', and dividing by y, a{ydz + ;2:6^) dy adz = (a;^; — -v/^« + l)Jj/ or 2/ Vl4-2^ (a;+ Vx'+y'Y — = (;S + ^/l + ««)« = 3^ r Cy-1 = (07 + >v/^'' + 2/^)''. _ y Ex. 6. ^^/c?«/ — «/56?a7 — [x + yYe ''dx. Substituting y = x% and dividing by x^, z{zdx+ xdz) — o 7 /I N 7 dx e^zdz , . . ^ ^ z'^dx = (1 + ^)2^-^6?.r .-. — = .-. (ch.i.art.42,cor.) ^ V^ + 2) I — = ^j— — = or {x + 11)1 — = xc c 1+;^ y ^ ^' c X — ^ dx ifdx ydy ydx—xdy dy ^ Substituting y = x^ and multiplying by x, dx + z^dx — z(xd;s + ssdx) + {zd:P'-'(xdz-\-zdx)] ^/l + z^ c?^; ^(i.s 8c?a7 , , xdz + 2- + -27 = 0or-^ zxdz- ^cZ^^/l+22 + — = 36?.r / 1 \ VOL. II. - I z or 114 FLUXIONAL EQUATIONS OF TWO VARIABLES. CHAP. IV. Ex. 8. (a;2 - ai/^)da: - a^dt/ = 0. Here fl - ""^^ V^ -di/ = 0; substitute -^ = dx tt2> y := zx ,', (1 — ass^)dx = zdx + iccZa or — = ^ *17 1. Z CIZ 11. Examples of fluxional equations which are not homo- geneous, but which may be rendered such by proper sub- stitutions. Ex. 1. Let (ax -\- by -\- c)dx + {ex -\~fy + g)dy = 0, which is a canonical form. Assume ax -\- by + c = v, and ex -\-fy -\- g —w .*. dy = — ^ — 7 — , .-. by substitution and multiplying by of— be, the proposed equation becomes v{,fdv — bdw) + *w{adw — edv) = or {fi — ew)dv + (aw — bv)dw = ; which is homogeneous. Otherwise. Assume x = v-^ m and y = "rw + ^ ; then the equation becomes {av -\- bw -^ am -\- bn -^ c)dv + (ev + fw + em + jfn -{■ g)dw = ; and if we suppose am + bn + c = and em + fn -\- g = 0, from which the requisite values of m and n may be obtained, there will result an equation which is homogeneous. Cor. 1. If 6 = e, this equation is immediately integrable without substitution, as will be shown in art. 22. Cor. 2. If «/* = be, then m = co and n = cc , and the example fails ; but in this case the proposed equation may be reduced to the form {ax + by)dx + (ax + by)dy + cdx + gdy = ; in which substitute v = ax -\- by and the re- sulting value of dy, and there will arise an equation in which the variables may be separated. Ex. 2. axdx -\- 2aydx -^ bxdy — abdy = 0. This equation being of the first order and degree with respect to x and y, is integrable by either of the methods of the preceding example. Assume x = v + m and y = w -\- n, .', (av 4- 2aw 4- am + 9.an)dv -{■ (bv -\- bm — ab)dw =^ 0, .*. we have am + ^an = ) ^ j .i Jm - fl6 = I ••• ^ = «' ^ = - 2 ' ^"^ ^^^ P"""^ CHAP. IV. HOMOGENEOUS EQUATIONS. 115 posed equation becomes avdv + 2awdv + bvdw = where V = X ^ a and w = y -{- --, Ex. 3. 2axdw — 2bydy — 4aj/da: + bxdy — a^dx = or ('^ax — 4ay — aP)ax + {ftx — ^by)dy = 0. If this be compared with the canonical form, it appears that qf'= be; but if we assume v=2ax — 4!ay (ex. l,cor. 2), 2ax—v . , 2adx-'dv ,, , . . , t/ = — T and ay = ; and by substitution, there ix / ox 7 bv ^dx—dv - , , . results {v — a^)dx + -- . ; and by reduction, , bvdv ax = Sa^v-^-^abv-Sa*' Ex. 4. V«*^^ -r ay^ . dx = y^dy. Substitute y^ = axP-^ and the equation is rendered homo- geneous. Ex. 5. x^dx + ^ ■ = dy, Va+ y Substitute v« + «/ = «?, .'• x^dx + %To'^d'w =■ 9>ivdw. Again, substitute x^ = v, .*. ^vdv H- ^vdw = 2wdw, Ex. 6. ^^07 + (bx + cy)dx = edy + (6a; + cy)ndx or ado: — ^(6^ + cy)dx = edy, if s = w — 1. Substitute v = bx + cy, and the equation becomes {a — sv)dx = — (Ji; — ^Jo:) .*. {ac + be — scv)dx = edv .'. dx = J .*. &c.' ac+be — scv Ex. 7. -^ = « + ^^-^ ^ dx a—x-\- 2y By substituting x = v — a and y = w — a, the equation becomes (2u — w;)(^t) + (^ — ^w^dw — 0, which being ho- mogeneous is integrable ; and there results c^ ={x - y)(2a + x ■\- yf. Ex. 8. e'dx ^ ^= dy - ydx. Substitute v =^ e'^ ,'. dv •\' — — dy — - — , which is ho- mogeneous; and the primitive is tan.-^^/.g-* —c-^l^fe^-\-y''• 1 ^ 116 FLCXIONAL EQUATIONS OF TWO VARIABLES. CilAP. IV. Ex. 9. e'^dx — '^-^ — dy — ydx may be integrated by the same substitution. Or thus, (f + y)dx = ( 1 + ~ Vj/ •*• e^dx == dy ,\ • • • ^ ■= y -f c. Here ^"^ + «/ = is an equation which satisfies the proposed, but it is not one of its fluents. 12. Equations may be sometimes integrated by sub- stituting .a; = Aj/ + t; + B ; and by assuming a and b properly, coefficients of the resulting equation may be made to vanish, which will render the remainder integrable. When the equation is of high dimensions, this process becomes very laborious. To illustrate the principle, we shall take Ex. 1 of the last article. Ex. 1. {ax -\- by + c)dx + (ex •\-fy + g)dy — 0. Let J7 = Aj/ + I? 4-B ; then, by substitution, {a{^y + v + b) + 5j/ + cj {Ady + dv) + [e{Ay -f i; + b) +fy -^g\dy = 0; or a(a« + 6) ^ ydy + a« } vdy + ( aa + h)ydv + ax:dv .... + Ae+/5 +e I -\-ABa + Acldy-\-{^a\-c)dv — 0. + Be-\-g 3 If the 2d and 3d terms could be destroyed, the equation would be immediately integrable ; but the equations of con- dition are Aa -\- e —0 and a« + 6 = ; or 6 = ^ ; in wliich case, as will be shown (art. 22), the proposed is integrable without substitution. If the two last terms be destroyed, the equation becomes homogeneous, and consequently integrable ; but the equa- tions of condition are a(bw-I-c) + b^ +g" = and Ba-|-c— ; wherefore Be -\- g =0\ and eliminating b, ag = ce, so that the two last terms cannot be made to vanish unless this condition obtains. But if we destroy the 1st and 5th terms, the proposed becomes (a« + e)vdy + {Aa -j- b^ydv + avdv+{Ba + c)dv = ; which is a linear equation, and is integrable by the formula of art. 15 ; and the equations of condition are A"a + A(b + e) -hf = 0y and B^Aa + e) -h AC + g = 0, which are two independent equations by means of which a and B may be determined. Ex. 2. abxHx + ¥yxdx + a'^ydx + orbydy -f a^xdy — 0. Let y.= AX -\- V + b ; then, by substitution, [ahx'^ + (b^x -f a^){Ax + r + b) j f/.r -f- CHAP. IV. HOMOGENEOUS EQUATIONS. 117 {a'b{AX-\-v + b) + aV^(Ada; + dv) =0; or {ab -\- A.h'^)x^dx '\-¥vxdx 4- (b62+2a«3 + A^a"h)xdx + {a^ + Aa'^h){Bdx + vdx) + {Aa'^h + a^)xdv + a-bvdv + Ba-6c?t? =0. First, suppose « + a^ = 0, or a = 7-; which will cause the first, fourth, and fifth terms to vanish. Next, suppose b/;^ + 2Aa^ + A:^a'^b = ; or eliminating a, bJ^ ^ = or B = -7^ ; then the proposed becomes til dv Qi dv bHxdx + a''h>vdv + -7— = or b''xdx-\-a''L^dv-i = 0, o^ v which is immediately integrable. 13. PRAXIS. 1. (6xt/-y^)dx + (Sx^'-2xi/)di/ = .-. nx^i/-i/x=:cK 2. xdx + ai^dx -h ydi/ = .*. l^x^ + axy+y' = -f— dz az+z- 3. y^dx — xi/dx + x'^di/ = .'. x = cev. 4. y'^dx + xydy + oj^rfz/ = .'. y^x = c^x + 2j/). 5. 3/^^+ (^ + ^j/)dx = .'. Z^ + ^ = 0. 6. ;r2^f?.r — y^dy — x^dy .*. a?^ = ^yH --. . ^3/ _ 7. dyVx"-{-y'^ — ydx .*. -^ = ^1 + ^^cZ^ + zcZ^f .-. &c. 8. (aj+j/) dx=.{x-~ay)dy .*. tan.~*— = aZ ^. 9. xdy — ^/c/o; rr Ja; Vx'^ — y^ .',1 — — sin.""^ --. c X 10. zrdy — ydx zz dx ^/x^ + y'^ .-. ^r^ zr c"^ + Scz/. dx dy dx dy ' X y ' y'^ X " 13. x^dyz=.f{dx + %) ,^ dx dy dx dy 11. ^-^ : — ^ : : w : 1 .-. (y— a;)"-'(^ + a:)"+' = c2". 07 2/ y X 118 FLUXIONAL EQUATIONS OF TWO VARIABLES. CHAP. IV. x^x'^y'^ •\-y ^ y^ •' y 1 13. xdy—ydx = yl^—dx .'. y "=■ xe'^. X 14. Fluxional equations of the first order and degree may be classed according to the number of terms which they contain. If the equation contain only two, the va- riables are always separable ; thus, let ax^y^dy = bx^y^'dx ; dividing by x"^i/'^, we have ay^~^dy = bxP~"'dx, If the equation contain three terms, it may be reduced to the form dy -\- hy'^dx = ax'^dx. For let the equation be cx^y'dy + bx^y^dx = ax^'y^dx ; then dividing by ca^y^^ we have i/~'^dy H if~^xP~^dx — — x^^'^dx. In this equation sub- stitute y^''my = % and x^~'^dx = = — i ; and there , , b{l-s + \) -J=^^ a(Z-5+l) -!:=±., results dw + — -, — -r{wi-^+mv = — ^ — r^t;/^^+icZu; c(^p-'k\-\) c{p—k-\-\) ' which is of the proposed form. If w = 1, the equation is linear with respect to y. If w = S, the form is 6^ + by"dx = ax^dx ; which is known by the name of Riccati's equation, who was the first that separated the variables in it. 15. The linear equation dy + Tydx = adx, where p and Q are functions of x, is integrable. For, assume y = xz% then dy = ^dz + zdyi ; and by substitution, i^dz + zdy. + PX2;J Sm+S ,„ ,., m"4-4 = — — rTT> = — ^ P? or m'" which = — „ . ^ . • • = — Q — TTt"' ^^ ^^* ^^ equal to or —2 or —4. The values of m which satisfy these conditions are — 4, 8 12 16 Q , — p,,— ^-, — &c. and are included in the form CHAP. IV. . llICCATl's EQUATION. 125 4i where i is any positive integer. 2i-i Next transform the original equation by substituting y zzL — and x'^^'^ r= x^ and the result will be of the form m di/ + b'y^'dx' z= aV'^'di' where m' = — : ; which there- fore IS mteffrable when ; z: — ~ . — r or mzz.^— — =- ; and consequently the proposed form is integrable when 42 ?w rr — 22 + 1* Cor. 1. The equation dy + ay'^-x^dx — bx'^dx, the form considered by Riccati, may be reduced to the form of the article by substituting x^dx z=. dv. Cor. 2. . When m == 2, ? i^ oo ; or it requires an infinite 1 z number of transformations of the form y ■=z -, — | rtoren- ■^ ox x^ der the equation homogeneous. Ex. 1. dy + y'^dx z=. a-x'^'^dx. Here m rz — 4, and the variables are separable. Substitute y=. 1 ^ ; then X X- dx X dx d% 9>zdx c \ ^ "> ^ , ^ + — 7- + \ — + -7 > dxzz a'x-Ux . . x^ x^ x^ X X x^ S dz z'^dx d'dx dx dz i , • , . or -T- H — = — — •"• -T = ~; :, *> and changms the signs x"^ x^ x^ X' a'' — z~ ^ ^ *= that the system of logarithms may be positive (vol. i. c. 2, 16), dx __ dz- 1 __ 1 c(z—a) ^ __c{z — a) ''~'~^^z'^—a'^ •** T ""2fl z+a ^ ~ z + a 2? (^H-a) ?^ x^y—x+a c — e^ ' = e" • -^ . z — a x^y — x—a Ex. 2. dy + yHx =z — a^x-*di. r^ 1 • * . . 1 , dx dz Substitutmgasinex.ljwe have— —7= TT~ 2 •*• * * ' X' a "1"^ a x^y—x \- c = tan.-* -^ . X a 126 FLUXIONAL EQUATIONS OF TWO VARIABLES. CHAP. IV. 8 Ex. 3. dy 4- y'^dx = a^x "^dx^ where m is of the form __ 4i ss'^dx * Here equation (a) is d% -\ — =a^.r ^ dx \ in which X' , . ^ , 1 dy^ dx -I, substitute y = — /. 5^ -f --— = a^'x ^dx or jy + J/'* . a^^ ^ c?^ = 0, which is homoge- neous and integrable. r »n— n-(-2 2m— n+3 ■> Ex. 6. 6?3/ + \ax'»y'' + 6j7 "-^ i/^ + c^r ^-i z/3 5cZa?=0 may be rendered homogeneous by substituting y = z^-^. 21. On finding a factor which renders an equation an exact fiuxion. An equation which is the immediate derivative of 2^ = r(c«', 3/) = is an exact fluxion. Equations of this kind seldom occur in calculation ; those which we generally meet with are such as have been derived from their pri- mitive by the elimination of a constant, and these are not exact fluxions, but may be rendered so by multiplying by some factor. Thus, let y — ax = O'y diff^erentiating, dij — adx — 0, which is an exact fluxion ; but if we eliminate a, the result is ydx — xdy = 0, which may be rendered an exact fluxion by multiplying by — . In general, all equations of the first degree which are not exact fluxions must have resulted from eliminating a con- stant from the primitive and its immediate derivative. It appears from the process of elimination that the resulting equation ceases to be an exact fluxion in consequence of the disappearance of a factor ; if then this factor can be restored, the equation becomes integrable per se. Thus, let the primitive be p -f ca = where p and a are functions of the variables ; then its immediate derivative is dp + ado, = 0, and eliminating a, qc^p — vda = 0. Now this must be the same equation as that which would arise from solving the primitive with respect to c, and then dif- p ferentiating ; in which case the primitive is c = ,and VOL. II. K 130 FLUXION AL EaUATTONS OF TWO VARIABLES. CHAP. IV. tne equation is o = an exact fluxion ; which shows that Q,dv — tdo, ceases to be an exact fluxion in consequence of the disappearance of the factor —7. If adp and vdo, have a common measure, this may also disappear in the process of ehmination ; but the above pro- position holds good, viz. that the function necessary to render adp — vdq, an exact fluxion is under the form of a facto?: We shall give a more direct demonstration of this pro- position in the following article. 22. An equation of the first order between tivo variables may be always rendered an exa^ct fluxion by the introductimi of a factor. By solving the proposed equation with respect to/;, it may be made to take the form p -|-/(^, 3/) ~ 0. Let the pri- mitive, when solved with respect to its arbitrary constant c, be y{x, y) — c =, by substitution, u where w is a function of {x, y) not containing c. Now, differentiating u = f{x, y), we have du du du u' ^^^ = ^ + ^^ = " +"'^ or^f~ = 0; an equa- tion which is identical with p i-f{x, y) — (ch. 3, art. 6) ; wherefore j p -{-fix, y) ??/, = ^ ; or the proposed equa- tion, when reduced to the form p -rf(x, y) = 0, may be rendered an exact fluxion by means of the factor -7- = -7-, ^ dy dy' the fluxional coefficient of c in c ~ f(^, y). Cor. 1. If the primitive be known, and it can be solved with respect to its constant, the requisite factor can be found. Cor. 2. If the process of separating the variables be known, the factor proper to render the equation an exact fluxion can be found. For the proposed equation is not an exact fluxion in con- CHAP. IV. euler's criterion. 131 sequence of the disappearance of a factor ; if then by any process we can separate the variables, and thus render it an exact fluxion, the factor will reappear. Ex. 1. yHx 4- ocydx — x^dy = 0. Since the equation is homogeneous, the variables are separated by substituting y ~ xz ; and it becomes f flic fl ^ \ xHz'^dx — xdz) == 0, or x^z^ ) — f = 0: or the re- ^ i X z'^ S quisite factor is — ^-^ = -^7-. Multiplying by this, the equa- X z y^x , dx dx xdy ^ , . , . „ . tion becomes 1 ^ = 0, which is an exact fluxion. X ^ y y'^ ' Ex. 2. dy + y^dx — a'^x-^dx zz 0. Substituting y = — + — ^, the equation becomes 6?^ z-dx a?dx z"~—a^ I dz dx ^ ^ . ^ + -i^ - -^ = "'"■• -^r- i ^Z^. + ^ I = 0. and the requisite factor is :orr— r. 23. Euler's Criterion of I ntegr ability. Let da = udx + -ady where M and n are functions of {x, y) ; then, when du is an exact fluxion, we have du .du 1 vm . • • n -J- = M and y- = N ; and, diiterentiating partially, dm d^u . 1 . I ^ ,^^ d-zi dN . d^ = d^y = (^^^- '' ^^^- ^*' ^^) ^ ^ ^ ^ ^^ '^^ ^^"^^^^^" in order that Mdx -^ ndy shall be an exact fluxion is that du __ d^ dy ^ dx' Co?\l. A formula in which the variables are separate is an exact fluxion. For, let du — xdx + Ydy : then -f- = = ^-. •^ ay dx Cor. 2, ^I^^/^dx. dy ^ dy Ti 1 y y 1 du , dHi ', . , For, let u = fudx ; then -p- = m, and =—7-, which dx ' dxdy k2 132 FLUXIONAL EQUATIONS OF TWO VARIABLES. CHAP. IV. dht dm . . . ... = J— 7-> = ~j- ; wherefore, integrating with respect to x, du dftidx dm _ -T- or ; = / -r-dx dy dy dy The theorem demonstrated in this corollary is called Leibnitz's from the name of its inventor. x Ex. 1. (2j; — y^dx — xdy is an exact fluxion. dif dx ' Ex. 2. ydix — xdy is not an exact fluxion. For -r~ = 1 and — = — 1. dy dx Fv q "^^4- '^^'"^'^ '^"^-^ j_ ydc^^-jcdy —-— dy an exact fluxion. This example may be put under the form \ X x^ x^ S L^y ^ S yS dM. _2y-\- s/x" +2/2 yi " dy~ x^ ^3 ^x^+y'' _ 2y ^/x'^ + y2 4. ^2 _^ 2j/« ~ x^ V^M^ rfN 1 X 9. . ^ du 24. Required to integrate a Junction of two variables which is an exact fluxion. Let du = yidx f Nc(i/ be an exact fluxion ; then du . du T- = M and -7- = N. ax dy Now integrating on the supposition that y is constant, we have u — fudx -f y (a), where the correction Y is a sole function of y. Next differentiate (a) partially with respect to y^ then du d.r^idx dv , ( d.fMdx 1 , CHAP. IV. EXACT FLUXIONS. 133 Y =y I N '"^ — \ dy. Substitute this value of y in (a), and there results u —Jmdx +J' ) N —^ > dy. Cor. 1. Since u =fiAdx + / ^ n ^ — > dy^ and also =fMdx + Y ; thereforey* ^ n — ^^-y — j dy — Y, and con- sequently N — -^^-7 — does not contain x. Hence Rule 1. ' First integrate one of the terms as udx on the supposition that y is constant ; to this add the fluent of as many of the terms in "sdy as do not contain x? In the application of this rule, the student must be careful to observe whether the terms in nc??/ which he neglects may not be decomposable into others, some of which may contain only y. If this should be the case, the rule will give an erroneous result ; because these terms which contain only y, when integrated, form part of the correction and may not be neglected. Thus, where n contains x is de- y./x^^y'^ composable into du\ — — — =^ v ; and if ( «/ Vx'^-y\x-\- Vx^+y'') ) we integrate according to the rule, we neglect ly which forms part of the fluent. If there is any doubt in respect to the nature of the coefficient n and none in respect to m, we may first integrate Nc?y on the supposition that x is constant, and to this add the fluent of as many of the terms in udx as do not contain y. When these two processes give the same result, the true value of u is obtained. Or the result may be verified du by ascertaining whether it gives the proposed value ^^-r- If both the coefficients be doubtful, we should integrate by the following safer rule. Rule 2. ' First integrate du with respect to x, add the correction y a function of y ; differentiate the result with respect ioy. Equate the resulting value of -p with its pro- lS4f PLUXIONAL EQUATIONS OF TWO VARIABLES. CHAP. IV. dY posed value, by which — , and consequently y may be found,' Cor, 2. To verify the equation of condition, let v =J%idx ; then, since n — 7- does not contain x, we have, diflPeren- .... fi?N d^v ^ dN d^v tiatinff with respect to x. -r- ^-^- = 0, or -r- = . , = ^ ' dx di/dx dx dydx dv d'v 'dx du . . . . . . p v • J — r = — , — = .— , which IS the equation ol condition, axai/ ay dy ^ Cor. 3. Since u — Jm.dx + y and also = f'^dy 4- x, therefore y*MC?^ + y ^f^dy + x, d^wAfvidx -f^dy = x — Y ; or the difference of the fluents integrated partially with respect to x and to y may be always separated into two parts, each of which is an explicit function. 25. In the following examples, having previously ascer- tained that the equation of condition obtains, we may either integrate by the rules ; or substitute for m and n in the formula u =J'Mdx -\-f -j n — -^ — 1 dy. Ex. 1. du == - + adx + thydy, C 1 } , 7 7 ^^ dm ^ d^ du — I — = V a ydx + 2bydi/. Hei-e -7- = = 3- - .-. by tberule, ?/ = Z(cr-f v/l+^®) -\- ax + by"^; which is the function which arises from integrating each term separately. Ex. 2. du — 2axydx -{- ax^dy — y^dx — Sxy'^dy. du ~ (2axy — y^)dx + (ax" — ^^xy'^)dy .-. (rule 1), u = ax'^y — y^x. When ^Zw is homogeneous and an exact fluxion as in this example, u may be found by changing dx and dy into x and y, and dividing by the dimension of the resulting func- tion (art, 7)» This method fails when du is of — 1 di- mensions. ^ ^ , dx y'dx ydu ydx — xdu du X x^ x^ x^ ^ 2y (art. ^S, ex. 3). ' , r 1 ?/- y yx'-{-yO , (1 y Vx''-\-y'0 , I X 1^ x^ 3 (2y x^ X- ) -^ CHAP. IV. EXACT FLUXIONS. 135 '—1--^.-^^^^^^.^% = ±Lxy( V^~Fi'-2/)- £,U^'+y'' + 2/)- Ex.4. c/. = ?^^:i%^^.-..=-^4-c. By changing the value of c, the same fluent may appear under forms which, though apparently different, are essen- tially the same; thus substituting in this example c + 2 = c, we have u — f- c ; substituting c + l = c, u = 4- c. X — y X ""■ V T^ ^ 7 ydy-\-xdx — 9,ydx Ex. 5. da = ^ -^ :-j^^ — . If we first integrate with respect to z/, w = Z(v — x) 1- c; which is the same result as before, the correction being increased by unity. Ex. 6. du = ^ ^r^^ ^ .-. u = tan.-i-^ + c. y-Vx^ X dx xdy Ex. 7. ^M = Here it is safer to integrate Nc??/; whence we have xdy , V^^ + y2_^ u=—r — — - — = — z ^ — . . . . yVx'^-^y^ y = I 1- c. y Or thus. Integrating partially, 136 FLtJXIONAL EQUATIONS OP TWO VARIABLES. CHAP. IV. du . y u = lix ■\- Vx^ 4- y2) -I- Y .*. -r=-^ -r- : ^ ^ ' dy ^x^ +y^{x + ^x^ +3/*) rfY , . - X dv 1 _ 1 4- T~ which .'. = , or — J- = — .'. Y = / — dy y^x^^y^ dy y y and u =1 =^. y 26. PRAXIS. 1. du = (3^2 _|_ 2^^^ _ Sy^)dx + {bx'' — bxy -{- 3ay^)dy .\ u =^ x^ + hx^y — 3y2a? -f ay^ -f c, % du = (2y2a7 + 3y)af^ + {^x'^y + 9a7j/2 + 8j/3)c(^ .-. ydx — xdy x 3. c^M = '^-i—r-\r ••• «^ = — T"- (oc+yY X^ry _ ydx^xdy . x y 4. aw = - — .*. M = sm.""^ — , or = cosec.~^ — . y^y'^—x'^ y X ydx — xdy _ x-\- 's/x^ — y^ 5. du = ^ -■ .'. u = l . ^-, yV^^-y' 'J e, du = _y^^:z^^ ,. u = ,/2 sin -A /El. {x\y)^ xy-if y x^y dx{x'\'^x'^-Vy'')^ydy , ^ -^ 7. du = ^ '^-4==L=^.\u = lc(x-\- Vx'^ + y'). y~ — xy _ x'^ — xy x -\- y S. du = -^^ ^dx + -^c/y .'. u = —=£=, {x^-^-y'^y {x^+y^y ^'^' +y^ 9. cZw = w = cos.~^ / ; 1 --y a;2+y2* 10. {ax •{- by + c)dx + (5a; + ey +f)dy =01 .-. c = \ax^ + Z>jy + ex + \ey^- -{-fy. > andyMc?;r -f^dy =z {^ax"^ + car) — {^ey^ +fy)'^ dx dy c ^ X \ ^ 11. +-^j 1 1=0 Va;^'+y' y ( Vx^-\-y'') .'. c = X -\- ^/o;* + y'. CHAP. IV. ON FINDING THE REQUISITE FACTOR. 137 X .'. c = a Va?* + j/* + tan.-^ — 4- by^. 13. sm.ydx + xcos.ydy -j- sin.jrc?y -}- ycos.xdx — ,*. ajsin.y + ysin.tT = c. 27. It has been shown (ch. 3, 13), that to every equation of the first order there belongs a primitive containing one arbitrary constant. It also appears that the primitive can be found, if the proposed equation is an exact fluxion : if it is not, there is some factor which will make it one; and could this be found in all cases, the inverse calculus, so far as relates to equations of the first order, would be perfect. We shall therefore proceed to consider the cases in which the nature of the factor can be ascertained, and shall add a few particular forms rather as specimens of Euler's manner of treating the subject than as being of any great utility. 28. Required to Jind the factor which mil render a form of the first order and degree an exact fluxion. Let the proposed form be vdx H- Q,dy^ and when mul- tiplied by the factor z, let it become Mdr + ^dy an exact fluxion ; then m = ps: and n = qj^. Since yidx + ^dy is an exact fluxion (art. 23), dm d^ . I . . P 1 1 ^- — -y- ; in which, substituting for m and N, v% and qs;, we , pJ^ -\-zdv Q,d^ + zdo, nave ^ = ; , or • • • • • • • • . u% ttz r ap oQ ^ dy dx d% dz X dp do. The proposed, if it be an equation, may be put under the form vdx -\- dy = 0; in which case (a) becomes dz dz dp _ dy dx dy ~ ' If we could integrate this equation, and thus obtain the value of ^ ; we should be enabled to render the proposed form an exact fluxion, and to integrate' it. But the equation, except in certain cases, is obviously more difficult of integration than the proposed; since it contains three variables and has two fluxional coefficients. FLUXIONAL EaUATIONS OF TWO VARIABLES. CHAP. IV. ss can always be found in the following cases. Case 1. Let ^ be a constant quantity. Here -j- = = -i-, and (a) becomes -j w~ ~ ^ ' which shows that the proposed form is an exact fluxion, and no factor is necessary. Case 2. Let pda: -|- ad^ be homogeneous. Let du = M.dx + -adi/ = vzdx + Q,zdy ; and let p or a be of 772, and z, which is also homogeneous, of n dimensions ; then since the degree of u is higher by unity than that of p^ or Q,z, we have (art. 7), vzx + Oizy — {m -\- n -\- \)u\ and dividing T^dx + o^sdy = du by this, we have rdx + Qdy du 1 1 . ^ , . , ; — ^ = — • — ; ; or IS a factor which Fx-^ai/ u m-\-n-\-\ p.r + aj/ will render ^dx + Oidy an exact fluxion. This fails when po; + c^ = 0. Otherwise, by means of separating the variables. Substitute as in art. 10, — = z. and let p = x^yz and X Qi = x'^fz ; then the proposed equation, eliminating dy = zdx + xdz, becomes x^'Yzdx + x^fz\zdx + xd%\ — 0, or ^^"{f^ + zfz\dx + x"-^^fzdz = 0; an equation which is rendered an exact fluxion by the factor — — n \ — -rr-. ; J x'^^^\Yz-\-zfz\ which = . P<2?4-Qj/ Case 3. Let ;^ be a sole function of one of the variables as x. Here — = 0, and (a) becomes a-7- = j-5 — >z, or ^=.-\^-^J-^Xdx' ^x^^lz ^ f-\^^-.-\d^ ^ Q.\dy dxS ^ ^ Q.\dy dx\ ^ z = e'^ is also a sole func- tion of y; but since the converse of this proposition does not hold good, it leads to a method of finding z which is merely tentative. Cor. When « is a sole function of x, it is of the form ^/xrf^ unlessy — } -J -T- ydx is also a logarithm ; and when a sole function of j/, it is of the form ^^^'^^. 29. Conversely f if ihe form ?dx -\- qJ«/ is rendered an exact fluxion hii the factor , it is homo£ceneous. p , \dx-\-^dii tdx dy Let — = ^ : then or + , -^ . . . a PX + Q.J/ tx-\-y tx-\-y is an exact fluxion, and consequently (art. 23), dt c dt ^ dt du ^rdy'^^i ^"^^^ dt dt ^ ^ -^ 1;^ • or y-r 4- x — 0" tx-\-y {tx+yY \tx+yY" »y dx ' wherefore (art. 8) ^ or — — f{ — |. Q \xj BO. There is an infinite number of factors which renders a form of the first order and degree an exact fluxion. For let x{^dx -f ^dy) = du be an exact fluxion ; then, multiplying by (pw, z(pu{vdx + ady) — (pudu which is an exact fluxion, whatever form we assign to (pu. Hence it appears that if z is a factor which renders in- tegrate pda: -j- Q.dy so that ^{vdx + adij) = du, then z and ydy^xdx'-2ydx . 7 \z = du IS an exact fluxion, and {y-xf u - f-^^ - -^~ (Art. 25. Ex. 5.) c y—^ Ex. 2. ydx — • xdy — du. Here p^ + ai/ = 0, and the method of art. 28, case 2, fails ; but since -5 -7- =2, the equation (a) becomes d% 2 . c 1 c , _ , ^ . , — = ax, .*. ^ = —; and— (2/0^ — xdy) is an exact fluxion. Ex. 3. xdy — 2ydx + bdx = du = 0, Since — )^ 'J' i ~ > ^ ^^Y ^^ ^ ^^^^ function of a:, and we have — = , or ^ = -^:and duhecomes % X x^ ^ dy %ydx bdx ,., . „ - —2 — I — ; which IS an exact fluxion, and the re- x^ x^ x^ . , . . . . y b - b quired primitive is -=^ — o~^ +c = ^j or cx^-\-y = -Q-' Ex. 4. aydy — by'^dx + cxdx = = du. p = co; — by- 1 c dv da~i 2b Here — > -j j- J = •'• • • ■ • Q, Idy dx y a dss 2bdx , J^ , , , — = and z=e « ; and du becomes ss a ce 'a'xdx + ae ~^ydy — be ~^y^dx an exact ifRdx . CHAP. IV. ON FINDING THE REQUISITE FACTOR. 141 J^^ a -?^ ac fluxion, ,'.Jce « xdx ■{■ -^ e « ^/^ = c, or /^^/^ — c^ — ^* • • = ci^. (Vid. 18, ex. 8.) Ex. 5. The form dy + siydx. Here — is the requisite factor, and — + yidx is an exact y ^ y fluxion = du\ .-. u — ly +,Jxdx = Le-^^'^^'y .*. (30) the canonical form is — (p.e-^'^y. Ex. 6. The Unear equation dy + itydx = sdx is inte- grable by the introduction of a factor. For the equation is (ny — s)dx -^ dy = 0, and conse- . I cdv do,-) , ' ' r . ' quently — ^ -^ -j- > = R, or the requisite ractor is and multiplying the proposed by this, we have e^'^^dy + y&^'^'Rdx = e^^^'^dx ; and integrating &'^y —fe^^^'^dx + c, and j/ = e--^''^'' {fe^^^sdx + c], Ex. 7. aydx + bxdy + x'^y''{cydx + e;rcZ^) = 0. — is a factor which will render the two first terms an exact xy fluxion, and the fluent is alx + hly or l.oify^\ wherefore — Y.xf^i/' is also a factor which will render the same an exact xy ^ fluxion. Similarly ^^^^ ^^^ renders the third term an exact '^ y fluxion, whose fluent is I .x^y^^ and consequently —r-r-z-r:fx''if renders this term an exact fluxion; and to find a factor which renders the whole integrable, make — F.^y^ = ^ , „ , f.ofii^ or F.j^?/ = ii fx'^y^. xy ^ j.mt\yn,iJ if u x'^y'^ ^ Now, suppose F.^^^ = ^kaykh and/.jT't/* = x'"'y''\ .: we have ka = re — m\ nc—me . na — mb A6 = r. - « I ••• ^ = JTZj^ and r = ^^-^. 142 FLUXIONAL EaUATIONS OF TWO VARIABLES. CHAP. IV* Multiply then the two first terms by ^'^^a-y^'-i, and the third by a:'''^'"""y^~"~i, and there results ay^^ai^^-^dx + ha^^y^^-^dy + x''''-^y''^-^ (cydx + exdy) = ; and integrating, -j-a^'^y^^ -\ a?'*y^ = c is the required equa- tion, in which the values of k and r are known. Cor, 1. If either nc — me =^ 0, or na — mb = 0, we have A: = 0, and a logarithm enters into the resulting equation. Cor, 9,. I£ ae — be = Of k = CO , and this method fails ; but m this case, smce e= — , the equation is aydx + bxdy + -^x^y^aydx + bxdy) = 0, or (aydx + bxdy) \ 1 — y-jc'"/* ? = 0. This equation is satisfied either by aydx + bxdy = ; which integrated, gives x^y^ = const. ; or by (^ 1 j-x'^y^ = ; of which the latter is a mere algebraical solution of the original equation. Ex. 8. axdy + 2aydx = xydy. T=z9ay Q=(«-i/) 1 cda dp^ 1 Here — } :j t" > = 7. — (a — y — 2a) V (dx dy ^ 2ay^ *^ ' ^— ^ ; if then 2 is a sole function of y, d% ady dy ^ ^ 1 y — = — rr-^ — ~ and Iz = — 7 — 7?-. ss 2ay %a j i. 9a The requisite factor in this case is not a sole function of . . 1 .... 1 y; it is — ; and the primitive is x-y = ee^. xy 3% An equation may be sometimes integrated by dividing" it into exact fluxions by means of factors^ and substituting new variables for the fluxions. Ex. 1. x"'(ydx + xdy) — y'^^ydx — xdy). CHAP. IV. ON FINDING THE REaUlSITE FACTOR. 143 Here x'^d.xy = i/''+^d. — . Let v =■ xy X w — — y '^1 - /T ^ > .-. a? = Vvw and yzz / — ; and by sub- n+2 m m V~^ dw m—n—l _mj-nj-2 stitution, v^W'-idv = -r- .•. v « dv=w 2 dw ,\ in- n+2 W 2 wt — n tn+n grating, — -1 ; = 2c. or ♦n— n m — n m+n m+n X 2~«/ 2 J? 2~2/ 2 .? 4. ^^ 3^ ^, m — 7i m+n Ex. 2. a(a; =^^w dw or :; J l-f-u^ 2 * l+i;2 L » = .'. atan~^v = w — atan.~^ — , or atan. ^^ w+-a^ a ' X s/x^ + y^—a^ —a tan. .^ Vo^+y^—a'^ ay + x s/x'^ + y "^ — a' ^x'^ + y'^ — a'^ — — = tan. . ax—y^/x'^+y'^'-a'^ a Ex. ^. ydy ^ xdx -^-^ + ^^ = 0. Here-L^.(«/2-a;2) = A^.J^. X^ X Let r; = ^^ — 072 ^ a; = — 4 (o'^- 1)2 144 FLUXIONAL EQUATIONS OF TWO VARIABLES. CHAP. IV. by substitution, dv = — - — ^ -^— , or vHv = 26(tt;^ — 1 fdw .«. t;3 _ g^ > ^ _|. ^ < or (y'--'y=^^\t ^' ' X 'Y" ky"" ' Here y^dl . ^^^^ = -r x'^dx. n na Let x'^i/' = V .-. 3/" =:v^x~'b; and by substitution, ii '^ dii 1 i! 1 1 m+— i;6 07 6 — = -^r^'^dx : ov vb dv :=i-T'Oc t>dx; and inte- bk It bk "11 x^ ~b ffratmff, — v «>, or — x^if =. f- c. ^ ^ n n ^ na ^ m + -^ + 1 Ex. 5. aydx + ^^riZj/ + aTy^^cydx + ^076/2/) = 0. Here dl . ^y + x'^y^'dl . o;y = 0. 2 i -i Let a?*'?/^ = u -i t;« w^ ^!f:±' 22;c ^ V .*. = .♦. «/ ac = .*. • J7V = J^J - ± ^ - — c a e -6 Similarly a: = t;ae-6t x xv^e-bc .•, by substitution, JL. me — nc na — mh ^y^, nc—me na—mb ^ -|- ^ae— 6c X Z£J«e— 6c IT 0, Or t;"^~^<'~ Wz; + Z^ae— 6c t?iK? — V W nc — me na — mh 1) ae — 6c 1V^^ — ^'' .*., integrating, 1 •=. c, in which v and w ° ° nc—me na—mb are to be replaced by their values x^y"" and x^y\ (Vid. art. 31, ex. 7.) Cor. If either nc — me = 0, or na — mb = 0, the fluents of the final equation become Iv or Izv. This very useful method of integration is nearly allied to the one given in art. 30, cor. The proposed should be di- CHAP. IV. ON FINDING THE REQUISITE FACTOR. 145 vided into such exact fluxions that the new equation may be under a form the most convenient for integration. Ex. 6. xdx + aydx + 2xydy — 9^a~ydy — 0. If we divide this into xdi^x + if) -f ayd{x — S«z/) = 0, the equation between v and w will be as difficult to integrate as the original. But, since {^x-\-ay)dx-\-iX'-dP)2ydy — ^y .•.0 = ^-±f-, and equation (a), art. S8, becomes — w(x?/ + x') + y.nfx + x(y ■\- foe) = 0, or (?i — l)xi/ — ?2/"'a:z/ + wx' — x/ir = 0, which obtains what- ever be the value of y ; wherefore {n — 1 )x — rfx = and wx' — xfx =■ are the equations from which the values of X and of x' may be determined. Cor. Conversely, if x and x' are given, jTir may be found from either of the above equations. Ex. -J- — -- xdx -f {m + \)ydy = 0, or m xiidx xdx CHAP. IV. TRANSCENDENTALS. 147 Here x = and x' = —^ ; wherefore (772 + l)>v/l+a:2 w-fl from the second equation, we have nx moo ,. ^ , =- — fx — \J. OY n = — m and W2+1 (m + l)Vl+^- fx = Vl + X' .'. z - {y + -v/1 + xr^y ^ If m be assumed any integer, 2 for instance, and the binomial be expanded and multiphed into the proposed equation, the result will prove to be an exact fluxion. 35. It has been shown at the beginning of this chapter, art. 2, that the algebraick relation of transcendentals may be sometimes found. There are other more complicated forms in which the variables are separated, which admit of alge- braick fluents, though the fluent of each part is a tran- scendental. dx dy ^ Ex. 1. — =— • ^ = 0. ^/« -\-ox-\- ex'^ s/a -\-by-\- etf- dx , dii . Suppose : zz dt — :- ; then *^ a-\-bx-\- ex"^ ^/a -\'by-\ ey- dx'^ dy'^- — = a -\- bx-{-ex' and -^ = a -\- by ■{- ej/". Diff*erentiating on the hypothesis that / is the principal variable, —z~ = b -{- 2ex and --7±- = b -{- ^ey. Adding d~p these and substituting x -\- y =p^ we have -~ = b + ep\ 0.1 and to integrate this equation of the second order, multiply - by 9.dp, and it becomes — J = ^bdp + 2epdp ; of which the complete fluent is ~ = c + 2bp + e}f~, or -^ = v'c+ ^bp + €p\ But dp dx dy df ~ dF ^ dt ~ ^^' + bx -^ ex^ +v/« + % + ey-', where- fore the required flu ent is ^/a + bx + ex- + Va -\- by + ey"- — ^/c 4- 2bp -\- ep' = ^/c -t ^6(^ {- y) + e[x -\- yf, l2 148 FLUXION AL EQUATIONS OF TWO VARIABLES. CHAP. IV. Similarly it may be shown that the fluent of dx dy — + ^ — ^ = IS Va-\-bx + ex"^ — y/ a -\-by -\- t^y'' = Vc-{-2b{x-\-i/) + e(x -^yf, dx dij Thus, if rr--{ ^— - — 0: considering a -{• (3x and a-{-(3x cX'+py ° a + /St/ as the square roots of a -\- bx -\- ex'^ and a + by-\- ey", we have a — a^^b = 2a/3 and e = /3% and consequently the fluent is c = a(^x + ^/) + (3xy ; which is the same as that which would result from considering the proposed as the fluxion of the sum of two logarithms. ^ ^ dx dy Ex. 2. .y ^a-^bx-\- cx^ -{- ex^ -\-fx'^ ^/a-^by^ cy^ + ey'' -{-Jy* = 0. dx du This by substitution is — = = = 0, where :i. — a-\-bx-\-cx'^-\- cx^ +/^* and Y = a-\-by + cy- + ey^ +/z/-*. r. , ^ dx ^ d?j , dx'^ iSuppose, as before, — - = dt =■ — ^ ; then — - = x Vx VY "^- and^=Y. Substitute x-{-y=p, x — y = q. Now "^^'Jt ^hich ~ ^^i^ ^'^~^^ - ^ . . . ^^"""^ ^f^ dt^' ^^'""^ - dt ' dt - dt^ ' = X — Y = b{x-y) -f c(.r« — y'^)-\- e(x^—y^)+f(x'^—y*) = bq-\~ cpq + ^(3^^ + g')+'^(p' + r)rOV Differentiating on the hypothesis that t is the principal variable, we have ]^d^x dx ^ --p;- = -j- = b-\-2cxi-Sex^-i-^fx^f ^^' ^^ y ; wherefore, adding CHAP. IV. OF THE FIRST ORDER AND DEGREE. 149 ^ =b+cp + -^{p-- + (fl.37 + 5^/ + c) + «. 15. -^ = :i — ^— .'. St'di; — wdv + tcZa; — 4reJ^zy =0, ax a-r^y—x a %a where v = x =-, w = y — =-. 16. xdx + dy = ay'^dx), , , ^ 2 V become homogeneous by y = z-, 17. dy = dxV^^ + ay ^ ^ -^'^ 18. flfw H — —, -t7*j7=0. becomes homogeneous by e'=z> 19. ^6x'^dy — Saxdy = aydx .*. o^j/^ = c(a: — a). 20. «df«/ =ydx 4- ^^r-Jo; .*. ?/ =ce^ — 6(a;- 4-2<7a7 + 2a"). 21. ^j/ + «/^^ = (1 + x)dx .'. y = X -{• ce~'' , 22. Ji/ \ydx = (1 + oiP'yix .♦. «/ = a;^ — S-r^ + 6^ — 5 + c^-'' . 23. 6[y — o^^j^ + xydx = «Ja; .-. y =iHlx ■\- c^/l — ^^. , {n—\)ydx (n — l)dx mdx xdx 24. dy + ^^^— = ^^ '— + + .-. ^ X X l—x^(l^xy 1,1c ^^'"'dx _ x^'dx , ^ 25. . J- = ttsin.o? ■\- by a .'. y = ce^'' — - — — (6sin.^ + cos.^). ax X y y -^ 4 7 36. rf„ = 53^£|r^_._„^'^^l^. 37. xdx ■\- ydy — «/*. Substitute y r= xz ; then, dividing by .r, z—p=. a/1 -\- p\ or z =. p + Vl +p-. ^, , , , ^ dx dz —dz Also pdx :=. ay ^=. zdx -\- xdz .*. — = =r — = ^ P-- VI +/?^ = -— -- - frS .-. ~ = v\ k- pip + vl H-p ) and a^ = Vl-\-pKp+Vl+p^) To eliminate p, we have 1/ zi xz = v/l+i?« 1 ^ p'i = ^ and p =: .*. by substitution, vr = -^ = ^ or a/«'^ — 7/2 rt « + Va~— y^ CHAP. V. OV A HIGHER DEGREE THAN THE FIRST. 157 .r a/^^ — y'^ ■=. if- — ax^ and the required equation is zr 7/* — 2axy^ + x'^y'^, or y^ -j_ ^2 — 2ax ; where a is of one dimension, because the proposed equation, when cleared of radicals, is of the first degree. Ex. 2- Required the curve whose length is a mean pro- portional between the ordinate and twice the abscissa. s = ^^Zxy^ or the fluxional equation of the required curve xdy + ydx is ^/dx"^ -i- dy'^zz. — '^ or a/1 -f »^ ^^^Zxy — xp + y, Ay2xy Substitute 3/= ^js; then a/1 + p^ V^^^—p + 2. Here 2 may be found in terms of p by the solution of a quadratick and the equation integrated ; but it will be more convenient to find^ in terms of ^ ; for which purpose we have P' - 2F^iP = ^^andp= _c>,_i — ; .-. P -^ _ g^-2:z"- + (l-z) ,/^z _ (2;g+ V9. z)(\-z ) _ { \-z)^2z dx ^ . ^ dz ( v2z—l)dz dz dz .'. — , which — — — X p-Ti (l-z)v/2^ 1-2 (1-2) v2<^ _ . d^ 7 1. lo integrate = = du, substitute 2; = f^; then (1-2) V, 22 V2dv ... 1 , 14 u 1 , l + v/^ 1-v^ ** V2 1-t^ V2 1 — ^^** Ix =. la — 1(1 — 2) ^ Z ^, from which there re- a/2 1 — a/^ suits x-^i/ = J^^~^H ^. (Euler, Calc. Int. vol. i. p. 450.) xdy —ydx Ex. 3. 7 ^ : = F Vx^ -i y-. x/dx'^ -\-dif '^ Substitute x = rw) and zo = a/^- + j/^ ; -'-J/ = v'^e''^ — ^' ,1 o 7 vk;c?i; , = t£) a/1 — 'y- .*. dy = ____ -I- ^1 _ ^2j^ and dx = tet/t; + vdw; .*. 158 FLUXTONAL EQUATIONS. CHAP. V. xdu — ydx = ^= — i€'^ Vl — v^dv = «— ■ ., , , -, , v"xv^dif- ,, ,. , ^ Also, dx''- + Jz/2 = re'^cZu^ + v^-dzo'' + , + {\—i:)diif- = w^Jr;°- + -j 7 + ^w- = r^ — -r^ ; and the . w^dv equation becomes =i = rrc, or ^^/zv'dv- 4- ( 1 — v'^)d'w"- ,N 7 ^'^ Yw.dw w^dv"^ = Yza\w'^dv'^ + ( 1 ~ v)dw"), or an equation in which the variables are separated. (Paoli, Elementid' Algebra, torn. ii. p. 152.) 4. Clairaufs Form. dy y zz. px +,fp, where p zz -—- and fp contains only p. Differentiating the proposed equation, dj/, which =z pdx^ 7 /• = pdx + xdp H ^dp; wherefore — dp ix + ^-^ I ; or dp — 0, and x + -f^ = ; of which the second, eliminating p by means of the proposed equation, will give a solution ; but this is not the re- quired primitive, since it does not contain an arbitrary constant. From the first we have p ■=. c^ and consequently the required primitive is y = ex •\- fc. Hence all equations included under Clairaut's form are integrated by substituting a constant for the fluxional coefficient p. Ex. 1. ydx — xdy = h ^dx--\- dy\ Here y = xp ^- h ^^X + />', and the required fluent is y ■=. cx-\-hx/\ \ cK If this equation be differentiated, and c be eliminated, there will result the proposed equation. Also, to obtain another solution, we have bp ^ X ^ X H ==:. = 0, or p = : and ^/Hjr- ^¥~'X'' CHAP. V. CLAIRAUT's FORM. 159 h ^1 -j- p2 — ; and by substitution, y = ^/Z'- —x^, or x'^ ■\- if- — h'^. If this be differentiated, we have X p =. ; and substituting this value oi p in the example, it is satisfied. These two solutions are essentially different ; the first is the equation of a right line, and the second belongs to a circle. This example contains the solution of the following pro- blem. ' Required the curve in which the perpendicular drawn from the origin of the co-ordinates upon the tangent is a constant quantity.' Ex. 2. ydx — xdy — -r — -—, CLX Here y = xp -\- a{\ + p°) ; and the complete primitive \'&y=.cx-\-a{\. +C-). Another solution is x" = ^a{a—y). Ex. 3. ydx — nxdy = a^/dx^ + dy'^. Here y = nxp -f- a a/1 + p'^ .'. dy or , , apdp , 7ix dp pdx = fipdx + 7ixdp -\ ^ .*. dx H r -^ . . . _____«_ dp ^ ,,^_i_L_JL ri^^l 71-1 ^i+y. •- ^_!L^ n-V^ ^n^p^V 2m +1 n Suppose n = —^ — - ; then — -j = 2?« + 1, and the form to be integrated becomes - ^du ; .*. (ch. i. 22, ex. 5.) -^ ^^\2m-\-\ (2w + l)(2m-l) ^ 2w(2m-2)p2m-4 ^ {^m-\-\){^m-\)i2m-2>)' * ' | ^ and consequently r"^^ -^^277^ + 1 (2m + l)(2w7.-l) 160 FLUXIONAL EQUATIONS. CHAP. V. {2m-\-l){2m-l)(2m-S) 5 ' If we suppose 2m = cc in order that n may = 1, in which case the example coincides with ex. 1, we have — fill 1 ap ^ \p P' P' J ^^i-f■f' and y = xp ■\- a^\ + P% and as before a solution is I Ex. 4. ydx — xdy — a(djc^ + Ajf'Y > y z=z xp -\- a{\ + p^)^ .'. the complete primitive is • • • J/ = C<2? + «(1 + C^)^. Also to obtain another solution ; we have ap^ ^ ap"^ ^ -\ 1 = 0, or a? = .-. • • • ♦ • •, (i+P'V ' a+fv y= ^ + fl(l i-pr = :-, .-. p^' = - -. ... {i+P'y (i+/?T ^ J. 1 J. ButyXl + f'')'= a' .-. 1 +P' = ^, or ^0*^= ^IZ^ .-. ... 3 3 \-j)= J ^ ^^ -^ + («^ -r )^ = 0. 5. Required to integrate y = xf -\- a, where p awcZ q ar^ functions of p. Differentiating ; dy^ which = pdx^ = ^dx -)- xd^ -j- c?q ; /Wherefore (p — p)dx -f- ^^p + cZq = 0, or «?j: + 07 . — - = ; which is a linear equation, and — /!^ r fJL. do. -) consequently 07 = e p-p) c —Je^-p j-. From this equation p must be eUminated by means of the proposed, and there will result the required primitive. Ex. bdx^- — xdydx — ydy'^ = 0, or b — xp — yp^ = 0. CHAP. V. CLAIRAUl's FORM. 161 X U Here y :=. — 1 — r .*. dy. or ^ p ' p2 ^' ^ dx oodp 9.hdp , xdp 2bdp pax — 1 r- or ax 7 ^' . = — — ^ , , ♦ which is a linear equation, where b cp bp 1+ v/^ + l . , . , — "IT ^ T . / ; rrom which /? Vp"-\-l Vi>« + 1 i> j9 is to be eliminated by means of b~xp — yp"^ — 0, and the required equation will result. 6. PRAXIS. ^bx^ 1. dy'^ — bxdx^ — .-. {y — «)- = — — . 2. df - dxdy V oc'^dx^ ,\y ^ azu |- ±fs^x''^ + J-Jor. 3. ;rJ^"- — e/J^% — = .-. l—zz±f — ^^^ 6^^ + c. xdy"^ dy 1 ( , '^^ ^ = -^ + ;^ •■• ^ = (7---iT» J « - i' + ^^ I ' f™'" which ^ is to be eliminated by means of ?/ iz xp"^ -\-p. 5. 2/fZa; — X ^/dy^ + d^'* = .*. p(p + \/l +/?-) = I- — ^ — —, from which p may be eliminated by means of— = \/ i+p^. 6. xdx + a£?y = b ^dx^ + ) ~ rr w; where (p denotes \. Here we cannot equate the two first terms ; but if we suppose that dv' — 0, or v' rr constant, or v' =: 1, for v' is not to contain an arbitrary constant ; and also that fx. -— m, we shall have c/v'' — v'dx^ or v" =Jr^dx ; and all the remaining terms may be determined in the same manner. But if m < 1, we cannot by any supposition whatever make the two series identical ; for // and all the succeeding in- dices are (ex hyp ) greater than unity. In this case, there- fore, the equation z/ = x does not admit of any such quantity as k, and it is a singular and not a particular sohition. Hence the rule. ' In the equation -^ — j9, substitute x-f ^ for y, apd develope the resulting value of— in a series as- cending by the powers of k. Let the series be p + pX"* + vVc"" + • • • ; then, if m = or > 1, 2/ = x is a particular solution ; but if ttz < 1, it is a singular solution." Cor, 1. The value of -^ corresponding to a singular so- lution is infinite (vol. i. ch. 5, 4). Cor. 2 Singular solutions belong to those cases in which p fails to be developable by means of Taylor's theorem. dj) Cor, 3. The value of -y- corresponding to a singular so- lution is also infinite. For let the proposed equation clear of radicals be ¥{x,y,p) = = 2/ ; then, il" it correspond to a singular solution, u is a function of two independent variables x and y ; and consequently we have r. _du du dp^ ~ dx dp dx! ^ dp , n du ^ . J 1 J >. But -7- = X ; wheretore -7- = 0, and — — 4- — ^ k 6?y dp ' ~ di/ dpdy) consequently y- = oc . 20. In the preceding investigation it is supposed that y is a function of .r ; but if 7/ does not enter into the given solution, i. e. if the given solution is under the form x = a, we must suppose x to be a function of^. Let the proposed CHAP. V. SINGULAR SOLUTIONS. 173 fluxional equation be dx — 'pA.y-> where p^ — -.- ; then it may be shown, as in the preceding article, that if « + Ic be dx substituted for a;, and the resuUing value of y- be developed in a series ascending by the powers of k of the form P, + p',A:'" 4- T!\k"- + • . •, a; = a is a singular solution only when m < 1. 21. It appears then that whether we proceed by Lagrange's method (arts. 17 and 18), or by Euler and Laplace's (art. 19)t we arrive at the same property of singular so- lutions, viz. that they render infinite both the fluxional co- efficients o£ p =f(x, ^). They will therefore be found among the radicals contained in this value of/?. To apply this property, it is necessary to solve the pro- posed equation with respect to p ; but by means of the following article this may be avoided. 22. Required to find the singular solution of ajiuxional equation whose primitive is unknown. Let f(^, ?/, p) = — ?< be the equation clear of radicals and fractions; then, when it corresponds to a singular solution, w is a function of two independent variables {x, y), du and it may be shown as in art. 19, cor. 3, that -7- = 0. Eliminate then p from -7- = and u — 0, and there will result the required singular solution. It is supposed in this as in art. 19? cor. 3, that the re- sulting solution renders -7- and -7- finite; otherwise -j- may not = 0. 23. The second fluxional coeflicient corresponding to a singular solution appears under the form of -^, For differentiating m = totally, we have du , du . du ^ 174 FLUXIONAL EQUATIONS. CHAP. V. du da dx ^ dif -' da ^ ^ , dp -^r~dx H T-du — 0: wherefore o', which = — r-, is under dx dtj ^ ^ dx the form of — . Hence the following rule. ' Differentiate the proposed equation ?/ = 0; and find the value of q. Let p ^ = — ; then w = 0, p = 0, and a = 0, are three equa- J tions which contain p ; and if there be any the same so- lution which satisfies all three, when p is eliminated, it is a singular solution." The truth of this proposition also appears from ch. 3, art. 13^ for if the second and the higher fluxional co- efficients do not become indeterminate, the value of y in terms of .r will involve arbitrary constants, and consequently will be its complete value, (Lagrange, Calc. des Fonctions, p. 223.) 24. All that is known of the properties of singular solu- / tions and of the theory of arbitrary constants upon which they are founded, is due to the German and French mathe- maticians. If the reader is desirous of further information, I must refer him to their works. The principal authors are Euler, Lagrange, Legendre, and Poisson. In his Calcul des Fonctions, Lagrange has dedicated the fourteenth and the three following chapters to this subject ; and Poisson has made some important additions in the ' Journal de I'E'cole Poly technique, 13^ cahier, p. 113.' This branch of the subject is as important as any other ; for the problem which has given rise to the fluxional equation frequently depends upon the singular solution, and not upon the complete primitive. 25. EXAMPLES. Ex. I. y = xp -{- h Vl ■\- p"" (art. 4, ex. 1). Here '-y-\-xp-\-b^/\-\-p'^==(} = u • CHAP. V. SINGULAR SOLUTIONS. 175 du bp ^ X .'. -- = X -^ — = 0, or » = , and dp a/1+P^ Vb'—x' ^ 1 4- pi = - • and eliminatinoj p by means of —~ = 0. x'^ 4 ?/- = 6^ is a singular solution not contained dp ' ^ '^ in the complete primitive ?/ = ex -{- 6 ^/i + cK Ex. 2. y = xp •\- fP' (Clairaut's form.) The fluxion is under the form of 5 07 + --^ J 9' = (art. 4) ; of which y = gives p = Cy or y = cr 4-yb, the complete primitive ; and X + -y^ = 0, which is the same as -^ = 0, will give the singular solution, if jo be eliminated by means of ^ = a:j9 -\-fp- Hence all equations included under Clairaut's form have a singular solution. dx d)i Ex. 3. (a + 3/) -^ = (^ + y) - X -^^, or a ■\- y = {x ^ y)p — xp\ (Cambridge Probs. p. 405.) Since dy — pdx, the fluxion is under the form of = (jp + y)q — ^xpq = (x -\- y—2xp)q\ of which q = gives p =z c, OY a + y ~ c(x -{■ y) — c'v the complete pri- mitive ; and to find the singular solution, we have X ~i~ 11 X ■\- y — ^^P = 0, or p = ^ ; and consequently the sin- liX ffular solution is a \- y — — —^^ -^ — -^^^ = -^^ ^^, or ° ^ %x ^x ^x X — y — ^ \/ax, Ex. 4. p^ + AJ9"-1 4- Bp"-=^ ... 4- GJ9 4 H = = Zf, where a, b, • . • g, h are constant quantities. du Here — ~ np''-^ 4 [n - l)Ap"-2 ••• 4 g; and the values da of p m -^ — = are the limits between the roots of m = 0, and we cannot eliminate p. If w r= has equal roots, we may eliminate^ ; but since p is constant, the resulting equa- tion is a particular solution. 176 FLUXIONAL EQUATIONS. CHAP. V. It is observable tbat in tbis case --j— appears under tbe form of -77-; since — r- —. and -t-=- 0. (Art. 19, cor. 3 ) dp dy ^ ' ' Ex. 5. d'-dy -\- y'^dx — a-dx + xydx. Here y z=. x \s a. solution. And to find whether it is a singular solution, we have -r- zz. —^ — ''^— zz p. Sub- dx cti" -, c- , , , ^(-^^ + fc) - (^ 4- ^)- stitute x^k lory; then p becomes 1 H — — 1 — ; or «/ r: ^ is not a singular solution. To find the complete primitive ; we have , ■, -, , , K T a^d(y—x) , _, - a\dy — dx) ~ (xy — y^)dx, or zz — ydx. Sub- 1/ stitute V = _ ; then a^dv — vxdx = dx ; and integrating, •J II c 1 -_iL 1 V =: e2«2 > c H 7/^ 2r«2 j^;^ f which, when c = go , gives uzzx,orj/=ra particidar solution. Here «/ =z a is a solution. And to find whether it is a particular or a singular solution, let 2/ become a + k{ then di/ -~~ becomes bk"^ ; or 3/ z= a is a singular solution only when n < 1. The complete primitive is -=-:j — bx -{■ c. If w < 1, -^, which r= p -r-— r= x when «/ zi « ; but if w > 1 ; or if n zz 1, in which case ~- zz—zzb (vol.i. ch. 5, ex. 10, cor. 2), in either case -7— is finite, and ^ =: a is not a singular solution. CHAP. V. SINGULAR SOLUTIONS. 177 Generally, if -~= ^/y where y is a function of j/, we have CLX 3- = — = -J-. Let {y — aY be a factor of y ; then f -7- j = X 5 and there is a singular solution only when < 1. Ex. 7. dx{\ — y'^y = dy{\ — x'^y. solution and -¥- =7t ax (1 dy. c\-(x-{-T<:y ^ 1 du (1— T/*")" Here y = x'issl solution and ~ = )]_ m\n ' Substitute y "=. X + A; ; then -7^ becomes J — ^ __ — i * ' * ^cX — x^—mx'^-'^'k X" _^ Ci 'nix'^-% -^ ^\ n^^ \ ~ X T^^'s ' ' * nmx^~'^'k . .71. = 1 1 m + • • • » .'. 2/ =: ^ IS a particular solution. Ex. 8. e'dx — ^ = dy -- ydx (ch. 4, art. 1 1, ex. 9). Here y = e'' is a solution. Also -j- =z e" = y .*. when ^ becomes y -{■ k, p becomes p -\-'k\ or y = e"" is a particular solution. ^ ^ , ^dx Ex. 9. ^y= . ■^ Vl-x^ ^^ dy . . ^ , . da^ Here -r^ does not contain y. Substitute p, = -7-, Vl~^' then 0,=: — * = Vx~^ — x'^, Substitute ± 1 + ^ for x"^ ; t hen p^ becom es y/ ( ± 1 + ^)~^ - ( ± 1 + A;) • • • = ^ — 2A; + A;2' or 1 + ar^ = and 1 — o^^ = are both singular solutions, and they are not included in the complete primitive x'^ = sin.(2y + c). Let x^ — sin.(32/ -{- c) = = v; then dv —- = — cos.(% -I- c) = ; from which we obtain VOL. II. N 178 FLUXIONAL EQUATIONS. CHAP. V. 2y + c = — , or — , or • • •, and consequently c = — — %/, lit 4^ ti or = -^ 2y, variable quantities; from which also it fol- lows that these are singular solutions; and if they be substituted in the primitive, there will result as before x'' = ± 1. Ex. 10. {xdy — ydx){xdy — 9>ydx) + x^dx = 0. Here (xp — y){xp — 2y) + o;^ = = w .-. • • . • du ■J- = x{xp — %y) + x{xp — y) = = 9.xp — Sj/, or xp =z -^ ,\ = — ^ — h x^, or y = 2x^ is a singular so- lution. Or thus : — = 0=xq(xp — %/) + {xq -^p){xp—y)-^ Sx% or 2x^pq — Sxyq — xp'^ + j/i? + Sx^ = 0, or __ xp'^—yp — Sx'^ ^ ~ x(2xp-3y) ' ' Assume q = -~; or xp^—yp— Sa:^^ and ^xp-—Sy — 0. Of these, the second gives j? = ^, which substituted in M = 0, gives =— ^-\- x^. If this value of p be also substituted in the first, it gives -^ — ^ — Sor^- z= 0, or ^X /CX T" + J^' = 0, the same result as before ; and consequently ~ X + ^' = 0, or y = 2.^'^ is a singular solution. The x" complete primitive is y — ax — — . Ex. 11. ydx — xdy = x^^ dx^ + t/y^. Here (j/ — ^^)^ — J7'(l + p-) = = m .*. • • . . du CHAP. V. . SINGULAR SOLUTIONS. 179 z=. —%eyq — 2jf(1 +p^), or qi=. ^- — ^— which becomes xy ~ when ^ = unless p = co ; but this is not a singular solu- tion, since it is a particular case of the complete primitive 2/2 4- ^3 = <^ax (art. 3, ex. 1), viz. that in which « = x . The rule of art. 22 fails here, because .r = renders infinite y-, which = — ^yp — ^x. Ex. 13. i/'^dx^ — xydxdy^ — 9.xydxHy-\-x"dy^ i-x^dx^dy"- = 0. Here x'^p^—xyp^ •{■x'^p'^ — 9>xyp + //^ = = w, and y = x\s a solution. To find whether it is a singular solution, we du have — = = xp^ — yp^ -\-{fox^p —6xyp^ + 9>x'^p —2xy)q^ yp^ — xp^ is not a singular solution. T. , . df/ I x^ du ) d^y d^y"" Ex. 14. y _ ^ / + I -- _ / i -4 + X -f- = 0, or •^ dx \ 2 dx \ dx^' dx* ' ^^ + — 2 — ^ ~ ^^ ~^^ = = M. This iexample and all equations of the second order may have two different kinds of singular solutions; the one, an equation of the first order whose primitive contains one arbitrary constant; and the other, a primitive which does not contain any arbitrary constant. The first is to be ob- du tained by eliminating q by means of w zr and ~ = 0; and the other, by eliminating both q and p by means of m =: 0, du ^ . du ^ - , 1- . ^ . ^ = 0, and -J- = ; and the result, if it satisfy u — 0, is in either case a singular solution. TT d^ ^ x'^—^p ^ 2p-x- Here ^ = 2xq-]- —^ =: 0, or q - -^—- ; and by ... (^p-x'-y {%p—x'^Y substitution .^ =^— - xp + y zz 0^ or ibx ox n2 180 FLUXIONAL EQUATIONS. CHAP. V» p^ + ^x°p ■\- -7 ^xy = (a) ; which satisfies the pro- posed, and is a singular solution. In this case we cannot obtain a singular solution of the second kind ; because u = being linear with respect to p, we cannot eliminate p by means ot -r- = 0. dp x^ It is worthy of remark that «/ = - -^j though a singular solution of (a), will not satisfy the proposed example. It may be called, by way of contradistinction, a double singular solution. Lagrange has shown that this is a property which double singular solutions, in general, possess. The complete primitive is 3/ — \ax'^ — bx — ab = = v. If we eliminate both a and b by means of -^ =r and •^ da -Tj^=.0, we have = ij® + b and zz x -\- a; and there results 1/=. — , the singular solution of (a). ^^•1^- " 5^ - jf d^ + " = ^' "^ ^^ - %.^ + ^ = = It. du v Here -^ = = 2xq — 2», or <7 = — , which substituted dq ^ ^ ^ X »^ in II = 0, gives — ^^— -I- a? = ; or a?^ — p^ = is a sin- gular solution. The complete primitive of this example is x^ -^ r- 2ay -\- d^x + 6 = 0, and the two first primitives are x^ — 2ap 4- a'^ = = r; (1) ; and, eliminating a, (^x\ / 'ilx\^ ^ ~ T/-^"^^^^"^ V ~ Ty * * * = = K' (2). or CHAP. V. SINGULAR SOLUTIONS. I8l dv From (1), we have — = = — 2p + 2«, or a = p, which substituted in x^ = 0, gives x^ — p^ = 0. From (2)~ = 0=- 4(^ - a:p)p + ^(b - ^), b — = 2{i/ — ocp)p, which substituted in k; = 0, gives ^x^{i/ — xpY — 4(^ — xpYp^ = 0, or (a?^ ^p'i){y—xpy - 0. Of these, ^ — a;j9 = is not a solution; for it gives, when differentiated, .r^' = 0, or g = 0. The other factor gives 07^ — ^2 — 0. It appears then in this example that the two first pri- mitives each give the same singular solution. Lagrange has shown that this is a general property, and that it be- longs to equations of any order whatever. The same writer has also given a rule by which the singular solution may be obtained from the complete primitive without know- ing either of the two first primitives. (Calcul des Fonct. p. 190.) 26. PRAXIS. \. y •=. X -^ {a — Vf-x^ \ «/ = tt is a particular solution. 2. y z= a; + (a — !)'(« — xf\ « z= 1 and a-^ x give X -\-\ particular solutions ; and a •=. — ^— is a singular so- lution.- 3. xdy"^ — ydxdy + adx"^ = 0. The complete primitive IS y :=! ex -\- — ; and j/^ rr 4iax is a singular solution. 4. (b-x)df' + (y — x)dxdy + ydx^^—O ; {x -{- yf—^by is a singular solution. 5. dy^ — dx^ =. 0; y =: .r is a particular solution. 6. ady — adx z=. a/j/^ — x^dx \ y •::z x is a singular so- lution. 7. a^dy ■- aHx — (j/^ -> x'^^dx-^ y — x \% s, particular solution, where c 1= go . (x^\ x^ 4ixy J- jdx'^ ; .y = S" is a sin- gular solution. 9. dy zz x/y — adx; y =. a is a singular solution. 182 FLUXIONAL EQUATIONS. CHAP. V. ] 0. dy — (2ar + v'^^ — y)doc \ y z=. x'^ is a singular so- lution. 11. aydx z=. {ax — Vx'^ + y^^)dy ; x^ + y' = is a sin- gular solution. 12. ydy^ + xdxdy rr bdx^; y =— Tr ^^ not a solution. 13. ydy^ — bdxdy r: ac/a?^ ; y =: — — is not a solution. 14. a'^dy^ — xydxdy z=. x^dx"^ ; j/- 4- 4«* = is not a so- lution. 15. ydx — x Vdx^ + dy^ ; a; z= is a particular solution. 16. xdx + ady ■=. a s/dx^ + dy'K What kind of solution is a: n .'^ 17. dx"^ ■=. x{dx^ + dy^) ; ^ z= is a singular solution. 18. adxdf — y{dx^ + dy"^)^. What kind of solution is y = o? _J 19. ^/2xy Vdx^ -f- dy' = xdy + ydx ; y "=■ x is a par- ticular solution. (Art. 3, ex. 2.) X E ^dx 20. dy — — ; 07 ~ « is a particular solution. ea—e dx 21. xdy = J- — — \ xy=. a is a particular solution where The complete primitive is ay zz c{x — a) ■\- a ^/b^ + c^ ; and the singular solution is y = — ^/Sa^ — x'^. «-£.(l-"M(l-)-»-"=»- The complete primitive is x\c-ay- b^c-a)* y — ax ~ H 7 r= ; and ^ = a^* is a par- 2 4 46= X ticular, and y = ax + -rr^ is a singular solution. CHAP. V. SINGULAR SOLUTIONS. 183 '^^' daf'^ Idx ""dx"^ S 2 dx^^ ^dx"^' The singular solution of the first order is — + xjp — Tg - (1 + ^")^ = (a)> whose pri- mitive is / . (^ + v^TT^) = A/T6yT4rM^ —^ vTT^+ c. ttX^ The complete primitive is y ^ bx—a'^ — b^ziOzzv. The double singular solution is 4^ + a:^ -|- r* n: ; and it will not satisfy the proposed example. It may be deduced from the primitive by eliminating a and b by means of ^ dv ^ ^dv V = 0, 3- =:0, and -77 = 0. da db ^5. ^ {X - 1)^^^ dxs ^ dx' ^"^dx y- "• The two first primitives are |p_(._l)6|__=y(l),andf, + |+^... ■f - \, ^^ + - ^ ^, = y (2). They each give the X X same singular solution, viz. {x—\){x^9) -^—■:r^-=.{x — ^Yy \ a linear equation, whose primitive is x{x-9.)c 4 -^ dy^ dy The singular solution of the first order is (2x'-3x2+4;r)« x' 4>(x''-x-\-l)u , , , ' P' + S = 12 + "^ 3 ^'')' ^"""^ ^^^ complete primitive is «/ = i^j^ + 5^? + «2 _{_ ^j _j_ 52, "4^+"ill'=''- 184 FLUXION AL EaUATIONS CHAP. V. A first primitive is yp^ — 2axyp + aroc =: ; and a sin- gular solution \s xy ■:=. 1. 28. y ^ xp -\- -^ p"^ -\- 2,vpq — x'^q'^ — x"pr + The singular solution of the second order is x^qr -T-- = n M. 2j7» x^q x"^ . .... y = -g 6 ~ ^' ^"^^^ primitive is y = cx3 + dx^ — —(1 + Z.r) (a). This example does not admit of a singular solution of a lower order ; since the equations -r- = 0, -r- = and ;i-= " are such that they cannot be simultaneous. rr^i 1 .... ax'^ bx^ i he complete primitive is y = -x ex ^ c®. CHAP. VI. OF THE SECOND ORDER. 185 CHAPTER VI. Fluxional Equations of two Variables of a higher order than the first. 1 . The general form of an equation between two varia- bles of the wth order is F(a;, y,p, q, - ^ ' s, t) = 0, where . , dy d'y d^^-^y d'^y p,q,'''S,t respectively = ^, ^., ' ' ' -[^.^ 5^«. It has been shown, ch. 3, 13, that every equation of this form has a complete primitive containing n arbitrary con- stants; but we have not the means of finding this primitive except in certain cases. We cannot even integrate the general equation of the second order f(^, y, p, q) = 0, though all the combinations except those which contain both the variables x and j/, admit of easy solutions ; or at least are reducible to a simpler form. We shall give these integrations in the next article, and shall conclude the chapter with such other forms as are suited to the nature of an elementary treatise. In the following articles the fluxional coefficient of the highest order, viz. q^ is in general supposed to be of the first degree. 2. Required to integrate thefcxrmSy (1) F(^, q) = 0; (2) F(y, q) = 0; (3) F(p, ^) = ; (4) f(^, p, gr) = ; and (5) ^{y, p, q) = 0. (1) F(^, q) = 0. d^y d^y Here -^^ =fx = x, wherefore -^ = sidx ; and integratmg, -p =fxdx + c, or y\=fdxjkdx -\- ex -^ d. To decompose this into single fluents, since Jdxjkdx = xf^dx — fxxdx, we also have ^ = xfxdx —fxxdx ■}- ex + d, Ex. d^y = ax'^dx^. 186 FLUXIONAL EQUATIONS CHAP. VI. Here -^ = ax'^dx /. ~ = r + c /. ax ax n+l Cor. 1. If w = — 1, -^ — — .*. du — alx.dx + cdx and ax X axlx — ax -\- ex + c' = axlx -{■ ex -\- d, ^ . Tn ^ d^y ^dx ^ « _ , , Lo?\ 2. n n =— 2, -f- — —-- .*. ay = ax-\- cdx and ax X X y =— alx ■\- ex -\- c\ (2) F(2/, q) = 0. Here —- —fy — y ; wherefore, multiplying by 2dy and integrating, there results %^ = ^f^dy = y- + c .*. • • • Ex. 1. a'^di-y + «/r/a?°- = 0. Multiplying by My, 9.aHyd^y -f- 2ydydx'^ = ; and inte- grating, a"dy'^ + y^dx"^ = c^dx% or dx = . .•.••• ^c* — y^ X = «sin.~^ — + c', or 2/ = csin. f h c' ) I which may be made to take the form of y = csin. — + c'cos. — . ^ a a Cor, The primitive of d'y + Aydx"^ = is JL J- ' y = csin.A^J7 4- c'cos.A^iT. Ex. 2. a'^d'^y — ydx^ = 0. Here rfa? = .'. — ■ = I .±J—L^ .-. .... ^^24.^2 a c' ,V X ji de^ ce ^ c'^« —y\- ^y"^ + c^ and 3/= — —r'^ which is of the T. X form y = ce « + c'e ". CHAP. VI. OF THE SECOND ORDER. 18T Ex.3. dx^ (a-yy 2dud'y 2mdy df 2m Here ;; / = , ^ .'. -/^ = + c dx^ {a-yy dx- a-y L if from the , ^ 2m f and = 4- c a nature of the question dy — Q when 3/ = ; whence we tZ?/* 2m y J /a Va-y.dy have -/- = -^— .\dx= —- ~ — • • • dx- a a—y v 2m ^y ^ /~ { a-y)dy ^ j~ r [ yi-y)d}J_ ^ \ady^ ) This solution is a particular primitive. (3) F(p, 5) = 0. Since g- = -^, we have rfp, y- j = 0, which is an equa- tion of the first order between p and x. Solving this equa- tion with respect to -p, let dx — - ; then x = Fp -\- c. pdn Also dy = pdx = — — ; wherefore^ —fp + ^'i ^^^ eli- minating j!? by means of these equations, there will result the required primitive. Ex. 1. d^y = dx vdx^ -f dy'^. Vl-hp' Here -f- = ^l -\- p^ .-. dx = dx ^ ./lJ-n2 ^= c '''P-Y-2^''''^^ = '''^^^^''' Ex 2^-^'^ + a-O ^''' '^' dx^ + dx'^ + a - 0. TT ^P . , dp Here -~ 4- 7np- —— a,oY dx — — ^ dx ^ ' mp^-\-a^ 188 FLUXIONAL EQUATIONS CHAP. VI. 1 , m'^p I X =■ =^tan.~^ — -f- + cf Vnia fjj% f . and let = =^ ' -k -^ ^ \ X = — cotan. * L- ,\ v z= / a — cotan. sfmax ^ma ^ ^ s/ m ,\ dy — — cotan. A/ma^ X ^madx * ,•. y = — /, sin. y/max + c, Ex. 3. Required the curve whose radius of curvature is constant. (Vol. i. ch. 11, 5, cor. 1.) — ttdt) The equation is (1 ■{■ p'^Y = — aq = -p .*.••• dx =^ ^ ^ 3 .-. (vol. i. ch. 2, 13, ex. 9) x—c "'^ Also dy = pdx = — — 3 .*. 2/ = . -f t/; and (l+p2)^ VH-p"" eliminating p, the primitive is a"^ =. (c — xy + ( j/ — c')% which is a circle whose radius = a. The two first primitives are X = c and y = — ■ + d, which will each Vl +p^ ^/l +p^ satisfy the proposed equation. They may be integrated by solving them with respect to p, which will give dy in terms of X and dx ; and the result in either case is a circle. Ex. 4. ;7^ + A /' + B / + c = 0. dx'^ dx'^ dx Here-y- +A»2 + Bp + c=0, or —hdx^ S- . dx ^ ^ „ B c B C Let— a and — 6 be the roots of p^ H p H =0; then A A CHAP. vr. OF THE SECOND ORDER. 189 , I r dp dp 1 1 . Aff ; and inteffratino' a-b\p+a p+bS ^ ° p+a , ,, a— 5<7e^(«-«')^ a-b . ^. ^ dx , 1 C^A(a-6)^_l .-..?/ = (« - 6)/^^j^^z^jj--Y - 6^ = — I ^,^(,_,). -^* 1+i^' .'. ^ = (1 — m-)x^ -\- 2mcx — c^ • • • • • • x'^ .-. 1 + p2 ^ .•.p^ = (1— m^)^^ + gmcjr— c^ (1 — m2)a:2-(-2?wc.r-c"- TTzr — c p = — ■ Vil — m^)x'-\-2mcx—c' 1 mx — c , . , = — = . — : which a/1 — m- / 2mc c^ / 0^2 + = X — z V 1— m2 l-m^ may be integrated by substituting « = :r + j (5) F(y,p, gr) =0. Let the equation be reduced to the form q —f[y^ p) ; then, since dy = pdx and dp = qdx, we have, ehminating , dy dp pdp dx, -^ =-^, or o =/V-. Substitute this value of ^ in g' z=ij\y^ ^) ; and there re- sults pdp =y(2/? p)^3/» vvhich is an equation of the first 192 FLUXIONAL EaUATIONS. CHAP. VI. order between y and p. Suppose that it can be integrated ; and let its primitive be ^(3^, p, c) = (a). Case 1 . Let p z=.y\ then dx z=. -^ and x = f~. Case 2, Let 3/ iz p; then p, which = ^, = ^; where- fore dx n — and x =iy — , from which p may be eliminated by means of ^ = p. Case 3. If (a) cannot be solved with respect to either 7/ or Pf we must find some third function v, so that z/ := v, p =z w; then, since dj/ zz pdx, we have dv zz wdx and dv X — / — ; from which X) is to be eliminated by means of Ex. 1. yd'^y + dy'' + dx'' = 0. HereqorP^zz^l±^,..^-l = -pP- . • • ^ dy y y i+p^ c ,*. y "=. — : — , which is one of the first primitives ; and for the other, we have p =z — . .*. dx. which Jp —cdp , cp zz — , = ^ .*. X zz d P (l-^p^y VI +P' To obtain the complete primitive, we may either eli- minate p fwm the two first primitives, or by means of y = — , we have ^ V I +p^ Vc^ — y^ , ydy , , . « = —. or dx = ^ ^ and x = d - VC^ — y\ y VC -y'' ^ which is of the form 07^ + 2dx -{- y^ = c". Ex. 2. {ydy + adx)d'y = di/{dx^ + ^y^). Substituting g = —-, there will result dpij/p + «) = dyil + p''), or ^j/ - ^^^^ = T^^^ ^^"^^'* equation /. (ch. 4, 18, ex. 3) «/ = ap -f c^/ 1 -f- /?^ • • • CHAP. VI. HOMOGENEOUS EQUATIONS. 193 .*. dx = — ■=. — - + ■ ; which beinff integrated, and V p v\+f- ° ^ p eliminated by means of the value of 3/, will give the re- quired primitive. 3. Certain equations of the form F(a;, 3/, p, q) — 0, which are to he integrated hy particular artifices of calculation. Ex. !.«<& = 5^^±f^. ax ' T adx yd^y-^rH^ • .• 7 U^h , Here — = \i **• ^'^^^^g^'^^i^o' ^^-^'^ d' '^ ^ ''' alxdx = ydy + cdx^ and integrating, -^ -^ ex \ d - • • = aflxdx = alx . x — ax \ or changing the constant, -^ + ex -\- c' = axlx, 4. Homogeneous equations of the second order are re- ducible to the first order. Equations of the second order are said to be homogeneous when they are so with respect to d^j^, dy^ dx^ y and x. Or, expressed in terms of p and q, if they are homogeneous, the dimensions of;? not being taken into the account, and q con- sidered as of — 1 dimensions. Assume y — vx'\ which substituted in the proposed, di- = vx'\ wnic = — \ vidii q = — i viding by x^ will give an equation be- tween w^ V and p. Let it be w zz (a). ^. dx dv Smce du or pdx zz vdx + xdv, we have — = . Also , - wdx dx dp dp dv dp or qdx = , or — = — ; whererore — = (p). ^ ^ X ' X TV w p — v ^ '' Substitute in this expression the value of re' deduced from 11 =0, and there will result an equation of the first order between p and v. If this can be integrated, so that p may be expressed in terms of u, we shall have from the equation dx dv . ... — = , X m terms of v ; and consequently ?/, which = vx^ may be found. VOL. II. o 194 FLUXION AL EQUATIONS. CHAP. VI. Cor. 1. The proposed equation is always in tegrable when (/3) is integrable; and the result is the complete fluent, because the double integration will introduce two arbitrary constants. Cor. 2. The proposed is integrable when w \s sl sole function of p; for then (/3) is linear with respect to v. Cor. S. The proposed is also integrable when te; is a function ofp — v. For substitute s = p — v; then p = s -\- v and (/S) be- comes sds + sdv = sdv .'. v = f .-. &c. ^ s -5 Ex. 1. x^d'T/ ■= xdxdy + Sydx'^. Assume?/ — vx'\ w > .•• xw — xp 1- 3r^, or eg; = p + 3i; ; 5' = — S wherefore by substitution, " p-\-'6v p — v or {p — v)dp =z {p -y ov)dv, or pdp — vdp + pdv -f- Svdv ,'. p^ = 2vp + 3v'^+ c^ and p = v -{- -v/4i;^ + f?- .*. by sub- dx dv dv . . stitution — = ' = : .*. mteffratmff, reduction, ?/ = — ttt^ — = cx^ H Ex. 2. (c^or'^ + di/'^)'^ = wc?^^^ v'^^ + y'- By substitution, (I + p-)"^ = wt^?^/! + u^ .♦. w = — ==^; and by substitution, __ . — dp= n(p-^v)dp dv ..... , , or — = ; which is integrated by means or {1+p")- Vl-i-v^ circular arcs by Euler and by Garnier. Calc. Int. p. 234. . Ex. 3. Sydx" -— Zxdxdy + ^y"d~y = 0. By substitution St; — 8p + 4y'ty = 0, or 22; = — ^-r— — - CHAP. VI. HOMOGENEOUS EQUATIONS. 195 //DN u 4 be the factors of 4u^- — 4^3:; + 3 ; then dx a 2dv b 2dv a — = ^^ — f ; or c\2y + hxf = {2y + a^)'' ; which is the complete primitive, because c is contained in a and 5. Ex. 4. nx^d^y = (y^^o; — xJ?/)^. By substitution we have nxw = {vx — xpY, or tv — — iv — pY ■=z — : and (B) becomes = — , n^ ^^ n ^ s- s lids s — n .. dx dv nds or dv = .*. V = 111 . Also — = s — n* c X p—'v s(s—n) ds ds s—71 ^ , s—n , or .r = — ; — . But y = vx = nxl . S -11 s cs '^ c ndx = nxl .. , .. c(\—dx) Otherxvlse. Since 7ix^q — {y — xpy, substitute y—xp=v; and differentiating, we have — xq = -j- ,:hy substitution, nx^dv ndv dx \ \ n \ n — v'^y or — dx ' v"^ x^ X c V c y — xp* ncx ydx — xdii ricdx . . or y -^ xp = .-. — ■- = — r; and, integrat- '^ ^ X—C X" x{x—c) ° y _ x—c *' X c'x 5. Equations of the second order which are homogeneous only zvith respect to j/, dy, and d^y are reducible to the first order. fj ._ ^^£ f \ then, since (ex hyp.) the proposed is ho- mogeneous with respect \.oy^ /?, and q\ the same power of ^ o2 196 FLUXIONAL EQUATIONS. CHAP. VI. will be found in each term of the resulting equation ; which therefore by division becomes an equation between v, w, and X, Let it be w = (a). Since di/ =. pdx = vydxj we have -- = vdx. Also dpj 1 • 1 7 7 n 7 1 ^ ^y *^dx — dv yfhich = qdx = wi/dx, = vdj/ i-ydv; wherefore -^ = , , wdx — dv , , , ^s or vox = , or dv + v~dx = wdx (3). V Substitute in (/3) the value of w deduced from (a), and there will result an equation which when integrated will give V in terms of x ; and to find i/ in terms of x, we have dy , — = vax, or y = ce-^"''". Ex. 1. yd'y — dy" — xydxdy^ ^^ y^ — p^ — ^J/P- Substituting^ = vy, q = wy; and dividing by 2/% there results w — v'^ = xv, or w = xu + ij^ (a). Substituting this value of w in (/3) it becomes dz) dv + V^dx — (xv + v'^)dx or — = xdx .\v = ce-l'"'^ . • • V ... A == ce^^d.^x and y = cW'^^'"'"'^, Ex. 2. The linear equation d^y -\- vdydx + oydx"^ = 0, where p and a are sole functions of x. Here q -\- vp -\- ay — Q \ which by substitution and di- vision becomes w -\- vv + o. = 0, or z^ = — Pf — a ; and (/3) becomes dv + (i;- + Pt; + Q)dx = ; from which v may be obtained in terms of ^, and consequently y, which — . f^g/vdx^ jjj^y i^Q found. Ex. 3. Required the spiral whose curvature varies in- versely as the radius vector. Here „^ ^ = ny, where the anejle described by the radius vector y is the independent variable. (Vol. i. ch. 11, 10.) Since this equation is homogeneous with respect to y, p and q, substitute^ = ,:7/; then, since q = ^r^, we have, di- . 1. , , /, A « n^izdy -\- ydz) ,vidmg hyy\ (1 + ^') = w + %i%'^ \j ' * ' ay CHAP. VI. HOMOGENEOUS EQUATIONS. nyzdz 197 dy dy _ —nzdz nzdz Cor. If re ^ 1, there will result Or if ^ = cos -'^^ , we shall have y = :; • and %y—c y — c — + cos.-*-^ . y y - 1 — cos.( ;r = a + 1, let 6" = 1 + c^; then — = 7 rrr- — - and X (Z — 1)2+C- Zo; = — tan -^ h Zc'. c c Eliminating 2 from these by means of y = x^z, the three equations which include all the possible cases, and which together constitute the complete primitive are c c cx" p 2y Next take v — 2z = ; which gives ^ = 0, or -^ = 0, and mtegratmg, 7/ = ex- ; and smce this, IJ X when differentiated, satisfies the equation, but is not in- cluded in the complete primitive, it is to be considered as a singular solution. By means of the forms of art. 2, equations of a higher order than the second may frequently be integrated ; of which we shall give instances in the following articles. d^y 7. Required to integrate , ^ = x, i,e. tojindy^xdaf'. This is to be integrated in the same manner as form (1); for J'^xdx^ —fdxfxdx = xfx dx —fxxdx Ix 202 FLUXrONAL EQUATIONS CHAP. VI. y^xda^ =J'da:/^xdx'^ =^fxdxfiLdx —fdxjxxdx Similarly,y'*xcZa?* = t-^. \ xlfxdx — Sx^^xxdx • • • • 3.2 > + zTT^xfx^xdx — fi^xdx J . 1.2 •^ S Generally it may be demonstrated by the method of in- duction that «/ =y"xc?.r« = T"o~q 31 j x'^~l/xdx • • • - (« - l)a?"-y^XCZa? + (^ - ^^) (^^ - gl^n-3J^2xc7^ . . . ±yx^~^xdx > J according as n is odd or even. The arbitrary constants are involved in the n fluents which compose the value of «/; and the part to be added is manifestly of the form c + c'x -\- c''x^ ...-{- c'^'—^x'^~^. The fluent may be also developed in a series in which no integrations are necessary; for, integrating by parts, we have (vol. i. ch. 4, 46.) ^ cZx , x^- dx ^ d^x^ fxdx =. XX -fx ^dx =^x--^ +lfx^ —dx XX x"^ dx x^ d^x x^ d^x 'd^-'^2jA XX x^ dx 1 2 dx^23dx'- 2.SAdx^^ "^ f XT X fitX "^ _ XX- x^ dx x"^ d^x 2 2.3 dx 2.3.4 dx^ 1 c xF' dx ^* ^'^x ^ "Eli "3 5^"" 3T4 5^2"^' "5 1 c a'^ ^ _ 7 '^TJ^X'^dx'' "5 2.3 , - &c. — x XX'' 2x'^ dx ^ chi L2 ~ L2^ dx ^ 1.2.^4 rf?- "~ ' CHAP. VI. ** OF THE N*^! ORDER. 203 Similarly it may be shown that 2A •^'^'^■^' = iii - lib i + TJ^Ai i^' - • • ■ ' """^ " * may be demonstrated by the method of induction, that J xax - i2.S..,n \.2,..{n+l) dx^ 1.2...{n-\-2) dx^ n{n + \){n + 2) xjT^ Saj-^ Jx 2"^ J2x M+3 — + . • • to which is to be added, 1.2...(//, + 3) dx^ as before, the part which involves the arbitrary constants, viz. c H- dx -{- d'x'^ • • • + c'"~^^'*~^ The form -~ = y where y is any function of j/ can be integrated only when w = 2 as in art. 2. (2) d^ij Cor. The complete primitive of t^^ = is • • ■ • y ^ c -\- dx + c"a;- ... 4- c'"-^^"~^ (d"''ii d'^ — ^^\ This may be integrated as form (2) ; for, taking a par- lar case, let the equation be f f -^— ^, -7-^ j = 0. Substitute q — -j— ; and the equation becomes F (-7^,2') = 0; and proceeding as in form (2), we have dx = f and JT z=f—=^ + c'. v/q' + c -v/Q +c Also ^ = yrf. = -1^-= .-. t =/-^ = <^" + ^' .', y —fol^dx + A + c'" = q'" + c"a; + c"' ; and ehminating q from these values of x and «/, there will result the com- plete primitive containing four arbitrary constants. ticu 204 FLUXIONAL EQUATIONS CHAP. VI. d^" dx"' Here -j^ = q; wherefore multiplying by ^dq, and inte- dq^ , dq ^ grating, ^, = ?' + c^ ■: dx = -== and c Also -JL^qdx =z ^ ^ .•.3^= a/^MTc^ + c"... ,'. dy = dq -\- d'dx ,\ y = q -{- c"x + c"'. To eliminate q, we have de'' = q + Vq'^ + c^ .-. • . . c'2e2^ _ ^qde" = ^ .-. q = —prrr- = "t: Trry or the re- quired primitive is under the form 1^ = ce'' + de-" + c"a7 + c'". 9. Eeqmred to integrate f(^^^, ^ j = 0. -T-j, -7^ J = 0, which is the same as F f ^, 5^ J = ; and proceeding as in form (3), dx = — and X = q! + c. Also -^ = qdx= ^, .-. -/ =z a" + c' and j/ = q'" + c"; and eliminating q, the result will be the complete primitive. Ex, <» = 1. dx^ _. d^y dr -,•,■, *'^ jLet r = -j-^ .\-r x r = 1, or dx = rdr .\ x — -z^-\-c. dx^ dx 2 Also dq — - rdx = r^dr .*• 5' = -«- + c' ,*. CHAP. VI. OF THE N*^ ORDER. 205 dp = qdx=rdr^^ + d^ /.p = ^ + '^r^ + c^' .'. • " in which substitute r = ^/2(x — c), and there will result the complete primitive. 10. Required to integrate Fi^^^, ^^, ^^;^j = 0. This is integrable in the same manner as form (5) ; for dr let the equation be F(r, s, t) =0; then since s = -r-, and d^r ^ dv d'T \ ' = ^. ^e Lave ^[r, ^^, ^j = 0, of which the pri- mitive may be found as in form (5). Let the result be d^y d^'^y r = X ; then if r = ^-^, we have -r—^ = x, which has been integrated, art. 6. 11. An equation of the nth order containing only the in^ dependent variable is reducible to one of the ii — 1th order which shall contain both variables. For let the equation be f(x, p, q, > » - s, t) = 0, where t = ^-^, s = -^—^i .... Ihe equation becomes by . . / dp d^'-^pK substitution F ( .r, /?, -^ • • • T~^ j == 0, an equation be- tween X and p of the w — 1th order. 3 Ex. (1 — x)d^i/-Sdxd^7/ = 0, or r- q = 0. 1. ~~ X dq . da 2q dq Qdx Since r = -T- , we have -7 - = r— ^— , or - = -z .•. mte- dx dx 1 —X q \ — x c dp c ^ cdx , grating, ?= ^^—y„ or ^ = ^^-^ .: dp = ^-j^^^and c cdx . _ FLUXIONAL EQUATIONS. CHAP. VI. C y = z + 6-'.r 4- c", which may take the form . . • . (I — ^)j/ = ex- + c'x + c", 12. PRAXIS. 1. md^y = (a^ + bx)dx^ .\ my = "2" +2~3 "'' ^^ "^ r — o ^- ^-^ +^ = «•••«"- *^ = ''^""'' + '• 5. xd^y + c/j7J^ = .'. j/ = cZ^ + c'. 6. xd-y — dxdy = .\ dy = x''- — c'^. _b_ 7. (1 — ax^)d'^y = hxdxdy ,'.y = cf[\ — ax'') ^dx. 8. 2 ^a'^ - axdxd'^y + (dx'' + r/y^)^ = .• 2^ y = ^/cfo: — a?2— -^tJ.S." 9. yd^y + 6(2/2 -f £/.r- =: 0; a circle, rad. = \/«^ + h~. hxdifi 10. xyd^y = ydxdy -)- o^Jy'^ + — - .*. • • • \Olr — x" Ach+ x/a'^ — x'Y _ vct^~ x'^ L — dy h 11. x'^d^y + xdxdt) — ^y^dx"- .*. — = 13. The above, together with hnear equations, are all the forms which are integable by general processes. Before we proceed to the discussion of linear equations, it will be ne- cessary to consider briefly the method of rendering equations of the higher orders integrable per se by the introduction of a factor. This is Euler's favourite method of integra- tion ; but, since we can seldom find this factor without knowing the primitive, it does not possess, in the present state of the science, any great advantage over that of the separation of the variables. 14. It was shown, ch. 4, 21, that when an equation of the first order is not an exact fluxion, it is in consequence CHAP. VI. EXACT FLUXIONS. S07 of the disappearance of a factor by the process of dif- ferentiation and elimination ; and that if this factor can be restored, the equation is integrable. Since an equation of a higher order than the first is derived from one of its first primitives by the same process of differentiation and eli- mination, it is obvious that the same must be true in this case; and that if the flictor can be found, the proposed equation is reducible to one of an inferior order. We shall, however, give in the following article a direct demonstra- tion of this proposition : it is the same as that contained in ch. 4, art. 22, generalized. 15. An equation of any order whatever between two variables may be always rendered an exact Jluxion by the introduction of a factor. Let /?, ^, • • • 5, ^ be the 1st, 2d, • • - w — 1th, wth fluxional coefficients of// z=:fx\ then, by solving the equa- tion with respect to ^, it may be put under the form i ^A^^y^V^ . . • r, 5) = 0- Let c be the arbitrary constant of one of its n first pri- mitives ; and let this primitive, when solved with respect to c, be c = r(a7, z/, /),••• r, s) =, by substitution, ?/, which is therefore a function that does not contain c. Now in u = F(.r, y, /?,••• r, s), we may suppose that there are only two variables x and s, which must be made to vary so as to satisfy the equation ; and differentiating on , . , , . , du . . , _ du du ds this hypothesis, we have -j-, which rz 0, r: -j- + ;t- ^- • • • du du , .n , du . du — -T- -\- -r t =^ IV ■{- ujt, if ?/ zz -- and Wo = -,-, or • • • dx ' ds dx ds W t H rr ; an equation which is identical with w, t +f{oc^y,p, •••?•, 5) — 0, since it contains the same con- stants ; or ^ + f{x, y, p, ' ' - r, s) = t + Ur 1 , , . I du r /./ ) du which multiplied by dx is an exact fluxion, whose fluent is of the form ti — (p{Xy y, p • • - s). It appears then from this investigation that an equation, whatever be its order, when under the form FLUXIONAL EQUATIONS. CHAP. VI. t +y\^} ]/f p ' • • ^, <^) = may be always rendered inte- grable per se by the introduction of a factor ; and that this dc factor = Uq = -j-y where c = f{x, i/, p - ' ' s) is the arbi- trary constant of one of the first primitives. Cor, 1. Since an equation of the second order contains two first primitives, there are two factors essentially dif- ferent which render an equation of the second order in- tegrable. Similarly there are n different factors which render integrable an equation of the nth order. Co7\ 2. If z is a factor which renders the equation an exact fluxion and = du, then z(pUi where

^"^ ^^^ t and s be the two factors each of which renders it an exact fluxion, so that ^, . ^ du , , . dv . , , . ndu ^^"^^^ "" di ^^^^^^"^ d^' ' ^^^'(^+^)'"";^ d.(pu , , , ^ d, V Let ;// denote the function ;//[fw, x^}; then, we have { ^S^''' + 4^ x'^ }(? + ") •••••••• \ cdyh ^ d\b ^ ^ ... d\p = -Y-} -r-d* •'• CHAP. VI. EXACT FLUXIONS. 209 X X dc ae "■ - dc^ ae« ... , „ , T- = -77- and -T- = w ; which are therefore the re- dp 2 dp 2 quisite factors, 3! Multiplying the proposed by e ^dx, it becomes —^/ ijdoc\ e '^[dp — ^—^ \ which must be an exact fluxion ; and it ap- X p a, pears from differentiation that it is the fluxion of — (y + ^p)* CL X If the proposed be multiplied by — e^dx^ it becomes X e~4 dp — —^ \ which is the exact fluxion of — (y — op). Any factor included in either of the forms e «0j ^ «(j/ + ap) j", e''yXe"-{y — ap) 5 will render the proposed an exact fluxion. Ex. 2. xd'y 4- dxdy = 0, or 9 + — =0. (Art. 12, ex.5.) The complete primitive is y = clx + c'; and the two first primitives are c — px and d = y — pxlx; .\ • • . • — — X and -p = — xlx, which not being included in x(^.px is the second independent factor. Multiplying the proposed by these factors, it becomes xdp + pdx, or d.px = and — xlxdp — plxdx, or diy — pxlx) :=z 0. Dividing c' = j/ — pxlx by px = c, and substituting ,, c' - „ y ^ dc" y , . , . c' = — , we have d' = ~ Ix .•. -j— = — -^ ; which is c px dp p'^x a third independent factor, since it is not included in either of the forms x(p.px, and xlxx{y — pxlx] ; it must there- fore be included in tlie more general form d^P , d-^ , 'dil^^^'d^^""' To verify this ; we have t zz x^ s = — wlx, u = px^ V =. y — pxlx. Suppose (pu = u, x^ = "v •'• = dx^ is the re- quired primitive as in art. 4, ex. 1. Ex. % xyd'^y + (4.??/ + y)dxdy + xdy'^ H- ^^j^a — q. Here n := xy and dm ■=. 2y"dx + ^xydy, which being homogeneous and an exact fluxion, ^i/'^x -\- 4.27?y^ M = ~ — ^ (ch. 4, 25, ex. 2) = 2y^x ; and the first • • • • /^ 7 7 7 7^7 ^^^ primitive is 2y'^xdx + xydy = car, or ydy + 2y'^dx zz ; and by substitution, dv 4- ^'odx = - — '- ; and integrating, X V or w"- — e-^' \ I + d i= 6'^-*^+ ce'^^ f , ^ I -^ X S -^ X ' which can be integrated only by developement (ch. 1, 39). Ex. 3. y^-x^d^y + (% + ^xY)dxdy + ^x^ydy'^ + t/'^Z^o; The equations (a) and (/3) obtain ; also dM = ^ydy^ or M z= y- ; wherefore this example is an exact fluxion, and it is the fluxion o^ydx-^-y^x^dyrzOzidu, where z = —^, X y- LINEAR EQUATIONS. 19. T)ef. A linear equation is one in which the function or dependant variable as well as all its fluxional coefficients p2 212 FLUXIONAL EQUATIONS. CHAP. VI. are of the first degree. The canonical form is and X are functions of a?. Linear equations frequently occur in almost every de- partment of physical science. It has been already shown that the primitives of equations of the first order may be exhibited by means of a very simple formula. Equations of the second order are always reducible to those of the first, which however are not linear ; and they possess some very peculiar properties which facilitate their integration ; but yet there are examples of which the primitives cannot be found by any known methods. 20. Required to integrate ^ + a^, + b^, + • • • + D ^ + E^ — . 0, where a, b, • • • d, e are constant quan- tities. Substitute i/ =. ef""; then, since -^ = «e«^ -rz ay i the equation becomes, dividing by «/, ""^ \.a^ + A«"-i + Ba'^-^H +Da + e • • • dx"- ^ J &c. = &c. If this can be solved by the Theory of Equations, we can obtain n values of «, and consequently of 3/. Let them be y -z: e^'"" y y ■=. e**'*, ••• ^ = ^''"''; each of which is a solution. Represent these conjugate values by y', y, • • • z/'""; then shall the required primitive be of the form y-dy^ -\- cY + • . • + dy\ For assume y = dy^ + c'y + . . . + c'y« ; then dif- ferentiating this assumed value of 3/, and substituting it in the proposed form, the form becomes , (dZV c?'^-y dy^ ,, ^''{■d^'^'d^^-'^^'h^-y'} \ + . . . dx"" dx''- ' ' *•' ' dx + Ey'-y CHAP. VI. LINEAR EQUATIONS. 213 which n independently of the values of c', c'' • • • c'", because each of the forms inclosed in the brackets is (ex hyp.) = ; and consequently the assumed value of ?/ is its true value ; and it is the complete primitive, since it contains n arbitrary constants. 21. If the equation (a) contain impossible roots, th ey lie under the form a! — x -\- ^ V — \ and a" = a — /3 \/ — 1 (Alg. 355), ory = <'+^'^~'^^ and / = /-^^"'^ ; and the corresponding part of the value of y, which denote by {y)> =ce -{- c'e n e (ce -f ce j". ^ (ixJ^ ^ — - . ^ - — /3xV— I But e — cos./5x\. Cor. 1. The conjugate values of y are particular so- lutions ; viz. those which result from supposing that one of the constants of the primitive = 1, and all the rest = 0. Cor. 2. When two or more roots of the equation (a) are equal, the coefficients of (y), as developed in art. 20, become infinite. Thus, when two roots are equal, c'' :=■ —r- and ^'' d •=. e' r-, both which ■=. x when ^ zr or fl' = a". h The proposition contained in this article may be also de- monstrated in the following manner. 23. Required to investigate the form of the piimitive when the equation (a) contains equal roots. * As this demonstration, which is D'Alembert's, contains a fal- lacia suppositionis, the student may assure himself of the truth of the result by ascertaining that y = e'e"'^ -j- e xe^''^ satisfies —^ -|- A -^- + Bj/ = Q when a = — 2a and b =. a\ CHAP. VI. LINEAU EQUATIONS. S15 In the proposed form substitute y = xe% where x is a function of x whose form is to be determined; then it becomes x{«" + Aa"-^ + Ba"-2 4- . . . + Dflj + e} + j-| ^«"~' + {.n - l)Aff"-2 + . • . + D J + Y;^,{<^-lK-' + (^-l)(w-^)Aa"-'+-+c| + &c. where each succeeding term inclosed in brackets, considered as an equation, is the limiting equation of the preceding (Alg. 307). If equation (a) contain no equal root, this form corre- sponding to any of the values of a cannot = unless x z: a constant =r c' ; in which case j- n 0, -7-^ =: 0, • • • But if two roots a\ 2b; then 1/ = ^~v"2^"/ and y = e • ^ ' , and the required primitive is 216 FLUXIONAL EQUATIONS. CHAP. VI. Let n^ be negative or the roots of (a) impossible. If w = /3 V — 1, it may be shown as in the article that ax Let n" = Q ov the roots of a equal ; then ax y — e ^(c + dx). Ex.S. ^-^ + ?^-« = 0. 6?^^ J.r'^ dx ^ Substitute y — e^ .-. a^ — Sa^ + Sa — 1 = 0, or (« — 1)"^ = 0; the roots of which being equal, the primitive IS y = e^[c -\- c'x + c"x^ \ . Substitute y = e'''' .'. a^ — I = ; and the conjugate values of «/ are y = e"", y — e'"-', y "=- ^ and y = e ; and the primitive \s y = ce'' -^ c e~ '^ • • • • + ce + c^g = c^ + c'e"-^ + (c 4- c')cos..r •(§• + (c — c') a/— Isin.^ = c^ + de-^ + c'sin.(a^ ^f- c'"). 25. T/i^ /m^ttr equation -~ + Ar; — ^ + D -^ + E?/ = X ?,s always integralle. To integrate a particular case ; let the proposed be d-y , d^y . dy Assume y = e'^'^'fy'dx ; then p = e"''' (7/1/7/' dx + y) q = e^^Xmfy'dx + ,^;7?y + p'), if p' = ^ v^ . ^^^ r = em^(ml/y'dx + S/w^j^' + Swj?' + ^X6/a; ; and Now to determine m, m\ m" ; we have the equations (1), (2), (3) which are the same as m^ + Am2 + bwz + c = (1) w'2 f (3r72 + A)m' + 3m2 -f 2aw + b m'l + 2m' + 3/« + A = (3). To find the values of m, /w', m" which will satisfy these equations ; let a, a', a" be the roots of ^^ -r az^ + bz + c i= in the order of their magnitude. Assume m = a the least root ; diminish the roots by a by substituting 7/ — z — a, and the resulting equation is y-j-(3a + A)y + (3«2-f 2Aa + b)z/ + a^ -{- Aa^ + Ba + c = (Alg. 282); or, since the last term =. (ex hyp.), y^+(Sa + A)y + (3^2 + 2Aa + B)y zz 0, where the whole is divisible by 2/, because one value of ?/ = 0. Dividing by 7/, the resulting equation is 7/- + (Sa + a)j/ + 3^^ + 2a«+ b =0, which is identical with (2), since m — a. The two values of y are a' — a and «" — a. Assume as before m! — a! — a the least root, and diminishing the roots by a' — a by substituting x zz y — [a! — a)^ the equation becomes x'^ + { 2(a' — a) + 3« + a j a? + (a' — af + (3a + A){aJ -a) + 3^2 + 2Aa + ^ = 0, or X + 2(a' — a) 4- 3a + A = 0, which is identical with (3), and consequently m" = x = a" — a — {a' — «) = a" — «'; or a, a' — a and a'' — a' are values of m, 771' and m" which satisfy (1), (2) and (3), where a, a', a" are the roots of an equation of the form a^ + Aa" + Ba + c = 0. = (2)| 218 FLUXIONAL EQUATIONS. CHAP. VI. Substituting these values in the expression for «/, we obtain the complete primitive, y zz e''''fe'^'~'^^''dxfe^°'" ""'''''' dxfe-°'"'yidx^ involving three arbi- trary constants. The complete primitive of the proposed equation is where a', a", •••«'" are the roots of a" -f Aa"-i + B«"-^ . . . + Dfl + E = 0. If the equation contain equal roots, this solution does not fail. If any of the roots are imaginary, they may be ren- dered real as in art. 21. Cor. 1. If X = 0, we have/e^^'^'xria? =/0 xdx =. c, be- cause dc:=iOx dx ; wherefore y=z ef^lfe''"''~^^''dxfce^"'"~^'^''dx • • ■ — ^''y^(«'-^)^J^{ c^(«"-«')^ +c' } —e^^'lfce^^'-^^^dx +fc^e^"''-^^^dx ] r: e"^ [ cg(«"-^)^ + c'e*"'-"^* + c'' } = ce'''' + c'e""'^ -j- d^e''"'' as in art. 20. Cor 2. The value obtained for y when separated into single fluents is ^a'xy^-a'^xf^a: e''"^fe~^"''y.dx y -~ («'-^'0(«'-«O---(«'-«'^"^(«''-«')K-«''')---K--«^*'* ^■Y(^"''^dx + (a'« - «')(«" - «'')...(«'"-«'"-') ' 26. EXAMPLES. Let a and & be the roots of a* + aa + b = 0; then the primitive is y — e^fe^^-^^'dxfe-^^^dx^e,''''fe^^^^='dx\fv.e-^'dx + c} = e^'^fce^^-^^^'dx 4- e'''fe^^-''^''dxfKe-^''dx — ce^^ + e«^ > ^ fv^e-^'dx — /-^ + c' J- = ce«^ + c'^-^ + -— ^ ie-'lpRe-'^'dx — e^^fKe-^'^dx |. If « = 6, 2/ n e^'/dx/Ke-^'^dx— erfdx\fKe-''''dx -|- c} = c^e''^ + e^'fdxfKe-^'dx =r cj^^'-f e''''[.r/k^~''^J^ —fB.e~^''xdx + c'} _ g = a -f /3 v/— 1 ; then it may be shown, as in art. 21, that the first part of the value of y is ce"^(sin./3a: + d) ; which is its value when r n: 0. The remainder = < e jRe ax J Re ax > = } e fe e Rdx • • je e B,dx >. — e Substituting e zz cos./3;r ± ^ — I sin./3^, this part becomes ^(cqs./Ba' + v^ — lsin./3.r) x 2/3 ■ax y(cos./3.r-- a/— lsin.jS^)e nrfa; — (cos.jScr — \/ — Isin/Sor) x y(cos./3a? f v^ — 1 sin.|3er)^ rc^jp J =< —2cos3xf ^/ — l s'm.3xe Rdx • • - 2/3 ^-U + 52 a/— 1 sin./3a;yco§./3;r^ "' r^/oj ( ^ ~~ "fl"] sin,|3.r/cos./3^.e Rfix — cos.^xfain.jSx. e Rdx > ; and the complete primitive \% y =. c^*^(sin./3^' +. To give this a real form, we have a z:: 0, /3 ~ n .*. j/ = c + c'sin.wo? -\ — -\ ^yxi.nxfco^nxvidx — co^.nxJ'^inMXTidx i. If R is expressed in a series in terms of sin.oj and cos.a7, the value of 3/ may be found by means of the series demon- strated, ch. 1, 49. Ex.3. J + «=^/ = 0. This may be integrated, and a real form be given to the value of y as in the preceding example. Professor Airy, in his Mathematical Tracts, gives the following integration, which is effected without the introduction of imaginary quantities. Multiply by co?,.nxdx ; then cos. nx . -, - + tfy cos.nxdx = ; and integrating by parts, dy . _ _ . - cos.nx ' -^-\-nysm.rix—c, or cdx = cos.nxdi/ +?ii/ sm.nxdx. The factor necessary to render this an exact fluxion is (ch. 4, 28, case 3), and we have and integrating, cos.%a; cdx _ dy yd.cos.nx cos.-nx cos.nx cos.'^nx c y , c . — isiw.nx = c or V = — sin.f:x 4- c cos.nx. n cos.nx *^ u This will take the form either of j/ = csin.(w^ + c'), if Gcos.c'= — and csin.c' = c' ; or of ?/ = ccos.(w^ -f c'), if CHAP. VI. LINEAR EQUATIONS. ^21 ccos.c' = c' and csin.c' = . n Similarly, if 7^ + ^!3/ = ^^ it may be shown that c . k k y = — sin.?ia7 + c'cos.wa? -\ -~ ■=. ccos.fwo? + c') ^ — -. Ex. 4 Tl + J/ + kcos.mx = 0. Here (a) — and (/3) = 1 .-. ?/ := ccos.x + c'sin.x • • • + 1i:cos.xfs\}[i.xcos,mxdx — ksm.xfcos.xcos.mxdx • . • • ,. A:cos.^('cos.(m + l)-3:' cos.(m— l)^;') 2 I m + 1 w— 1 j ^'sin..r Csin.(w — l).r sin.(/w 4-1)^7 i- h cos.mx k cos.mx , . kcos.mx — -^ r~r + ~a r = ^cos.o? + c'sm.crH — --. 2 w-fl 2 771—1 m^—\ This example may be also integrated without the intro- duction of imaginary quantities. These examples belong to the Problem of the Three Bodies; for other instances, vid. Woodhouse's Physical Astronomy or Airy's Mathematical Tracts. Here a ■=. a! =. ci)^ :=. 1 .\y=. e'^fdocfdxjRe~^ dx. liB. — 0,y — celfdx[x + d\ c x'^ + dx + c'' ( zze' \ cx^ + dx + d' 27. We have not the means of exhibiting by any known formula the primitive of the general linear equation of the d^y d^^^ii ^th order ^ + p^^^ ... + sy = x (a). It may be made to depend upon the integration of dx"' "^ ^d^^ '" + »2/ = (^) ' but we cannot integrate this latter when p, q, • • • s are variable quantities. Even I S2i FLUXIONAL EQUATIONS. CHAP. VI. the linear equation of the second order d'^y dy -, 2 + p%- + Qj/ = 0, though always reducible to one of the first order, can be integrated but in few instances. The equation (/3) is always reducible to one in which only one of tlie variables enters. For, substitute J/ =: e'' ; then dy = e^dz = 7/dz d'^y — yidz" + J'^«) d^y — y{dz^ + Mzd'^z + d^z) 7iin—V\ \ d^'y zz 2/(d^« + -^- — -dz''-''d'z...+d"z) ) and the equation becomes, dividing by t/, -j—;^ + r-j——^ • • • + R-j- + s zr 0, an equation which con- tains only X and the fluxional coefficients; and which must be integrated as in art. 11. The equation (/3) is also reducible to one of the next inferior order. For, substitute y zz e-'''-'^'' ; then ■~~ ■=. e-i'^^Z •= yz, d^y __ c d;s ^ dx^~^r''^'d^l dhj r , ^ dz d-z ■) dx^ ^ ) dx ^ dx'^ S and if these be substituted in (/S), there will result an equa- tion of the n — Ith order; which, however, is not Hnear. 28. Required to integrate the linear equat'i07i .... d^y dy __ dx^ dx y ~~ * Substitute y — e->''"^% or % =: -— ; then ^dt ^ dt 7 r. Assume 2 3- + p^ = 0, or -— + \T?dx = 0, or dx '' t ^ t = e ; then, we have d^!^ cdH dt > ^ . 1- , -P '5^+ I d^^ + "^ + ^'r = ^' ^" ^'^^^^' ^^ ^^ ^"^- stitute t = e '^ , there will result -:; h < Q — -^ ^T~ J 2; = ; which is of the form dx^ i 4> dx S 30. EXAMPLES. ^ , d^y «^-l Substituting y = ^•^^'', the equation becomes «^— 1 c?2 + z"dx=- — 7 — x~^dx> This is rendered homogeneous 1 , . , dv dx a^—l (^^ by s = — ; when it becomes -f -— = — or «'-! > — x'^dv 4- x'^dx = — -. — v'^dx. Substitute v ^ rvx; then, dividing by ^i-", we have fl2 - 1 — (wdx + .rrfz^') + c?^ = — - — zv"dx, or c?a: 4 <^?^ ~ a; "a^— 1 „ 4re? 4 ' ^^ H — : a '— 1 a^ — 1 Let m and — ?i be the roots of w- + — — — 7 = ; then n = r and m = r- and a — I a + i dx I € dw dw 1 ^ e ^ r^ — 7w __ 1 — mxi^ X ~ a I w—m xv-Yny ' ' x"- '~ w-in ~~ 1-\-tix% CHAP. VI. LINEAR EaUATIONS. 22S maf-^^ + cnx a+X , and the complete primitive is 1/ z= ex ^ -\- dx 2 . Ex.2. a.^g + ^|d^-«^ = 0. Substitute y = e^^"^^'; then dz+ \ z° -\ idx =^ 0, or x'^dz = (n — xz — x-z'^)dx. To integrate this, substitute xz = v, or z = — .*. • • • ndx dv xdv — vdx = (n — V — v")dx, or = r .'. • • • Tl ^ ~\~ X^ * ^« _ ^_ 1 . from which a value of z may be obtained, n^—xz which must be substituted in y = ^•/■-<^^. dz Assume «/ — m = e-^'^'' ; then J" + ^^ + ^^-^ + x^ = 0, d^ , X ^ ^Z^ + f^o; or (z + x)^' = - — ... (2 + o:)^- - 1 = ;t— -, or or dx ' ^ ^ dx d;s-^dx . . , 1+2;+^ dx= - — ; — ; — r- .'. mteffratinff, x = U-p-. — ; — ; — -r o ,, , xN t ce'^(l-x)-{l+x) e,^.(l_(^ + ^)) = 1 +^ + ^ ... ^ = ^ ^^J_ jrj ^ = — ^ -4 r 7 .*. /^aj: = / -— -ax pr • • • • = l{ce'' + c'^-^) ~-^ .'. y -^ m — ce 2 4- c'^'v'^ 2/. This last step may be also thus deduced. Let z' and z" be the conjugate values of ^, and we have x^ x'^ x^ jydx^ l.e'' — -— = X — axidjk"dx= — x — -^ • • • X-2 / XS \ .'. (28, cor. 1)3/ —772 = ce^'^+de ^""^^K VOL. II. Q. 226 FLUXION AL EQUATIONS. CHAP. VI* ax- X ax x^ - If we substitute y — e^''^% the equation becomes -_ 4- ^2 J j — 0, which may be rendered homo- dx X ^ x'^ ^ geneous by substituting x — — . r^dx ^,y zdx Or thus. Substitute y = e * ; then — = , or - ' y ^ dy %y d^y zdy-\-ydz zy z^y ydz zy , dx~~ X ' ' dx^ xdx x" x"- xdx x"^ ' . . si"^ d^ ^ a^ ^ h the result is -— + —^ rH -4- — = 05or x^ xdx x^ x^ x^ dx ^ dz d% , , — = TTt T"^ — 7\ — — 7 — ; — \7~. — f: ■» whence the conjugate values of 2; are z^— —m and 2"= —ni!, and the re^ quired primitive is ?/ = cx~"^-\- c'.y~"*', where w + 7/i'=« — 1 and mm' == b. Ex.5. (l+^^)g + ^|-4n^3,=0. dss ■x^)}z'' + zxdx 4n'^dx l+x^~ } -^x^' Substitute 3^ ^^•^^'^ ; then (1 +^^) 5 s;^ + - ? + 0^2 = 4w'^ or dz + z-dx + To separate the variables, assume ;3r = — == ; then ^ -v/l+^' dv v'^-dx ^riHx dv dx 2n--v c.{x + a/1 + x^y =, by substitution, cz0*" 4^ .« ^ ^ 4w = cw^"" + 1 .•. 2n—v czv^'^' + l dw dx But w = X -j- VI -i- x^ .: — = ■ .-. zdx. which ^ a/1 +0^2 GHAP. VI. LINEAR EQUATIONS. 227 _ 'vdx ___ 2n(c'w*'' —l)d'w 2ndzv 4incw^''~'^dw y = e-^'^ = -^ — ^ = C2£>2" 4- c'w-^% where ay = ^ + ^1 +072. T^ ^ d^y vdy , « , A , B Q = Substitute 3^ = eJ'-^ ; then C^^ A B It is manifest from the form of the proposed equation that y is a function of — ^-7- ' assume therefore Cb -f" OX m . . . ^ z rz 7—, where m is a constant quantity ; then dz zz — - — ' , .„ : wherefore, by substitution in (a), we have m^ + (a — h)'m + b n 0. Let the roots of this equation be m and rriy and we have two conjugate values of Pmdx m ' m y, y = /^^zz^^'^""^'"^ = (« + hocy, and m' m y zz (a + ba:)^ ; and consequently 3/ — c{a + hxy • • • + d{a ■\- hoc) 5 which agrees with ex. 4. If w =: m', this ceases to be the complete primitive ; and to obtain it by D'Alembert's process, assume m :=.m^ + h\ then 2/ = (« + hx)b ic + d{a + bx)^S - c h h^ '\ = (a + 5^)0 c +c'(l+Z(a+5.r).y +Z(a+6^)2._ + &c. i dli Substitute c -^ d z=l d and -7- = ^'' ; q2 FLUXIONAL EQUATIONS. GHAP. VI. ,y = (a+ M M ^ + ^"^(« + bx) + Z(« + bxY-^^+ &c. ^ ; and if we suppose ^=0, y =. {m+bx) b(e' -\- e"l(a-\-bx) j. If the roots of the equation are impossible, a + ^a/— 1 and a — 13 V — 1 ; then = (a + bx) b lc(a + bx) b + c'(a -{• bx) b \. y But since e ^ = cos.a7±-v/ — Isin.o:, substituting for Xy -rl{p' + bx)^ we have (a + ^^) 6 = cos.-^/(a+&a;)± V — lsin.-r-/(a + &^); ^ c /3 . i(3 ^ and y = (a + ^.r) 6 s ccos.-rl{a + 6a;) + c'sin.-7-Z(a 4 bx) r = D(fl + 6^7) & sin.( -T-/(« + bx) + d'J. 31. Required to integrate the linear equation .... d^y dy It may be made to depend upon the integration of d^y dy -p^ + v-j- + Qy = by the same substitution as that by dy which the linear equation of the first order -r- + i'y=-Q. was reduced to the form -^^ + p = 0. (Ch. 4, art. 15.) For, assume y — xz; then the proposed becomes {d^z d^ ^ ^dx dz dx d^x , rfT« + ^di+'^j + l^ di+''T. + 'd^' = '^' """^ since z and x are undetermined, they may be so assumed d^z dz that -7-^ + p-j- + QZ = (a) ; and consequently CHAP. VI. LINEAR EQUATIONS. 2dx dz dx d^x .^. Integrating (a), we shall obtain z in terms of x ; and to determine x, substitute — = p ; then j^^—-^ and (/3) be- comes dp + j P + — p \pdx — ; which, since p, z and R are functions of x, is a linear equation of the first order ; and consequently we have •=. — ~ > c + fRze-^'"^''dx I ; wherefore x, which zzfpdx + d and ?/, which = x^ may be calculated. Cor. If we suppose r in 0, the proposed is d^^^^Tx^^^^^'" ^ ^^ ^ ^ "~i^ «/= ^\fpdx + (^']' The value of ;s in this case becomes a conjugate value of 1/, viz. that which results from supposing c=0 and c'=:l. ^ ^ dh/ dy Here (a) is^r-r + a^- + b^ = 0. Let a and 6 be the roots of a^ -). A« + B = ; then one value of 2 is z =: e"' ; and consequently p = -:^^ f^e^^+au _^ ^ j • . • • — e(6-«kfyRg-6*^^ ^ c| .-. j/=^''y^<''-«''^t/a7{yR^-^^ti)w + B = 0. Art. 30, ex. 6. Substitute 2 = (« + 6a;) * ; then _AH-2m J A+m ^ p = (« + 5a;) ^ } f{a + 6^) ^ "Sidx -\- d\^ wherefore lid _ A+2m — h _ A+2m A+w +y(a + 6ar) ft dxf{a-\-hx) b Rdx, yvhere a— b = —{m-\-n) n—m h , n — m A+m = c4-c'(«4-6^)~~i~ 4- _^ ] (« + hxy~b~J{a 4- hx)~b~Kdx • • . — y(a4-^^) ** Rofa; 5 ; consequently y= c(« + 60:) ft • • • H-cY« + &a7)'ft+ \ia-\-bx)Y{<^-\-^^) '^^^'Rdx • • • ^ m — nl n n 1 — (« + bx)1f{a + 6a:) 'ft"*'*R£?a7 5 . The same value of?/ would have resulted, if we had taken n 2; = (a + bxy. If m = w, jo = (a + bxY^ \f{a + 6a:)~ft^^R(?a7 + c' > , where- m fore X = c + c7(« + ^^) +/ (« + bx)-^ dxj{a -f 6.r) ^ iic?^ = c + c'/(a + 6a;) + -T-J /(« + 6a7)/(«+ So') ft^^R^or.- -;//(« + Ja?) .(a + bxY^'^'^dx \ and 3^ = (a + 6a:) * ^ c + c'/(« -\- bx)s - . • . • • CHAP. VI. LINEAR EQUATIONS. 231 m + i^±MlUi^a+b^)f(a-^bxri^\da: ni "J We may also realize the value of ?/ when m and n contain It appears then that we can always find the complete fluent of a linear equation of the second order, — ^ 4- p-r^ + Qw = R, if we know one of the conjugate dx"^ ax ^ A^% d^ ^ . . values ot -J— ^ + p— + qz = (a). In the present state of the science we have not the means of exhibiting in a formula z in terms of .r from equation (a); and consequently we cannot exhibit the primitive of the general linear equation of the second order. Thus far we have succeeded; that if we know the conjugate values of . . d'^y dy y, or even one of them, m the equation -^ + p-t — h a = 0, we can always find the primitive of the proposed equation. This property belongs to linear equations of any order what- ever, and we shall make it the subject of the next article. 32. The complete primitive of the linear equation • ■ • y = x'^/' + x"y'' + • • • + x'"«/'", w?ie7'e y\ «/", • • • j/'" are n values of y, which satisfy the equation when x = 0, and x', x" • • • x'" are certain functions of x, which may be de- termined. For, taking a particular case, let the proposed equation be d'j/ + vdydx + (^dydx" + Rydx^ = xdx^ ; and let y, y, y'" be three conjugate values of y which satisfy d^y -i- vd^ydx + adydx- + Rydx^ = 0. Assume that y = py' + qy" + ry'", where p, q and r are functions of x. Diff^erentiating, we have dy = pdy' + qdy" + rdy'" + y'dp + y"dq 4- y"'dr. Now since there are three undetermined variables p, q and r, we may assume for them three equations. (Alg. 145.) FLUXIONAL EQUATIONS. CHAP. VI. First, let y^dp + y^^dq + y^"dr = (a) ; which gives dy = pdy' '\- qdy" + rdy'". Differentiating this, we have d^y = pd\y' -\- qd^ + rd'^y'" + dpdy^ + dqdy" + drdy'". Next, let dpdy' + dqdy" + drdy'" = (/3) ; which gives d^y = pd^i/' + qd^y" + rd-y". Differentiating this, we have d^y = pd^y' + qdy" + rdy + dpd'y' + dqd^ + drd%"' Lastly, assume dpd"y^ + dqd-y^' + drd~y"' = xdx^ (7) ; which gives d^y = pdh/ + qd^if ^- rd^y''[ + x(i^. Substituting these values oi dy, d-y and c?'y in the pro- posed form, it becomes y.dx^ + pld^y^ + vdy^dx + ady'dx'^ -f Rj/'c?.r3| .... + 9'l^>" + Fdydx + Q.dy"dx^ + rj/"o?^3 | . . . which obtains independently ofp, 5' and r, because the forms which are inclosed in the brackets are (ex hyp.) each = 0; and consequently y = py' + qy" + ry^" is a solution of the proposed equation. -if To determine p, q and r, we have three equations of con- dition (a), (/3) and (y) ; from these the three quantities dp^ dq, dr may be each obtained in terms of x by means of eli- mination; and thus we shall have dp = x'dx, dq — x"dx and dr = x"'dx ; and integrating, p =Jx!dx-{-c, q =Jk"dx + c' and r =Jx"'dx +c", and con- sequently y =y' ^fx'dx + c j + y"\^fx"dx + c'} -f- /' ^fx"'dx + c" I . The same process being applicable to an equation of any order whatever, it appears thi^t the integration of ^ny d^-^y dy ^ "^ ^d^' + . . . + R ^ + sy = X may be always re- duced to that of ^^ + v-^^i + • • • + R^ + sj/ = 0. Cor. The complete value of y in the proposed equation = its complete value in the equation d^y + vd^ydx + otdydx"- + Rydx^ = + yfx'dx + 7/'fx"dx -{-y"fx>"dx. CHAP. VI. LINEAR EQUATIONS. SS. If only two values of y in dhj + vd'ydx + adydx'^ •f nydx^ = b be known j the proposed equation may be re- duced to one of the second order. Let y', 7/" be the known values of y. Substitute in the proposed equation y = py' + qy" ; then (]y = pdy' + qdi/", assuming y'ap -{- y"dq — ; and d^^ = pdy' + qd'y" -\-xdx^, assuming dpdy'-\-dqd?/" = xdx\ And as we have only two undetermined variables, we may not make a third assumption, so that in substituting for d^i/ we must take the full value without any equation of condition, viz. d'y =pd[i/ + qd^y" + 2dj)d-y' -{-2dqdy + d^^pdy' + d'qdy"- Substituting these values o£ d^y, d^y, dy andj/, the pro- posed equation becomes, omitting the forms which are (ex hyp.) = 0, 2dpd^-y' + 2dqdy + d'pdy' + d'qdy" + T.^dx"^ = xdx\ From this equation we can eliminate dq, and consequently d-q by means o^ y'dp + y"dq = ; and there will result an equation of the second order containing d-p, dp and func- tions of X, which is reducible to the first order by art. 2, form (4). If only one value y' is known ; substituting in the pro- posed equation y = py\ the equation for determining p is y'd^p + ^dy'd^p + ?)d^y'dp ^ + ^dx\y^d^p + 2dy'dp] V= Jidx^. -\- adx'^dp ) This is an equation of the third order containing only p and X, and consequently is reducible to one of the second order. If we suppose x = 0, the conclusions deduced in this article will give the method of integrating the equation d^y + vd-ydx + adydx^ + Rijdx^ = when we know only two, or even only one of the conjugate values of y. Since the same process is applicable to an equation of any order whatever, it follows that we can always reduce the general linear equation to one of the first order and degree, if we know ti — I of the values y',y'', • • • y". This pro- position is due to Lagrange ; and the artifice adopted in its demonstration is technically called the Variation of the Parameters. 34. Required to exhibit the complete primitive of - > • FLUXIONAL EQUATIONS. CHAP. VI. -~ -f vj- + 0,1/ = R in terms of the conjugate values of^. doc CLOC ■7^. + p -# + ay = 0. dx'^ dx f Assume y — p}f + q_y^\ and let ly = -^ ; then by the me- thod of the variation of the parameters we have dy = pdy' + qdj/"; if y'dp + i/''dq = (a) dy = pdy 4- ^cZy + Rc/.r- ; if dy'dp + di/^dq = ndx'^ {13). From (a) and {j3) we have (y'dy" — i/''d^^)dp =— 2/"Rda;^ vRdx^ , . . .vRdx'^ ""^^P^- Ydh ' ^"^ integratmg, ^= c -fy^- Also (yc?y — 'if^d\/)dq — j/RdxS or c?g' = —^ and n =z d -X- r ——— ; wherefore ^ "^ ydv 35. EXAMl'LES.' The complete primitive of d'^y + t/dx- = is 1/ = csin.aj + c'cos. x (art. 2, ex. 1); or the conjugate values are y = sin.^, and i/" = cos..r. Assume 2/ = psm.x-^ qcos.x .*. ^2/ = /?r/sin.a: + qdcos.x + sm.xdp + cos.xdq = ^c/sin.^ + qdcos.x, if sin.a:c?p -j- cos.xdq = (a). Differentiating, d"j/ = p^-sin.o; + qd^cos x -{- bdx'^, if co&.xdp — sm.xdq — hdx ((3). It is manifest that these values of d-i/ and of j/ satisfy the example. Now, eliminating dq and dp by means of (a) and (/3), we have dp = bcos.xdx and dq =— Hin.xdx .•.*••♦ p = 6sin.^ + c and q = bcos.x + c' and the required pri- mitive is 2/ = csin.o; + c'cos.a; + /;. (Art. 526, ex. 3.) CHAP. VI. LINEAR EaUATlONS. 235 Otherwise. Substitute y -^ h ~ z% then -r—^ + ^ = 0, whose primitive is ^ = csin.^ + c'cos.o:, or y — csin.o; + c'cos.a" + 5. Ex.2. ^ - y = 6 .-. 2/ = ce' — c'e"^ - h. Ex.3. ^ + — ^~-^=-^-. * dx" X dx x'^ x^^—V lhepn.„mveofJ+-^-|,=0>s ^ = cx + dx~^ . (Art. 30, ex. 4.) Assume y = px ^ qx~^ and we have xdp 4- 3C~^ dq — (a) and dp ~ x-^dq = -^^ (/3), .-. dp = ^^ and , ^ax^dx « , 07 — 1 dq = ^ — - .\p — ~l :: + c and ^ t- — ! ^ 4 x^\ q = c'—- g I ^ + i^-TT r (^^^- ^- ^^- ^y ^^' praxis 5), .-. y = cx-i-cx-' - 2" "^ 4'*^ —^-')l ^:^j . . . . , ,. . a(x^—\) x-l ^x x-\-\ Ex. 4. y?, — T-T-y = ^^"^ ^ • dx' 4x^ ^ Ihe primitive of-j^ J~rj/ = is ?— 1 y = c^ =» + c'o;'' 2 (art. 30, ex. 1). « : 1 «— 1 Assume y = px '-^ -^ qx ^ ; and we have a-l X ^ dp •\- X ^ dq = 0, or x^dp -\- dq = {a)\ and Q X ^ dp -— X ^ dq — mx 2 dx^ or {a + l)^''(i^ — (a - Ije/gr = 2mcf^ (/3) /. (//? = -— - and 236 dq = mdx FLUXIONAL EaUATIONS. m CHAP. VI. .-. p c ^ -3 a— 3 - and q = c /. 1/ = C,T 2 -{. c'X 2 mx Ex.5. a(a— 1) « _^f«— 3 X 2 f/j74 y = a\ The primitive of -~ — y = is For y z= ce" + de''' + c"sin.^ + c'"cos.^. Assume y = joe"" + qe~^ + rsin.o; + ^cos.^ ; then we have by successive differentiations and assumptions e'^dp + e~''dq + sin.^^ir + cos.^c?^ = 0. (a) e^'df — e~''dq + co^xdr — sin.a;^*' = (/3) e'^dp + e~''dq — sin. xdr — cos xds = (y) e'^rfp — e~^dq — cos.xdr + sin.^of* = a^ («J) dy ^ — ^ = ^g'' 4- '=^,g/=_,H = ^...,wehave FLUXIONAL EQUATIONS. CHAP. VI. 1.2.3 f whence, by sub- ,A; .^'^ , A:3 ^stitution, 7 f , ^ , , ^' , ^ 1 + ilji'* + ?'#'•••} + &c.; and arranging the terms according to the powers of Ic, and dividing by A;, l=;?pj+(p2' + ^/^)— + (p-' + 3(7/?V + ry^) j-^ + • • • But li is an independent quantity; and consequently (Alg. 347), 1 = ^y or^ = — as in vol. i. ch. 4, 14, =;>(?' + yp'^ or ^' = -^=^^ Q _ ^yj ^ 3^py -f- rp'^ or r'= ^ ^^ ; 8cc. From the same equations /?, <7, r . . . may be found in terms of/?', §'', r' . . . and there will result __}_ _ 4_ _ M^'-y'r ' Cor. These equations may be also deduced immediately by the process of differentiation. d^y dx For, from the notation -7^ = —r: — — , li dy and dx both dx'' dx -^ dxd^y — dyd^x d-y dy d^x . ^ , , ^'"•y- — d^^ — = d^^-fx - - coincide with those deduced in the preceding article. The values of p, q, r, > • - might have been deduced im- mediately by differentiation without the aid of Taylor's Theorem. By means of these values the principal variable may be changed from x to any proposed function of (x, y). If the two equations between y, x and t are under the form t = f(^, J/) and t =J'{x,y)^ it will not be necessary to eliminate t in order to obtain the values oiy\y^\ . . . and of x\ x'^ . . , Before we proceed to the consideration of formulae which have no independent function, we shall give one or two in- stances of changing the independent function by differentia- tion solely as in art. 37, cor. 39. Required to transform a formula of the second order in which A/dx^ -(- cly^ = ds is constant into one in which dx shall be constant. Let d'^y enter into the proposed formula ; then d's = ^ -^ . = --—- ; wherefore Vdx^-\-dy'^ ds Id'y) d'^u du'^d^y dx^d'v . i i . o u ^4^ = T^ — r / = , . ; whence the rule. ' Sub- ds"^ ds^ ds^ ds^ ' dx'^ stitute for d'^y its equivalent -p^d^y' If d'^x also enters into the formula, we have CHAP. VI. ON CHANGING THE PRINCIPAL VARIABLE. 241 -jT^ = —77- —'i'y^dx be constant and ds vary, dxd's dxdyd^y pq (d-x) pq " ~d^ ~ d7~' ^ ~ (T+^' ^^ ~dx^^ ~iTV' d^x which must therefore be substituted for -y—;, dx^ I^d-x enters into the proposed formula and not d-i/, and it be required to transform it into one in which d?/ shall be constant, the rule is, * Substitute for d-x its equivalent 40. Required to transform a formula of the second order in zvhich dx is constant into one in ivhich ds shall he con" stant. First change the formula into one in which dx shall not be constant : this is effected by substituting d^y d'y dyd^x .,^ , ^^ But, since ^5 = ^dx'- + dy^ = constant, we have = dxd^x + dj^dy, Pr — = - ^ ; wherefore, for ^ in the proposed formula we are to substitute d-x dyd'X . ds"d^x — -j—j T~i-j which = 5— TT— . Also, since y is to dxd?/ dx^ dx'dy ^ continue to be the dependent variable, we must again sub- stitute d'^x = - -^^ ; and there results ^-~ = ^ / , dx dx"^ dx^ ' which agrees with the preceding article. By a similar process the independent function may be changed, when there are only two variables, in any"^way that may be required. 41. EXAMPLES. Ex. 1. dfd.y - dy^ =: adxd^y + xdxd^y =(a + x)dxd"y = xdxd^yt if the origin be changed. VOL. II. ji FLUXIONAL EQUATIONS. CHAP. VI. Transform this equation into one in which y shall be the independent variable ; then by art. 37, cor. rule 1, we have doc'^dy—dy^ = — xdyd^oc, or xd^x + dx" = dy--^ an equation in which di^ is constant. Let dy = c\ then xd'x-\-dx^ = cdyf and integrating, xdx = cy -\- c' -— ydy -f ddy ; and again integrating, x'^ — y- -\- cy -{- c', and the required primitive is x^ H- ^ax = y'^ + cy -\- d. Ex. 2. xd'y - dxdy -f- ^dif = 0. (Art. 2, (4), ex. 1.) dud^x Substituting for d^y its equivalent — -^ — , there results xdyd^x _ , , d-x dx ^ , . — ^ + dxdy = xay^, or -^ 1 =«?/; and mtegratmg, I- -J- = y'"* foi' ii^ the first integration dy is constant; whence xdx = ce^dy, and again integrating, .r® -f- c'- = c^^. In the following examples s is the independent variable. ^ ds'^d^y 1 X Ex.3. —7-^= COS.-T-. dx^ a o __ d-y 1 cT c/.r® , ^ , 1 • • Here -j-^^ = — cos. ^ — 7-^; whereiore, by substitution, dx^ a b ds"- ' ^ ' ^^V 1 X dx'^ d-y I X . dy b . X , , ...... T- = — ^^"•"1" + ^> ^"" t"® required primitive is ay + o^cos.-^- = ex + c«. Ex. 4. dsdy'^ = adxd^x. Interchanging the co-ordinates for the sake of conve- , , adyd"y . ... adyd^y mence, we have ds = •: ^^ = , by substitution, "y ^ > where x is the independent variable ; whence (l+pO^ = «/^.=$,or.. = -^... . . /. (1 +p^)-^ V\+p"~ + c. CHAP. VI. ON CHANGING THE PRINCIPAL VARIABLE. 243 Also dy — jydx = ^ .*.••. y = a\l{jp ■\- a/1 +i?^) — \ ■\- d \ from Avhich two t v/ 1 + p^ J equations eliminating p there will result the required pri- mitive, remembering to restore the' co-ordinates. Ex. 5. adsd^y + ydy^dx = 0. If we were to substitute for c/*«/ according to the rule, there would result an equation between ?y, p and q^ from which both y and x may be obtained in terms of/? ; but as the elimination of p is in this case difficult, we shall integrate this example by a method which is independent of the pre- ceding rules. Dividing by dydx^ we have adsd'y adsdy 7 /^ • -^ +ydy=0, or _-^===^+3,rfe,=0 ; .-. mtegraUng, ds — dx y^ ^ J -, --'^ -, , 1 • al. — ^ 1- ^ = .*. ds — dx = ce ^ady; and reducmg /2 this, ^07=) — ^ ce 2a i^, from which x may be ob- tained in a series in terms ofy, (Ch. 2, 14, ex. 11.) Ex. 6. J.r^Jy — xds'dH/ — adxds^d~x- + «-«/-. Here d^y and d^^o: must both be adapted to the hypothesis that dx is constant; in which case the equation becomes xdp adp dx dp p-xq=aq,orp- -^=-^or-—=-^~.'. • • • Hit U/X u -f- X p «f^ , , 7 (a + x)dx p = and dy — pdx ~ .*. cy = 2ax + x^-\-c"^. c c Ex. 7. ydyd^v — ydxd~y — dx'^ — dxdy^ = 0. Here y{dyd^x — did-y) — dxds"^ — 0; but since ds is constant, d'^x — =^-^ .-. by substitution /dyW^y , \ yd'y - y i ^^ + dxd'yj - dxds'^ = 0, or - ~-^ = ^/or, or yd'y - J«/^ 4- ^5^ = 0. dy d^y , Let p = ^, ^ = 5^' ^^^" 7/7 - iP^ + 1 - 0, or R 2 244 FLUXIONAL EaUATlONS CHAP. VI. ? = '-^ ; but (art. 2. (5) ) j= ^ .-. J = ^. ; and integrating, «/ = — . ■— —-, which belongs to a circle. 42. ^7«. homogeneous formula of a higher order than the first in which the fluxion of neither variable is constant, may he always changed into one which ahall contain only p, g', r, . . . and dx, whatever he the function which is taken to he the independent function. To show this by examples. First, \eX.Jydx be the independent function. Here ydx is constant, or yd-x -{- dydx = 0, or d^^=:^^^=.^^dxK ^.^ . . , qdx^ 2pdx p , p-dx" DifFerentiatinff, d^x = - + — — - ^dx^+ - — ^- • • • y y y y \ y y s Similarly d^x = • • . I y y'' y^ S i y y'' i y I y y^ «/' J &c. = &c. If these values of d^x, d^x, • • • be substituted in the pro- posed formula, and the whole be divided by the highest power of dx, it will be changed into one which shall contain only p, Qi r, • ' ' and dx. Next, let f^^dx'^ -j - dy'^ be the independent function. Here dx x/ L + p^ is constant ; and it may be shown that d^X =— J^2^'272 PS (lOp'^-S] d'x=^ I ps_ , (I0p^-S)qr (I5p^- lS)p'g j ^^^ &c. = &c. CHAP. VI. WHICH HAVE NO INDEPENDENT VARIABLE. 245 Whatever then is taken to be the independent function, whicli is always decided upon at the outset of the calcula- tion, the formula may be changed into one which shall con- tain only p, q,r ' • ' and dx. 43. A formula of a higher order than the first, which has' no indepe7ide')it function, has, in general, an uncertain and indeterminaie value xvJdch is different if different inde- pendent functions he taken. The truth of this proposition appears from examples. Ex. 1. Let the formula contain d-x; then if ^ be taken to be the independent function, d-x = 0; if a?~, 2xdx = a constant, and differentiating xd-x + dx"^ = 0, or dx'^ d^x = ; \fx% x^'-H'^x + (w — l^x'^-^dx'^ =z 0, or X . (n — l)dx'^ ,. , . ,.«, T 1 v/. d'x = , which IS different accordmg to the dit- X ferent values of n. yd-x+xd^i/ Ex. 2. ■^- — —I— — ^ = du. dx^ xd'V If X be the independent variable, du = -r-j ; if j/, • • • yd~x Now these do not represent equal quantities ; for if they did, they would be the same whatever function // is of x. Suppose 2/ any function whatever oi' x, ij = x excepted, and it will appear that the values are different. Thus, let , xd'i/ , 7jd~x y = x-: then — r^ = 2x; bufS—r = — x. ^ dx^ dx^ Hence we conclude that the value of the proposed ex- ample is uncertain and indeterminate, and that it depends upon the function which is taken to be the independent function. 44. There are, however, some formulae which possess the same value whichever is taken to be the independent function. ^ dyd^x—dxd'^y Ex. -^ — :j- ^ = du. dx^ d^y If X be the independent variable, du =^ — -r^ ; if ^, • • 246 FLUXIONAL EQUATIONS CHAP. VI. dyd^x .« ^ , . , , , ^ du = •% 3 ; ixJydXi since j/a^^^ + dyax = 0, .... __ J^ d^ ~ dx" ydx'^' Now all these possess the same value whatever function y IS o^ x\ for let ?/ = x"^, then — -rr-^ — — ri{n — 1)07**"^ Also dy = nx''~^dx ; and differentiating, y being the inde- pendent variable, = nx''~^d^x + w(n — l)x''~^dx% or (;z — l)d^2 c/yr/% dy d"x d'^x = - ^^ ; wherefore ^ ^ =, -j- -7-^, • • • X dx^ ' dx dx^' — — n{n — l)x^~^. Again, since dy = nx'^~^dx, we have ydy — nx''~^.ydx ; and differentiating,^6?a7 being the inde- pendent function, yd^y+d7f = n{n-l)x^^-\ydx\or-£^-^^ . • • -—n{n — l)x"-2. Suppose y z=L e-^ then d'^y = e^'d^x + ^r'^a;^ ; and we have 111 1 <^^^ dyd'x on the three hypotheses — e^= — -r-^ = — r-y .... _ ^ i^ (/;r2 ydx"^' Suppose y — ^/X—x"-'^ and there will result as before, (1 - ^^-)~--- = ^ dhj _^dyd^x ^_d\y dy^ ^ ^ dx"^ dx^ dx"- ydx^' And whatever function y is of x, this formula possesses a certain value which is not affected by the independent variable. Hence it appears that there are certain formulae in which the fluxion of neither variable is constant, which possess a determinate value ; and that in order to find this value, we are at liberty to suppose either variable or any function of them to be the independent function, according as it may be most convenient for the integration. In the next article we shall investigate a criterion to di- stinguish equations which have, from those which have not, a determinate value. The latter, though not derivable from a single equation between two variables, may be deduced from two equations between three variables by elimination. 45. Required to find nschether a formula of a higher order CHAP. VI. WHICH HAVE NO INDEPENDENT VARIABLE. 247 than the first, in lohich thefiuxion of 7ieither variable is con- stant, has a determinate value. It follows from the nature of differentiation that in the equations dy = pdx, dp = qdw, dq — rdx, • • • the quan- tities^, q^r, ' ' ' are solely dependent upon the form of the function y =^, and that they are not affected by any sup- position that may be made with respect to d-x, or the higher orders o^ dx. From these equations the values of d'^y, d^y^ • • • may be obtained by differentiation ; and if they be sub- stituted in the proposed formula, it will, in general, contain p, 5', r, • • • dxj d^x, • • • ; of which the value of d^x will be affected by the independent variable. But if the formula should be such thot after these substitutions, and without making any supposition with respect to the independent variable, d^x and the higher orders of dx should of them- selves disappear, it contains onlyjo, q^ r, • • • and dx, and will therefore possess the same value whatever function be taken to be the independent variable. „, 1 • , , . . . yd-x-[-xd^y . in us, makmg these substitutions m *^^^ p^ — =^ and dyd'X—dxdhf . _ ,^ , i , -j—^ ~, since ay = pa^x + qdx-, they become -7—, h xq and — q\ of which the first is indeter- minate, being dependent upon the hypothesis made with re- spect to d'X\ but the second possesses a determinate value. Cor. The equation of condition that the canonical form of the second order, vd~x + Q.d-y + B.dx- + sdxdy 4- •sd'^y may have a determinate value is vdx + Q.dy = 0. For, since d'y = pd"x + qdx-, the coefficient of d^x be- comes V -\- Q.p which is to = 0, or vdx + Q,dy = 0. The equation vdx-^Q.dy=:0 is not necessarily identical. In the following article we shall give a few examples to illustrate the preceding propositions. It appears from the criterion that these examples possess a determinate value. 46. EXAMPLES. Ex. 1. vdy^ = dx'^dy — xdyd^x + xdxd-y where p = ra7. 1st. Suppose X to be the principal variable; then vdy^ = dx'dy + xdxd'^y, or — ^ = _^_^_^_5__^ . . . '^ X" x'dj/- 248 FLUXIONAL EQUATIONS CHAP. VI. ^ dx dx vdy dx dec _ vdx ~" xdy xdy ~~ x^ "* ' ' xiiy xdy x^ dx'^ c „vdx , , dx ,'.mtegrat.,i,.-^j- = -^ -f~-^ and dy = 'd7f ^ ^ x' "^ I ^vdx ^J^--/^ 2dly. Let dj^ be constant ; then PcZ«/~ = dx'^ — xd~x ,'. Tdy^ xd^x — dx'^ dx dx dx _ „ vdx . . dx'^ cdif , vdx - ^ T = ~ ^^'-TT ••• integrating, i, — - = _— -dyy—^ XX X , dx or ay — 3dly. Let xdy be constant ; then, dlfFerentlatlng, dxdy + xd^y — 0; and the proposed equation becomes vdy" — — xd^x. Suppose xdy = a ; then dy — — .-. — - = — xd'-x .*. • • • X X va^dx , 7, . — ~ = — dxd^x .*. integrating, X dx^ ^vdx , , ^ vdx dx 2 = ca^ - a"/-^ = f^ rf<,= - ar%»/ ^ dy = - ^J^-^f-^ dx 4thly. Let — be constant ; then xd'^x — dx'^ = ; and the proposed equation becomes vdy^ = xdxd'^y. dx Suppose — = a ov dx = ax .'. vdy^ = ax^d-y = dx vdx a^d^y . . a'^ c „vdx .,_ = __ ...,ntegratmg,g^^ = - -/_, or ay = — = = =- ; a result m which / ^ pa^ / ^^vdx CHAP. VI. WHICH HAVE NO INDEPENDENT VARIABLE. 249 the variables are separated, and which is the same in all the four cases. Ex. 2. dr-i/dx — dyd-3c — ydx^ = 0. This equation has been obtained on the hypothesis that both dx and dy vary ; and in order to integrate it, suppose X to be the independent variable, or d'x — 0, and it becomes d^ydoo — ydx"^ = or -^-^ — ?/ = ; .*. integrating. X = i J ; the same that would result it y were the independent variable. Ex. 3. xy{dxd\y-^dyd-x) = ydydx'^ — y'^di^dy'^ — xdxdy'^ where q = ry ; or xydxd-y + y'dQ.dy'^ = xydyd'^x + ydydx'^ — xdxdy"^. If the right side of this equation be divided by y'^dy, it becomes the fluxion of — . Let then = a be taken for y y the constant ; then the proposed becomes 7 7« 7 7 r. 7 ^^^ d'V ^ y xydxd-y + y^aQ,dy^- = ; or do. =- — - ^^.= -a- ~; cL xdx .*. a= -7- + c = ■— — [- c .'. Qiydij = xdx + cydy, and the required primitive is 2p^ydy = o;^ + ci/"- + d. Ex. 4. xH^x + x"yd-y + (a^ — y'^dx'^ + x'^dy'- = 0. Here x^{xd~x + yd^y) -\- («- — y')dx^- + x"dy' = .-. assume d{xdx + j/^J/) = ; and by substitution, we have - xHx^ + {a- — y')dx-= 0, or v a^ - y- — x'^ - dx ~ \ and since tZ.r cannot = 0, v/e have «'^ — y"^ — x'^ = 0, or a'^ = ^^ 4- 2/2 . gQ i^j^at this method in this example does not give the complete primitive. (Vid. Lacroix, tomei. p. 219). Ex. 5. xd'x t- yd^y + dx"^ — 0. Assume d{xdx + z/cZj/) = ; then, by substitution, we have — dy'^ — 0, which is impossible ; or there is no single relation between the variables which satisfies the proposed equation. If both in this and the preceding example there be sub- stituted dy =pdx, dp — qdx according to the rule contained in art. 45, in each case, in order that the coefficient of d'^x may vanish, we must have ^ + ?/p = 0, or xdx -f- ydy = 0. Now this satisfies the first, but not the second ; which 250 FLUXIONAL EaUATIONS CHAP. VI. shows that the first has, and the second has not, a deter- minate value. 47. MISCELLANEOUS PRAXIS. ^' 5^ "*" "'-^ "^ ^ •'• ^ "^ csm.(ax + c'). d-y dy ^ , dx^ dx ^ _ d-y dy ^ ,x dy , ^d'^y X 1 (x''-\-c'^)dx 6. nxd"!/ + Vda:' + dy' = ,: - ncx " nx "^ , ^ = 2(71-1) ~2c(n+ 1) "^ ^' beloi 8. 2 ^/a'^— aadxd^y = (dx" + ^^O"" belongs to the cy- cloid. 10. dsdy = dx^ where ds = const. ....... 1 2j/ = ce' + -— + c\ 11. ds'd^y = xJ;^'* where ds = const, .-.j^ =Jxdx. .r. ^'y 3 c -^-fe-o^ds i or -— — ^fi-^'ds + c. This method of deducing the complete primitive from a particular solution is applicable to all equations of the form dy + y'^dx = xdx. 11. If the proposed equation is of a higher degree than the first, the preceding methods fail ; and we must search for some other criterion of integrability. Now it follows from the rules of differentiation established, vol. i. ch. 1, that the first fluxion of any function whatever of three or more variables must be of the form Mdj;-\-'i^dj/ -\- rd :: -\- • • • ; if therefore the proposed equation be solved with respect to dz, which we will suppose of the second degree, when it is independently integrable, the result must be an equation in which djc and di/ are not under a radical sign ; which fur- nishes a new condition of integrability. Having then solved the quadratick with respect to dz, the equation is not independently integrable, unless the result consists of two equations each of the form dz — mdx-\-7idi/\ and applying to these the equation of condition (y), it will be seen whether the required primitive consists of one or of two equations. The primitive, when it is a single equation, is the pro- duct of the primitives of the two equations dz = mdx-^ndj/, Ex. vdx^ -i- ad?/^ + ^.d^'^ + sdxdy + Tdxdz + \di/dz = 0. If this equation be solved with respect to dz, the quantity under the radical sign is (t^ — 4:VK)dx'^ + 2(TV — n^yixdy + (v- — 4aR)iy2 ; and that this may disappear as a radical, we must have (t2 — 4pr)(v^ — 4qr) = (tv — Rs)'. Unless this equation obtains, the primitive is not independently integrable. 12. Total fluxional equations of the second order. Equations of this kind seldom occur. They appear under a form in which the first fluxion of one or more of the variables is constant. We have considered the formula of two variables in which dx is constant in ch. 6, 17. Next let the formula be ^dxdy + sdy- + Tdx'^ = d^u where both dx and dy are constant. (1). To find the condition that d-u may be an exact s2 ^0 TOTAL FLUXIONAL EQUATIONS CHAP. VII. fluxion, let it be the fluxion of Mdx + T^dt/; tlien the term "Rdidy is the sum of two terms ; the one, arising from the differentiation of nJj/ with respect to x ; the other, from the (liflerentiation of Mdx with respect to y. Let then r = r' + r", and d'u becomes {Tdx 4- ^^dy)dx + {^I'dx + sdy)di/. ^, . . ^„ . , d{Tdx-^B!dy) d{B?'dx-\-&dy) This IS an exact fluxion when — -, ^^ = ~ = dy dx dr , ^r', ^r" , <^s , ^ , , or -^dx + -T-dj/ = -j—dr + -r-dt/. But x and y, and con- sequently dx and dy, are independent quantities ; wherefore , dT Jr" , dR' ds we have 3- =-r-, and -j- = -^. ay dx ay dx To eliminate r' and r" ; we have r'' = r — r', and con- r/jj" da d'iJ sequently — = -1 t- ; and by substitution, dT __dvi Jr' dvi^ dR ^T dy dx dx^ dx ~ dx dy ^ Also -T- = ^ (2) ; wherefore, difi'erentiating the first with respect to y, and the second with respect to x, there results ^~S d"T d R , . , . . n 1 • n /7~9 + -r-^ = , . ; which is therefore the equation or con- dition that d^u may be an exact fluxion. (2). Required to integrate d^u when it is an exact fluxion. Let du = udx + NtZy ; then vdx^ is the fluxion of udx and sdy^' the fluxion of Ndy ; wherefore du = dx/rdx -f- dyfsdy where y is constant mj^rdx and or inysc/?/. In correcting the fluent, we must introduce at the first integration two arbitrary constants of the form adx + bdy ; for if the complete primitive contain ax -|- by, both a and b will be eliminated by the second differentiation. Similarly, if a primitive contain ax- + by^, both the constants will be eUminated by the third differentiation ; and so on. (3). The equation of condition deduced in (1) may be verified by the result obtained in (2). For, differentiating, du = mdx + ^dy, and comparing the result with the proposed formula, we obtain CHAP. Vir. OF THE SECOND ORDER. 261 e?M c/n C?N C?M ^ , t/N , B. = -} — I- -7-, or -J- = R J-. Also -7— = s ; whence dj/ ax ax dy ay ds _ d^N d^N dn d^m _ dn d"fvdx dx ~~ dydx ~ dxdy ~ dy dy'^ ~ dy dy"^ again differentiating with respect to x^ there results (/2s d'w d'T . . . . . . „ J. . , -j-^ = T— 1 7— 2 vvhich IS the equation of condition de- duced in (1). Ex. d^u — y-dx^ -\- A^xydxdy -f x'^dy"'. Here the equation of condition obtains, and du —y-xdx -f x'^ydy + adx + hdy and u = — ^ -f «^ + ^y + ^» (4). Next let the formula contain three variables. If 2 is a function of {x, y) ; i. e. if the proposed formula, considered as an equation, is independently integrable ; we have, differentiating on the hypothesis that both dx and dy are constant, d.^ = fdx + qdy 1 where p, y, r, s and t are d~z — rdx' 4- 2sdxdy -f ^c??/^ ^ the fluxional coefficients of S2=:^{x,y). (Vol. i. ch. i, 18.) If dx and dy both vary ; then d^^ — r^^- + ^sdxdy -{- tdy"- + pd'x + 9<^'j/ ; which, by making d'x or (/-j/ = 0, is aolapted to the hypothesis of dx or of dy being the only constant. Now it may be shown, as in the case of two variables (ch. 6, 45), that when the proposed equation is independently integrable, if these values of flf^ and J% be substituted in it, the quantities d~x and d"y will not be found in the result, which will contain only the variables and their fluxional co- efficients. This therefore furnishes a criterion by which we can ascertain whether an equation of three variables and of the second order be independently integrable. Ex. {xdx f zdz)d^y - zdyd~z - dy{(lx'\ dy"^ -f dz^) - 0. Here d^z = rdx"' + 2sdxdy + tdy- + qd'y ; and by sub- stituting for dz and d'^z, the equation becomes (x 4- zp)dxd'y — ^dy(rdx^ + ^sdxdy -f tdy^) ? _ r» - dy\ (1 + p'^)dx^ + 2pqdxdy + (1 + q")df/^\ § ~ This cannot be Independently integrable unless x -\-^p=0. Integrating, we have x^ + j^^ == 2y : and differentiating in order to find y, xdx + zd;s = dY ; and again difleren- 262 TOTAL FLUXIONAL EaUATIONS. CHAP. VII. tiating on the proposed hypothesis, viz. that dx is constant, we have dx"^ + dz'^ -\- zd-^ = d-Y ; and by substitution, the equation becomes dvd'^t/ — di/d^Y — dy^ = 0, or dyd'Y—dyd-y , . . dY ^ ^5 = -dt/', .-. integrating, ^ = a - ?/ and Y = 5 + «j/ — -^5 and consequently a:'2+y^ + ^" = 2a2/+ 25; which is the complete primitive of the proposed equation. 13. Required to investigate the equations of condition iliat a jiuxional function of any order and of any number qf variables may bean exact fluxion Let the proposed function be of the ?2th order ; then, if the fluxion of one of the variables as x be supposed constant, it may be put under the form vc?^*", where m depends upon the decree of the function, and v is a function of (o;,^, ^,«") and of tlie fluxional coefficients as far as the wth order. Substitute dy = pdx, dp = qdx, - - • ds = td. dz = p'dx, dp' = q'dx, * ' ' ds' = t'dx > then = tax 'I = t'dx > V is a function of x, y, z -• • ; p, q »" s,t; p\ q'- • -s', t' \ &c. The first fluent of \dx"' is of the w — 1th order. Let it be udx'^~^^ ; then du = vc?.r5 where u = f(^, y, ;? • • • p, q-'^s; p',q' ' -- s' ; &c.). , J du , du , du , du . Now vax = du = j~"^ + JT ♦^ "^ TT -P * * ' + T du J du , , du- , ■_ du du du du du dx ~ dx di/^ dp^ ds du , du , du , ^diP'^-d^'^-'-'^d-/ + • • • Let dv = udx + adi/ + vdp + adq . . • + Tdt + ^'d^ + v'dp' -i-oldq' . . . + tW + .. . Then we have CHAP. VII. EXACT FLUXIONS. 263 N _dy_ ~~ dy~ \ du for V =-rdii dx P dv ~" dp~ d^u ' dxdp ( du dHi dy'^dyd^'^ d'^u dp^"^'" d^u dsdp d^u , + d.dpP' + d^u , dp'dp^ ' ' ' "^ d^dp* + ••• Q du 1 - dy"" d^ Similarly d\ du 1 dq~ dp da 4" : dp r dq ' T dv - dt~ da " ds' If the proposed function is of the^r*^ order, u does not du contain p ; and we have p = -7- ; wherefore n, which .Idu 1 . ^P n z= -7~ d -J-, = 3-«P} or N — ' J- = U. ax dy ax ax If the function is of the second order, p = ^r + -rda; dy dx , p \ -du dF d^Q dv d^Q. ^ wherefore n, = -7- a -^, = -7—, or n — ^-+-7— =0. ax dy dx dx^ ax ax" Generally, an equation of condition that a function of the wth order may be an exact fluxion is dy d\ -|- . . . =0, where N = ^r-, p = 3-, • • • ' ' dy dp^ N- ^p d'^Qi dx^dx^ d^B. dx^ a dv -dq^ A similar equation may be deduced for any other of the " I du du 1 du variables as.; for N'=^cZ^;p'=^ + ^rf^,; , du 1 .du , . , ^ o! = Tl I j~, ^ ;7"'' ' * • ^"d it may be sliown that an equa- 264 TOTAL FLUXIONAL EQUATIONS. CHAP. VII. . . A" • , ^'^' ^'^' ^'^' JL - n tion or condition is n' ~ + -= y-x +•••—". Hence it appears that the number of equations of con- dition that a function of the 7ii\\ order of any number of variables may be an exact fluxion is equal to the number of the variables exclusive of the independent. Next to find the equations of condition that udx"*'"'^ may be an exact fluxion. Let dtJ — udx^ where i^ is of the n — 2th order ; then it has been shown that the equation of condition corresponding to the variable y is d'd 1 du \ du _ dy dx dp dx'^ dq ~~ ^ . du 1 ^du , nut, since p = -^ + ^r- d -r-, we have dy dx dp du _ -^ C J ^ ^ ^^^ > _ ^^ ^^^^ ^'^ _L dy~~ dx\ dx dq 3 ~ dx dx- dx"^ dp^ dx\ dx dr y~~ dx dx"- wherefore by substitution the equation becomes 2dQ , M'K 4>s _ ~ dx "^ dx'^ ~~ 'da^ + • • • — In the same manner by supposing du" = u'dx, where u'^ is of the n — 3d order, it may be shown that the equation of condition, corresponding to the variable y, in order that wW^'""^may be an exact fluxion is Q — h-r-^ — i7~3~+ * * * Hence collecting all the equations of condition corre- sponding to the variable y, they are ^ - i""- + i'''<* - i'^''' + i*^ - • • • = « 3,6, CHAP. VII. EXACT FLUXIONS. ^65 • - • &c. Similarly there may be deduced n similar equations cor- responding to any other of the variables, the independent excepted. Cor. In the article it is supposed that the function has an independent variable : if it has not, assume t to be the independent variable which connects all the others; then the proposed function is of the form Ydf^, X . . , dx d"x Adoptmg Lagrange s notation, let -rr = x', —=:x", • • • -^ = y, -^ = y . . • &c. ; then assuming dv = udt + N6?, it may be shown + ... • J as in the article that the equation of condition corresponding to X that the function may be an exact fluxion is ^1 1 dv dv .. rJut -j-dp = -r- d -r-, = d-j-f IT X, = dx dt dt dx dx' 1 JO ^jo^^ jo^^ -e J. ^; from which it appears that the equations may be expressed independently of t in the followinor more convenient form : o dv dv dv^ — (\'\ dx dXj dx^ f 1 7 ,„ , , , f where a^i =«.r,ar2 = a-^,' dy dy, dy. s Similarly we may express the equations of condition that the proposed function may be an ?ith exact fluxion. Ex. 1. Adx + j\dy where a and b each = F(ar, y). Here da = vdx = (a + yip)dx\ .*. n, which 266 TOTAL FLUXIONAL EQUATIONS. CHAP. VII. Jv dk dB ~ dy' ~ dy dy^' Also p = B ; /. -r- = — + -j-Pi and by the equation of ,. . dh dB 6?B dB dA dB condition, _ + ^p = - + -^p, or ^ = ^. Ex. 2. A Jo: + Bdy + cdis = {a + bj) -{■ cp')d.i\ Here v = A + Bp + cp' = ; I a B "I \ dy dpdx dydp^ which, since a and b do not contain q, may be separated into the two 2dA d-A d^A d^^ _ (\ f \ dy dxdp dydpP dp^ ~ \ ) ds d^B d^B d^A d^A d^A „ _ ^ ^~ d^p ~"d^pP "^ 5^ "^ d^y^ "^ d^P^^ ~ " ^^^' which are the conditions in order that the proposed may be an exact fluxion. When these conditions obtain, since du = \dx = Adp + Bdx, we have by partial integration, u = fkdp V v, where t; is a function of (x. y) not containing p, Vid. ch. 8, 4, cor. Diffe- rentiating this with respect to x and y, and comparing it with the proposed, we find the value of --rdx -\- -j-dy, and by in- tegration obtain the value of v and consequently of u. If it be further required that the proposed be a second exact fluxion, another equation of condition must obtain, 2 , ^ dB 2rfA 2dA dA VIZ. p j-dQ, = 0, or 3 J T-p — -j-q = ; dx dp dx dy^ dp^ 268 TOTAL FLUXIONAL EQUATIONS CHAr. VII. . cZa ^ _ ^ whence -j- = (y), or a may not contani p. The three equations (a), (/3) and (y) must obtain in order that the proposed may be a second exact fluxion. Thus {2x7jdi/ + ochjdx)d'^y + xdif + (y + x'^)dy'^dx -f (2 + Qy)xydydx'^ -\- y^dx^ = \dx' satisfies (a) and (/3) but not (y) ; and we have u =J'Kd'p + v =y{^xyp-\-x^y)dp-\-v = xyp^ + x^p + V. But u — fsdx'\ .-. differentiating partially with respect to x and j/, there results dv dv ~fdx -f -j-dy + p'^d.xy + pd . x^-y = (xp^ + {x~ ■\- y)p'^ + (2 + Sy)xyp + 3^^)c?a:5 or -r-dZ^ + -i-dy = Qixy"p -\- y^)dx =y^dx + Sxy^dy .*. v =^y' + c and u=xyp'^ -f ^*j//? + ory^ + c; and the required fluent, which = udx"", =xydy^- + x^ydydx + xy-dx- -h ccZo:®. Ex. 4. j/J2.r — xd'^y = v. ^v ^^ dv ^ dv . dv , -v +»*'■ 1 he linear equation —^ + —f^ + t = 0, where t = sj/ — x, is an instance of this case ; but we shall not apply the method here pointed out, since the proposed equation has been already exhibited by a shorter process. As examples of the above formulae, which we have not de- monstrated on account of the length of the calculation, the student may take the following. Ex. 1. d^y + "^-^dydx + ?^ + 9.xdx'' = 0. ^ xij ^ y This is independently integrable ; 2 = x'^y- ; and the re- quired first primitive is 2oc-y~dy + cdx + x^y^'-dx = 0. Ex. % d^y-\ ^dydx + > x ~ —+3/- [dx'^"> X C 3 = 0. Here a = — ; ^ = .r ; and the first primitive is X ^xdy + (c + x2y- - Sxy + 2fKxdx)dx = 0. Euler has also integrated several equations of the second order by means of factors of given forms which render them exact fluxions as in ch. 4, art. 33 and 34; but we shall entirely omit these, and shall conclude this part of the sub- ject with integrating the general linear equation of two variables. 15. Required to integrate the linear equation of the nth order hy means of a factor. ^ , . , d''y d^Si d^'-^y Let the equation be ^ + p^^ + a^l, . . . + Ty = x ; CHAP. VII. INDEPENDENTLY INTEGRABLE. 271 and let zdx be that factor which renders both sides of this equation an exact fluxion, z being a function of x. c d^'ii d''-^y d"-^y ^ , . S'"^« n ^ + 'i^' + ^^' ■ . . +Ty I rf^ IS an ex- act fluxion, we have (art. 13) £ - ^''^"- • ^^ + rfi^^*"^ • ^"^ - • • • ± ^-^ = "^ (")• If one of the conjugate values of this equation be known, one of the first primitives of the proposed may be found. For the first primitive is of the form (jn—iy d^~'^y d^~^y and to find p', a', • • • t', diff'erentiate this and equate its terms with the corresponding terms of the proposed ; and there results pf = ;kp _ ax dp' d.^T fc dx ~ dv dx~ d.zQL d^.zv d^z q! =z za — -r- = ^0. -J— + dx dx^ dx^ d,!ss d''~~^.ZF _ d'^-^ss '^ = ^"^ "" ~d^ + • • • ± -^-;^ + ^-^y Similarly if the n values of (a) are known, its n first pri- mitives and consequently the complete primitive may be found. If only n — I values of (a) are known, we may eliminate all the fluxional coefficients except -p, in which case there dy will result an equation of the form -^ + M2/ = x' ; from which the complete primitive may be obtained. The complete primitive may be also obtained, if « — 1 conjugate vakies of the proposed are known when x = 0, For let i/' be one of them ; then, substituting it in (/3), it becomes ^ ^ + ^d^^ + ^^ • • • + Ty = c where 272 TOTAL FLUXIONAL EQUATIONS CHAP. VII. p', q', • • • t' contain the n — 1 fluxional coefficients of;;?. Similarly there may be obtained w — 1 equations, each con- taining a different arbitrary constant; and if all the n — 1 fluxional coefficients of z be eliminated from these except the first, there will result a linear equation of the first order, from which ^ may be found. This value of ^ contains n constants, and consequently we may obtain from it n con- jugate values of ^ ; and, as before, the complete fluent of the proposed equation may be found. Ex. ^ + -- ^ - -^ = -^. r d^y ^ dy y -) . Let ^ ^ -ri -\ -J -^ > OtT be an exact liuxion ; then ( dx- 3c dx x^ S d^;"^ dx ' X .r^ ~ ' dx^ dx x ~ ' d^ss dx . .J , , -j— = — .*. integrating, az = cxax and z = ex''- f c .'. ^' = 1 and 2" = X". Substituting these values in (/3), the two first primitives of the proposed are ^ + ~ = / - — - + c and ^ ^ dx X -^ x^ — \ du ^,ax^dx , • . ^ n x'^ ^ —J — — =- + c'; .-. the required fluent is ^ adx ^ax'^dx xy = cx'^ — c -\- x""- f-- — I — / — — r-. 16. When the equation (y) of art. 4, does not obtain, the proposed equation neither is nor can be rendered an exact fluxion ; and, as has been already observed, the re- quired primitive cannot be an equation, in which one of the variables as 2; is a function of the other two x and y con- sidered as independent quantities ; but this hinders not that it may be a function of x and y connected by a second in- dependent equation ; in which case the required primitive shall consist of two independent equations. We shall con- sider then in the next article, whether by establishing a new relation between the variables, we are enabled to find two equations which by differentiation and elimination will satisfy the proposed. In these cases the proposed is not independently inte- grahle; and it is manifest that the new relation which is to CHAP. VI r. NOT INDEPENDENTLY INTEGRABLE. 273 be introduced, as it does not form part of the data of the question, must be to a certain degree arbitrary ; or that the primitive contains arbitrary and indeterminate functions. 1 7. Required to integrate an equation of three variables^ which is not independently integrahle. Let mdx + ndy 4- pdz = be the equation, where m, n and p are functions of (^, 3/, z) ; and let t be that factor which renders tlie two first terms an exact fluxion, z being considered constant; then, if it rendered the whole function an exact fluxion, we should have du = t(nidx -}- ndy -h pdz) — an exact fluxion. Substitute dv — t(nidx -{- ndi/) and R = ^/;; then the equation becomes J?^ =r t/v + wdz = (1) ; and integrating on the supposition that z is consiant, u = \ -|- ^vhere

= Oi (p + p'9 + p»,/, - (r 4- r'9 + r''0) = 5 * It appears from the process of elimination that the equa- tion for determining 6 and <^, each rise to three dimensions. Let then the values of S and be 9, 6', 5'' and , 0', 0"; and let the corresponding values of p + p'^ -j- p"^ and of T + T^ + t'V be m, 7n', m'' and v, v', v'' ; then integrating the linear equation dv + 7nvdl = Ydt and replacing v by its value X -\- S^ -\- (pZf there results ^ + 9^ + 0z = e~"''\fe"-'vdt + c } ^ X + Qy -\- (i>'z = e-"^'^ \Je"''*V dt -{- d \ > from which Xj y and X + ^hj -{-(l>"z =er-"'"'\fe^'"*y"dt-{-c" ] S IS may be each obtained in terms of t. The same method is manifestly applicable to any system of the first order and degree in which the number of the variables exceeds by unity the number of the equations. 31. Required to integrate a linear system of any order. The proposed system is always reducible by substitution to one of the next inferior order in which the number of the TOTAL FLUXIONAL EQUATIONS. CHAP. VII. variables exceeds by unity the number of the equations ; consequently by continued reduction the integration of the proposed may be made to depend upon a system of the first order; and it is therefore integrable. Thus, let the proposed be under the form d^x 4- (jdoo + Q.dy)dt + (R.r + S2/)dt- = Tdt^ d^y + (P'c?^ + (idij)dt + \s!x + ^y^t^ = ^'df^ Substitute dx — -pdt, dy — qdt ; then we shall have four equations between the five variables x, 3/, t, p and q ; viz. dp + (p^ + (^q -\- Rx -\- sy)dt = -rdt ) , dx—pdt = } dq + (P> + a'q 4- n'x + s'y)di = T'dt S dy ^qdt=OS''' which, being integrated, will give p and q, and consequently X and y in terms of t. 32. Required to integrate a linear system of any order in wh'icJi all the coefficients are constant, and the function of the independent variable = 0. (Vid. ch. 6, 20.) Let the proposed consist of three equations of the second order between the variables x, y, % functions of t. Substitute x = e^\ y = ae«*, % = /3^^; then the result is divisible by e''\ and there will be three quadra ticks for de- termining a which contain the constant coefficients and the quantities a and /3. Eliminating a and /3 by means of these quadraticks, there will result an equation of six dimensions, from which six values of a may be obtained. Let them be oHy «''••• a^. Also let the corresponding values of a and /3, as determined from the same equations, be a', a" • • • a'^ and /3', |S" . . . /3^; then it may be shown as in ch. 6, 20, that the primitive is X — cV' + cV"' -f . . . + c^e'^' 1 y = dale"'' + c"a'"e«"' + • • • + cV%''®' >-. % = c'ft'e^'' + c"/3"^''"' + • • • + c'6/3 V' ) The same method is applicable to a system of any order ; and the cases of equal and of impossible roots may be pro- vided for as before. For other methods of integration, vid. Garnier's Calc. Int. p. 345. 33. EXAMPLES. Ex. \. dx + {ax + hi/)dt = > dy + {a!x + lly)dt =0y Here dx + % ■\- {{a + a!^)x -\- (h + ()d)y)dt = 0. Substitute z ~ x -]- ^i/; then, eliminating x, we have d^ + (rt + a'^)zdl - y 1 (/5 + ( {a + a'O)^ - (A 4- b'S) )dt \ , CHAP. VII. SIMULTANEOUS EQUATIONS. Assume d^ = and (a + a'9)9 - (6 + h'S) = 0. Let the roots of this quadratick be S and $' ; then, since dz -\- {a -{- a'Q)^dt = 0, we have, integrating, z = c^-(«+«'^)^-=c^~'''^* (for instead of 9 constant and t va- riable, we are at libeity to suppose 9 variable and t constant) ; and consequently the required primitive consists of ^ + 9j/ = ce-^'^^ I ^ = — rrr — ^"^ y = — r^' — • If 9 = 5', these values of x and 3/ become infinite and fail ; but in this case there is only one value of 2, viz. ;^ = ^ + Q7/, which therefore = c^"~"'^^, or a? = c^"*"'^' — 9z/. Substitute this in the second proposed equation, and there results di/ + (b' — d^)ydt -\- ca'e~^^^*di = 0; wherefore integrating, *^ ^ I i'-^a'O j-^^ ^ The value of a; in terms of t may be obtained from w = ce-"'^' - O1/. If the roots of the quadratick are impossible, x and z/ are to be expressed in circular functions of t as in ch. 6, 21. It appears then that x and y may be each expressed in terms of ^ and of two constants, which may be determined if any corresponding values of the three variables be known, Ex. 2. d'X + {ax + bij + c)dt'^ = ^H d''y + {a!x + 6^ + c')dt'' = J * We shall integrate this not as in art. 31, but by a method similar to the last. Multiplying the second equation by 9 and adding, there results d^x-^Wy ^[{a -\-dh)x -V {h-{-U^)}j -^ c -\-c^\dr- = (1). 01 6 + 69 c + c^ substitute z =^ x -\ -u -\ ; — n ; then ditrerentiat- a-\-a!Q^ a-\-a'r ing and assuming dS = 0, b + VQ , ,, cb + b'^ a-\-a(} ^ ^ \ a Substituting this in (1), and assuming d'^z = d'x H- -^^—id^y^d^x\U^y\- \ -^, - 9 \d;Hj. ^ a-\-aO ^ ' J/T- ^ rt+a9 S '^ 284 TOTAL FLUXIONAL EQUATIONS. CHAP. VII. b + b'Q -I-- -9 = (2), there results d^z + (a + a'^)zdt^ = (3), (Z ~J~ Cb 9 whose primitive is ;2 = csin.A*^ + c'cos.a^^, if a = a + a'9 (pp. 186, 220). If it be required to obtain if as a function of (a?,^), we have, multiplying (3) by 2d^, —7- f- 2Azdz = .'. integrating, dz^ dz _ + A2^ = c% or dt = — ^ I which, when integrated, ^t /v/C^ — as'"* will give the two requisite values of i by finding 9 and fi' from equation (2). Substitute x — ef"* \ y = ae^* \ and there will result 2a2 4. « + 1 + a(a2 + « + 1) = > a^ + 96« - 9 + a(a^ + 50« + 15) = r and elmimatmff a, we have —^ - — — -^ r-, or ft* + Aa^ ~ 7«^ — 22« 4- 24 = ; whose roots may be found by the method of divisors (Alg. 336) to be 2, 1, —3, —4 ; and the corresponding values of a arc 11 4 16 29 ^ ^ "" "^^ "~ 3" ' ~ T' "" 13 ' wherefore llc'^2. 4cV 16^'^^-=^' 29c'-^^- - 2/ = -Tr- + -3-+ ^T- +-13- 34. miscellaneous praxis. ydx xdy xydz xy a — z a — z {a—%Y~ ''a^z~ 2. dx + dy + dz -\- {x ^ y-\-z)dz = .*. {x-\-y-}-z)e^'=c, 3. (ay — bis)dx + {cz — ax)dy + (bx ^ cy)dz = .*. ay—bz cz—ax' CHAP. VII. MISCELLANEOUS PRAXIS. S85 4. (9.x + z)dx + (2y -{. z)dy -\- {x + y)dz = .-. • • • x{x-\-z) +2/(^"+«) = c. 5. (2/« 4 .?/i2f)t?a7 + [xz + ;^2)c?z/ + (?/*— a:j/)£?z = .'. y^z ' 6. 2(y + 2)g?^ + (a; + 3^ + 22)4y -h (.r + y)^^^ = .-. {oc + 7/)M j/ + 2) = ^'. 7. (,z/^ + ;s )z = x'^ -^y^ \ 2 ' } xn J TOTAL FLUXIONAL EQUATIONS. CHAP. VII. 18. ydy -|- ydx -|- xdz = .*. y + x<^{x\y) = > 2=0' — 19. dz — ^dx-^xdy.\ | 02+^.^^ = and a70'2;+l=a^ ^. 20. J^ = ^ydy-\rxdy-\ydx .'. z = y(x + y) \ a-\-x-\-y 0'^ = a -\-x-\-y ) ' 21. zdx + xdy+ydz = ,'. y = x ^ ^2. dyd^z - dzd^y = xdx"dy .-. ^ =yf^ ^ -f \/x^-\-X(h'x — (i)xdx J 23. xdxd^z-^yd\i/-\-xdx^ = \ y =fVx^-j- xcji'x — <^xdx . 24. dx + {5x +y)dt = e'dt > dy + (3j/ - ^)6Z^ =e^'dt j *** 25. 4dr + 9di/ + (Ux + ^9y)dt = tdt \ Sdx + 7dy + (34a; + 38y)^^ = e'dtj ''' y y -4>x= e'er-' -{■ 20e'' - 31t + 31 3 26. c?2a7 - {2x - 5y)J^2 = ) = 0| d^y - (x-\- 2y)dl^ ^ =ax. Ort==f-^-~^-^^, where 9 = ± v'^S. ^ ^HAP. VII. 28. X di MISCELLANEOUS PEAXIS. ce" + dr-"" ^ %/ - 3a; - 1 ^ 287 27. d^x — (3a; + % - 3y^2 = q ^2^/ + (^ - Sz/ + 5)df' 17 dt' ^"^dt^ 5x + 20?/ — 3 = csin.^ ^75 + c'cos.^ x/5\ 29. 7c/2^ + 5c/2?/ - (Wx + 23y - 46)^^^ = ? 11(^^0: + 8d^7/ - (25:r + 36z/ - 7S)dt' = J 30. 31. y = cV+ d"e-'- -j{ce*- de-') + 18^ cd^x ^ d^'ii ^ d^z ^> 2^ dt dt~ dt d'Z Sd^ dt' + rf/f -^^^^ dt^ ^dt M^x ~dF d'z d Udx + 250^, + -^4- 17^ dt 36^ = dt ,^_ c?^j2: 13c?^ 2/ = ce« + c'e^' + 2c V- cV +26-5 (Paoli, Elementi d'Algebra, torn. 2, p. 192). 'ye-' + c^'c-2* ■) PARTIAL FLUXIONAL EQUATIONS CHAP. VIIT. CHAPTER VIII. Partial Fluxional Equations. . 1. Def. A PARTIAL fluxional equation of three or more variables is one which has arisen from differentiating par- tially on the hypothesis that one of the variables is a func- tion of the others. The form of the equation indicates the hypothesis on which the differentiation has been effected ; and our only restriction in integrating is that we make not any suppo- sition inconsistent with the hypothesis. 2. Partial equations are attended with difficulties peculiar to themselves; and they are said to be integrated when their integration is made to depend upon that of a total equation. 3. An equation of two variables has only one fluxional coefficient in each order ; but an equation of three variables, when differentiated partially, has two fluxional coefficients of the first order, three of the second, four of the third, and so on. Hence \? z=f(x, «/), the canonical form of a partial fluxional equation of the first order is d^ 4. Required to integrate ^ = r, 'where r = ¥(x, «/, z). It appears from the form of the proposed that it has arisen from differentiating^ z=:f{cc, y) partially with respect to x\ wherefore in integrating, the arbitrary constant is to be a function of y. dz Now ^— = R or {dz) — v.dx, if d% be enclosed in brackets to denote its value when y is constant ; and since this is an equation between two variables % and x^ it is to be inte- CHAP. VI 11. OF THE FIRST ORDER AND DEGREE. 289 grated as in ch. 4, and a correction is to be added which is an arbitrary function of z/. The function of y in this case is as arbitrary as possible; for it is not under any restriction whatever. Cor, If more than three variables enter into the function R, the correction is an arbitrary function of all the variables except z and x. Ex. \. -J- = a. ax If this be a partial equation, we have {dz) = adx ; and integrating, z = ax + (y — ax). u 2 292 PARTIAL FLUXIONAL EQUATIONS CHAP. VIII^ — ^ dz dz Ex.6. xy^-.;^-=y-. s Here ado: — ^dy is xydx + x^dy .'. 8 — xy ,\y — — •*. -, wh.ch = - ^, =__.... = _ +^. = 1^ + ?...y. ^ ^ c?2 dz Here adx—vdy is e/c?^ — dy ,'. s = ly — x, or y = ^"*"^; and the primitive is « =f(x + e*+^)c?a7 + 0.9 = — + e*+^ + (l>{ly - a;) • • • x^ = ~ + «/ + 4^ . y^""''; for, since I .yer" =z ly -• x, there- fore 0(Zj/ — x) = I.ye~'' = ;// . j/e""". # Here s ~fxdx—y, or y =fxdx — s /. R' = ¥(x,fxdx^s) .*. z =J'F{Xffxdx — s)dx + 05, where s ^J'xdx — z/. 7. Required to integrate p -^ — h q-t- = r, where p «wcZ ci are functions of{x,y) and r i^ a Junction of{x, y, z). Eliminating p by means of dz — pdx + qdy^ we have iR,dx a d% = -{Q,dx — Poz/), or ^ q{Q,dx — "Pdy) — (r^o: — vdz) — 0. Let t as before be the factor which renders Q.dx — pdy an exact fluxion = ds, and t^ the factor which renders ndx — -pdi^, when y has been eliminated from r, an exact fluxion = da^ ; then ~ds -rds' = 0, or ds = —rds^\ which cannot be unless t f qv -J = ^V and s = 0^; which is therefore the required pri- mitive. If we eliminate $', there will result p{Q.dx — T^dy) + Rdy — adis = 0. Let t' be the factor which renders Rdy — adz, when x has CHAP. VIII. OF THE FIRST ORDER AND DEGREE. J293 been eliminated from r, an exact fluxion = ds' ; then it may be shown as before that the primitive is 5 = (ps'. Hence it appears that all equations of the above form are integrable, i. e. reducible to the integration of equations of the first order between two variables. We shall subjoin another mode of integrating the same form by means of an indeterminate function. 8. Required to integrate v- + o,—- = R, where p and a are functions of (x, «/), and R is a function of {x, y, is). Assume Q such that z =f(x, 6), where Q is a function of (x, y) hereafter to be determined. \^y represent a fluxional coefficient of ;s —fi^^ 0? which is enclosed in brackets to distinguish it from the fluxional coefficient of 2; = ■f{x, y) ; then, differentiating z =Jlcc, 9) partially, we have g = Q + J ^ and d% dz d^ dij~ dQ ^~ ^ wherefore by substitution the proposed becomes (dz\ dz c dQ dd } Assume 9 such that 1* -7- + a-r- = 0(1); then P f -T- J = R (2). The equation (I) can be integrated as in art. 6; from which we obtain 6 = (/)(^, y) an arbitrary function of (x, y) of a known form. Also in integrating (2), fd%\. , „ . , smce ( "T" ) IS the fluxional coefficient which has arisen from supposing to be the only constant, the correction must be %0 or x^Kpc, y) = i//(^, y) where -^{x, y) represents an arbitrary function of a known form of {x, y). ^ dz y dz Ex. -7- + — -7- = x^'z^ ax X dy TT .• /,x . d& y dd Here equation (1) is "^ + — "J" = .-. (6, ex. 2) PARTIAL FLUXION AL EQUATIONS CHAP. VIII- y 6 — (/)— . Also idz) = x'^z'^dxj or z-" dz^x^dx ; and inte- 1 , y a;"*+^ grating,^— fy^^ = ^^--^^-^. 9. Required to integrate p-% — (- q-j- = r, where p, a and R are ea^h functions of {x,y^ %). Eliminating both p and q by means of d^ = pdx + ydy, we have bining these, ^{iidx — pj^) — qijady — q^^;) = (3). Suppose that any of the three functions Qdx — vdy (a), Mdx — PcZ^ (/3), R^y — Q<^^ (y) contains only two of the variables. Let it be (yj ; then if t be the factor which ren- ders it an exact fluxion = ds^ we have from equation (3), piJLdx — pc?^) — 4-^s = 0. ^ Let R and p become r' and p* when y is eliminated by means of 5 which is r. function of (e/, ^), and let ^' be the factor which renders "^dx — y'dz an exact fluxion/ = ds' ; then ^ry — -^ds :-. 0, ^r dj ^ ^dsf, and consequently the requirsd primitive iz £ = (^d. It t.r^?2ars then that the proposed is always integrable if either "[if tha three functions (cc), (/3) or (y) contains only tv/o of Llie vailabiec. The case in which each o2 the three equations contains the three varlabl::^ shall be integrated in art. 16^ Cor, If p.— '^ ; the ec :ation ^dx — vdz = gives c?2;=0, or ;/ = «. Giibstllute this value of 2 in Q.dx — vdy, which then becomes an ec/^atior beUveen two variables. Let it be rendered an exact : uzion = ds ; then, the required pri- mitive is 2 = (pSf v/here s results from the integration of Q,dx — z^dy = 0, in which z is considered constant. 10. 7t Is observable that the equations which result from assuming y and x to be separately constant in the form of d?i the preceding article are p ^ = r, or i^dx -^ i^dz = and CHAP. VIII. OF THE FIRST ORDER AND DEGREE. 205 Q -7- = R, or Rd^/ — ad^ ; and by combining these, ada! — Tdj^ = ; which are severally the same as (a), (jS) and (y). Either of these equations may be assumed ; and if it contain only two of the variables, the integration may be effected as has been shown ; but we may not assume any two of them to obtain simultaneously and then integrate, for the hypotheses of integration would be inconsistent. In (y), X is constant, but in (/3) it is variable. Also in (a) both £c and 7/ are supposed to be variable. Cor. 1. Since we may assume either ds = or dsf = 0, we have s = a or s' = 6, where a and b are the arbitrary constants of two of the equations (a), (/3) and (y). Cor. 2. Conversely, if a and b are the arbitrary constants of two of these equations, the primitive is a = (pb where a and b are each functions of tv/o of the variables. 11. EXAMPLES. d^ d^ Here (a) is 2/d^ — xdy ,-. s =—. Also (/3) is nzdx — xdz ,'. s' = I—, and the primitive is — = — , or z = x'^ij, — = an homogeneous function of (x, y) of n di- X mensions (ch. 4, 6). ^ ^ dz dz Ex. 2. az^ xzj — h xy = 0. dx dy ^ Here (a) is xzdx + azdy = ; (|S) is xydx + azdz = and (y) is xydy — xzdz — Q\ .'.the primitive is «^ — 3/« = «^(a?2 -1- 2«j/). ^ ^ dz dz -— Ex. S. xi/-^ + 2/^^ + a:^ = a.27j/ ^/a;^ +«/\ Here (a) is y^dx — ^3/^(3/ = 0, or 5 = — . Also (^) is (a^^ A/a;2 + 2/2 — a?^)c/jc — a7?/6?2 = 0, or zdx , , ,. . . dz H = a x/x'^ + y'^dx, and elimmatmg t/, ^96 PARTIAL FLUXIONAL EQUATIONS CHAP. VIII. %dx dz -\ = a ^/l + s^xdx ; whence sx z = a;~r> (f,s + a^/l -i^ s'^Jx^^^dx > axy ^x'^-Yy'^ S y x-{-2y ^ X 12. All homogeneous equations of the first order and degree are integrable. dz dz Let the equation be reduced to the form -3- + Q-r- = r. Assume x = 'vz,y = wz\ then, by substitution, it becomes p + vg = w, where v and w do not contain z. Also the equations (/3) and (y) of art. 9 become wfi?. vz — dz = 0\ {vvf — \)d,z -1- zwdv = (|8) > vfd. isoz — \dz = 03 (WW — y)dz + zwdw= (7) 3 ' and eliminating dz, there results {ww — v)c/tJ — (tJw — l)dz0 = (a), an equation which contains only two variables. Let (a), when rendered an exact fluxion, be = ds; then, eliminating either v or w from (|3) or (y), there will result an equation which contains only two variables and s. Let this be integrated on the hypothesis that s is constant ; and replacing v and zz; by their values v = — ,w= — , there will result the required primitive. 13. Required to integrate the linear equation dz dz -T- + ay- + Rz = s. ax ay Here jo + ag = — R2 + s ; and (a) and (|S) are (^dx — dy — and (s — Kz)dx — dz = 0. Let (a) which contains only two variables, when ren- dered an exact fluxion, be = d^ ; and suppose that s and R, when y is eliminated by means of s =J'(x, y), become s' and r'; then (j8) is dz -\- Kzdx = s'dx a linear equation; and consequently the required primitive is z = e-/^-''^'^ ty^''^'^^ s'^^ + 0j/)di/ = 0. Sub- stitute 3/ — tx; then (a) becomes (c + g^)c?a7 — (« + bt){tdx + xdt) — 0, or 9>dx 2{a^-bt)dt f- y n / ^ = = dS, X bt^+(a-g)t-c dt To integrate this, let lu =/ ^^,^^^_^y_^ , or ^^ __ ( ""^\("»— w), where w and w are the roots of \t-nj bt' -\- (a — g)t - c =^ 0; then 2^^ 2btdt-\-(a-a)dt , ^du ds — h TTo — r-^^ — ^^— + (« + ^) — •*• integrating, X bt^-{-{a'-g)t-c ^ . ^' u ^ ^' e' = x^(bt^ 4- (a - g)t — c)ii«+^. Also (,5) is A;sc?:r - (fl^ + bi/)d% = 0, or kdx , y ^dz ^ dz hdu ^ . , . . * (a -{■ bt)— =0 .'. 1- = = ds'; and mte- X ^ z z u grating, zu^ — s' ; and the required primitive is z = u-^((>{ {x\bt^ + (« — g)t — c)w«+«^) = u-^(p{t^^^(by^ 4- {a — g)xi/ — cx^) ), where \j/ — nx J ,.„.,,., ^ dz dz dz 14-. Kequired to integrate i*;t- + a;^ — h Rt- = s, where p, Q, R awd s are functions of{x^ y^ t, z). The equation becomes by substitution pp + Q^+Rr=s. Since z — f(^, 3/, ^), c?;s = pdx + ^-c^y 4- rc?^. Eli- minating p by means of the proposed, there will result 298 PARTIAL FLUXIONAL EQUATIONS CHAP. VIII. qiodx — vdy) H- r{vidx — pdt) — (sdx — vdz) = 0. In this equation q and r are independent quantities, and consequently (Alg. 347), we have Q.dx — vdj/ ;= (a), Rdx — vdt = (^) and {sdx — prf^) = (y). First, let (a), (/3), (y), each contain only two variables, and be rendered exact fluxions = ds, ds\ ds" by the factors O 7" 1 kf k', k" respectively ; then -^c?s + -jjds' — tjj ds" — 0, or ds — rf^5'+ -ttA^^' qhi ^ qTc^ Since q and r are not only independent but also inde- terminate quantities, they might be so assumed that kf k "" P' "^ '^'"^' ^"^ oF ^ ^^"' ^" ^^'^^ ^^^^ fl?s = (ji's'ds^ + ^s"ds" ; and integrating, ,9 = ^5' + x^'. But we can also obtain ,!? as a single function of (^', 5'^ by as- kr ds , k ds . ... suming - _ = _ and ^, = ^„ in which case ds C?9 ds = -ffi^ + -^fds" ; and integrating, 5 = =■ \J. To verify the first result, assume z — — {y -\- t) -{• x'^v, where v ■= ^(p, q), p = xy — xt, q =■ xz — xt\ then dx \^P dx dq dx j dz ■ c dv dz dv -) . — - = — 1 -|- ^2 1 -j; ^ ^ f and dy i dp dydqS c dv / dz \dv > dz dss ^ dz dz , •■•'^s + (' + ">^ + (^ + ^'rfr-(^ + ') • ■ • • . ^ 1 , <^2; dz , .dz IS satisfied by ^^ + (^ + ^) + (y + ^)_ =- y + ^, whatever be the form of 0. = a: + y + ^. Three of the equations are (x + y + %)dx — (y + t -\- z)dt = 0) (^ + 3/ + z)dy ^ (x + t + z)dt = V, (x + y + ^)dz - (^x -\-y-Jr t)dt = 0) ^^ ^ 7T7~rJ^'^ ^y ^ "■; 7~^^ and dz = —-^ dt : x^y-\-% ^ x\-V'¥% xA-7i-Vz or y-\-% ^ x\-y-\-z x-^y-^-z t—x t—-y — ■ —dt \dy ^ dt— — dt and x-\-y^z ^ x+y + z dz — dt = ■ — dt, x+y-i-z PARTIAL FLUXIONAL EQUATIONS CHAP. VIII. Also dx + dy -^ dz = ^; r "^ 3t; Substitute iJ = a:+3/ + ^ + ^; then dv = at, dv dt dx—dt ... dt dv ov ^r =^ .*• -; , which = — ■ r- , = — or Sv x+y+z t—x ' x-\-i/-{-z Sv dv S(dx — dt) ^ . . / ^v, a- • — + -^ = /. integrating, v{x — ty = s. bimi- V X ^~ t larly v{y — tf = s' and i {iS — tf = s" ; and the primitive is v{z - tf = ^\v(x — ty, v{y — tY}i which may be put under the form -^ \ v{x — ty, v{y — ty, v(^ — iY } = 0. 16. The primitive of any partial equation of the first order and degree where one variable is a function of two others is a = (j)b ; where denotes an arbitrary function, and a and b are the constants of any two of the equations (a), (|(3), (y) of art. 9, considered as simultaneous. For let the proposed be, as before, vp -\- oiq — r. Only two of the equations (a), (/S), (y) are independent: inte- grating these simultaneously (ch. 7. S8), let their primitives , ~ 7 >■ , where 5 and 5' are functions of (x, y, z) ; and a and b are the constants introduced by the integration. The following demonstration consists in showing that a — (pb shall satisfy the proposed independently of the form of (p. Assume a = ^6 or s = 0s'; then, since this is an equation between three variables, we may (ex hyp.) obtain by dif- ferentiation, ds ds , , c ds^ ds' ^ . . _ + _.2 = ^V ^ _ + _ J I ; a„d consequently , ,ds' ds q = ^ — - — -T-, =, by substitution m ; which therefore con- eys; ^ £^z tains the arbitrary function 0. .1^ L } CHAP. Vlir. OF THE FIRST ORDER AND DEGREE. 303 Eliminating p from the proposed equation by means of dss =■ pdx + qdij, we have, as in art. 7, Kdx — vdz — q{ ddx — vdy) — 0, or vdz = ndoc — m{Q,doc— vdy) . Now this satisfies the proposed independently of m ; for , , . ^ . , . , dz R mo: we obtam irom it, p^ which = y-, = ^^ ).;andby and g, which = j-> = ^^ substitution, the proposed becomes r— ?72a+wa = R; which obtains independently of m or the form of 0. It may be remarked that we arrive at the same conclusion, if we begin by differentiating s = (p's' either on the hypothesis that X and 5? are the only variables, or that x, y, z all vary. 17. The preceding theorem may he extended to equations containing any number of variables. Let the proposed equation be p/? + Q5' + Rr = s as in art. 14; and suppose any three independent equations of that article, when integrated simultaneously, to be * = a, s' = b and 5" = c, where s, s', s" are functions of (x, y, t, z), and «, b, c are the constants introduced by the integration ', then shall the primitive be « = 0(&, 6'). For assume a = (j){b, c) or 5 = v ; then differentiating on the hypothesis that 1/ and z are , dv ds dy dti the only variables, it may be shown that q = -j ;7~~» ass dz , dv ds u u ■ ■ .. ., , ^^d^'dt by substitution m. Similarly r = -j ^ = n; where dz ^ dis m and n contain the arbitrary function 0. Eliminating p from the proposed equation by means of dz = pdx + qdy + rdt, there will result sdx — vdz — q{Q.dx — pdy) — r{Rdx — Fdt) =: 0, or pc?^ = sdx - m(adx — vdy) - n(Rdx — vdt). This satisfies the proposed independently of m and n ; for . . s wa wR , , , . , It gives p — — and q-=i m and r zz n\ which 304 PARTIAL FLUXIONAL EQUATIONS CHAP. VIII. . substituted in the proposed renders it identical ; or a — (f)(b, c) satisfies it independently of the form of 0. These theorems, by means of which the integration of partial is reduced to that of total equations, are due to Lagrange. The reader will find an elaborate chapter on the subject of partial equations in the Calc. des Fonct. Le^on 20. 18. The Theory of Arbitrary Constants when applied to partial equations does not give the same results as in the case of total equations. The complete primitive of a total equation of the first order contains only one arbitrary constant, because more cannot be eliminated by differentiation ; but, since a function of two independent variables, when differentiated partially, gives two fluxional equations, it is manifest that two constants may be eliminated, so that the complete pri- mitive of a partial equation of the first order of three va- riables necessarily contains two arbitrary constants. Thus, let %rzax + bz/; then -r- —a and ~ = b; and V • . , , dz , dz elimmating a, there results z =: x-^ — |- y-7-. Similarly, if ;^ is a function of n variables, the complete primitive of an equation of the first order shall contain n arbitrary constants. Hence it appears that any equation is a complete pri- ^ dz dz . . , . ' . mitive oi z'=-x-T- ■\- y-r- which contams two arbitrary con- stants, and which is included in the form z = 070 — ; which X may be called, by way of contradistinction, the canonical primitive. bx Thus z ■=:- ay -\ — ^ is likewise a complete primitive of the if proposed, for it is the same asz = a7.| a \- b — v. If this be differentiated partially, and a and b be eliminated, there will result the proposed equation. If it should happen that both the constants disappear by CHAP. VIIT. OF THE FIRST ORDER AND DEGREE. 305 one partial differentiation, the equation is not a complete, but a particular primitive. 19. Given a complete primitive of a partial equation of the first order ; required the canonical primitive. First, let ^ be a function of two variables ; and let the given primitive be F(.r, ?/, 2:, fl, 6) z: =: u. Since h •=. (pa, if «^, = be differentiated with respect to a, , . - ^ du du db du du , . . . there results = --r- + -r, -7- = -7- 4- -770^ — j by substi- da db da da ' db^ tution, A + B0'a. If the fluxional equation is of the first degree, the given primitive is of only one dimension with respect to the arbi- trary constants; and eliminating them by means of the given primitive and of (ba •=. b } ^y -n ,^ ^ , f /^ ^ • there will result an equa- tion between the variables in which ^ remains ; which is therefore the canonical primitive. Next, let z =/'(^, y, t) ; and let the given primitive be '£{x,y, t, z, a,b, c) — =: w; then, since c rr (^(a, b) rz, by substitution, (pv, we have by differentiation, ^ dtc , du ^^ du ^ = :7-oa -\- -frdb + -J- do. da db dc — i\da + Mh V C(p'v > -r-da + -jrdb I ^ Ida db S This, on account of the independence of a and Z>, gives A + c^'i) -J- = and b + ^(^a, b) and the given primitive will give by elimination the canonical primitive when the proposed equa- tion is of the first degree. The converse problem does not admit of a solution, since there is an infinite number of complete primitives belonging to the same canonical primitive. dz dz Ex. 1. ^ = w— , of which z zz a + b[nx-\-y) is a com- plete primitive. VOL. II. X 306 PARTIAL FLUXIONAL EQUATIONS CHAr. VIII. Since a + <^a{nx + ?/) — 2 i= = w /. • • • • • du ^ = = 1 + <^^a(rix -f y) .-. a =: x{nx + S/) •*• • • • h = ^{nx + «/) /. z, which — a ^- h{nx-\-7/)f —(l>[nx^y). The same result may be deduced independently of the article ; for, differentiating z -zz a -\- h{tix + y) on the hypo- thesis that a and h are the only variables, da + {hoc + 2/)db=0, or da= —{nx 4- y)db .*. —nx + y — (j)b, or b :=• x(^^ + 2/) .*. «, which zz 06, r:\|^(/2r + «/) •'• ^ = /? ; in which equa- tions, if any form whatever be assigned to 0, and p be eliminated, there will result a primitive of the proposed equation. „ ^ d% dz Ex.1. -T- :t--^' ax ay Here a z=i — .-. z, which zzxp+qy-r{xdp+ydq),=xp^^-f^ xdp -M-^ j ; or and z — xp — ~ — (pp (2). Since (p is an arbitrary function, suppose (pp — a; then (f/p = 0; and eliminating;? from (1) and (2), there results 2 -f a — ^^xy; which is only a particular primitive. If ({,p z= 0, then cp'p z=. 0, and the- primitive is z^ zz 4ixy. dz dz ^ -^ dz dz 'dx ^'%^'^ ]Jx' %| z zz px \- qy ~\"fv^ where /i? ~f(p, q) ; .-. differentiating, , ^, ( dv ^ dv 1 = xdp -hydq +/'^|^^P + ^^2' J This is satisfied either by x2 ^ ^ az az ^ i az az , Ex.2. . = .^+.^^ .,-/.; ^, ^L 308 PARTIAL FLUXIONAL EQUATIONS CHAP. VIII. dv ^ dp ZZ 0") dp^ f rr,l n e/^ _ 0/ » or by ^^ > . The first system gives ^ + dg-^'' = ^ > p ■=: a, q z= b : ov z z=: ax -\- hj/ +^(0, h) is a complete pri- mitive. The second, which is the same as the equations A n 0, B — of art. 20, gives the singular solution. (dz dz\ X, 2/, z, -7-, T'j ~ ^' Let it be under the form q —y^x, «/, ^, p) ; then, since z is to be a function of (x, y\ d^ — pdx -j- qdy\ and by the equation of condition (/3) ch. 7, 4, we have Differentiating q :=.f(x,y^ z, p), let dq =. Ddp + Adx + Bc?y -j- cd^ ; wherefore dq dp ^ dq dp , . , ^ . -7-= Dt- +A, and -7- := Dy- + c; which, substituted in dx dx dz dz (!>' S'^^ "S - 1 + (^" - !?^|- = - (^ + ^^) (^)- ' This, if q be ehminated by means of q :=.f(x, 3/, 2, /?), is a partial equation, in which ^ is a function of three variables (a?, «/, z) ; and the equations (a), (/3), (y) of art. 14, become dx + Dc?y r= 0^ (pD— q)dx~ Jidz — ^C (^^• (A + pCidx + Dc/^ = J Let the primitives of (3), when integrated simultaneously, be 5 := «, s' =: b, s" =. c; then the canonical primitive of (2) is ,9 = ^(5', s"). This is a functional equation between ;r, «/, 2: and /?. Take any particular value of p which satisfies it containing an arbitrary constant; call it (p). Let (g) be the corre- sponding value of q ; then, integrating dz =. {p)dx + (q)d2/ as a total equation between three variables, a complete primitive of the proposed equation will be found, from which the canonical primitive may be obtained as in art. 19. But it is not necessary to integrate the equation (1); for we shall obtain a complete and consequently the ca- nonical primitive, if by means of any of the equations (3), CHAP. VIII. OF A HIGHER DEGREE THAN THE FIRST. 309 or of their combinations we can find a value of jy containing an arbitrary constant. It is worthy of remark that if d be eliminated from the two first of equations (3), there results dz—pdx— qdy =0; which may therefore be substituted for either of those equations. If the proposed equation be under the form P =y(^, y, z^ q), the method of integration will be pre- cisely the same, interchanging a; and y. For the case in which ;jf is a function o^ three variables, vid. Lacroix, p. 567. ^''- ^- ^ ~ K ^ "^ ^ [ = 0, or g' = {px + z)\ Here dq = 9.{px + z)(xdp + pdx + dz) .*. D = 2x{px +2); A = 2p{px -\- z); b = 0; and c = 2(px 4 z) ; and the third equation of (3) becomes ^pdx 4- xdp = ; and integrating, px'^ = a ,'. (;,) = - and (?) = (^- + .) .-. dz = — -f (^- +. j d.j adx dz - ^ . . 1 or — = dy .'. integrating j/ -\-h ^ = 0, or {xz-\-a){y-\-b)-\-x = 0. From this the canonical primitive is to be obtained as in art. 19, ex. 3. ^ dz d% _ dz^ ' ' "" "^ dx dy ~ dx- ^' Here^ = ^ + -L... y p ex 1 > , p . ^p , ^p^ —y dq= \ 7idp + ^-dx - -\dy, or D = ^ , ; \y pM ^ y y yp' A = -^^ ; B = and c = 0. y y^ Eliminating dx from the first and third of equations (3), there results (a + pQ)dy = dp, or — "^ = dp .*. integrating, y p = ay .'. q which = - -\ , ~ ax -{- — .*. ^ ^ ^ y P ay a — -^ z X 310 PARTIAL FLUXIONAL EQUATIONS CHAP. VIII. dz = aydx + axdy H — - ; and again integrating, z = axy H 1y + 6, which is a complete primitive of the proposed , Co7\ This method will always succeed when the pro- posed form is linear with respect to either x or y, and z does not enter. In the following articles we shall apply this method to certain forms which become integrable in consequence of one of the equations (3) or their combinations containing only two variables. Lacroix has given an analytical in- vestigation of all the integrable forms of the first order. (Tome ii. p. 550.) The form may be considered as in- tegrated when a complete primitive has been found ; since the canonical primitive may be readily deduced by means of art. 19. 24. Required to integrate F{p, q, z) — 0. Here q =f{p^ z)\ and differentiating, dq = ndp + cd^, where d and c are functions of {p, ;s). Also a = b = ; whence the equations (3) become dx -\- Ddy = i (pD — q)dx — Dd^ = >. pcdx + ^dp = ) Eliminating dx froin the second and third, there results {pD — q)dp 4- pcdz — (a). Since this is an equation between p and z^ a value of^ containing an arbitrary constant, and consequently a com- plete, and also a canonical primitive of the proposed equa- tion may be obtained. _dz dz ' '^ ~~ dec dy' __ z %dp dz ~ J Here a = — .*. an = -f - — , or d = — — - and y p J p" p p'^ 1 , . 2zdp , ^ . . . €.-= — ; .-. (a) IS -\- dz = 0; ,\ mtegrating, ~ .-. (q) — u^z^ .'. dz = jjdx + qdi/, CHAP. VIII. OF A HIGHER DEGREE THAN THE FIRST. 311 z'^dx ± _L , d^ dx i- , , ^ . = — -^ H a^z'^dy^ or — = — + a dy^ .\ a complete pn- i_ i_ i_ mitive is 2^* = a '^x-\ a'^y + b. The canonical primitive consists of ^^ Y^i =x + mj + ^a\ jf ^„y f^^^ whatever be a ^;s^ — y = cp^a ) assigned to (p, fl may be eliminated and the primitive ob- tained. Thus, if we suppose (pa = Aa + b ; then ^'a =A, and the primitive is z = (y + a)(^ + b). / dz\ . 25. Required to integrate p = q, where p = f( ^> t" ) ^^ dz\ ^ . . C?P , rfp , G?Q , dOi ^ Differentiating, ^rf^ + ^d;, = ^-rfy + ^^. or c?p , rfp , f/a , '^ = di ■ Tq Assume as before dq = ndp + hdx 4- Bdy + cJ^ ; then dv dot dp da ^ r, D = :7--^:^-;A = 3--^-7-; and c = 0. dp dq ax aq Substituting these in the last of equations (3), there will result -7-dx -i- -.-dp = ; wherefore p = a, an arbitrary dx dp ^ constant. Assume then F(jr, p) =/( j/, q)=a\ and solving these equations, let p = x(«? ^)j ^ = »//(«, y) ; where x and ;// are known forms ; then c?2?, which = pdx + ^'^(y, Let V =fx(^> ^)dx and z^j =Jlp{a, y)dy ; then, since a is the constant, the correction is in each case an arbitrary function of «,' and incorporating the two corrections, there results ^ = w + jc; + (/)« (2) . dz Also from equation (1), y- = ; wherefore the canonical 312 PARTIAL FLUXIONAL EQUATIONS CHAP. Vlll. primitive is ss = v + w ■{■ (f>a \ _, dv dzv , >: from which a is to be eli- minated, assigning any form whatever to (p. _ bdz dz Ex. —=- -- = ^*y'. ax ay -^ b"p z/- , ax"- , ?y'^ Assume ~ = ^- = « ; then p = -j^ and a = ^^ ,\ . ax^dx vHy , .... d% = —r-^ h "^ — ^ .-. the primitive is ax^ «/3 ^ If we suppose • Substituting in the second of these for px -j-dx its value deduced from the first, viz. =xJz, there results, n — 1 S n ) p < zdx H =-xJz > + nxzdp = 0, or •^ ^ w — J ) dp dx dz IP- j h 7 rr— = ; wherefore integrating, p tix (n — l)z ^ ^ j_ 1 1 i_ _ 1 px^^^i = rt, or /) = ax "z "~^ and ^ = «"yz "-». CHAP. Vlll. OF THE SECOND ORDER. 313 1 __J_ _ I Also dz = pdx + qdy = ax "z "-iJ,r + a'^YZ "-'Jy, I __}_ or z'^-^dz = ax "dx + a^Ydf?/ ; and the required primitive is Jz^'^^dz — a/x "da: — d^jYdy = (pa I _i (* EQUATIONS OF THE SECOND ORDER. 27. In a partial equation of the second order the same quantities enter as in one of the first ; and also, if ;^ be a d^z d^z d^% function of only two variables, -^, , , , -r-^; for which we shall substitute r, j, t. The canonical form is r(a7, j/, z, p, q, r, s, t) = 0. We shall begin with the simplest examples; those, namely, which are of only one dimension with respect to the fluxional coefficients. 28. In order to eliminate n arbitrary functions, the re- sulting fluxional equation must be at least of the nlh order ; conversely, the canonical primitive of an equation of the 72th order contains n arbitrary functions. It may contain more ; for it sometimes requires a higher order of differentiation to eliminate the functions, as in ;2 = as before. 314 PARTIAL FLUXION AL EQUATIONS CHAP. VlII. Ex. -^= ^j/ .-. integrating, — = — + - any two of the fluxional coefficients as dq = sdx + tdy ) , . , dp— sdy dq — sdx r and t, we have r - — ; — - + S5 + t— ^— 7 = v, or ax dy ' ndpdy -\- Tdqdx — vdxdy - s{ ndy'^ — sdxdy + ndx") =0 ( 1 ). Since s is indeterminate, this is satisfied by the system R^y^ — sdxdy + Tdx'' =0 {2) \ lidpdy + Tdqdx — vdxdy = (3) ) * The equation (^), which is the same whichever of the 318 PARTIAL FLUXIONAL EQUATIONS CHAP. VIII. fluxional coefficients be eliminated, is of the second degree with respect to -~; and consequently (ch. 5, 1) is to be re- solved into the two, di/—Mx — 0, di/ — k'dx=0, where k and 7,1 n ^V^ s ^y T ^ k' are the roots oi -f- r t~ H = 0. dx^ li ax R Hence, if to these we add dz = pdx -{- qd^, we shall have four equations which obtain at the same time between the five quantities a; y, z, p and q, by whose integration, a first primitive is to be found. But the resulting form of the primitive being inconvenient, the following theorem has been proposed and demonstrated by Monge. Substituting the two values of dj/ in equation (3), we have the two systems nkdp + Tdq — wkdx = J ^ -' dy — k'dx = > .ws nk^dp + Tdq - Yk'dx = j ^ ^' Suppose that by integrating (4) we obtain m = a | N = 6 r then shall one of the first primitives be « = (pb. For assume a = (pb ; and let da = Adx + ^dy -h cdz + T>dp -{- Edq = db = A'dx + B'dy + c'dz + i>'dp + B'dq — 0. In these equations, substitute dy = kdx ; vkdx — nkdp dz = pax + qdi/ — [p ■{■ qk)dx ; and dq = ^ ; and they become ^ A + bA; + c(;?4 g-A;) + ldx-{- ^d ldp=0'^ ^^ + B^k + c\p^qk)+^ldx^-^iy^--^\dp=o\ These, since dx and dp are indeterminate, give the fol- lowing, EvA; ^v erA: >. A + bA: + c(;. + yA;) + — = I d - -^ = Oi A'-f B'A; + €'(;;+ 5A:) + 5^ = 0^^" i)'-?!^ = 0^ Also (ex hyp.) da = (p'bdb; or CHAP. VIII. OP THE SECOND ORDER. 319 hdx -r ^dy •\- cdz + T>dp -f ^dq = {y -kx) ^ ;//(«/ - k'x). . ^. J% . d^z d^z ^^' d^ + ^^ "^ ^^ =^ ^ ''^'''' ' =^^'^' ^^- Here the system (4) is the same as dy — kdx = 0) dp + k'dq - vdx=^OS'" CHAP. VIII. OF THE SECOND ORDER. 321 and one of the first primitives is p + h'q =yp'c?a7-|- (f>{y — lex) (1), where p is accented to show that previous to the integration, y must be eliminated from it by means of y — kx = s ; and after the integration, y is to be restored by means of the same equation. To integrate (1) we have from art. 9. the system dt/ — /c'dx = (a) I dz - {fp'dx + (p'(y - kx)]da; = (/3) ] ' Eliminating ?/ from (j3) by means of 7/—k'x=s', let it be- come dz = dxj'p"dx + ^'(5' + (k' — 'k)x)dx ; then, integrating, z = jdxfs'^dx + ; and a first primitive is dp- -jdq bp — aq = (p\ax + hy). To integrate this as in art. 9, the system is adx + bdy = > 1 a _ ^{ax + hy)dx - 6 J2 = S *'* «^ + ^3^ - « •'• (p's,dx — bdz = 0, .'. integrating, x(p^s — bz =■ ^5, or z = x(p[ax + hy) -\- \p{ax -|- bi/). ^ ^ d^z ^ d^z , d'i VOL. II. Y SS2 PARTIAL FLUXIONAL EQUATIONS CHAP. VIII. Here the system (4) is dy — Mx = 0) , y _ f^^ ^ ^ . dp + k^dq — xydx = 0\' '^ dp + ydq — x{s + 'kx)dx = .*. integrating, sx'^ kx^ , J, x'{3i/—kx) ^ = p + k'q---^ ^ = p + k'q--^ ^.•. the two first primitives are > ; and the required -to)) p + kq = -^ ^ + ^p'(y 6 x^y AX* fluent is 2 = -^ - ;j^ + 0(y - Jcx) + Hy - k'x), c?z/2 e?jt;2 c?^ cify ^rf^dy '^ dx^ dy"^ ~ ' i Since r -s + ^t = .*. ^ = /c' = — — ; and there gr q^ q is only one system, di/ + ^dx = 0^ _ dp-jdq=0^ ^^ ^^ p f / •'• a first primitive is — = (^z. There is no direct method of integrating this ; but, since p dy — = — A: = — -p- .*. dy = — ^%dx = — ^"''

{y+x) + xj^iy -.x) -^ x{(p'(y -^ x) -y^'iy — x)]. If we reverse the process of integration, we shall see the reason of this circumstance ; for, eliminating \ps from sz =e '^ ixps + Je~dx(p\s — 2x) J by means of two partial differentiations, we obtain (^+2/)(^ — q) -{- 2z = ^^^ fluent is z = 0(aT — ay^ — 1) + ^(^ + oy V— 1) ; and to give this a real form, assume , ^x—ay\/ — l (p(x-ay^/-\) = x(.^ -ayx/^l)+ Jc ==- a/ — I x-{-a?/ V — ^ \p{x + ay^ — 1) = x(a7 + ay / — I) — 7c /-I then we have z = xi^'^oy V — 1) + x(^ + «y \/ — 1 ) — ^kay, which is always real ; and it is the canonical fluent, since it contains two arbitrary functions. 48. LINEAR EQUATIONS. The canonical form is r -\- ss -\- Tt + vp -\- aq + MZ = 'if, where s, t • • • n are functionsof (o;,^). This is not generally integrable even when s, t • • • n are constant quantities. We shall not investigate the equations of condition for the general equation, but shall consider only particular forms. 49. Required to investigate when d^z dz dz . . , , 7 . -j—f + P-j — I- Qt- +M2;=n e'y>dy\ di) Having found f, we obtain ;t-» or ;,' =_ (fl 4. %){x 4-.v)-«-^{0a; +f{^+yr^^yAy] I . substituting in (1), mz = - (« + 2)(a; + «/)-«-i [.^^ +/(.r ^yT-^'y4y\ + (07 + 2/)-^{0'^ + w/(^ +yY-^'y'dy] ■^Ux + z/)-'^-^ }0^ +/(^ + yY^yd7j\ ,-.^=-{x -{■ 2/)-«-^ I 'P^^A^-^yTi'^y-dy | + ' -^^ ^ -^ + (07 + y)-y(o; + yT'^'y-dy^ Cor. When a = h or the equation is ' + ^(^ + ^) + -^^:f^^ =0'^^havem = 2. But /(or 4- yy^'y^dy =. (x-V yf^y - y{x-^y)^yAy, if x"y = ^«/, = (^ + 2^)'»/'«/ — 2(^ + y)yiy + ^xy- Similarly/(a7 + y)^'y.dy = (x + yy^y-^ly^ •*• by substitution, is = ^-^^*^f^-(^4-^r-v^- + (^+3/)-«x:^ • • • -2{x +yy-^Xy = ^^^*^|^(?'^ + liy) - (^ 4-i/)«-^(?>a: + xy)' PARTIAL FLUXIONAL EaUATIONS. CHAP. VIII. 52. Euler integrates the general linear equation by the introduction of v and w, two new functions of x and 7/ ; and by making certain assumptions, he transforms the proposed equation in which 2: is a function of (x, y) into another of the form of the last article in which ^ is a function of The following equations will be required. We shall sub- stitute p', (a) ; then the proposed becomes Rm'^ -f sm'n' + Tn'^ = 05^ ^ ^ A^ + Bp' + eg'' -i-Mz = N (3) ; which is to be integrated as in art. 51, and z will be found in terms of (v, w). The two equations {a) have the same roots; let them be k and U so that k = — ,/[;' = —7- ; then k and k^ are func- tions which contain only x and j/. CHAP. Vlir. LINEAR EQUATIONS. 3^9 To determine v and w, we have the equations m — kn=0 and m' — k'nJ = 0, which are the same as -7 Jc-t- =0 and ik' -J— = 0; which, when integrated, will give v and w in terms of arbitrary functions of (x, y). Take the simplest forms of these functions; and calculating by means of them the values of a, b, c, there will result an equation which, when integrated, will be the required primitive. We are at liberty to take particular forms of these arbi- trary functions ; because in assuming for v and w, the only restriction is that the equations {a) shall be satisfied. ^ , d'^z d'z d'^z dz dz Ex. 1. 3-- + A;j— y- H- B;t- + C;t- + Dj- + T.Z = F. da;^ decay dy^ ax dy Let k and V be the roots of "k^-^-xk -j- b = ; then the dv dv . dw dw ^ . equations are -j- — ft';T- = ^, -1 W-r- = ; whose primi- tives are v = (f>{y + kx), w = \\j(y + h'x). Take v = y -\- kx, w =y + k'x ; then by transformation fj^^ (J 7* Hz the proposed becomes -7-7- + a-r 4- b-. H c<^ = e', where ^ ^ dvdw dv dw /cc 4- D ^ A'c + D E , , F a = -, b = -, c = and f' = 4b— A^ 4b — A^ This equation, and consequently the proposed is inte- grable when c = ab. ^ ^ d^z d'^z ,d% ^,c?s? Ex. % -r-„ - a^-r-^2ab^ + 2«2Z,— = 0, or dx" dy'^ dx dy r - aH + 2dbp + ^a^'bq = 0. Here s = .*. A;^ — a^ = 0, or A; = «, A/ = — a\ and the equations tor nnding the new functions are -^ a-r- = . dw . dw ^ . ... and -1 1- tt-7— = ; whose primitives are v — (j)(y + ax) and w = ^(y — ax). (Art. 6, ex. 5.) Take v = y •]- ax, w = y — ax ; then, we have 330 PAllTIAL FLUXIONAL EQUATIONS. CHAP. VIII. p =z ap^ — acf = a{p' — q') I ZtJ'^^a^s' + an' = a^V - ^^ + ^' '"^"^'''"' ^^ t = r^ + 2s' + H substitution, - 4aV + 4>a'^hp' = 0, or s' — bp' = 0. dp' To integrate this, we have -^ bj)' — 0; /. integrating, pi--^bw^f^ and ^ = e^"'0t; + \//!X' = e^<*'"^'''^0(2/+a^)+;^(y— aa;). Ex. 3. -^-- - -J— • = — J-, or r - ^ = — . a;r2 CM/2 X ox x Here A:« — 1 = .*. ^ = 1 and k' =— 1 ; and the equa- dv dv ^ ^dw dza i i . , tions are -^ = and -, — r j^ j ^i^d taking the pri- mitives w = y + 57 and w = i/ — w, we have p = p^ ^ gr', r = r' — 2s' + ^', and ^ = r' + ^s' + ^' ; .*. by substitution, 2 1 ■^ 4s' = — (p' - 9% or 5' + ^^n^Cy - q) = 0; whose ... . X'— WJ r , ,, ■) primitive is ^ = 2M; y^iodw -\-(l)V=rp + 2/) d.xQ, . . dp do. dx dy dx TT^ =~J~ ; whence, since — = ;=-, p-jt-^-jt = Qt". dt ds dt ds^ dt dt ds . dx d-v dy 'd-v dx d^v . ^ d'v d^v d^v ^ ... . . , , rf<^ + ''5^ -"^rf? = °' ''^"'^' "'^'^° ">tegrated by Laplace''s method, will give v in terms of (5, t). Having found v, we may obtain x, y, ^ each in terms of (s, t). CHAP. VIII. NOT LINEAR. 333 For a? = — (!);«/ = -77 (2). Also to obtain z ; we have p = rx -^ Sj/ —f{xdr ~\- yds), in which xdr + i/ds being an exact fluxion, and expressed in terms of (5, t), it may be integrated, and p be found in terms of {s, t). And q =^ sx -\- ty — V, wherefore ^, which =z px -\- qi^ ~/{xdp + ydq) may be also found in terms of (5, t) ; and eUminating s and t by means of these three equations, there will result the required primitive. We shall give but one more instance of a general form ; and then terminate this part of the subject with some ex- amples that are to be integrated by particular methods. It appears by differentiation that the proposed is satis- fied by z = • ^'" be the n roots of this equation; then it may be shown as in ch. 6, 20, that the primitive is z = (?'(// + Jc'x) +(p"{i/ + k''x) h (/>'"(3/ ^k'^'x), which contains n arbitrary functions. If /c' = fc-", ^=^'{i/ + k'x)-\-x(p"{i/+k'x) . . . H-0'»(2/ + A;''^^). 57. EXAMPLES. Ex. 1. £-(^«+^)^+(^«^+«\^.-«{y + ax) + ^'/'(j/ + ax) f x(j/ + ^^). Ex. 2. 3— = a"-z. dx^ Substitute t- = kz .*. -7^ = k-^- = A;%, which .*. = a^^. dz _ c?^« ,rf^ -J- = kz .*. 3— = A:-j- a^ aj;^ ax .'. ^ = 1, ^' = - 1 and 2; = ^«^^y +'^^^;//y. Ex.3. ^ = a«^. dis d^z Substitute -r- •= kz .'. j—- = k^z = a^;^ .*. k = a, J<^=ma and /d" = na, where 1, m and n are the cube roots of unity ; and ;? = ^^0j/ + f '^cos. ^ x^ + g ^ sm. ^ ;//,//. PARTIAL FLUXIONAL EQUATIONS. CHAP. VIII. Ex.4. (« + 2&>-(2a + 36)g+c| . . . . . Substitute z =■ e'v^ then the proposed becomes Substitute w = ^-^ ; and this becomes /x / rtix ^dw dw = (« + 3i)«, + 5- + c^. If this be integrated, there will result e a(p(^cx — Sj/) + xxy + ^J/ 5? where »z = a + 3&. ^"- ^; 5^ + *s^ - ^»d^. - «*<^ + "''-' = "• Substitute z = e^'^v ; then _ _ cZ^^ d^z Ex. 6. -5-- = a^TT- dx^ d.T^ It appears by differentiation that the proposed is satisfied by d^^ dss -i— = ^T-, where h'^^a^ 0x1:= ±a\ which divides it into d^z ^^ _ n 1 dy^ "^ dx ~~ f 2 = p7, . ... ^, i4 da ( • Let ^ = ^ J be the pnm.t.ves of these : ^ + "^ = ^ 3 then the required primitive is^ = p -ha, which contains four arbitrary constants. ^ ^ 6?% d^ss ^ .d% This is reducible as in the last example to the two, d^% , dz ^ For more examples of similar reductions, vid, Euler's Int. Calc. tome iii. ch. 2, p. 859. CHAP. Vlir. CONSTRUCTION OF EQUATIONS. 335 tZ-^ d^z d"z^ Ex. 8. ^ i— r = -^ — r-» or rt = s"^; which is the equa- dx^ dy^ dx'df^ ^ tion of developable surfaces. v s Substitute — = — = u .: r — ^cs^ $ ^=. vt ,', s t dp = rdx + sdy = v{sdx -\- tdy) = vdq, or p = (pq, which is therefore a first primitive. The canonical primitive obtained as in art. 25 is y — ipa — {z — a)(pa I ^here a is to be eliminated, having X — \pa= (^ — a)\l>'a j ' & assigned a form to (p. 57. We shall omit the cases in which ;^ is a function of three or more variables ; though they frequently occur. The truth is, that the student must consider this chapter as containing a very imperfect account of the doctrine of partial equations. It would require a distinct treatise to discuss fully this branch of the subject. When the equa- tions are of the first order, they are always integrable; or at least they may be considered to be so ; but of those of the second order it cannot be predicated that to each equation there belongs a fluent containing two arbitrary functions. The student will find numerous instances of the integration of particular forms in the works of Euler and Monge and in Poisson's Memoire on Sound, and various other Memoires in the volumes of the Ecole Poly technique. 58. On the construction of equations; and the deter~ mination of the arbitrary constants. The arbitrary functions in partial equations are to be determined as the arbitrary constants are in total equations from the conditions of the question. If, after having used all the conditions, any remain, the problem is in some re- spects indeterminate ; but the result will show in what sense and to what extent it is so. 59. Required to construct the surface to which belongs I . dz the equation -j- = a, where z is a Junction qf[x, y). The primitive is z = ax -{- (by (a). Adopting the same notation as before, dz = pdx + qdy ; and, referring to the diagram of vol. i. ch. 9. 17, it results 336 PARTIAL FLUXIONAL EQUATIONS. CHAP. VIII. from the notation that dz = pdx is an equation which be- longs to the curve tpv ; and dz = qdy, to the curve spr ; for y is constant in the one case, and x in the other. dz Here -^ =z a or z : x \s a constant ratio ; and conse- dx quently tpv is a right Hne. The same results from (a) ; K>r, substituting c = 9j/, it becomes z = ax -\- c, which is a right line inclined to the plane xoy at tan.~'«. Suppose X = 0; then (a) becomes z = (pi/ = c. As- signing a particular value to c, this represents only one point in the curve zy ; but in general z = (py represents a curve ZY, which is as arbitrary as possible. It may be discontinuous^ i. e. consisting of two or more different curves; or it may be discontiguous, in which case the parts are not only different curves; but at their junction, they have different or- dinates to the same abscissa as iu the annexed diagram. Suppose the discontinuous curve to consist of an indefinite number of small arcs ; then, in the limit, it may represent an irregular curve, i. e. one which is described at random not subject to any law. Describe then (libera manu) any curve whatever in the plane zoy; then a right line moving along this curve parallel to itself and to the plane zox and inclined to xoy at dz tan~^a will describe a surface which belongs to -7- = a. 60. Required to construct the surface belonging to dz —- z=z YX, dr The primitive is 2: = fpx.dx + ^y. The surface is con- structed in the same manner as in the last article, except that the generating line is now a curve, whose equation is ^ =jFx.dx + c, and whose nature depends upon the form of F. The problem is as before indeterminate, i. e. the curve ZY may be described at random ; but, when once described, the surface is traced by the same curve z =fFx.dx + c moving along it parallel to itself and to the plane zox. CHAP. Vlir. CONSTRUCTION OF SURFACES. 337 61.' Required to construct the surface belonging to ^ = p, where p =f{x, y). The generaiing line tpv is, as before, a curve whose equa- tion is z =fvax + 0y. Let y become successively y, y, y • • • ; and let the corresponding values of p be p', p", p"' • • • ; then the generating curves are z =fp'da: f (py' ; z —Jp'dx + 02/";... These are all of the same Mnd, dif- fering only in magnitude and position in consequence of the parameters, which contain y, varying. When ^ := 0, let the equation become z = y + (/)j/; which therefore is the equation of the arbitrary curve zy. Having described this, the required surface will be traced by a curve whose equation is z —Jpdx + c moving along it parallel to itself and to the plane zox. The same construction obtains if p =J'[x^ ?/, z). These constructions also follow from the consideration, that when the equation is -7- = F(a7, y, 2), since -7- does not enter, it cannot be determined from the equation, and consequently the transverse curve is arbitrary. If both -T- and -j- enter into the equation, the problem is still indeterminate ; for in partial equations -7- and -1- are independent of each other ; it is this which constitutes the characteristick difference between partial and total equations. In the latter the fluxional coefficients are not wholly inde- pendent, for they are connected by the condition -j- = -j-, nr, r^ ' J 1 n d% dz 52. liequired to construct the surjace p-y- + Qt" = 0, zvhere p and a are functicms of{x, y). Let t{Q.dx — ^dy) be an exact fluxion — ds ; then the equations (a) and (/3) of art. 9 give * = cr, 2 = a constant = /; ; and the primitive is ;^ = 0.s. (art. 9- cor.) The equation s = a between x and z/ is of a known form. Describe the curve which it represents in the plane xy, and describe upon this curve a cylinder whose axis is parallel to z or perpendicular to xy ; then since s — a belongs both to VOL. II. z 338 PARTIAL FLUXIONAL EQUATIONS. CHAP. VIIT. the required and to the cylindrical surface, and that at their common section we have z = b, one property is that this common section is parallel to xy, or perpendicular both to xz and to yz. Hence to construct the required surface ; in the equation s = a give to a all possible values from to qd, which will not alter the nature of the curve. Describe upon the curve a cylindrical surface perpendicular to xy. Let a plane move at random along z parallel to xy; then the inter- sections of the plane and of the cylinder will generate the required surface ; which may or may not follow the law of continuity. 63. Required to construct the general equation, dz d% dx dy " ' Let the primitive be s = 0s' where s = a, ^ = 6; s and 5' being functions of (x, y, 2) ; and a and b the constants in- troduced by the integration. By eliminating z and y, two equations may be obtained of the form y(x, y, a, b) = = u, f{x, z, a,b) = = u; or w = 0, u = may be considered as the projections of any arc of the required surface on the planes xy, xz re- spectively. Hence to construct the surface ; describe upon these planes the known curves u ~ 0, v — 0, and also per- pendicular cylindrical surfaces ; then their intersection is an arc of the required surface; and if the curves' parameters a and b be supposed to vary as in the last article, this arc will trace the required surface, which may be regular or irregular. It appears then that the problem of determining the arbitrary function, of which we shall soon give an analytical solution, is nothing more than the making a surface pass through a curve which is drawn arbitrarily in fixed space. Similar constructions obtain where the equation is of the second order ; but in this case, since the primitive contains two arbitrary functions, the problem of determining them amounts to the making a surface pass through two arbitrary curves. We shall give only two instances, and those of the simplest form. d^z 64. Required to construct -; — 7- = 0. ^ diVdt/ CHAP.VIIL DETERMINATION OF ARBITRARY FUXCTIONS. 339 The primitive is z — x; and if a: = 0, r = \p/j. Describe then tlie arbitrary curves z = (px, z = \pi/ in the planes xz, yz respectively. Let z' be the ordinate cor- responding to a;' in ;^ = ^^, and z'' the ordinate corre- sponding toy in z = \pi/; then at the point (x', i/) of the plane XY erect a perpendicular = ^' -f- z''; which evidently traces the required surface. 65. Required to construct -j—^ — 0. The primitive is z — x*py-\-^if\ whence, if ^ = 0, z = \py. Describe in the plane zoy the arbitrary curve s = ;I/y; let it be zy. Tlirough any point t in zy draw a plane tmv perpendicular to xoy; this will intersect the required sur- face in a right line ; for, y being constant, z — ax ')r b, where b =■ ■^y. It does not appear so immediately as in the last article that we have the means of determining the position of this line from the other function

y -t '^j/. Draw then a plane snr parallel to zoy and at a distance from it = 1 ; describe in it the arbitrary curve z = (py -{' \py, differing from z = \py only in this, that the ordinate instead of being z = \py is js: — (py ^ xpy^ then the right line joining the corresponding points of the two curves shall generate the required surface ; which may be regular or irregular. 66. On the determination of the arbitrary functions. The arbitrary functions are to be determined from the conditions of the question ; and it is obvious that we must have as many conditions as there are functions to be de- termined. The conditions consist in the function z taking a known form, when certain relations are assigned to x and y. We shall illustrate this by only two instances; the one, of the first ; and the other, of the second order. 67. Required to determine the arbitrary function in 1 = 505', where s aiid s' are certain Junctions of (x, y, z). Suppose the condition is that f{x, y, z) = when f{x, y, z) = 0, where f andy' denote given forms. To de- termine (j>; we have the three equations ¥{x, ^, z) = ; f{x^ y-, z) = ; and 5' = function of (^, y, z). By means of these, each of the variables ,r, «/, z may be expressed in 340 PARTIAL FLUXIONAL EQUATIONS. CHAP. VIII. terms of s'. Substituting these values in s, let it become s' ; then the proposed equation becomes 1 = s'(ps' or (j>s' = — ; which determines 0, and a definite form may be thus given to 1 = S(j)S'. 68. Required to determine the arbitrary Junction in 1 = s^^ + s'x/'/, where s,s', s" are Junctions of £c, it/, z. Here there are two functions to be determined ; and we must in consequence have two conditions. Suppose that ^{XfT/, z) = gi\esj{x, q/^ z) = 0; and that r'(^, y, z) =0 givesf(x,7/,z) = 0. By means of F(a7, i/, %) — 0,f{x, y, ^) = and / = funct. (x, 3/, ^), the variables x^y, z may be expressed in terms of 5". Substituting these values in s and s\ let them become s" and or a ^ = -T- Va"^ — y^ ; but when ^ =0, we have from (2), a? =0^ ; whence 0j/ = -^ ^a^ — ^2 . ^nd the required equation is a The canonical primitive may be also obtained in the fol- lowing manner. The generating line is always parallel to the axis ; let it be a: = wz^ + a ^ , a = x —mzl 7/ = n^ + (3 y (3 = 1/ — n^ y Now, if the position of the generating line be changed, m and n remain unaltered, but a and /3 vary ; i. e. a and ^ are quantities such that a varies when /3 varies ; and is con- stant when /3 is constant; and consequently a = 0/3, or X — mz = (p(i/ — nz). Prob. 2. Required the equation of conical surfaces. Def. A cone is generated by a right line, which always passes through a given point called the apex, moving along a curve called the directrix. A characteristick property of a cone, which is independent of the nature of the directrix, is that a tangent plane always passes through the apex. Let («, bf c) be the apex. The tangent plane is !2 — 2' = {x — x')p -\- {1/ — y^)q. But this passes through («, b, c) ; wherefore the required equation of condition is % - c =^{x - a)p + (?/ - b)q (1). qj Q rjQ __ ft The primitive of this is = (Z> : and to deter- ^ z—c ^ ss—c mine , let the directrix be f(^, ^) = ^ By means of these two equations combined with 842 PARTIAL FLUXIONAL EQUATIONS. CHAP. VIII. x—a , y — h ,. . , = a and = (3a, we can eliminate x^ y and z. Z-C Z~C ^ ' ' J/ > and thus determine the form of (p ; let the result be F(a, 0a) = 0, where F represents a known form; then the • 1 • . T^ ex — a y-h > required equation is Jb ? , - — ^ = 0. The canonical primitive, as in the first problem, may be found independently of the calculus. Prob. 3. Required to find from the same data the conical surface which touches a given curved surface. The cone touches the surface in some curve whose equa- tions are to be found from the data; and this curve is the cone's directrix. Now at every point of the directrix, the cone and the given surface have common co-ordinates; and also the values of JO and q are the same, since they have the same tangent plane ; find then the values of p and q from the given equation ; substitute them in the cone's equation % — c = {x — a)p + (y — b)q (1), and let the result be y\x, j/, 2r) = 0; in which x, y and z belong to the common section of the cone and surface, %, e. to the directrix. Let f'(^, y, z) = 0, when projected on the planes xz and yz, give F(a:, z) — ) ^ x* u- u *i A i n ^ ; by means or which the co- fiyyz) =05; -^ nical surface may be found as in the last problem. If an eye be situated in the apex, the directrix separates the visible from the invisible part of the surface. Ex. Let the surface be of the second degree, whose general equation is ao;- + a[2/- + a'z'^ + ^Bxy + 2b' xz + 2b"«/2 V = d, + 007 + dy + c^js 3 _ Aa? + By + B- - + c J _ BoH^^y^^^^z-\-d '''P =~ B'x' + Bl'y + A'^TT^' ^"^ 2'- - :B'x^B"y + A"z+d''' and if these be substituted in (1), and the original equation be added, there will result an equation to a plane surface, which shows that the points of the directrix lie in the same plane, and that it is not a curve of double curvature. If the surface be a sphere, it is manifest that the di- rectrix is a small circle of the sphere at right angles to the line joining its centre and the apex of the cone. It may be shown in the same manner, that if the curved surface is of the wth degree, the equation y(^, y, 2) = is CHAP. VIII. DETERMINATION OP ARBITRARY FUNCTIONS. 343 of (m — 1) dimensions, or the directrix, when a cone is the touching surface, is a curve of double curvature drawn upon a surface of the (m — l)th degree. PiioB. 4. Required the equation of surfaces of revo- lution. Def. The plane drawn through the axis of revolution and any point of the surface is called a meridian plane. A characteristick property of a surface of revolution, which is independent of the nature of the generating curve, is that a tangent plane is at all points perpendicular to the meridian plane. Let the axis he x = az + a) ., ,, . j. i — h 4- i8 i ' meridian plane h 1^ — ss' = — (x - x') -^ -T-{y — 2/') (vol. i. ch. 7. 75) ; cl o and the tangent plane is ;^ — ^' -= p{x — x') + q{y — «/'), where a;, j/, z are the co-ordinates of any point in the required surface *. But P Q these are normal planes; wherefore — +-t-+1=0(1) (vol. i. ch. 7. 70, cor. 1) ; which is therefore the equation of condition in order that the surface may be one of revolution round an axis which is parallel to the line .r = «^ > y = hzy _ Next to find an equation contingent upon the position of the axis ; we have the equations of the normal f Z y + ^(^ Z f') Z I ' ^"''' ^^"^^^ ^^^ normal passes through the axis, these equations are simultaneous with ^r Z f ^f I S ( ; whence, eliminating ^, y, z\ there results (i/-(3-bz)p -(x-a- az)q=.b{x-cc) - a(y-(3) (1); which is the equation of condition that the surface may be generated by a line revolving round the axis a^ = az' -{- a I The primitive of (1) is {x - af + {y - ^y + 2j2 = (f>(ax + &j/ + z) (2). If the * It is here supposed that the tangent plane meets the axis in a point {x, y , z). 344 PARTIAL FLUXIONAL EQUATIONS. CHAP. VIII. axis passes through the origin, this becomes ^2 _j_ yi 4- 2« = ^i^ax -\- by ■\- z). If it coincides with z, we have a=0 and 6 = 0; x"^ -f j/^+«^ = 05:, or a;"+j/^ = ^z; which is therefore the equation of all surfaces of revolution whose axes coincide with z. To determine -^ ; let the generating curve be an inverted parabola, whose axis coincides with z ; then, when the curve is in the plane zy, y'^ =.^:z\ but (ex hyp.)^^ — ^^ . whence i\i% = mz^ and the required equation is x"^ ■\- y"^ = mz. Prob. 5. Required the curved surface in which all the normals meet a given plane in a given right line. Take xz for the given plane; and let the given line pass through the origin, its equation being z = ax. The equations of the normal are ^ I y t ^5^ 1 3 = (2) I ' ^^^-"^ ('^' V' ") '' ^"y p°>"' in the surface, and (x\ y\ z') the corresponding point in the given riglit line. Suppose y = ; then (2) becomes y + q{z — z') = 0, or y % — ;^' = — -^ ; and by elimination from (1) we have X — x^ = — , and consequently %^ =. — + z and J?' = X — ^-^. But (ex hyp.) «' = aaf \ whence — ■ -\- z = ax ^-^, or ayp + (z — ax)q = — ^. This may be integrated as in art. 9 ; but if we change the axes so that the given line may coincide with z, we have a = GO and the equation becomes yjp — xq — 0, whose pri- mitive is ;? = <^{x^ -V y^) (Art. 6, ex. 3.) This result shows that the problem is to a certain degree indeterminate, as we have not the means of determining ^ ; but it also shows that the proposed property belongs alone to surfaces of revolution *. * These problems properly belong to our chapter on curved surfaces; but as we shall be compelled to omit this chapter for want of space, they are inserted here to illustrate the theory of arbitrary functions. CHAP. VIII. MISCELLANEOUS PRAXIS. 345 70. MISCELLANEOUS PRAXIS. -dss = ai .'. ^ = «sin.~' 1. -v/a^ — w« — j;^ _ ^ . . ;^ — asin,"^ — ==l + ^ dz 2. a:-T^ = nz .'. ;2 = ^ 0^* ^ ^ ^dx ^ 1 -x^y dz dz — _— — — tf 5. x-z- + y— - = n^x^ + y .'. ;2 = n^x'^ + «/^ + 9~« ^ dss dz w^ . , i ^ dz dz xy „ y ,- J^ d« / c o\ 10. xyj- + a;2~ = yz .-. z = ^^(y^ — x% .^ dz dz y'^ 12. ^ J- + zj- +y = .-. a:/^°' /s/a^+2/^ = <^{x- + y^). 14. a;«3- + «V =■ z^ .\ =(3-! V. ax '^ ay z x L x y j dz dz ^ -1 y—x dz dz Az dx ( Ay ^07^ y 346 PARTIAL FLUXIONAL EQUATIONS. CHAP. VIII. — /»3 dz dss . /, dz" . d^^ z = ^n'^ — x'^ — j/2 is a singular solution. dz dz /d;^ dzY gular solution. dz dz^ ^0. -T- = ^3- .-. 2^ = 2ax + ^a^y + ^>. a' IS a sm- , dis d;^ dz^ t. — -rr- z= mx + w-r- n ' z = X x/2mj/ — a + —(9.my — d)^ + b. dz dz dz dz x dfa?^ oc+y dx~ ay ' ' / x^ \ x^ 4-3x1/^ (/^j; a dz b dz ab _ dxdy y dx X dy xy Z = y^i^x + x~^'\iy + y-'^x~^fy^dyfvx^dx. d^z a + ^2 dz b dz ab-\-^b—a—2 ' dxdy x-\-y dx x-\-y dy (x + yY^ .\z = {X + j/)-«--^{ **-^ = '^i;^'-7r'- ,^ <^2^ . , , d'^z d'^z X dx''^^ ^ dxdy dy"- x^y (m — n)z = ^(j/ — mx) -\- -^[y — nx) .... (m—n)xy Cay^ , (2b—a)x'^l^^ ay^ (2b—a)xu , 1 1 and b = («+l)2 (m + l)«* CHAP. JX. FLUXIONAL EQUATIONS, &C. '549 CHAPTER IX. Tlie Integration of Fluxional Equations by Series and by Approxima lion. THE METHOD OF APPROXIMATION. 1. When all other methods of integrating equations fail, we must endeavour to develope the dependent variable y in a series in terms of x. This method has been already applied to two examples (vol. i. ch. 4, art. 11). If the law of the exponents be known, a series is to be assumed with indeterminate co- efficients ; this is to be differentiated, and being substituted in the proposed equation, the coefficients may be found by making each term =0. If the law of the exponents be unknown, there is fre- quently considerable difficulty in ascertaining it. From the same function, there may be sometimes deduced very dif- ferent series, as has been already shown. We can in ge- neral succeed in the approximation only when x is either very small or very great : in the former case, we must en- deavour to obtain an ascending, and in the latter, a de- scending series. To obtain an ascending series, the indeterminate ex- ponents must be assumed in an increasing arithmetick pro- gression ; and if the terms can be so arranged that both the coefficients and the exponents may be determined, there will result a series which converges the faster the less x is. If the exponents cannot be arranged in this manner, we may conclude that y cannot be developed in a series whose ex- ponents increase in arithmetick progression ; and recourse must be had to what is called the method of Successive Substitutions. Both these methods are due to Sir I. Newton. Before we apply them to the integration of fluxional equations, we shall give a few instances of their application to algebraical functions, which may be sometimes more readily developed 350 ' FLUXIONAL EQUATIONS CHAP. IX. by their means than by the theorems of Maclaurin and Lagrange. For examples in which the law of the exponents is sup- posed known, vid. Alg. 348. Ex. y -f- a^y - '■Jla^ + cixy - x^ — 0. To obtain an ascending series, assume 7/ = Ax'^ + Bx'^'^'^ + 007*'+^^ H ; then arranging the terms, we have a^^^"+3a^b^^«+^ + 3ab2^3«+2/3 ^ ^^^3.+33^ ^ — ^3 J equation which is satisfied by assuming « = 0, /3 = 1 ; in which case (Alg. 347. cor.) a-^ + a-A — ^a^ — 0; 3a"b -I- a^B 4- ttA = ; 3ab- + «-c + «b = ; • • • or 1 1 • or a = ft, B = ■ 4 ' ^ ~ 61a' X x^- 131^3 ^ 64a ^{'20^ If the exponents do not increase or decrease in an arith- metick progression and the law be altogether unknown, we must assume y = ax'^ -{- bx^ -\-cx'^ -j. . . . The following is a useful rule for finding the first term, whichever method of approximation may be adopted. ^. Sir I. Newton s rule for finding the first term of an ascending series. Rule. * Suppose the term of the proposed function where X is separately of the fewest dimensions to be T>x"' ; compare it successively with the other terms as with Eo^y ; and ob- serve where — — is found the greatest ; then shall be the required exponent.'' For, let Yx^y"^ be any other term of the function; then, ... m — t . ,, ^ m — p substituting a = , we have (ex hyp.) a > -, or qcx. + p > m. If therefore we assume an ascending series for y^ viz. CHAP. IX. INTEGRATED BY SERIES. 351 a , m—t..... y = Ax"" + Bar + . . . where a = , m the limit we s shall have 1/ = kx'^ ; and if this be substituted in the pro- posed term f^^z/*^, it becomes of the form vx^'^'^P, which may be neglected when compared with dx"^. The same is demonstrable of every other term of the function except i£.x*y\ or its equal e'^*^^"', which does not vanish, since (ex hyp.) it is of the same dimensions as d^'^. It appears then that y = Ax^ causes all the terms to vanish except two, which it reduces to the same dimensions ; and by means of them A may be determined. If a be assumed either > or < , there will be only one term left in the limit, and a cannot be determined. We conclude then, if y is developable in an ascending series of the form ax"" + bx^'-\- cx'^ • • • where b, c, • • • are functions of A which can be determined, that a = . s In some examples three or more terms of the functions remain in the limit, but the same rule obtains for deter- mining both a and a. m — t ^ Cor. 1. If it be observed where is found the least ; s it may be shown as in the article that is the exponent of the first term of a descending series. Cor. 2. Similarly if any term of the function, as c ly, be compared successively with the other terms, as with Ex^y% n — t and it be observed where is found the greatest or s — r *» the leasts it may be shown that the first term of the required n — t series is kx^ , where a = , and a is to be determined as s — r before. If both the numerator and the denominator of are s—r negative, the reverse of this is true; for in multiplying both 352 FLUXIONAL EQUATIONS CHAP. IX. sides of the symbols > and < by the same negative mag- nitude, the symbols are to be reversed. Before this rule is applied, the example should be cleared of fractions, and if any of the exponents are fractions, they must be made integers by substitution. Sir I. Newton has also invented for the same purpose a rectangular diagram, which the reader will find explained in The Method of Fluxions, p. 9. 3. Given an implicit function ; required to develope the dependent variable in an ascending series of the independent bij the method of successive substitutions. Let y ■= A.r* + boc^ + cx'^ • • • be the required series. Having determined ax'^ by the above rule, an approximate value of j/ may be obtained when x is small. To obtain a nearer value ; let y — ax°' -\- y\ where?/ = b^ + cjr'^ + • • • =? in the limit, ^xF. Sub- stituting this value of y there results a function from which the first term of z/' or b^^ may be determined as before ; and we thus obtain the value of y = a^" + bo?^. Similarly, substituting y = bjf'^ + y, cx'^ will result ; and by repeating the process, the series may be calculated to as many times as may be required. Ex. 1. To develope y^ + a'^y — 9.a^ + axy — a;' = in an ascending series. Since is the least exponent of x^ the quotients are 0, 0, — 1, — f ; of which is the greatest; and the equa- tion for determining a^*", or that which arises from sub- stituting y = Kx^^ and neglecting the terms in which x enters, is y"^ + ci^y — S«^ = 0, which shows that y =^ a when X is indefinitely diminished. Let y = « -|- y ; then by substitution 4ay + 3«y2 + y + ax + axf — x^ = (1) ; where the quotients are 1, 4, i, 0, — | or /3 = 1 ; and to find B, we have, neglecting the terms in which .r^ enters X ^a^y^ + a'^x — or y = y- ; and a nearer value of ^/ is X «/ = « - V X Next, let y = — ^ -1- y'; and if this be substituted in CHAP. IX. BY SERIES. 353 (1), it will appear that bjt = ^7- ; and so on. Ex. 2. ax^ + x^y — a'jf = 0. The quotients are and 1 ; and the equation for deter- mining Ax^ is ax^ — ay^ = 0, or y = a:. Let 3/ — X + y ; then, by substitution, ^ .+ ay — 3«xy — 3cf^y2 _ ays = o (l), where the quotients are 1, 2, |, 4^; and the equation for determining bo; is a* — 3a^-y = .*. y or Bar = -^. ^2 Next, let 3/' = -^ f- 3/" ; then ( 1 ) becomes x^ x^y-^ 1 g;^ + ^axY + ^y + -g^j- f ^ 0, where the quotients + Saxi/'- + xy^''+ ay^'^ J are 4, 3, 2, |, or y = 4 ; and to find c, we have ^, + 3a.y' = 0ory' = -gg5;.-. x"^ a?* X^ ^ = '^'^3^~3^'^3^^~"* To obtain a descending series ; a = 0, and to find a, we have ax^ + x^y =: or y = — a. Let y = — a -\- ,y' ', then, by substitution, a^y + ft* — 3ft3y -\- 3«'j/'2 — flty^ = O ; where the quotients are — 3, 0; .*. ^ = — 3 ; and to find b, we have a?y + tt* = or y = — a*x~^ ; and, repeating the process, there will result y=^a— a* x-^ - ^d'xr^ —VZa}^x-^- • • • To find other series, compare the second term with the first and third ; then (cor. 2.) the quotients are — and ly the same as before. But if the third term be compared with the first and second, the quotients are 1 and \ ; and substituting y — A.r% the second and third terms are of the same dimensions and greater than the first; or hx"^ is the 3 first term of a descending series; and Aar" = ± —7. Let y = ~r + y ' then, by substitution, aJ J. 1. ax^ - 2j:y — 3a ^^y 2 _ ^ys. ^j^gre the quotients are VOL. II. A A 854 FLUXIONAL EQUATIONS CHAP. IX. 0, i, 1 /. j3 = 0, and to determine bx^, we have ax^- 2j?y = 0, ory =-|-. Similarly, assume j/' = — + y, and there will result This developement may be also obtained by the method of Indeterminate Coefficients, by substituting y = Aa;« + Bx"-^ + cx"-'^^ + Bx^-^l^ + • • • Ex. 3. y — OTy^ + ^^ = 0. Substitute w = w = 2/^3 t;^Z£> + tj* = 0. To develope this by the method of Indeterminate Co- efficients ; the first term of an ascending series is v. Assume then w = V + BV^"^^ -\- cv^+'^f^ + dz;^+^^ • • • /. we have » ^4 - Bu4+^ - cz;^+2^ - Di;^+3^ . . . ^= .-. ^ = 2, + t)* 3 B = l, c=6, = 6.6, • • • ; or aj=t? + t;' + 6u^+6.6tj''^ • • • and _l 1 5 _7 replacing x andj/, y^ = a?"^ + jr + 6x^ + 6.6^7 ^ • • • When the equation consists of many terms, this method of finding all the possible developements becomes laborious ; we shall therefore give another method of finding all the possible first terms ; and the others are to be obtained as in the article. (1) (2) (3) (4) (5) (6) (7) Ex. 4. x^i/^ + axi/^ + bx^i/ + cx^ + exy +/^^ 4-^2/=0. When X ■= — ; (1) may be neglected compared with (2) (3) (5)1 (4) (6)< (5) (7) In which equation y must be either finite, go, or — . Now y cannot be finite ; for then we should have gy — 0. CHAP. IX. BY SERIES. 355 Suppose 2/ = 00 ; then aooy^ -^ gy ■=^^or axy + g = 0. Suppose y = — ; then axy'^ may be neglected compared with gy orfx^ -\- gy = 0, When a; = 00 ; (2) may be neglected compared with (1)"^ (5) • • • (3)1 . (6) (4)f-- O) (3)) ar^y + hx^y + cx^ = 0, or y'^ + by ■]- ex = 0. Here y must = oo ; and the only possible equation in the limit is y^- -\- ex = 0. It appears then that this example admits of only three developements ; the two ascending series are ^ g g' g^ y=- ^x-^ -^ r— ^a; The descending series is of the form y = + c^x'^ + b; and consequently the curve to which the equation belongs has an asymptote which is a parabola. Ex. 5. a;7 — a^x'^y-\-a?x"y^ + a^y^ — 9^a^xy + aV = 0. Or x"^ — x^y + x'^y^ + y"^ — 9>xy + ^^ ; for a may be at any time restored by rendering the terms homogeneous. When X = — ; (1), (2), (3) may be neglected, and there results «/2 — 2xy + ^^ = ; or there are two ascending series whose first terms each = x. When X = oo ; the equation becomes x"" — x^y-\-x'^y'^ = 0, ox x^ — xy + 2/2 = 0; where y is necessarily infinite and of a higher order than x^ or x^ -^^ y'^ — 0, which is possible only when x is negative. To find the ascending series ; substitute y — x + «/.'. x'^ + x^y^ + y2 _ (a), where the quotients are 4, |^; .*. x'^ 4- xy = 0, or y = — x'^. Next let «/' = — a;* + y ; then, by substitution, x^y"-\- x^ — ^x^y" + y"^ = 0, where the quotients are 5, 4 ; /. x^y" -\- x^ = 0, or y" = — ^^ . ^^^ gQ qj^ To find the other ascending series ; compare in (a) x^y' with the other terms, and the quotients are — 4, 3 ; and x^y' + y'^ = 0, or y = — x^. aa2 356 FLUXIONAL EaUATIONS CHAP. IX. Let y = — iT^ + y ; then, by substitution, x' — a?y + y'2 _ ; where the quotients are 4, |^ ; .*. x' - xy = 0, or 2/" = X*. Next, let 2/" = a?* + y" ; then we have — ^syw + ^ + 2a:*y" + y''^ = 0; where the quotients are 5, 4, .-. — ^^y H- o;^ = 0, or y = ^' ; wherefore the re- quired series are X* x^ x^ x"^ x^ Ex 6. Required to find the asymptotes of the curve «,r* + bx^ + ca^ + ex + ff ^ mx^ + nx^-i-px + q ax'^ ax - . , . When X = co ; v = — •, = — ; or the equation admits •^ mx^ m ^ of only one descending series. Let 3/ = Ao:" + b^-*"^ + c^'^'^^ + • • • WA^"+^ + mBa;°'~^+^ + mcx"'-^^-^^ + • • + pAX"-^^ + . . — o^c* — 6^3 — cx^ — . . .'. C6 = 1 and /3 = 1 . Also, 771A — a = ; ?nB + «A — i = ; mb—na i , , . i B = 1 — ; or the curve has only one asymptote ; viz. the .... ax mb'^na right line y = \ — . The curve has as many infinite ordinates as there are possible roots in the equation mx'^ + noc^ -\- px + q = 0. 4. Continued fractions. Similar to the preceding is the method of continued fractions. For, kx"" being the first term, A^* , B^ „ cx'y assume successively y = j-^^^,, y = y:^^^^/' = fq^, • • S then if «, /3, y • • • be an ascending series, approximate values of ^, when x is indefinitely diminished, are t/ = ax'', CHAP. IX. BY SERIES. y = ax" y = hx" 1 + B^' &C. = & "l+B^^' 357 accurate than that which precedes it. The value of bo;^ is determined by substituting AX^" y = in the proposed function, and taking the value of y in tHe limit ; and in the same manner the succeeding terms may be found. If an approximate value of y be required when x is inde- finitely increased, ax"^ must be the first term of a descending series. Ex. Required to develope ^ in a continued fraction. Let y — 6"= = \ -\- X -\- ^- •\- ' ' ' \ then i/ = 1 + j: is a first approximation. X Assume e/ = 1 + ^ , which, substituted in the pro- X x^ posed function, gives .. , = a; + ^-^ -|- • • • or 1 +y = Tl + ^+ . • O =, in the limit, 1 _ -|- .-. . . . ^ = 1 -^ is a second approximation. 1- - a: a; X "" ¥ ~T Assume y or - ^ = j^„ /. 1 + j-^' =' ^ + V' - 1 + y'= (l - ^ + • • •) =, in the limit, 1 + ^ ••• X y :=: 1 -] is a third approximation. 1-1 ' + 372- ^8 FLUXIONAL EQUATIONS CHAP. IX. By repeating the process, it may be shown that >-o 1 + 3.2 I - X 3.2 1 + 5. X 1— ... PRAXIS. ' ^5 y* y3 y'^ ^ y •^"■"^ "^ '2""^3x^ "*■ 35x1"*" 3.4.5 x^"*" '" S. a:y + 3a^^3^2 _j_ 3flt2^9^ _ ^3^2/ — fl^ar + ^5 = may be developed in three descending series, first term of each =— a. 4. 6a;7— 2^5^'^— a^a:2^^+4ttVy+2a»a:2— 3a'j2/+ay =0 .-. 3^^ 15^* 4;r6 3/ 5.^_i^3^ + |!.^+y'^y_^'=0... . . . y •=. -^x^ + Ba? • • • i %:. CHAP. IX. BY SERIES. 359 nx 6. (i+^r=i+ ^ (n-l)x 1.2 1 ^ (^+1)^ 3.2 {n^^)x 3.2 J _^ (n + 2)a7 5.2 1 6. Reqim-ed to integrate in a series dy + ydx = mx^dx. Assume 3/ = A^" -I- Ba?"'+^ + Cd;"+^ + • • • ; then ^ = akx"-^ + (« + l)/3:r" + (a + 2)c^+^ • • • Substituting these, and arranging the terms, we have a^x-^ + (a + 1)B^« + (a + 2)cx«+J • • •) ^ o ; which — mx"^ + Aa;"' + Ba;*+^ • • • j may be satisfied by assuming a — 1 = w, in which case aA — w = 0; (a + 1)b + A = 0; (a + 2)c + B = 0; ••• mm A —m or A = — = =- : B = ^ = (. + l)(. + 2)(. + 3) -'-'"^^"^^ 2/ a;"+^ 0?"+* 0?'^+^ m ?? + l (w-hl)(7j-l-2) "^ (w-hl)(w + 2)(w + 3) Since this method does not introduce an arbitrary con- stant, it cannot give the complete primitive, whenever, as in this example, the constant enters as a coefficient of some function of the variable. The following method has been devised by Euler to supply this defect. It does not in- troduce the arbitrary constant itself, but two corresponding values of x and y by means of which it may be determined. (Vol. i. ch. 2, 6.) 360 FLUXIONAL EQUATIONS CHAP. IX. 7. Required to integrate an equation hy a series so as to introduce the arbitrary constant. Let a and b be two corresponding values of x and 2/. Substitute x = a-{-t,y = b + Vj where v and t vanish to- gether. Assume a series for v in terms of t with indeter- minate coefficients and exponents ; and arranging the terms, make such assumptions that each line of terms may vanish when ^ = : then, if by means of these assumptions, the series can be determined, there will result v or j/ — 6 in terms of t or x — a; which is the complete value of y. The series may be also found by means of Euler's Theorem, vol. i. ch. 4, 45 ; in which a and b must be sub- stituted for £c and y in the values of the fluxional co- „ . dy d'^y d^y dy Ex. \, dy -{■ ydx — mx^dxt or -— = mx"^ — y. 0g By substitution the proposed becomes dv -^ = m(a + ty - (6 + v). Assume r; - a^ + Br+^ + ct'"'^^ + • • • ; then dv , ^ or m{a + ty - (6 + v) = aA^^"" ^ + • • • (a -I- l)/3r + (a + 2)cr+' ; and arranging the terms, we have aA^*-' + (a-f l)Br + (a + 2))cf+l + " '\ + b + At" ^- Br+^ + •••( = m — ma'' — mna^'-^t -^ 1- • • • J In order to satisfy this, we must have a — 1 = or a = 1 ; in which case, aA + 6 — ma^ = ; (a+l)B-f-A-7Wwa»»-^=0; (a+2)c + B ^^ — —^ =0; ma^ — mna^~^ ~ b or A = Twa" — 6 ; B = — =-pi ; 1.2 ' __ ma'' — mna''~ ^ + w?2(w — l)6t"~^ — b ^"" 1.2.3 GHAP. IX. BY SERIES. S61 md"- — mna n—\ h Ex. 2, ydx + ydy = dx or z/f 1 + -^^ j = 1. The law of the exponents being unknown, assume y = A.x'^ + Ba^ -\- cx'y + ' ^ • ', then -^ = axx"-^ + (iBx^-^ + yca;^"' + • • . ; and (ax"--\-bx^ + cx'^-{- • • •) X { 1 + aAx"'~^ +/3B,r^~^ -\-ycx'^~^ + . . . } = 1 ; and arranging the terms, we have -1 + Ao;* + /3B«a;2^-^ +...V=0. + nx^ +•••) ' This is satisfied by assuming 2a — 1 =0; a + /3-- l = a; 1 3 a 4- y - 1 = /3 ; . . . or a = — , /3 = 1, y = — , 5 = 2 ; • • • in which case we have -^ 1 = 0; -^ — h a =: ; 2ac + B® + B = ; • . • or A = V2 ; B n -- — ; • • • .-. o _ ± ,»72 i. If the complete primitive be required ; substitute x=a-\-t, j^ = 6 -j- r; ; and the proposed becomes Assume v = At"" + Bt^ -{■ cf^ -\ — ; then, by substitution, we have (5 + a^ + b^^ + c^^ + . . •> x {1 + aA^-l + /3b^^-» + yc^^-' + . ..} = 1 ; and ar- ranging the terms, -1 +aA^^2«-l_^^^^^)^^^«+/5-l+(ct + y)AC2J^+^-l... J + 6+ Ar + B^'^ + cfy ' ' ' (^^ + &aAr-^+Z»/3B/^-^ + bycfy-^ + ISd^'^ ... 4 FLUXIONAL EQUATIONS CHAP. IX. Whence a=l ; /3 = 2; y = 8;-- Also 6a-|-^>-1=0 ; 26b + A + a2 = 0; 36c + b + 3ab = 0; 46d + c + 4ac + 2b^ = ; • • • from which a, b, c, • • . may be found, and the value ofvori/ — b obtained in terms of ^ or 07 — a. Otherwise : dy 1 , \-y dx y y dx'^ 1 y' dx~ 1 t 1 _y-i d^^_^ 2 3 >rfy^ (y-l)(2y- 3) ^ 1-6 , 1-6^ ••• y - ^ = "t;-!^ -«) - ■i:26"3^'*' ■" "^^'^ -* (l-6)(3- 26) + 1.2.36^ ^"^ "^ 8. The above method is also applicable to equations of a higher order than the first ; in which case it sometimes has the advantage of giving the complete primitive at once, with- out substituting new functions for the variable. Ex. d^y + axydx'' = 0. Here x and dx are of dimensions in the first term, and of w -I- 2 in the second ; from which it may be inferred that the exponents increase or decrease by 7i + 2. Assume 3/ = ax'' + 6^^*+^+^ + c^^+2n+4 , . . ^jjgjj ^=a(a-l)Aa7«-2^(a+w + 2)(a+w-fl)B^'^+'* + (a + ^n + 4)(a + 2w + 3)007'*+^''+'^, and the terms must be arranged in the following manner ; a(a - l)Aa?'*-2 4. (^ + ^^ 4. 2)(a + n + l)Ba7«+'*) ^ q + aAa;'*+"J Whence a(a-l)A=0; (a+7z+2)(a-f 7i+l)B + aA=0.-. This may be satisfied by supposing a an indeterminate quantity ; and either a = 0, or a — 1 =0. Let A and a' be the values of a corresponding to these CHAP. IX. BY SERIES. 363 two suppositions ; then, when a = 0, we have B = — -, TT-. ;:rr, C = (7i+l)(7i+2)' (w + l)(w + 2)(2w+3)(2w + 4) Also, when a — 1 = or a = 1, we have a^Al from which we obtain two conjugate values of z/, each of which satisfies the proposed; and consequently (ch. 6. art. 20.) the complete primitive is their sum ; or - V 1 _ ^^""^^ ?L^m (/2+l)(w + 2)(2w+3X2;z + 4X3w+5)(3«+6) ^ "^ ■^ "^ i "^ ~ (w + 2)(72+3)+(/i + 2)(7J+3)(2w+4)(2wT5) ~(;2+2)(w+3)(2w+4)(2w + 5)(3«+6)(3w+7K 3 * There is no descending series. Cor. If the proposed equation be of the third order, we shall have a(a — l)(a— 2)a = ; and making three suppo- sitions with respect to a, a complete value of 3/ may be had containing three arbitrary constants. 9. The first of the above series fails when either w + 2 =0 or m + 2i — 1 = 0, i. e. when w = — 2 or = : — ; and the second, when n = — 2 or = : — ; and conse- quently they both fail only when n = — % In this case the equation is d-y + ax~^ydx^ = 0, which is homogeneous, and may be integrated as in ch. 6. 30, ex. 1 ; or in the following manner. Substitute y — x"" \ then we have a(a — 1) 4- « = 0. Let the roots of this be a and a', then «/' = x"" and y'' ■=x"'\ wherefore the complete primitive is^/ = kx"" -f a'o;*, where 1 I- Vl-4a , , 1— sfl—^a ^ = ^ and a'= ^ . If a > i^; let a = 4 + /3 J^\, a' = i - /3 V - 1; then 364 FLUXIONAL EGIUATIONS CHAP. IX. But a^^-^^ = i'^'^^-^j^ cos.iS/^+ v- lsin.|3/a:? ^-/3V-l =e-^^-/3v/-A= cos.filx- V— lsin./3Z^5 (Vol. i. ch. 2. 36.) ; whence ^ = ^^{(a + A!)cos.(ilx + (a — a')a/ — Isin./S^Ar • • • = cj7*(sin./3Za7 + c')} where /3 = ^ . If « = i-, the roots of the quadratick are equal and /3= 0; whence y — ^=^(a + a') = c^^, which is only a particular primitive. To obtain the complete primitive, proceeding as in ch. 6. 9S. cor. 2, substitute ^= tJy', where y = ;r^; then the proposed becomes v(d\y^ -\-ax~^y^dx'^)-{-y^d^v-\-2dvdy^ = 0; wherefore y^d^v + 2dvdy' = 0, or -7- + -^ = ; and integrating, ddx y^'^dv = ddx or (Zu = ; and again integrating, v = c'lx ; or the required primitive is i_ I I y = cx"^ 4- c'x^lx = x^(c + c'/o;). 10. When n = — — : — = — 2/ + -7-, one of the series of art. 8 fails, viz. the first when the sign is + and the second when it is — . In either case then we obtain only a particular value of j/ ; and if from this we deduce the com- plete value by substituting as before y = vy\ the result will be V = c[/* -J, and consequently ^ = «/' < ^+^7* -7 \ • This value of^ is under a very inconvenient form, inas- dx much 3ls f—T contains a series in the denominator. Euler, y to obviate this difficulty, makes use of a very peculiar trans- formation. He observed that one of the series which ex- press the value of y fails whenever the primitive contains lx\ and he attributes the failure to the property of the function Zjt, that its fluxional coefficients become infinite when x = 0. He therefore substitutes yzzp + glx, p and q CHAP. 11^^ BY SERIES. * 365 being two indeterminate quantities. By certain assump- tions, p and q and consequently the value of y, may be determined, as will be seen in the next article. In order to introduce a constant, we may substitute 2/ = p + ^7 + (l^x =. p -\- ql.cx. 11. Required to integrate d"y + ax'^ydx" =. when 2e + l Substitute j/ = p -\- cq + qlx :=: p + qlcx ; then the pro- , , , 9Axdq qdx"^ , posed becomes d-p ^ — -— y + apx^'dx^ .... X X + {d"q + ax''qdx^)l.cx = 0. The terms of this equation which involve Ix may be as- sumed to destroy each other; and consequently we have d'q i-ax''gda;^=0(ly, andg + |- ^--^^+apx"=0 (2). Of these (1) is of the same form as the proposed; and consequently we can find a particular value of q. Sub- stituting this in (2), and integrating it in a series, we shall find p and therefore «/ in a series in terms of x, Ex.1. ^i _ b' - c'^ -...(= ^ ' ^^'^^ g^^^«' ^^ ^ ^^ aAOT-^ + aB + flc^ -f . . . 3 assumed = 0, a' + «a = ; 2c + Sb' + as = ; 2.3d + 5c' + ac = ; . . . or S66 ♦ FLTTXIONAL EaUATlONS CHAP. IX. A = — — ; and b is not determined ; a 3b' «B 3a a' "1.2""*" P.22 «B 1.2 5c' "2.3 ■""" 1^.22.3* ao, 7d' 3.4 ~ ' P.22.32.4. flD 9e' «E 9«*a' flE ^ - 20 " 4.5 ~ 1^2^.32.42.52 4.5 In this series we may suppose b =: ; because the term Bx, which we shall neglect by this supposition, may be in- corporated into that part of the value ofy which is repre- sented by cq. Having thus found the values of p and q, y will result from substituting them in 2/ = p -|- qlxx^ which contains two arbitrary constants a' and c. _, r 1 3« 14«2 -J ThuSy=A'|--+:f^,~p;^:r^-...| If B remains undetermined, we must substitute y = p + qlx ; and the resulting value of y will be the same as before, making b = a'/c. Ex. 2. d^y + ayx~^dx'^ = 0. The series which does not fail is the second. Assume 9= A'l^-j^^ +12^3^-1:2^3^4"^ +-7 or = a'^ + b'^^ + c'a;2 + • • • In order to integrate in a series (2), which is d^p ,- ^ dq q ap "TT + — J ^+-^ =0, assume dx"^ ^ X dx ^2^3 ' x"^ p = A 4- B^"" + co; + do;* H- • • • ; then CHAP. IX, BY SEEIES. B X * + O.ar-1 3d - "^ + 2e + 15f ^ 4 ^ + + 2(a'j:- , 3b' ■ "^ + 2c'+ 5d' i- + - A'a;-1 - b'*- "■^- c'- d'^^ + •\-aKx "^ +ajix~^ -\- acx *+ aD-t- flEa;^ -|- This gives cta — 7- = ; ab + a' = ; 3d ac + 2b' + ^ = 0; aD + 3c' + 2e = 0; «E + 4d' + — r- = ; • • • or 4 b a' '^ A = T- = — r— , ; c is undetermined ; 4a 4a2 D = 8b' - 1.3" 4«c "1.3 ■ = + 32flA' 12.3^ 4ac 1.3 E = 12c' 2.4 4«D 2.4 = - 12.16aV 1.3.22.42 4aD "2.4 F = 3.5 4«E 3.5 = + 16.64fl^A' 1.3.2.4.32.52 4«E 3.5 367 =0 We may suppose c either to remain undetermined or to = ; in the former case y = p -\- qlx\ and in the latter, y = p + ql,cx; and in either case there will result the same value of?/ containing two arbitrary constants. 2. Ex. 3. d^y + ai/x "^dx = 0. Here the series which does not fail is the first. Assume then g = aM - Ys^ "^ 132 4 N = a + Ba7 ~^ + ca?~* + • • • In order to integrate (2), which, is 368 FLUXIONAL EQUATIONS "<#' t^z assume J9 = a'x + b'^^ -\- c' + d'x * + . • -ir ' + 6^- —3 2. — A^~^ — BX '^— CX~^ — • • + aA'x~'^ +aB'^-2 + ac'a;~'^+ aD'x~^ + • • which gives b' 3d' aa' — -J- = ; as' — A = ; ac' - 2b + -j- = ; «d' -3c + 2f' = 0; . • . . « 1 , a , b' A Suppose c' = 0; then b' = — ; a'=: -r- = 7-1;; 2.4b 8.4aA , f d' = -y-^ = - j;^ . . . ; where 3/ = /? 4- V»+^ + A'l^-zrT-dr + + 1 (w + ^)(2m + 2)(2w+3)(3m-{-3)(3w.+4) Now one of these series will always terminate when 2i , , 4>i . - m = — ^. . , , 2. ^. when n —— ^rr—r. ; and consequently a 2z±l' 2z±l ^ J conjugate value of g or of 3/ may be obtained in finite terms; and having obtained this, the complete value of?/ may be found as in ch. 6. 28. cor. 2. The proposed form is reducible to Riccati's by substi- tuting 7/= ^^'^j when it becomes dz -^ z^dx = —aw^'dx. Euler has integrated the proposed by changing its form into c?*z/ — ¥yx"dx = 0, and developing ^f in a descending series, as in the next article. 13. Required to integrate d'^y — Wyx^'dx = when _ 42 By the last article, if we substitute y — ^*q, where t = — -— r and 2m — n, the integration of the proposed is reduced to that of d'^q + ^bx'^dqdx + mbx'^-^qdx'^ = 0. To develope 5^ in a descending- series, assume q = Ax"^ + Bx^~^ + cx'^~'^^ + • • • ; then, arranging the terms in a reversed order, +2^>(aA + (a-i8)B +(a-2/3)c +••) + a(a-l)A + (a-/3)(a-i3-l)B +•.• = 0. Suppose A to be undetermined ; then Qbx + mb = 0, or a= g-; a — |3 + m~l=a — 2, or (5 = l+m=l—^a,; and VOL. II. B B 370 FLUXIONAL EQUATIONS CHAP. IX. the developement of q is of the form qz= ax'' -\- BX^'^-^ + cx^"-^ + Bx'^"-"^ + • • • To determine b, c, d, • • • we have -2^/3B + a(a-l)A:=0; ~46/3c + (a-/3)(a-/3-l)B=0;... a(a-l) (3a-l)(3a-2) 2/36 4/36 {5a.-mBQL-^) (7a-3)(7a--4) ° = Wb ' ' ^ ^ 8^^ ' • • • ^ ^^"^^ ' which will always terminate when (2i ± l)a — i = 0, or i . a = rr-— 7 , i, 6. when w, which — 2m — — 4a, 2i±\' These being substituted, there will result a(a-l)(3a-l)(3a-0) ^ ^ . "^ 8(2a-l)«62 "^ ••*5' But we have only one arbitrary constant, viz. a; and to introduce another, we observe that the proposed may be integrated on the supposition either of h positive or of b negative. Let A, as before, correspond to + 6 ; a', to — 6 ; then, when b is negative. l'I x' a(a-l) 1 ^ %^a-\)b a(a-l)(3a-l)(3a-2) ,_^ ) 8(2a- 1)^62 ^ -t---.| Let A.p = sum of all the terms in which 6's exponent^ is even ( . A.6a = sum of all the terms in which 6's exponent L ' is odd 3 then the conjugate values of g' are g = a(p + bo) and ^ = a'(p — boi) ; and the complete primitive is y = p(a^^ + a'^-") + 6a(Ae^* - a'^-^O where t =— XT' Cor. L When a in the last article is positive, b contains \/—\, and y may be expressed as a circular function of .a:'. CHAP. IX. BY SERIES. 371 Forjet b = fi ^/"'^ ; then (vol. i. ch. 2, 36) g-^V-l=: COS./325 - v/- Isin. (it)" J/ = P[ (a + a')cos I3t + (a - a') v /"^s in /3zf J . . . . + ial(A ~ a')cos./5^ + (a + a') v— lsin#| = p{ccos./Sif + c'sin/3^}. f /3q jc'cos./3# -■ csm.(it\, where c = a + a' and c' = (a — a') a/ — 1 = cv(sm.fit + c') + c/3Q(cos./32f + c]K where A + a' = csin.c' and a — a\/ — 1 = ccos.g'. Cor. % Riccati's form is integrable by this method. For let the form be dz + z^'dx — ax^'dx ; then sub- stituting % —~^ ^i^d differentiating, it becomes d^y - ayafdx^ = 0. Ex. 1. d'^y - bydx'' = 0. Here n = .'. a = — m ,'. t = x; and q = a, q^ = a' .'. y = A^'^^ + a'^-^^. If 52 _. _ ^^2^ 2/ = ccos.a^ 4- c'sin.ao: = csin.(«j; + c'). Ex. 2. d'-y — bHjx-^dx'- = 0. Here m =— 2; t and a = I .', q = ax and X ^ b b_ _6 _b_ y = e ^Ax + e^^x — x(A!ex -\- Ae •^). If 62 = _ fjfi^ y — Qx%vci.{ax~^ + d). Ex. 3. eZ^y - ¥yx'~^dx'' = 0. Here w = — ^ ; ^ = 3a;^ and a = -f •*• ^ =a) ^^- gjf and^' = a'> ^^ + 3^ f and I 1 1 y = a?"^(A^^' + A'^-^0-gT(A^' - A'e-"), where t = 3^^. If b''z=z-a\y = x { (a + a')cos.«^ f (a - a') ^/^-[svn.at \ — 5 (a — A')cos.a^ + (a + a') ^/ — lsin.«^ ^ • • • 3fl V — 1 1 3 1.1 = x^ (ccosMt ■\- c^sin.at) + ^(c'cos.a^ — csin.a^) I 1 Q 1 = ca:^sin.(3a^^ + c') + «-cos.(3«a:^ f d). BB^ 372; FLUXIONAL EQUATIONS CHAP. IX. Ex. 4. dry — ¥yx '^dx^ = 0. Here tw = — * . ^ _ 5^5 and a = f .*. i « = ^ 1 "^ - ^ + 5i3^ I ••• ^ = r^ + 55j-« f K'+ *'«-" — -^ (a^' -- a'^-^'), where t = 5.r . r - 3 > i. If 52 __ a^^y _ c> x^ ^—- [sin.(5«a;^ + c') . . . 3ca7^ ^^ 4. ,^ + -^ — cos.{5ax^ + c). oft Ex. 5. c?^j/ + «^a;""^c?a72 = 0. To integrate this without referring to any formula, sub- stitute y = e-''P^^q ; then it becomes 8 d'^q + 2pdqdx + qdpdx -\- {p^ + ax '^)qdx^ = 0. Suppose ^2 _j_^ -J. 3^=0,orp=x ^^/ — a=bx ^, by substitution ; then -j^ 4- 2p^ + q-r- = 0, or c?^2 d^ 3 Assume q = Aa?* + B.r*"'^ + co?*"^^ +-"then, reversing the order, - 3 ^ 3 ^ 3 ^ + 26aA +25(a-/3)B +26(a-2/3)c +.., + a(a_l)A4-(a~/3)(a-/3-l)B +... = 0. To satisfy this, we have a undetermined; a=-|; /3= — ^. Also B =: ^ and c = d = . . . = ; wherefore This furnishes only a conjugate value ; but since ft^z: — «, CHAP. IX. BY SERIES. 373 q^— Al we have b "=. + V •— a, and consequently another value is Now j!? •=. ± hx "^ ,\Jpdx = + Sbx ^ = by substi- tution Tbt; then i/=x^{Ae-^' + A'e^')-\-^{Ae-** - a'c^O* where b = \/ — a and t =■ S^r"""^. When a is positive, this is under an irrational form ; and may be expressed as a. circular function, as in the former examples. We shall give only one more of Euler's examples, which is the developement of the form of the article in a series in terms of i. 4/ Ex. 6. d^i/ — b^x^'^dx^ = where n = — a-^y 1 __L_ Substitute ^= e^*q where t = — „._, x 2t-i, or dt ai -r- zz X 2i-i ; then dx d^q ^, __2Lc?o 9,Hx ^±11 — i + ^bx 2i-i-^ - _ — - ii-iq zz 0. dx^ dx ^i—l ^ Assume q:=: Ax^i-^ + Bjai-i + cj?2i-i + . . .; then da i -r± i-\-l _i±l i-\-2 _i=i -7^= ■—: — rAa: 2i-i-f.-- — -b^ 2»-i + Tp; — -ex 2'-» ....'. f/^ 2z-l 2^ — 1 22 — 1 AX 2j— 1_ — -T^x 2i— 1— — rC^ 2'— 1... >2-r 22-1 9.1-1 2bi 26(i+l) , 26(i + 2) f-^rr— 1-B + -ST-l-c !i-l ' 2i-l ' 2i-l = 0. ^(^•-l) _ 0+1)0- 2) (2i-l)2^ (2i-l)" "" 2(2 -1)a 26b Whence -p3yy:, + g^—f = 0; 0>1)0-^ )B , 46c ■-(27ir[)r- + 2-11=0, ...or Sl4f FLUXIONAL EQUATIONS CHAP. IX. 2{^i-\)b ' "- ~ 4;^/ -1)6 ' 6(2i-l)b ^'"' 2.4.66^^ ...^ ana i(,._l)(,-._4)(i^-^9)(i-4) >.^^.^^,^.. ^ 2.4.6.8Z>*^4 + . . . ^ iA^ H- A ^ ) I 2bt ^ 2A.6bH' +...^(A^ A^;. 42 If w =— p-^i> the same series obtains, making i ne- gative; and t = (2 j+ 1)^-2^71. It may be shown, as in the former examples, that the pri- mitive of d-^i/ + a'^i/a: ^i—^doc'^ = is ^^,,-.fl_^-(^--l)(^-g) •^ ( 2.4«^^'^ ' 2(7'^-l)(/2-4)(22-9)(z-4) ) . , The same series will give the primitive of U ] 1 dy -\- a^yx ^i+^dx^ = 0; if iJ= —. — tx^^^, and i be made negative. Euler has integrated by this method other forms which are more complicated ; but he observes, that it is a very in- convenient method unless the example is linear, i. e. of the form d'^y -\- M.dx'^dy + i^iydx'^ — xdxK The following form, of which we shall give but a brief perfect account, he has fully discussed. 14. Required to integrate in a series x\a + bx'')d'y -\- x[c + ex'')dxdy + (/+ gx'')ydx"- = 0. Substituting ^ = ^% we have CHAP. IX. BY SERIES. 375 or «%a+bz)d^y+^(^^+'^d^d +(^Z+^)yB +... + ^A + gB +... = This is satisfied by supposing a indeterminate, and the value of a obtained from the quadratick equation a(a- \)a + ac+/=0 (1). Let the roots of (1) be a and a! ; and let a and a' be the corresponding values of a : then, by means of this series, two conjugate values of j/, and consequently its complete value, may be found as befoce. The equations for determining b, c, . . . are {(a + l)a«4 (a+l)c+/|B+{a(a— l)& + ae+gJA=0 {(a+2)(a + l)«f(a + 2)cH-/}c + [ (a + l)a6 + (a+ l>+g j B = {(a+3)(a+2)«4(a-i-3)c+/iD ^ +i(« + 2)(a+l)6+(a + 2)e+g}c = . . . ; which will terminate when (a + 0(a + i - \)b + (a + i)^ + ^ = 0. If a descending series be assumed for y^ y = Aocf" + Bx"'~^ + • • •> the equation for determining a is a(a - 1)6 + ae + ^' = (2). 15. There are two cases in which the first solution gives only a particular primitive, viz. when a = 0; and when 4iaf = (a — c)-, in which latter case the two series become equal. The second solution fails when & = and when ^hg = (b -- e)^. If both solutions fail, or if the difference between « and a' is divisible by n, Euler substitutes y = p -{■ ql.cx. FLUXIONAL EQUATIONS CHAP. IX. If, when « = 0, we have also c = 0, the first solution fails to give even a particular primitive. Ex. 1. xd'^y + dxdy + gx^-^ydx"^ = 0, or x^'d-y 4" xdxdy + gx^ydx'^ — 0. Here a = \; b = 0-, c = \, e = and/= : .'. a = a!. Substitute y — p ■{■ ql.cx ; then xH^p + xdxdp + ^xdxdq + gx'^pdx"^ + (x-d-q + xdxdq -\- gx^qdx'^l.cx — ; which divides into x'^d^p + xdxdp + ^xdxdq -\- gx^'pdx'^ =i (1) af^d'q + xdxdq + gx"qdx'^ = (2). Assume q = \ + Bx^ + co:®" + bx^"- +...), ^ p = a' + b'^" + c'^^« + d'^3« + ... I ' ana irom (2) it will result that B =- 1^; c =- g= + -g^r, To integrate (1), we may assume a' = 0, because it is included in cq and we have b'= + -^: c' = — , ~, : f 2>S' 62-^ 222-3 ^ 3^ = A J ^^^'- rf--.^'" + r^^r^-^" 100^ ,^''" + 1.8.27.64«9"' "^ • • • I (^ 71^ 1.4w* 1.4.9«^ ^1.4.9.16?^« •••5- Ex. 2. ^(1 - a;2)d2y — (1 + x^)dxdy -f a:y<^^ = 0. Substitute «/ = p -\- ql.cx; then CHAP. IX. BY SERIES. 377 d^ dp -s To integrate (2), when reduced to the form of the article, it is x2(l — x'')d^q — ^•(1 + x'')dqdx + a;2^Jj?^ = 0. Since n here = 2, assume ^ = Aa?" + B^"+^ + cx"-^^ H then a(a— 1)a^* + (a + 2)(a+l)B^"+^ + (a + 4)(a + 3)c:r'*+* + - a(a-l)Aa7'^+2-(a+2)(a+l)B^'^+^- — akx"^ AO; .+2 2^'^+*- = 0. This gives (a^ — 2a)A = ; or A is indeterminate and a = 0, a' = 2. Assume then g' = a^^ + Ba?* + cx^ + . . . i^ + b'^2 ^ c'^* + 1.3 2.4b - 1.3a= 0; 4.6c = S.Sb; . . . or b = ^a; 1.3^.5 _ 1.3^.5^.7 ^ ""2.4^.6^' ^^~ 2.4^.6^.8^' ••• To determine p, we have from (1) Sfi'a? + 4.3c'.r3 + . - 2b' -2b' - 4^ - . - 2b' - . + a' + b' + . + 4a + 8b + . - 4a -. -2a -2b -. ^ 1 q Q 7 a' = — 2a • b' — • r' — — r' a. — -A A- x:a,b_u,c- g^B + g^^^A , 3.5 , 2.21 ^ = 1:6^ +¥:e^^' = ; which gives the following : 378 FLUXIONAL EaUATIONS CHAP. IX. , 5.7 , 2.43 ^ = 6:8° + 6^^" • • • Having thus determined the values o£p and q, they are to be substituted m i/ = p ~\- ql.cx. Euler has shown that this method will always succeed when the difference between the values a and a! is divisible byw. Cor. If the roots of the quadratick contain V — 1, let a = /3=i=7\/— 1; then, substituting for x'^'^^^^ which = o^^+^'^iyV-^ the expression afi'^^^ (co^.ylx ± ^/ — Isin.y/^), the V— 1 will disappear. If this fails to give a complete primitive, we may sub- stitute y = pcos.klx + qmx.klx; and deducing the values of dy d~y -T- and of -T-^, and substituting these in the proposed, it will divide into two equations, if we make the coefficients of cos.klx and of sin.klx each =0. 16. When the law of the exponents cannot be ascertained from any arrangement of the terms, which is the case when the equation is of a higher degree than the first, recourse must be had to Newton's method of successive substi- tutions. To introduce an arbitrary constant, let a and b be two corresponding values of the variables; then by Euler's Theorem (vol. i. ch. 4. 45.) «/ = ^ + (JP)(^ - «) + (2') ^^ +. . • where (p), (q),... denote the fluxional coefficients corresponding to a; = a or 2/ = &. Let X - a == h; then 3/ == ^> + (;?) y + (y)— + . . . ; and if 7i be supposed small, the first approximate value is 3/ = ^ + (P)y Let this = y, the corresponding value of a being aJ or CL 4- li. Considering these as new values of y and x, a nearer approximation may be made ; and by repeating the process, we find a value of ?/ corresponding to any value of X, CHAP. IX. INTEGRATED BY APPROXIMATION. 379 If the equation is of the second order, let it be under the form q —f{^,;f/ip)i then, in order to find the com- plete primitive, we must not only know corresponding values of .r and «/ as before, viz. x — a^y = h\ but also another condition must be added, viz. that the corresponding value of p is^ = c. This agrees with what has already appeared, that when the solution of a problem depends upon an equation of the second order, we must have two conditions in order to de- termine it: 1st, that when x — a^ y = h% and 2dly, that when X — a^p — c. Generally, if the equation is of the wth order, the n — \ first fluxional coefficients must be considered as arbitrary constants. This method of approximation, which is Euler's, is in- convenient in p-actice, not only on account of the length of the calculations, but also because the errors of each opera- tion influence the succeeding, which is not the case in the method explained ch. 2, art. 3, where the operations are independent of each other. Lagrange therefore proposes to develope ?/ in a continued fraction, as in art. 4. When this can be accomplished, an approximation is ob- tained which never fails, and which in general is sufficiently rapid ; the equations, however, for determiningAa?^, b^^, cr^ . . ., after the two or three first approximations, become so complicated, that we shall subjoin but few examples. Cor. This method, it is obvious, furnishes the means of representing any proposed function in a continued fraction ; for we have only to differentiate the function, and then in- tegrate the resulting equation in a continued fraction. 17. EXAMPLES. Ex. 1. mydx 4(1+ x)dij = 0. Let the first value of 2/ be «/ = a^*; then, by substitution, {m + a)Kx'^ + aKx'^~^ — 0, or {m + a)KX + aA = 0, which, when a; = — , becomes aA -- ; which may be satis- fied by a = 0, A remaining indeterminate. Let y = ; then, by substitution, 1 + BX'^ W (1 + Bx^) - (1 + x)^Bx^-'^ = 0, or FLUXIONAL EQUATIONS CHAP. IX. m — jSea?^"^ + (m — /3)Ba;^ = 0, which is satisfied by (3 = 1, B = m. Next let 1/ = ; then, by substitution, it will appear that y = 1, c = ^ ; and by repeating the pro- cess, we shall find A m—\ X 1 2 m-\-\ X 3 2 m—2 X 3 2 m + 2 X 5 2 m—3 X 5 2 1 + . . . Integrating the example, (1 + x)'"i/ = a, or A , mx (1 + xY= — = 1 + 1 1 1 2 1+.. * Ex. 2. dx-(l + x^)dy = 0, or (1+^")^ -1=0. Lety = kx"", then (1 4- a;")aAa;"~^ = .*. a = 1, A = 1. CHAP. IX. INTEGRATED BY APPROXIMATION. 381 (1 + x-){l + (1 - /3)B^) - (1 + Bx^)"" = 0, or - (1 + i3)B/ - B2^2/3 + (1 _ ^)b^+« + ^« = .-. /3 = w, B = -j ; and repeating the process, a; I/ = - 1+ "" + 1 1 + (7i-hl)(2w+l) ^ (n+1)^^'^ (2w + 1X3/1 + 1) "^(37i+l)(4w + l) "^ (4w+l)(5n + l) 1+- dx Cor, Since 3/ =y ^ ^ , we have, 1 + ^ ifw= 1, Z(l +^) = - ^ i + ra ^x ^ "^ 3:2 207 ^+5:1 307 1 + ... S82 PARTIAL FLUXIONAL EQtTATIONS OHAP. IX. X if 71 = 2, tan.-* a? = ^-^^:5 9^^ 1 4- ''"' 7.9 1+ ... 18, There are other methods of integrating by ap- proximation total equations, which we shall entirely omit : one is Lagrange's method of the variation of the para- meters, of which we have seen the use in ch. 6, art. ^32 ; another is Newton's method of successive substitution com- bined with the known processes of integrating equations of the first degree. It is this which is so much used in the solution of the problem of the three bodies ; but it cannot be rendered intelligible to a student who is wholly ignorant of the science of Physical Astronomy. Omitting these, as well as the geometrical construction of fluxional equations, which is also a method of approximation, we shall conclude this part of the work with a few examples of integrating partial equations by series. 19. Euler observed of several partial equations of the second order which are not integrable, that an infinite number of solutions may be found belonging to them. The following example, for instance, though simple in its form, is unintegrable. Ex. J— 7- - - az. dxdy - Substituting ^ = xef^^Y^ where a and a are indeterminate and Y = fuf it becomes a 3- = a y or — = — ^ ; whence, in- *^-^ dy Y a — ax-f- — tegratmg, y = e « and z = Ae « . Since a and « remain undetermined, it is manifest that CHAP. IX. INTEGRATED BY SERIES. the primitive of the proposed is „+"* ^x+f y.^'^l ^ , . %= he '^ + Be f^ + ce y + • • • where the series may be continued to any number oF terms ad libitum. Thus, if three terms be taken, and it be differentiated, it will appear that this value of 2; satisfies the example. Cor. We may express 2; as a circular function of x and 2/ by supposing a to contain V — 1 ; or thus : Suppose — aj3 = a, which leaves one of the quantities a and ^ indeterminate; then, it appears from differentiation, that z = Asin.(ad7 + /Be/) and z — Bcos.(a^ + %): each satisfies the example, Suppose a to have an infinite number of values, a, a', a" • • • ; then the corresponding values of /3 are a a a , , , r, — —r — . . . and we have a a' ' a ' z = Asin ( a^ — — ) + A'sin.fa'^ f j 4- A"sill.(a"^ - ^V . . . + Bcos.fa^ — -)-\- B'cos/a'a; j\ + c"coS.ra"^ "" ^)^ >•, where A, a', • • • B, b', • • . and a, a', • * • are all indeterminate. SO. Partial equations may be integrated in series by means of Maclaurin's theorem. For by fhat theorem, ^ = z+^Y + ^2i;2+*" respectively when x = 0, and consequently are functions which contain only 3/ and constant quantities. First, let the proposed equation be of the ^rst order ; and let it be reduced to the form -^ = fIx, 1/, z, ^j ; then by successive differentiation we may obtain the values of — , ^3 . . . ; and consequently we may find 384 PARTIAL FLUXIONAL EQUATIONS CHAP. IX. -p, -j-^5 ~r-j ... in terms of z, which may be considered as an arbitrary function of 3/. If any of the coefficients become infinite when x = 0, the failure of the theorem is to be prevented by substituting x — a for x; or, which is the same thing, substituting a; + « for x, and continuing to develope the series by the powers of x. Next, let the proposed be of the second order ; and let it , TIP d^^ f d^ dz d"z d^z \ be under the form ^ = f(^^, y, z, ^, ^, ^„ ^ j. Adopting the same notation as before, let z = 0j/ ; then d^ f J d^^ ti A 1 • dz , . -7- = (p'l/, and -7-^ = 0"?/. Also assume -j- an arbitrary dz d^z d^z function of y, or ^ = ^y ; then ^-^ = ^^Z^, ; or ^ may be expressed in terms of ^ + b(^ + «/)''+ V^ + c(x + j/)^+Va? WA(^ +^)>.r+WB(a7 -!-«/)*+ V^ + '^^(^+«/)''"^V''^ + — + 2maA + 2m(a+l)B+ 2?w(a + 2)c + ... + 7rtA+ WB + -"^=0 + a(a-l)A + (a+l)aB+ (a + 2)(a+ l)c + . + aA+ (a-J-l)B + - This is satisfied by a, indeterminate ; and l-^m /I ] I n + 2ma + a^-a=z0, or a = — - — ±\/ X""^ -\-m-—n; and the remaining coefficients give VI -\- a m + a + r a; d = — _ m + a+1 _ m + « + 3 ^* which the law is sufficiently manifest. VOL. II. c c 386 PARTIAL FLUXIONAL EQUATIONS CHAP. IX. This series will terminate when m -{■ a -\- i = 0, i. e. when ~ -\- i ± \/~ — m + w^ — n = 0; which can only happen when i — m -\- rn^ — w is a square. Having found a similar series for z in terms of \p2/, the value of z will be under the form z = a{x -hj/T(x + ypy). Cor. 2, If a = — m — 1, i. e. if w = (w + \)[m — S), (x 4- v\~^ ^ = (^ + y)-''-\<^oc + ^y) - 1^ {^<^x + ^'2/). (Ch. 8. 51. cor.) Give these arbitrary functions any particular form, such as ^x = ax^ and \py = b?/'; and it will appear by dif- ferentiation, that the resulting values of z satisfy the ex- ample. ^ ^ d'v d^v d'v To develope v, which is a function of {x, y, ;s), in a series ascending by the powers of 2, assume t; = V + v's + v"22 4- v'"^^ -{-... where v, v', v" • • • are functions of {x, y) ; then d~v _ d^y d'y' d^y" ^ ^ "^ ^^ "^ d^^ "^ "^^' "^ d'v d^y d'y' d^y" , .. ^ ,-. r f=di^^di^'-' W'' + • • • >^-dbysubst.tution, ^ = 1 .2v"-|-2.3v'":2 + 3.4viv2 + d^y d'y , „ r c/^y' c?^v' ^ ^ ,„ > l_^d^^ d^v I c ay a^v > CHAP. IX. ' INTEGRATED BY SERIES. 387 ^'"^ ~ 3.4 i dx^ "^ di/^ S "1.2.3.4 ^ dx^'^ dxHy'^'^l^ \ • • • which are the required coefficients of z"^, ^^, • • • in terms of the two arbitrary functions v and v'. By the same method, as in ex. 1, Euler shows that fl2 ^ d'^% 2m d;2 _ d^^b^d^ ^^ ^ ^ ^^ are mtegrable when mis dy^ a^ dx^ y dy an integer, or 2m an even number. d^z d^% He integrates the form -r-^ = x-7-7 by substituting "K-d^^V z = e"^t;, where v =fx'^ then aH = —fi:' Again, substi- tuting V = e-^P^"", the equation is reduced to a^dx = dp -{- p'^dx ; which is Riccati's form. These, with other forms which Euler has integrated in tome iii. ch. 10, we shall pass over to notice a peculiarity which Poisson ob- serves belongs to all partial equations in which the highest fluxional coefficient of z with respect to one of its variables is of a higher order than with respect to the other. These cases form an exception to the proposition of art. 20, that the canonical primitive of an equation of the wth order con- tains n arbitrary functions. ^r» T^ • , . . . d^z dz 2». Required to integrate m a series -7-^ = -7-. To develope 2 in a series ascending by the powers of ar, assume 2 = y -f- y'^ + y^^x" + y%^ + • • • then we have 1.2y" + 2.3Y'"a7 + 3.4Yi^'a:2 -1- • dY dY' dY" S- = ; wherefore dy dy^ dy ^ -'■h- /f _ J_ ^ ,„__!_ dY' .^ _ J_ dY" ^ ~ 1.2 dt/'' ^ - 2.3 dy'' ^'^ ~ 3.4 dy '' ' ' ' ""^ ""'^ the coefficients may be expressed in terms of y and y'. Let Y = (py and y' = 4^^ ; then z = x* d'x^ 1.2.3.4 ' 1.2.3.4.5 d'x^ If- i , DX^ DOT ^ = X-^ + y x"-^ + fgX'^^'^ + r&3^'''^ + • • • ; which coin- cides with the second solution. Generally, if an equation is of the wth order with respect to one of its fluxional coefficients; and of the wth, with respect to the other, where n < m, its canonical primitive may be always reduced to a form which shall contain only n arbitrary functions. (Vid. Poisson. Ecole Poly technique, cah. 13.) It appears then from this result that the arbitrary func- tions in the primitives of partial equations are not in all CHAP. IX. PRAXIS. respects analogous to the arbitrary constants in total equations. 23. PRAXIS. 1. dy + xdy — mydx = 2. 2^(dr + (fe/)=&.-.j/=2^a;'--j^ + -g^- +g- L 1- ^X 2^.r* 2x'^ l7 B C . D where A = a^ + fi^. ^ = 2(« + a^b + 6"^); c = 2(1 + 2ab -h a" + 4(x + y) + \p{x — y) [ END OF PART SECOND. THE CALCULUS OF VARIATIONS. PART THIRD. CHAPTER I. 1. In that branch of the fluxional calculus which, for distinction, is called the calculus of variations, the form of the function as well as its principal variable is made to change. It includes a large class of problems, in which the nature of the function is to be determined from the conditions of the question. We shall first establish the general prin- ciples of the science, and then apply them to the solution of a few of these problems. 2. Def, The variation of a function is the limit of its increment arising solely from a change in the form of the function. Thus; if y = '; then i/ + ^7/ = <^x in the limit; and B.?/% which = the limit of ( {y-\-^yy—y'')i —ny^~^ly. CHAP. I. OF FUNCTIONS. ^95 Hence, generally, in taking the variation of any function of^, the coefficient of ly is the same as the fluxional co- efficient of dy. Cor. If u = f(^, y, py q, , . .) and we have du = udx + Nr/j/ + rdp -\- adq + . . . then du = M^x + N^y + F^p -f a^q + . . . . . • • . du du^ du^ 7. The variation of the fluxion of a function is equal to the fluxion of its variation ; or My = dly. For, let y = (i)x; then

- , .... — fdx ] Nw + -1 1- -7—^ + • • • J ; and by substitution 1 m Tdou Q.d'o (1) lfydx=vix +fda:^i^w + ^ + 1^ + . .. J (2). Butyp6?a; = rw —fu)dp ff^d^uo = Qc/cy —fdoiduj = Q,doj — vodo, -{■ foud^o. fsid^u) = Rfi^^o; ^fdRd^vo = nd^cv - dRdw -^fdoud'^R = Rd^cv — dRdoo + wd'R —fwd^R Generally /tc?«w = tc/'^-^w — d-rd'^-^uj + • • • ±/w^"t, where t = -77 and t is the nth fluxional coefficient. at CHAP. I. OF A FUNCTION. 401 Substituting these values in (2), and collecting quantities involving like symbols, there will result, ^ C da o'R a's ) {Jr d^y where l expresses the sum of all the terms of (a) except the last, v^^ being transposed to the other side of the equation. Cor. 2. If we suppose uj - 0, there results IJvdoc — ylx ; which ought to be the result, because in this case ^y ~ P^^ — ^5 o*' W6 establish the same relation between ^y and ^x as exists between dy and dx ; and consequently ywdx is the same as d/\dx or \dx. Cor. 3. If we replace w by its value 3j/ — plx, it will appear that the whole variation consists of two parts : the one wholly due to the variation of x\ the other, to the variation of «/. Cor» 4. If we suppose x to have no variation, w = ^y ; and the whole variation is found by substituting ly, dB^, ... for cv, duo, ... Cor. 5. If v contain three variables x, ?/, z and their fluxional coefficients p, q, . . . p\ ^', . . . ; then, substituting dw , dw d\ n' = y-, P — -T-,, Q, — -.— , . . . and also w ~ Iz — plx^ we may find the variation by adding to the ex]:)ression already obtained the part which follows vgj;; is, p, a, . . . and w being accented. Cor. 6. If V contain any number of variables, and be VOL. II. D D 403 VARIATIONS. CHAP. I. under the form v =F(ar, z/, z, v, ... p, q..., p\ q', ... p", 9",...), the variation may be found by similar additions. It is observable that m, which is the fluxional coefficient of V when differentiated with respect to x, does not enter into (a) : the cause of this will be seen at art. 25. 18. It is worthy of remark, that in the formula (a) the coefficients of the parts under the sign of integration, when made = 0, are the very equations of condition deduced Part 2, chap. 7, 13, that the function \daj may be an exact fluxion. These equations may be deduced immediately from the theory of variations. For, let du = \dx; then ljNdx = \fdu = dlff\dx, an exact fluxion ; which shows, that when we have brought from under the sign of integration all the integrable quantities, the sum of the remainder must =, ; or w, w', ... contain the indeterminate quantities hx^ ly^ ^?,...; dv dv' wherefore n — -z — h ... = 0; ^' — -j h... = 0;... The other equations of condition that the function may be an exact fluxion of the wth order may be likewise de- duced by means of the theory of variations; but we shall not enter into the calculation. 19. Required to Jind l(Jvdx) when v is a function of two variables and of their jiuxional coefficients. Let v^, y^^ Py, . . . represent the values of v, 3/, p, . . , at the first limit ; Vyy, y^n p^, . . . their values at the second limit. Then S(J\dx) = yu^^i, — ^i^^i (!)• r da,,d%, ^ i V"-d^^d^~'-r" (. {da, d% •) ( This consists of two parts, which are wholly different : CHAP. I. euler's formula. 403 the first depends upon the variation of x and y at the limits; the second is included under the symbol of integration : it represents the fluent of the variation, or the variation of the fluent between the limits; and it is enclosed in brackets to denote the whole variation between the assigned limits of the function. It is manifest that the former or the de- finite part is wholly independent of the latter or the in- definite part; which, as it includes a>, i. e. the general values of ^.r and ^y, may be supposed to vary, while the other re- mains fixed and determinate. Cor, If the two limits are given as in fig. 3, where they are J7 = «, a: = ^, we have ^.r^, dy^, and ^07^^, ^z//^, each = 0, and consequently 20. In the case of a maximum or a minimum, we have the whole expression — 0, and consequently each part = 0. dv d'^Q, The part (2) cannot = unless n — j~+ j^, — • • =0 ; because w contains the indeterminate quantities dx and ^y. This is Euler's formula *. The part (1) gives an equation which contains four inde- terminate quantities ^x^, ly^ and Ix^^, ^y^. If we can eli- minate two of these, Zy^ and ^j/^^ by means of the conditions of the question, there will resuh an equation which will divide into two by making the coefficients of Ix^, ^x^ each = 0. These we shall call equations of limits. Cor. 1. If V = f(^, y, z^ p, q, - ' ' p', q\ • • •)> the in- definite part becomes * Euler obtained it by considering only the indefinite part ; but it was Lagrange who gave it its present general form by adding the definite part ; and to the honour of Euler it should be recorded, that he was the first to acknowledge the great improvement which his formula received from this addition. " Neque tamen hae partes absolutse frustra sunt inventae, sed sin- gularem praebent usum, ad quem methodus mea prior, quae tan- dv turn aequationem n — |- . . . = suppeditavit, minus est ac- commodata ; quam ob causam haec methodus illi longe est ante- ferenda." (Vid. Woodhouse, Isop. Prob. p. 90.) D D 2 404 VARIATIONS. CHAP. Since cv = ^y — ^^.r and w' = ^:r— y^o:, this is under the form Adx + B^^ + c^z = ; which, when i/ and z are in- dependent of each other, divides into a^x + b^j/ = and Adx + c^z = 0. Cor. 2, If 3^ and z are not independent, but are con- nected by an equation of condition f(^, j/, z) — 0, we may by means of it eliminate both ly and Iz from hlx -f B^j/ -I- c3a? = 0, and there will result, as before, two equations which express the relation between Ix^ ly and Iz, Similar equations may be deduced when V = F(a?, y,z,v...p,'q.,.p\ci:,,. p\ /...)• 21. In the solution of problems, if the limits are two curves, we shall represent the co-ordinates of the curve at the first limit by a;', y' ; and at the second by x'\ y : these — x^, y^ and x^^^ y^ respectively : but it does not follow that -r-i and -, are equal ; for the one marks the position of the tangent of the limiting curve ; and the other, the position of the tangent of the required curve. We shall denote —^ -^„ which belong to the limiting curves, by /?j, /?2; and ~, -r-^, which belong to the required CtX , "^11 curve, by p^^ Pi, • » • as before. 22. Required to find the equations of limits of Kfvdx) = 0. The whole equation of limits is Vi^Bxn—Y^dx, c da^ ^ - }''-d^ +-^"' + ... Represent this by ^u^ ■- ^u^ = 0. Suppose the limits to be two curves; of which the first is l' = 0; and the second, l" = 0. Let x\ y' be the co-ordinates of l'= at the first limit ; and x'', y" those of l" = at the second limit. Then, since i.' '— F{x',y') and l" =f(x",y"), we have, differentiating CHAP. I. FORMULA OF SOLUTION. 405 + -1 = «? 7 „ d " ?> ' i ' vvhich equations belong to the re- dx dy Ix^^ dd dd _ . da; '^dt/P'-^f \ dy' ^ df dL" dL" ^C^'''P^ = ^^''^P-^=d^' d^^d^P^ = n Also, since l' = F{x,f y) and l" =y(^//, y/i)j we have, taking the variation, diJ dh' ly^ dx dy Ix quired curve, for it is the only curve which varies. Hence, by elimination, ly, - p,U, = n ; and combining these with ^v^ — ^Uy = 0, we can find either dx^, ^j/^, or ^Xfj, dy„; which will give the conditions that must be ful- filled at the two limits. If the limits are independent of each other — two curves, for instance, not connected by any equation — we have ^x,, ^y^ independent of lw^p ^y^, and consequently w^ independent of u),i ; wherefore the equation divides into ^u^— and BUyy ~ ; or v,dxi + Y^oo, — ' ' ' = and v^^^^^^ + p^^w^^ - . . . = 0. Euler has deduced from the general equation dp . . .... N — -T- + . . . = certain formula?, which in particular cases render its integration more easy : these we shall give in the next article. ^3. FormulcB of solution. (I). In dv — udx-{-ndi/ + vdp-{-Q,dq . . ., suppose all the coefficients = except n and p. Here dv = tidy + Tdp. Also the equation of the curve is N — -7- = ; or, since dx = — , n dy ~ pdv ; wherefore by substitution dv = ydv + idp — d.vp .'. \ = rp -{- c. Cor. If M is not = 0, v —fudx + pp -|- c. (2). Suppose all the coefficients = except p and q. Here dw — vdp + ^dq. Also the equation of the curve dv d^Q. .... do, 11 '' ~ ^ "^ d^ "^ ' ''''' mtegratmg, p = ^ + c ; and mul- 406 VARIATIONS CHAP. I. dp tiplying by dp, Tdp = -J-do, + cdp = qda + cdp ; and by substitution, dv = adg H- qdo, + cdp .*. y — aq -\- cp -\- d. Cor. If M is not = 0, v =fMdx + Q? + cp + ^'» (3). Suppose all the coefficients = except N, p and q. Here dv = Nc?y + vdp + Q.dg. Also the curve's equa- dp d^Q. tion is N — J-; + -r-^ = ; or multiplying by dt/, which pdcc, NO??/ — pdv +-^-7 — ~ 0. dx But pd^a = d.pdo, — dpda = d.pdQ, — qdadx, or adi/ = pdp — -^ h qdo. ; and by substitution, d.pdo. , dv = pdp -f joap -1 h Qog' + qdo. .-. V = P^ + ay _ — + <•. (4). Suppose all the coefficients = except p and R. Here dv = pdp + ndr. Also the curve's equation is dp d^R ^ , ^ d-R J- + -f—. = ; wherefore p = — -j—, + c, or dx dx^ dx^ od^R ' pdp = cc?p — ^-^ — . But qd^R = d.qdR — dqdR .-. pdfp = cdp — -^1 h t-^Zr; and by substitution we have dv = CG?/) + Rdr -|- rc/R -^ — and odR V =c'-i-cp+Rr— -J-, 24. We have hitherto supposed the required fluent to be expressed by a single equation ; but in problems which re- late to lines in space, it consists of two simultaneous and in- dependent equations. (Vol. i. ch. 7. 54). Let dv ~ udx -\- isdy + pdp • ' • ^ + ^^dz + p^dp' ' ' y To find the two equations in space between x, y and x, z between the limits ; for the first, we may suppose z con- CHAP. I. CONTAINING INDETERMINATE FLUENTS. 407 stant ; or the required fluxional equation is dp N — -7- + • • • = 0. Similarly, to find the second, y is constant, and we have n' — -j }-.•• = 0. The equations at the limits are to be obtained from the definite part of ^f\dx), which separates into two as in art. 22. Cor. Euler has given the following formula for finding the function between the limits in the case in which all the coefficients vanish except m, n and p. (Memoires de I'Acad. de Petersbourg, tom. 4, p. 23). Here ^dx — dp = and tt'dx — di>' = ; but -^ z= dx = —, ; whence we have ady — pdp = and p p' Mz — p'dp^ = ; wherefore, by substitution, dy = mdv + pdv + pdp + p'dp' + v'dp' = udx + d{\'p + v'p') . Substitute s = v — {pp + p^) ; then we have three equations ^dx = ^p, n'^jt = dp' and m^^ = dii ; by means of which X, y and z may be each expressed in terms of the same quantity p ; and eliminating p, the required equations in space may be found *. 25. The indefinite part of the whole variation is not {iffected hy the variation of the principal variable. For, let V = F(a:, y^ z, p, g, . . . p', q', . . .) ; and sub- stitute dp d'Q, ^"^^"di'^d^^' d^ d^ dx dx- /-; then, as in art. 17, the inde- finite part depends upon the equation aw -{■ a'cv' = (1). First, suppose ^x = 0; then oj = ly and oo = hz, and the equation becomes Aly -{• a!Iz = (2). Let the equation between the variables x, y, zhe v = ; * I recommend the student, before he proceeds to the more com- plicated forms of the function \, to apply the above formulae to a few examples : these are placed at the end of the chapter, that the subject may not be interrupted. 408 VARIATIONS CHAP. I. then (ex hyp.) -f^j/-\- -f^^ — O; and combining this with (2), there resuhs a ,- a'-t- = (3). a:^ ay Next, suppose x to vary ; then, dx ^ dy ^ d% ^ dv, ^ ^ dv , ^ ^ dv dv , ^ ^(Jy - pSa^) + jpz -pi.) = 0, or^„ + j^u^ = 0; which combined with (1) gives a -^ a'-^- = 0, which is the same result as before ; which shows that the variation of X affects only the equations of limits. Hence, in calculating the indefinite part, we are at hberty to suppose (5^7 — ; in which case cv = Sy and cv' = dz; and the equation becomes A^y + a'^z = 0. The same demonstration may be extended to the case in which V — f(^, y, z, V • ' ' p, (j ' • ' p', q' ' ' ' p', q" • • •)• Cor, If «/ and z are independent of each other, we have A = and a' = 0. 26. The indefinite part of the ivhole variation is not affected by the variation of the parameters. Suppose the function a curve, and let V =: f(^, y, p, q ' • • c, c' • ' '), where c, c' » - • are the para- meters ; then, these varying, let dv = mdx + N^z/ + vdp + • • • + cdc + ddd + • • • Suppose 8^ = ; then it may be shown, as in art. 19, that ^(rdx) ~ l +JHx(c^c + c'Bc) • Hence it appears that the required curve depends upon the same part as before, and that the two new terms intro- duced by the variation of the parameters do not affect the nature of the curve, but together with the part l determine the nature and position of the limits. 21, Required to find the variation qff\dx when dy = Tdt; CHAP. I. CONTAINING INDETERMINATE FLUENTS. 409 t being given by t—fidx ; and v beings as before j a function of{i,J/,p,q'")' It may be shown, as in art. 17, that ^vdx = wdx -\-f{lwdx - dylx). But c?v = idt ; where- fore ^v = ilt^ and consequently Isdx — d\Bx = 'iltdx — Tdtdx = Tdtdx — Tzdx^x, To eliminate dt , we have U = Ifzdx — zlx ■^fi.dzdx — dz^x); whence T^tdx — Tzdx^x = Tdx/\Hdx — dzlx) ; and finally ^\dx = Ycx -r fTdx/{dzdx — dzdx). For the sake of abridgment, substitute dh = Tdx ; then, integrating by parts, fdhfilzdx - dzlx) == hflzdx - dzdx) ^/h(Bzdx - dzdx). Also, if dz = udx -f- Nc?y + pdp + • • • ; then (art* 17, cor. 1), hf(^zdx-dzBx)=hL-\'hf'>'s - -f- + "' l(^dx{l). Again, fh{hzdx— dzdx) = hU ■^\ d.hF d^M > _, ,^^ Subtracting (2) from (1), and adding \Zx, there will result the required variation. 27. Required to find d{J\dx) when v contains, besides the variables and their Jluxional coefficients, a fluent of the form t =Jzdx. Suppose that f\dx is to be integrated between a; = 0, X — a\ which will not affect the indefinite part ; and let f^zdx = H, a constant quantity. Adopt the same notation as in the last article ; also, sup- pose that dY — ijAx + vdy + itdp -f * • • + '^dt. Then the variation = vS^ + s+/5^v-^ + dx d.hv ^ -/JAn- dx dx ; where s is the sum of the definite parts : and correcting from ^ = to a: = «5 there results 410 VARIATIONS. CHAP. I, If^ydx = yZx + S ■\-f% j y — — 4- ... C ujdx ^ r C?.HP ^ /!j An y — t- • • • S "^«^ vU + s +/« ^ *' "" ^ "*" * " ^ "'^^ d.kv d'Kko, where A: = h — ^ =flTdx —fidx. In the case of a maximum or minimum, we have dit , df./cp ,__ + ... +A;n-^+...=0. Ex. Let c?v = \i^dx + yc(y + tcZiJ. Here dt = ds = a/1 + j9^c?^ .*. z = Vl + p .*. «z = — IJut we assume c?z = ^dx + Nc/y -h 2dp + ac?<7 + • • • = .*. N = 0, p = , Q = 0, • • • and the curve's equation is y - J- a = 0, or vrf^ — d = 0. «^ A/l+i>'^ Vl+p'' 28. Required tojind the variation qfjy when v ** given hy ajluxional equation of the first order. Let the given equation be c?v + ^dx = ^dx, where u = f(v, X, y, p,q . . •) and x =j{x,y, p, ^ • • •)• . Taking the variation on the supposition that dx is con- stant, we have r dx^ dx -^ ^ Substituting — (p for the sum of the brackets, and t for -7-, there results c^^v — dx + T^vdx = («). CHAP. I. EXAMPLES. 411 To integrate this, multiply by the indeterminate quantity X ; then Xc?^v — \dx — 0. Find X from the first of these equations, and substitute its value in the second ; then, by integration, there will result the value ofj^v or ^v. The same process applies if u also contains c/*v ; and by repeating it, the variation may be found, when the given equation is of the third or of a higher order. 30. EXAMPLES. Ex. 1. Required to draw the shortest line. Since ds = vdaj^~+~dy^ = V I + p^dx .'.fVl -{- p'^dx = minimum. Here v = ^Z 1 + ^« .*. n = ; p = P Vl-^p^' 412 VARIATIONS. CHAP. I. Q = ; • • • and the equation n — -^ — ^- . . . = gives dp ^ p -J- = 0, or r = const. .'. — = const. .*. p — c, or da; ' Vl + p^ dy = cdx and y = ex + c', a right Hne. (1). Between two given points. Here, eliminating c and d, the equation becomes (2.) Between two parallel lines. Let the lines be parallel to the axis y; then, since Ix^—O, the first equation of limits is p^^j/y = 0, or p^ = .*. p^ — 0, for p = . ; or the required line is parallel to the axis X. We are at liberty to divide the whole equation of limits Uyy — Uy = iuto two, bccausc the limits are independent of each other (art. 22). (3). Between two given curves. The first equation of limits is y^x^ + PyWy = 0, or (vy — P/;?y)^^/ + P^aj/y = 0, or \ v/ 1 + p? zML= \ hx. + ^ Zy. = 0, or ^x, 4- Pt^i/i = 0. dx From this we obtain p. = — tt-^. But if ^,, v, are theco- ordinates of the curve at the first limit, ^r-' = -7—', because ^I/, dy! the increments of the line's co-ordinates arising from any change in the position of the line must be the same as the increments of the curve's co-ordinates; whence we have p^— ; or p^p^ +1=0. Similarly, p^^p^ -f 1 = 0; or the required line is at right angles to both the curves. By means of these two equations combined with y = ex ^ d and the equations of the curves, the position of the required line may be determined. CHAP. I. EXAMPLES. 413 Ex. 2. Required to draw the shortest line in space Here ds = x/dx'^ + dif- + dt' = ^1 + p^ + p"'dx .-. V = vl + ;)^ + y^ P Here M = 0;N = 0=rN';p= — ^ - ; p' = ^ ==r^ ; and the whole equation between the li- mits is -r-^u + -r- ^^ = ; which divides into dp — and ax ^ ax dp' = by considering z and ?/ separately constant (art. 24). Integrating, we have p = const, and p' = const, or P -, P' — = const, and — £== = const, or p = const. — a and p' = const. = b .'. the required line is ? Z ?^ T 5 P which is a right line in space. (1). Between two given Hnes in space. The first equation of Hmits is v^^^^ -f p^wy+p'^o;/ = 0, or (y, — p^p, — pIpD^^i + Py^j// + P/^Z/ = ; which by sub- stitution becomes Sx^ + pfy^ + pl^z, — 0. Since we may establish any relation ad libitum between ly and Iz^ this divides into Ix^ ■\- p^3j/^ — and Ir^-i-pj^Zf^O, I OY pf— — r— and pi = — -^—. Let the first hmit be the right line y — mx + i^-X, j z = rix + y 5 ' let ojj, «/i, Zi be its co-ordinates at the limit ; then dy^ h/, 1 1 C-. -I 1 1 1 m =L ~zz:— r= — — — . Similarly, w = r-jand CLx^ cXi Pi ci o consequently am -\- bn -\- 1 =: ; which is the condition that the line may cut the Hmit at right angles. (Vol. i. ch. 7. 67.) The same being true of the second limit, the required line is at right angles to both the given lines. (2). Between two given surfaces. Let ^1, j/i, 2| be the co-ordinates of the surface at the first limit ; and Xi n myi + nz\ + c be the tangent plane at that point. The equation of limits is the same as before. Also 414 VARIATIONS. CHAP. I. m — -r-^ =z ~ =: — a. Similarly, w iz — ft ; or the tangent plane is x^ + ay^ 4- ft^, = c ; which is a plane to which ^ ~Z. f T ^ ( is a normal (vol. i. ch. 7. Q^')\ which shows that the required line is at right angles to both the surfaces. Ex. 3. Required to draw the shortest line upon a given surface, joining two given points in it. Let a :=. be the equation of the surface ; and, when differentiated, let it give dz — mdx + ndy ; then, as in the last example, we have dv^y -t- dv^lz =: ; but here y and % are connected by an equation of condition; wherefore (art. 25.), since Ix may be supposed •=. 0, -yZy + —^^=0; and dy ' uz by elimmation -r-ap — ^dv z=. 0; which, smce •^ d^ ay ' p dy , dz du ^ du p =: =z= -f^, p'=: -J-, — = 1, — = — 71, be- ^I^p2|,y2 ds ds dz dy comes d -^ -Y nd-^ z=. (1); which therefore belongs to the required line. Since x and y may be interchanged in the equation dz z= mdx + ndy, it is manifest that dx dz d -r; -h i^d— = (2) also belongs to the required line. Let the given surface be a sphere. Take the centre for the origin ; then x- + y'^ + z- r= r- .*. X U X 1J dz = dx^ dy, or m = — '—, rt — — — and (1) z z z z and (2) become d^^ — — d-^ = and ^ ' ds z ds .dx X ,dz ^ w> 7 , d- a -r- == " ; or, it as be constant, ds z ds zd^V — yd^z ^ , zd~x — xd z — -^ ., = and ; rr 0; .'. mtegratrnff, ds ds ^ *'' CHAP. I. EXAMPLES, 415 zdy — ydz = ads and zdx — xdz = bds .^ zdy — ydz — c{zdx — xdz), whose requisite factor is — ; .-. y ex , , 1 • 1 • integrating, -^ = \- c, ov y = ex -\- c%, which is a plane passing through the sphere's centre ; or the required line is a great circle of the sphere. Ex. 4. Required the curve joining two given points, which within its own arc, its evolute and radius of curvature shall contain the least area. The radius of curvature = — (vol. i. ch. 11. 4.) .•. (\-\-pA^ds {l^p'^ydx (l+/?')2 d. area =- — Sr = ^ j or v = ^— . — % —2q q From the form \ = dq -^ cp + d (art. 23, form (2) ), (i+p2)2 {i+P^Y we have — — — -{• cp -\- c\ or q 9 ^ - ^\ + p-Ydx = {cp + d)dp ; or Ux = ^-j~^ ; ■ grating as in vol. i. ch. 2. 58. ex. 5, 4(^ + „)=__£_ +c'jj^, + ta„-.p[ . . dp — c = -f- — - + c'tan.->. Also, dy = pd.r = /^^^^y ..... cd'w cdp c'pdp inte- % + 6) =ctan.-'i>-|-|j^ + tan.-p j--^ ^„or CD -^ C 4(y + 6) = ctan.-^p - j-j-y^. Eliminating tan.~'p, the resulting equation is 4^c(x + a) - 4c'(y + b)=: — ^1— . 4>16 VARIATIONS. CHAP. I. Since a and h are merely parameters of position, transfer the origin to the point («, 5), which will in no wise affect the curve, and the equation becomes 4^cx - ^dy = iTp^'—' The curve is a cycloid ; for if, in 2c//^) = and q,,(1 + P/) = 0; but ,= _(i+^%hencef(i±^Y=Oand > il!LL. ) — ; or the radii of curvature at the limits 9u J each = 0, and consequently the limits are cusps, and the required curve is a complete cycloid. Ex. 5. Required the curve in vvhichy-T^ = maximum. Heref-j-^dx = maximum ; .*. v = q^; and since all the fluxional coefficients except q = 0, we have -r-^ = 0, or c?^ci = ; but a = 2q ,\ d^q — .-. dq = cdx .*. q or dp , cx^ , cx^ c'x"- ,, The values of the arbitrary constants are to be deter- mined from the conditions ; thus, to take a particular case, suppose that two points through which the curve is to pass are given, viz. {x = 0, i/ = 0) and {x = a, y — h\ and also the angles at which the curve at these points is inclined to the axis. Let t and t^ be the tangents of these angles ; then (x = 0, y = 0) gives c'" = 0. Also, .r = 0, p = t, give t =c'; (a:= a,y- b) gives 6 = — + — + ta and x = a, p = t' give t' = — -\- c'a + t; from which two equations, c and d may be found. Ex. 6. Required the spiral in whichyiirf.? = maximum, where r is a function of the radius vector and s = the length of the arc. Here Rds = ii x/dr- + f-db^ = r ^/l -j- rp'^dr, if VOL. 11. E E 418 VARIATIONS. CHAP. I. p = -J- /. N = and p = — ^ ; but Jp = .*. = a, or Jfl = — ., which is the required \/ 1 4- r-^'^ r a/ R®r'^ — w^ equation. Ex. 7. Required the curve in whichy -7-— = minimum. Here v = — .-. (art S3, form. 4) — = d ■\- cp A -qd — .•. = c' + c« + ^ ; and 'p P "^ P ^ p" taking one of the constants as negative that the result may -. dp — not contain // — 1, q or -^ = P\^c' — cp, or dp dx = = ; which may be reduced to a logarithmick p Vd — cp form by substituting %^ = d -- cp% and there results e 2 =z —^, cp Ex. 8. Required the line in space in which p{y±^)dydz . . . J -7 = maximum or minimum. Here v = (7/ + ^)pp^ .-. m = 0; n = pp^ = n'; P = (y -}- ^)p' ; p' = (y -f z)p. Also S, whicll = V - (p/7 + py), = - (7/ + z)pp'. The three equations are = — d . {y + ^)pp'i pp'dx = d,{y -{■ z)p' = d.(?/ -{- !s)p. (Art. 24. cor.) From the first, j/ + ;^ = — y .-. Wc?a7 = a.d, — = a.d. —f .-. — = —r -{- b .: p' = .. , .*. p p p p ^ 1 — bp l—bp p' p* a ab CHAP. T. EXAMPLES. 419 a(\ — bp)dp a ab Also, d^=pdx = - -f— .:y = ^,--+ ^• To find ;? in terms of 2?, we have a a(\ — hp) a From these we can find either of the three equations in space; thus, to find that between y and z, we have a ah , « , «6 ^^ 1 y = TT-o \- d — z + 2c ,-. p = -77-. and ^ 2p^ p V z—y-\-2c the required equation is 2ah\z -\- c') — {:s — y -|- Sc')^. Ex. 9. Required the Hne in space in which f \fx"^ + 2/2 + ^^ a/^^' + dy"^ + dTi^^ = a maximum or mini- mum. For abridgment, substitute r = V^" + 3/^ + 2% , sx sy 5 = ^ 1 4- ps + p'2 ; then V = r5, M = — , N = -^, r r t ^^ ^y , '^p' Ai u- u N' = — , p = — , p' = -!—. Also s, which r s _ s = V — (vp + p'p') = ^'5 — — ~ = — ; wherefore s s ' , . sydx , ^p,,, the three equations are — — = a.-^(l); s%dx ,^p',^v ,sxdx - r = d.—(2)',ar\d = d,— (S). r s ^ r s ^ T V From (3) we have xdx = — d. — .•. integrating, x^ -\- a"^ = — ; from which to obtain the equation between X and y, suppose ^ constant, or s = V'^ + p'^ .'. p\x^ + a^) = 2/2 + 6^ — a2 = j/2 ^_ ^2, if c^ = 6^ _ ^2 . dy ^^ r^ ' .*. — — ^=:rz= = .•• mteff rating, Vy'^+C' Vx^-{-a^ ^ ^ y±_x/^T^ — n .'. y — nx =^ 71 ^/x^ + a'^ — v/w^ + c E E 2 420 VARIATIONS. CHAP. I. 2w^a;« + a'^vy +- c* = Tfa'^ + c- + 2^/^:^ .*. 4w-(c2;r2 + a^'y') - {rfa'^ — c"y + 4!fi{nra'' + c")xy, which belongs to a conick section. (Vol. i. ch. 7. 18. ex. S.) Similarly it may be shown that the projections on the planes xz and yz are also conick sections. The curve lies wholly in one plane. Of this property Euler has given the following demonstration. (Memoires de TAcad. de Petersbourg, torn. iv. p. 3.) From (1) and (S)'-^ = d,^=pd.--^ —dp • • • \/ ^ ^ r s -^ s s ^ dy sxdx r , sxdu r , , , r^ , o- M 1 , , f" -, , dp ydx—xdy y—^P Similarly, zdx — xdz = -—dp' .-. ■j- = ~ /- = '' .«?« * dnf zd.v — .Tdz z — sen dp dp^ . . y — xp ... _ _ .^ mteffratinff, = w, which .*. y—xp z—xp' ° ° z^xp s^ ^ ' ' dp' zdx^xdss z—xp P dp^ xp z—xp' ^ dt) — ^f ''• P — ^p'+ f^i or dy = ndz-\-adx ,\ y=nz-{-ax-\-b, which belongs to a plane. Euler in his next example investigates the nature of the curve in which yR50?v/l+j9^ _ j?^ 1 1 1^ v> " vy ^/f+F' ^ " ^~y vT^P' " ^ ' ds c^ — =i~; which belongs to a cycloid, the diameter of whose dx -=• y\ generating circle = c. (1). Between two given points. CHAP. I. EXAMPLES. 421 Take the higher point for the origin of the co-ordinates ; then, since ■—= — = — ^, we have 1 + «^ = — , or Vc—y , i/'^du ydy . ^ p = —~ .\ ax = — ~ — - .-. nitegrating, X = -^v.s~^~ — ^/C7/ - y^, for c — 0: which is an in- = -^^^^^ — __— -i^i—— - .'. the required curve is 1^2 = -ffq^ + ^ ••• changing c, ^/;^ = c(l + /7^)^ or V ds^ The curve does not meet the axis ; for ^— = , , , , and c dy^dx since ds^ > di/Hx, we have y > c\ or the required solid is obtruncated. It can be easily shown that the angle wliich the curve makes with the least ordinate = the supplement of 45^ ; wherefore the fluxional triangle corrcsj^ondiiig to the least ordinate is isosceles, or Js* = \dy^ .*. {y) — 46*. If the least ordinate be given, and be denoted by («/), we shall have -Z^ = i^^ or y = ^ - -^(^)^ which is the construction given, Princ. vol. ii. p. 34. Scholium *. To find the curve which is a transcendental, we have yp^ = c(I + p-Y .-. y — c{p-^ + 2j)-^ H- p) .-. pdx = dy =— c(3/>~* + 2p-^ — l)dp .-. dx = - t(3p-5 -1- 2p-^ — p-^)dp .-. X — (^(—T—Vp~^~\-^Pj + c'; and combining this with j/j9^ ~ c{\ n'- p"-)", we shall have the relation between X and y. If the axis of the solid and its greatest ordinate be given, c may be found from the resulting equation, and the curve may be determined. The first edition of the Principia appeared in 1687. 424 VAUIATIONS. CHAP. II. CHAPTER II. Isoperimetrical Problems. 1. By means of the three formulae of arts. 17, 27 and £8 of the preceding chapter, we can solve all problems which involve only the maximum or minimum property. The first is to be used when v is a determinate function of (x, y, z » * • Pi g ' ' ' p'i q' ' ' ' )l the second, when v includes an inde- terminate fluent, such as s, the length of the curve, or even when V contains a double or a triple fluent ; and the third, when v is given by a fluxional equation of the first or of a higher order. There is a class of problems which, in addition to the maximum yjroperty, involve one or more of what are termed isoperimetrical properties. Thus the curve of swiftest de- scent from one point or curve to another, or the brachys- tochrone, belongs to the first class; but if it be required to find the brachystochrone when of a given length, the pro- blem involves one isoperimetrical property ; and if to this be added that it shall also contain a given area, it involves two isoperimetrical properties. We shall proceed, then, to adapt our formulae to the case of isoperimetrical problems. I'his reduction of relative to the case of absolute maxima or minima is accomplished by means of the following theorem. 2. If at the same time that one function, as u, is to be a maximum or a minimum^ other functions, as r, w, . . ., are given ; then the required curve is the same as that which ivould result from l[2i -\- av -\- bw + • • •) =0, where a, b ' ' ' are constant quantities. For, suppose that there is only one isoperimetrical pro- perty, viz. that the function v is given. Let lu, which = 0, = aIx f ^^y ; where, if there is no isoperimetrical property, Ix and ^y are wholly indeter- minate. CHAP. II. ISOPERIMETRICAL PROBLEMS. Next, suppose ^v, which = 0, = a'Bx + b'^^ ; then ^x and ^// in a^^ + b^^/ = G must be no longer indeterminate, but must also satisfy the new hypothesis or the equation a'^x + b'^«/ = 0. Now (ex hyp.) ^u + cw), =0, =(A+aA')a^ + (B4-aB'%, in which dx and 5y satisfy both Adx + B^y =0 and a'Bx + B'Sy = conjointly, and neither without the other; wherefore the proposed is reduced to the case of an absolute maximum or minimum, viz. d{u + av) = 0. The introduction of the constant a enables us to give to V the value required by the conditions of the problem. The same method is manifestly applicable, whatever be the number of isoperimetrical properties. S. EXAMPLES. Ex. 1. Required the curve of given length whose area shall be a maximum. Here u + av -fiy ■\-a^/\ -\-f)dx .'. v=j/ + a Vl +p« ; and from the form v = p/? + c, we have y + a .yl-\-p^ = p = c-y .-. dx = + c, or — === — c —y .', {c^y)dy _ . ^a'^-ic-yY X - c' ^ Va^ - (c - yY, or (^ - c')« -j- {y - c)« = a2. which belongs to a circular arc. (1). If thehmits be fixed, viz. (x'^y') and (^", y"), then we have (^1 - c'Y + («/' - cY = a^ I ^ ^^^^^ ^f ^^.^^ ^^^ ^ {x" — cf -f (y — cY = a'^S ^ may be each expressed in terms of the limits and of the constant «, the quantity upon which the length of the line depends. The chord of the arc — VC^ — ^'f + («/" -y'Y, and consequently we can find the radius a in order that the area included by y', «/" and the given arc may be a maximum ; or a minimum, if the arc be drawn convex to the axis. Cor. A curve of given length must be formed into a cir- cular arc in order that it may include with a given right line the greatest area possible. (2). Next, let only y and y", or y^ and y^^ be given. Here ^y^ = and ly^^ — ; and the whole equation of 426 VARIATIONS. CHAP. II. limits becomes (v^/ — p,iPii)^a;„ — (v^ — i?,pi)^^i — 0; which divides into v^ — p^^p^y = and v^ — Pfp, = 0. But V = p/? -f c or V — Tp — c; wherefore in this case c = 0; and the curve's equation becomes (a?, — c')* + 2/^ — a"; which belongs to a circular arc whose centre is in the axis x ; or the area intercepted by two moveable and given crdinates and an arc of given length is a maximum, when that arc is part of a circle whose centre is in the axis x. (3). Lastly, let?/' = = f. Here x,i — x^ is the chord of the arc ; and since the centre is to be in ar^y — x^, it a})pears that the curve of given length, which with its chord includes the greatest area, is the semi- circle. Ex. 2. Required the curve which shall include a given area with the least perimeter. Here n =^ V^ + p" ■\- oy\ and from the form \=vp-\-c. we have */ 1 -y rP- -\- ay — J- 4- c, or — == = c—ay\ which is a circle, the same as in ex. 1. Cor. This commutability of the properties, first observed by James Bernoulli, evidently results from the article; for, a being constant, ^{u + av) = must give the same curve as ^(v + au) — 0. Ex. 3. The surface generated by a curve revolving round its axis being given ; required the curve that the solid ge- nerated may be a maximum. Here v =z y^ + ay ^\ -\- p'^ .-. from the form v = P/7+ c, ayf^ we have y'^ -{- ay x/\ 4- p2 — — ^i£ — _|_ p., or •^ = c2 - y\ ordx= ^ -^ -^ Vl+p'' VaY-~{c'-y^'' This, when integrated, will contain three arbitrary con- stants, which are to be determined from the data. Thus, if the limits are two fixed points, the constants must be deter- mined so that the curve may pass through the two points, and that its surface may be a given quantity. If the extreme ordinates of the curve be given in mag- nitude, but not in position; then, since ^y, and hJ^^ each =0, CHAP. 11. ISOPERIMETUICAL PROBLEMS. 427 the equation of limits becomes (v// ~" ^iiPii)^^ii — (Vy — '^iPi)^^i — ^j o*' ^^^^n "~ ^^^^/ = ^' which, on account of the independence of hx^^ and hx^, gives c^ = 0; or the required curve is J — w-c/t/ -ydy ... a^ = — '^ = -^ '^ ; and mtesratinff, ^ 4- c' = ■\/cC^ — y' ; which is a circle whose centre is in x. This, together with the isoperimetrical property, determines the position of the ordinates. If y = — y, the curve is a semicircle; and the solid, a whole sphere. If we suppose that the cii'cular ends may form part of the given surface, we shall obtain the same result. For, if we include the circular ends whose radii are y^ and yyy, we shall have i{f(.y' + «y ^TTJ^dix + « J |! + ^ J) = 0. The indefinite part of this equation is the same as before; but the definite part is increased by -^'^{yf + ?//), which = a(y^y^ + i/n^yn) ; or the whole equation of limits is ^iMii - "^t^^i + P/;"^// — ^,^1 + a(?/ii^I/ii + ^ibi) = 0, or But v — pj9=:C'; for the equation of the curve is the same as before; also p = — .-. the equation of limits is cV^„ - ^^;) + \ -^i— + llayfy, S P, I VI -^p - 1 [ f^ypyi = 0- But Ixp dXji and dy^, ^J///are wholly independent. Also, the ]:)oints (a;^, y^) and (x^^, y^ not being constrained to move in any curve, Ix^^ ly^ and Ix^^^ ly^^ are wholly independent .*. we have c^ = 0; V ^^' - 1 ? 7v. = and 5 ' -^ lXy„ = 0. These are satisfied either by yi — 0, y^^ =: 0, or by /?y = x , p^y = — x ; both of which VARIATIONS. CHAP. 11. agree in giving the curve a semicircle, or the solid a whole sphere as before. (Airy's Math. Tracts, p. 178). Ex. 4. Required the curve of given length and including with its axis a given area, which by revolving round the axis generates the greatest solid. Here u =fy-dx\ v = f Vl 4- p'da; and w = fydx .'. u -{■ av +hw =y((yM^jfl_\/ 1 + jp^ + ^y)dx^ or V = y + ij/ 4- a a/1 + P^. Since v = pp + c, we have ap- y« + 6y + a a/1 + /?^ = ^ - f c^ or Vl -Vp^ y^ -\- by — c^ = , or 1 + p'^ = y~o — i 5TT, •'• •^ ^ ^1+^2' ^^ {y'^-hy-c'^Y which belongs to the elastick curve. When this is integrated, the primitive containsybwr arbi- trary constants : two of these are determined by the assigned values of the length of the curve and of its area ; the re- maining two depend upon the limits. Cor. If the isoperimetrical property of the given area be omitted, i. e. if b = 0, dx — ^^ ~^ '\1 — . Ex. 5. Required the curve in which {fy'^dx) is given, / y^{d^±dff\ . . . . and I / , o„ , — ) is a maximum or minimum. / y\dx-±dy'^r \ dy Here. =/-^-:7 QA THE UNIVERSITY OF CALIFORNIA LIBRARY m