Digitized by the Internet Archive in 2010 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/examplesofprocesOOgreg r / ^' / -x EXAMPLES OF THE PROCESSES OF THE DIFFERENTIAL AND INTEGRAL CALCULUS. COLLECTED BY u D. F. GREGORY, M.A., FELLOW OF TRINITY COLLEGE. CAMBRIDGE : ritlNTED AT THE UNIVERSITY PRESS; PUBLISHED BY J. & J. J. DEIGHTONj AND JOHN W. PARKER, LONDON. M.DCCC.XLI, 303 150204 PREFACE. The chief object of the present work is, as its title indicates, to furnish to the student examples by which to illustrate the processes of the Differential and In- tegral Calculus. In this respect it will be seen to agree with Professor Peacock's Collection of Examples ; and indeed if a second edition of that excellent work had been published I should not have undertaken the task of making this compilation. But as Professor Peacock informed me that he had not leisure to su- perintend the publication of a second edition of his "Examples" which had been long out of print, I thought that I should do a service to students by- preparing a work on a similar plan, but with such modifications as seemed called for by the increased cultivation of Analysis in this University. Accordingly I have not limited myself to the mere collection of Examples and Problems illustrative of Theorems given in Elementary Treatises on the subject, but I have also introduced demonstrations of propositions which, although important and interesting, do not usually IV PREFACE. find a place in works devoted to the exposition of the principles of the Calculus. I wished by these means to render this Collection, as it were, complementary to those works, and, with the view of allowing it to be read in connection with any of them, I have generally assumed as known only those methods which are to be found in all Elementary Treatises. To this, how- ever, there is one exception : it will be seen that I have made constant use of the method known by the name of the Separation of the Symbols of Operation, although the Theory of the process is not usually given in works which are likely to be in the hands of students. I have done so because I think it a matter of some importance that the use of this method should be extended as much as possible, since it shortens and simplifies many of the processes of the Calculus, while at the same time it offers to the stu- dent one of the most instructive examples of Analyti- cal Generalization. There seems to have been among writers on the Calculus an unwillingness to consider this method in any other light than as founded on an accidental analogy, and therefore to reject it as not based on a strict logical deduction. This idea I think is formed on a limited view of the nature of Analysis, and I shall be glad if the use which I have made of the Separation of the Symbols may induce others to examine the question closely, and so satisfy themselves of the logical validity of the process. The principles of the method are so simtjle that I think the short X sketch which I have given of them in Chap. xv. will be sufficient to make its application readily understood. PREFACE. V I have adhered throughout to the notation of Leib- nitz in preference to that which has been of late re- vived and partially adopted in this University. Of the Differential notation I need say nothing here, as it appears to be abandoned as an exclusive system by those who introduced it : but as the use of the suffix notation for integrals has been sanctioned by those whose names are of high authority, I may state briefly some of my reasons for differing from them. In the first place, on considering the subject, I could find no arguments against the use of the notation for Dif- ferentials, which did not apply with even greater force against that for integrals: indeed, although there may be some cases in which the use of the former is ad- vantageous, I know of none in which the latter does not appear to me to be inconvenient. In the next place, I fully agree with Professor De Morgan in an unwillingness to lose sight of the analogy to summa- tion which is implied in the old notation ; and if it were at any time necessary to consider integration merely as the inverse of differentiation, I should pre- fer to employ such a symbol as d^'^ which expresses the required idea better than f^-. But what I look on as a fatal objection to the suffix integral notation is that, like the corresponding one for differentials, it is not applicable to all cases. Of this any one may satisfy himself by attempting to use it in transforming a multiple Integral from one system of independent variables to another, a problem which is of frequent occurrence, but which I have not seen solved analyti- cally in any work in which the suffix notation is em- yi PREFACE. ployed. So long, therefore, as the old notation adapts itself to all cases in which it is required, while that which is proposed is not so accommodating, there ap- pears to me no doubt which is to be preferred. The sources from which the Examples have been taken are indicated by the references which will be found in the body of the work. For although I have not thought it necessary to cite an authority for every example, I have done so in all cases in which the student would be likely to wish for more information by consulting the original authors. It has always ap- peared to me that we sacrifice many of the advantages and more of the pleasures of studying any science by omitting all reference to the history of its progress : I have therefore occasionally introduced historical notices of those problems which are interesting either from the nature of the questions involved, or from their bearing on the history of the Calculus. From a fear of increasing the size of the volume too much, I have not done this to as great an extent as I wished, but these digressions short as they are may serve to relieve the dryness of a mere collection of Examples. TRrNiTV College, October, 1841. CONTENTS. PART I. DIFFERENTIAL CALCULUS. CHAPTER PAGE I. Differentiation of Functions 1 II. Successive Differentiation 10 III. Change of the Independent Variable 29 IV. Elimination of Constants and Functions 43 V. Application of the Differential Calculus to the Development of Functions 52 VI. Evaluation of Functions which for certain values of the Variable become indeterminate 79 VII. Maxima and Minima ■ 93 VIII. On the Generation of Curves and the Investigation of their Equations from their Geometrical Properties 127 IX. On the Tangents, Normals, and Asymptotes to Curves ... 142 X. Singular Points in Curves 160 XI. On the Tracing of Curves from their Equations 172 XII. On the Curvature of Curved Lines 185 XIII. Application of the Diflferential Calculus to Geometry of Three Dimensions 196 XIV. Envelops to Lines and Surfaces 220 XV. General Theorems in the Differential Calculus 233 VIU CONTENTS. PART 11. INTEGRAL CALCULUS. CHAPTER PAGE ' I. Integration of Functions of One Variable 245 < II. Integration by Successive Reduction 267 III. Integration of DiflFerential Functions of Two or more Variables 278 ~" IV. Integration of Differential Equations 287 V. Integration of Differential Equations by Series 336 -p VI. Partial Differential Equations 347 -|- VII. Simultaneous Differential Equations 382 VIIL Singular Solutions of Differential Equations 396 I ' IX. Quadrature of Areas and Surfaces, Rectification of Curves, and Cubature of Solids 408 I X. Geometrical Problems Involving the Solution of Differential Equations 436 .'XI. Evaluation of Definite Integrals 460 XII. Comparison of Transcendents 602 DIFFERENTIAL CALCULUS. CHAPTER I. DIFPERBNTI ATIQN . Fw^ctions of One Variable. If u be an explicit function of a?, which is of a com- plicated form, it may generally be reduced to the differen- tiation of simpler functions by means of the theorem du du dy dw dy doG y being some function of w, and u some function of y. This theorem may be extended to any number of functions, so that du du dv dz dy dx dv d% dy "' dx Ex, (1) Let M = (a + 6 «")'". Then y=:a + bx", u = y"", dv , ^ du , X , V . -^ = 7ibw''-\ — = my'"-^ =m(a+ bx")'"'^ ; dx dy die therefore — = mnbx"'^ {a + bx")'" '. dx „.i>i du \x + (l+x^)H^i (2) u= {x + il+ x'^y^lK — = L^ -IJ- ^ ' dx 2(1+ x)^ 1 DIFFERENTIATION . dll , _ (3) M = e" ; -— = /i.r"~ e' . UK du (4) u = e"" ' ; — = cos .» e ^'"'^. (5) 7/ = log ;.r + (1 +a'^)5[ ; — = , rfcT (1 + wy (6) w = log (log ai) = log^ (a?) ; dx X log %*• + mr cos ra?). do? DIFFERENTIATION . Implicit Functions of Two Variables. li u = Ohe an implicit function of two variables x and ?/, then du dy dx dw du dy (43) Let X logy = y log x ; dy 2//^-'^ log 2/ then - 1 dx X \x — y iogx (44) If dwy = X sin {a + y), dy sin (a + y) d X cos y - »e^ dy e'-^ e^ d/r 1 - xe^ 2 - y » DIFFERENTIATION. (49) Let ^(1 +//)^ + //(I +^)^ = 0; then il = 1, y + ~(1 + '^OMl +y)^' dw 07 * ct? + 2 (1 + w)l (l + y)l ' (.50) Let sin-^- + sin-i - = c; h k then ^^_(^lzl!)!. dos (h" - a;')i (51) Let (w' + yy=a\v'-b^y\ dy \a^ - ^ {w^ + y^)} dw [b^ + 2 {x^ + 2/^)| y' (52) Let (a + ?/)^ (6^ - y^) - w^f = 0, then ^ = - y' (^' - y'y^ dx y^ + ab^ ' Functions of Two or more Variables. {53) - ' ^ du 9,xy'^ du 2x^y dv {w' + y^)i{x^-y'Y dy {x' + y'')^ {x' - y')k' 2xy (ydx - xdy) du = -,--: (54) u = (t^ + y'^)^ {x'^ - y~)-^ 2 • ^ ••'^ x'^ sin 2 - 2/ (61) If M= ^'^ (cr^ + yy e"ydz xe" (xdy - ydx) (62) If u=^ - (x^ + 2/^)2 (x' + 2/^)2 2xydx x^dy 2x^yzdz a^ — z^ a^ - %^ (a^ - z^Y (63) u = (x^ + y^+%^)^ + ta,n-^ - + — z 2 DIFFERENTIATION, xdx + ydz + zdz %dx — xd% du = -— ^ j-i — H 5 1- ^a^. (.1? + y + ^ )2 ). Hence as before, d'^u - — ■ = (^a^ + n^Y e"'^ cos {nx + r0) ; (jLX Similarly, if «* = e*'*sinw ^^:t^/+ hc.y. 1.2 (m-r + l)(m-r+2) (c+.y-^ SUCCESSIVE DIFFERENTIATION. 15 (17) UV = /'■ COS 7101 . W'", In this case let u = e"^ cosnx, v = .1?"'. Then by Ex. (lO) if 71 d^ u ? -= tan0, = (a" + w^)^6"^cos(wa7 + «d)). a ^ dxP ' '^^' Therefore, expanding by the Theorem of Leibnitz, _p^ = e"'' (a' + ^')? [.i?'"cos(^a? + r^) . CJbW , cos (w<2? + (r-1) (i)| *■ (*■-!) , . „ cosJw<2?+(r-2)(i{)| + — '^ ~mim - 1)0?™-- *-- ^ — ^^-^ + &C.1 (18) Let wi) = e"-'' X, X being any function of X. Then making u = X, v = e'"', d'i^^v) „Jd'X d'-'X r{r-\) ^d'-'X doT (d'X d'-'X r{r-\) ^d'-'X ] I dV I dV-' r(r- 1) ^ / dy-' d,vj ' \dxj 1 . 2 \dxj Whence it appears that This result, when generalized, is of great importance in the solution of Differential Equations. 1 6 SUCCESSIVE DIFFERENTIATION. If the function to be differentiated be (a + bx + cw') , the general differential might be found by resolving the trinomial a + bx + cx^ into two factors of the first degree, as into {x + a) (a? + /3), and then differentiating the product {ce + of {cc + j3)" by the Theorem of Leibnitz; but instead of doing so we shall make use of two formulae given by Lagrange*. Let u = a -^ bx -V cx^, u = b + 2cx ; Then substituting x + h for x in u" it becomes (?/ + u h -V cl?Y\ and will be the coefficient of in the expan- dx' 1 .2...r sion of this trinomial. Developing it as a binomial, of which u + u' h is the first term, we obtain (u + u hy -^ n (u + uhy-' ch^ +— (u + uhy-'c'-h' + kc. Again, developing each binomial and taking only the terms which multiply //, we find that the term in (u + u hy IS ^ u" u \ 1 .2 ... r in (w + u hy~^ h^ is . {n — 1) ... {n - r + 9,) 1 . 2 ... (r - 2) u n-r + ■[ „,lr-2 . , (n — 2) ... (n - r + 3) in (u + iihY-^h' is ^ ^- ^ ^w^-'+^m''-^ &c. ^ 1 . 2 ... r - 4 Collecting these terms, and. multiplying by 1 . 2 ... r, we obtain for the r*^ differential coefficient of u" d''(r(") . , ( r (r - 1) cu dx' ' 1 . (w - r + 1) w" + '•0--l)(^-«-)(^-8) t^^^,l (^) 1 . 2 (w - r + l) (w - r f 2) «< * * Memoires de Berlin^ 1772, p. 213. SUCCESSIVE DIFFERENTIATION. 17 By developing in a different manner a more convenient formula may be obtained : u c (u + u'h + cfi^Y = u" (l + — h + - h~)" 11 u u , 4wc — u' ,.,s = wM (1 + — hf + — h~' \ But 4wc — u'^ = 4ac — 6^ = e^ suppose. Developing; u" \(l + — hY + h^l" by the binomial ^ * ^^ 2?i ' (2uy ' -^ theorem, we have u"\h +^hY" + n(l + — hY"-- j^h' '^ 2u ^ ^ 2w, (2uy n(n - l) ^ u , e^ . , 1.2 ^ 2m ^ (2m)* ' and the /** differential of u" is the coefficient of h' in this ex- pansion multiplied by 1 .Q...r. Now expanding each term by the binomial theorem, we have for the coefficient of . . „ /u\'' 1 2n(2n-l)...(2n-r + l) h' in the first term - - — ^^ ^^ ~ , V2/ u 1.2...r ^ (u'y~'' 1 (2w-2)...(2w-r+l) n ^ second — ^7— e\ \2 j 2-u' 1.2..,,(r-2) 1 ,. , /w'\'-* 1 {2n-4i)...{2n-r+l) ri{n-l) third — -—7 e', \2/ 2*M 1.2...(r-4) 1.2 and so on. Collecting these terms and multiplying by 1.2... r, we find -—~=2n(2n-l)...(2n-r^l) — 7A"-'h+ - -y- dx ^ ' ^ \2 I * \2n(2n-\)u' «.(n-l) r(r-l)(r-2)(r-3) e" — — + &c. >...(t>). 1.2 2n{2n-\)...(2n-S) u" ' -t- 2 18 SUCCESSIVE DIFFERENTIATION. (19) Let w" = (a'^ + a?-)". Here w' = 2.??, e = 4a'^, and if we make r = n, we find by formula (B), — -— =2w(2w-l)...(w+l).p" 1+ — . - \n{n-\)Y (n-2)(n-3} a* . (20) Let u" 1 .2 2w.-. (291-3) x* 1 a'' + <»^ The r^^ differential of this function may be found as in the last example, but the following method gives it under a form which is more convenient in practice ; _i L_J ' ^__l. Differentiating r times, Vrfa?/ ^ ^ ' 2a(-)i \{a?+a(-)i{'-+^ |^_a(_)ij'-+i/ ^"'' 2a (-)^ \ (a'+w'y+' i Now let = tan"' -, so that a? = (a^ + a?^)^ cos 0, a = {a^ + w^)^ sin 0, and therefore r + l {a?-a(-)i}'-+^=(a2 + a?2) 2 {cos(r+l)6'-(-)2sin(r+l)05, r + l {cr+a(-)i}'"+'=(«^ + .r^) 2 {cos (r+ 1) 0+ (-)i sin(r + 1) 0}. .SUCCESSfVE DIFFERENTIATION. 19 H ence we nave I dV 1 _ (-)'>(r- 1)...2. 1 sin(r + 1) {a~ + ^ ) '-i Liouville, Jour: de VEcole Poly technique^ Cab. 21, p. 157. (21) In the same way if we had the function 2 V ' we should find I d V ■ X , ,, , . cos(r + 1)0 \dx a- + X- ^ ., 2\^- (a- + X ) --i Liouville, lb. p. 156, These results are useful in the theory of definite in- 1 tegrals. ^ In the following examples the functions are reduced to the required forms by differentiation in the same way as in Ex. 11. X du 1 (1 -x')l'' Tw^ 0~^a?2)l d' X d'-' (22) Let u= ^^ .^^j, ^ - ,, ^^. Therefore dx' (1 -x')l dx'-' (1 -x')^' and by formula (B), d^u 3.4,..(r + l)c/-' S(r-l)(r-2) 1 d^^ " (I - x'Y'+i '^ ^ "^ 2 sTi X 3 . 5 (r - 1) (r - 2) (r - 3) (r - 4) l + ^ — + &c. 2.4 3 . 4 . 5 . & x" , ^ . , 'I? du 1 (23) ?< = SU1-' - ; — = a dx (a'^ - x^Y 2-^2 20 SUCCESSIVE DIFFERENTIATION. d' u d' ' dx"^ dx^ ' (a^ — ai'^y^ _ ) . 2...(r - 1) *'-' 1 (r - 1) (r - 2) «- \ ,S (r- l)...(r - 4) a' , , _ + 1^ 1 — ?^ 1 + &c. by (B). 2.4 1.2.3.4 .-p* 5 .7 V / . . -. , X dn a (24) Let w^tan"'-; = a das a^ + x^ dTu /dV-' 1 r - 1) (r - 2). ,.2 . 1 sin T u = (-)'■-' (r - 1) (r - 2). ..2 . 1 by Ex. 20. where y = tan"' - = tan"' - . a? 2 a The method employed by Lagrange may be used for the determination of the successive differentials of other functions. (25) Let u = e'""'. If a? become .^? + A, w becomes ^ccv+h)^ ^ ^c(^^ + 2.Tk + h^) Now 6^-* = l+2cxk+ ^^' h' + ^^ k' + &c. 1.2 1.2.3 2 3 and e"*' = 1 + ch'' + —h^ + — - — h^ + &c. 1.2 1.2.3 Multiplying these together, taking only the coefficient of h^, and multiplying it by 1 . 2 ... r, we find dx 1.2 - SUCCESSIVE DIFFERENTIATION. 21 (26) From this we can determine the successive dif- ferentials of cos.r'^ and sin ii;". Let u = cosir" + (— )2 sin x^ = e^"' '^^ Then differentiating by the preceding formula i^ = e<->*-"' {(-y (2wy + (-)~ r (r - 1) {2wy-' Now generally (-)2 = e^ ' -^2, and €^-)H<->*^l = cos [x-' + p-\ + (-)^ sin ix'+p- Therefore making these substitutions, and as -r-r=\T-] COSW^ + {-y[~\ Slnd7^ aw \dwl \awl equating possible and impossible parts, we have rf*" (cos '-^'-'> e'-'" + &c. (3). 1.2.3 ^ '^ The product of (2) and (S) must be equal to the second side of (l), and as this last consists of a finite number of terms having positive indices, the terms in the product of (2) and (3) which contain negative indices must disappear of themselves. Hence taking the terms with positive indices only ax ^ 1 ^ ' 1 1.2 ' -■ and therefore ^ (-n»-^--!2--^)l-ie'-"-+{3--^>2-+^^l-}e(--->' + &C.] ay-"""" ""^ (e' + iy+i — — — — .^ * Memoires de PAcadimie^ 1777, p. 108. SUCCESSIVE DIFFERENTIATION. Sec. 2. Functions of two or more Variables. If u be a function of two variables x and y, dy' doG^ daf dy^ ' Ex. (1) u = .t?"*?/" ; r = 1, s = 1, du , rfw rf,3? dy (Pu , , rf^M = mnc(f^~^y^ = 23 dy dx dxdy x'^ + v^ (2) w = ^ z'-, r=l, 5=1, ar — y~ d?u x^ + y^ d^u = — oooy-—^ dydx {/"'^ - y^y dxdy (3) u = f; r = l, 5=1, = f '{\ + a? logy) = dydx dxdy (4) w = sin {mx + wy) ; d'w , / 7r\ -— : = m^ sin I mti; + ny + r— j , d'u , . / 7r\ — - = n sm W.2? + 7i« + 5 — , df \ ^2/ dy'dx" \ ^ ' 9,] dx^dy (5) w = sm - ; r = 2, s = 1, y #w 2 .t' X X d?u — — -^ z= - sm - + -4 cos - = dydx' y^ y y V dx^ dy 24 SUCCESSIVE DIFFERENTIATION. (6) w = sin"^-; r = 1, s = l. y d^u V d?u dy dx {y^ — a?^)^ dx dy (7) u = tan — ; r = 1, 5 = 1; 2/ d^w d?^ —y^ d^u dy dx (y^ + x'^y dx dy ' (8) u = X siny + ysin x; r = 1, s=l; d'^u d^u = cos y + cos X = dy dx dxdy (9) u = sin X cos ^ ; r = 2, s = 2 ; d^«* . d*w rf*M g = sm a? cos t/ = dy dx^ dx^ dy^ dxdy dxdy Generally, in a function of any number of variables, the order of differentiation is indifferent. 2 X y (10) w = ^ a — »" rfw 2xy du x^ du 9.x'^y% dx ^-^da,^dy~ (m - 2) (m - 3) n(n-i) (n-2) , , , , , (m-l)(m-2){m-3) + —7^ r7 T^ ^ w^y^'-'dy'}. (14) W=6"" + *^ d^u = («^da?3 + 3a^bdx'dy+3ab^dx dy^ + b^df) 6"' + *^. (15) w = sin ma? sin n^; d^u = (m^^dx* + 6m^n^dx-dy^ + n^dy^) sin wa? sin w?/ — 4mw {m"dx^dy + n^dx dy^) cos mo? cos ny. (16) ?* = log (ax + 6?/) ; 1 d^u = - {d^dx^ + 2abdx dy + b'dy^) (ax + byy (17) M = ( = X— -y dx into a function of r and 0, having given x = r cos 0, y = r sin Q. In this case we consider r to be a function of 0; differentiating therefore x and y on this hypothesis, dm dr . /^ <^2/ ^^ • /i TZ = T^ c°^ ^ ~ '* SI" ^' TS = T5 s^" ^ + ** cos ^ 5 a0 a0 dd dO dr . -— ; sm + r cos d and therefore -— = , — — . dx d r . — — cos — r sin y d9 Substituting this expression for — , we find dx 34 CHANGE OF THE INDEPENDENT VARIABLE. »a (13) Transform p = da?2 into a function where 9 is the independent variable, having given a? = r cos 0, y =r sin B. Proceeding as in the previous example we find dy w (14) Express / sB-^-y aw dy in r and 0, having given X = r cos 0, y = r sin ; the result is t = r — . dr Sec. 2. Functions of two or more variables. I Let i« be a function of two variables, « and y, so that du du , then to express -7- and -— m terms of two new variables ax ay CHANGE OF THE INDEPENDENT VARIABLE. 35 r and 0, of which w and y are functions given by the equations a? = <^ (r, 0), 2/ = v// (r, 9), we proceed as follows. We have du du dw du dy dw dr dy dr^ du dd du dw ~ dw ' dd du dy ^ dy' dd ^,. . . du Ehmmating — we find du dy du dy du dr' dO dO'dr dw dw dy dy dw dr' dO dr' de ^,. . • du Eliminating — €t CO we find du dw du dw du dr' dO dO' dr dy~ dw d y dr' de dy dw ~ dr'de If r and 9 be given explicitly in terms of w and y, we have at once du du dr du d9 dw dr dw d9 dw du du dr du d9 dy dr dy d9 dy For the successive differentials we proceed in the same manner ; and if there be more than two independent variables, the only difference is that the expressions become more com- plicated. Such cases however seldom occur. If the independent variables enter into multiple integrals, , we cannot substitute directly the values of the original diffe- 3—2 36 CHANGE OF THE INDEPENDENT VARIABLE. rentials in terras of the new variables, because one is supposed to varj while the others are constant. To introduce this condition we proceed as follows. Let for example there be a double integral ffVdxdy, and let 07 = I (3) Transform —^ + —4- = 0, dor dy having given w^ + y^ = r^. d

dr a? d0 1 d(p dr so dso^ dr^ dw r dr r dr dx r^ _d^ n x^\ dr^ r^ dr \r r^j' dy^ dr^ r dr \r r^ Whence do^ dy^ dr^ r^ dr \r r^ ) ' , 1 ir d'd) Idd) and therefore — -^ + - -7^ = 0. dr'' r dr This equation occurs in researches on the motion of fluids. / . xr d'(b d'd) d'(h ' (4) If -T^ + -~ + ^ = 0, dd?^ dy^ dfs CHANGE OF THE INDEPENDENT VARIABLE. 99 when x^ + y^ + !s^ = r^i d^d> 2d(f) we find -TT + - T^ = 0- dr r dr d'V d^V (5) Transform ——^ + — — r = into a function of r and "^ do!^ dy^ Oy having given x = r cos 0, y =r sin 0. dV . ^dV cos0dV — = sin — — H ■ — — , dy dr r d0 d^V . d'^V cos^ d^V cos'0dV df^^^'^'d? "^~?~ d^'^"7~d7 SsinOcos^/ (fV dV\ "^ 7 V drd0~~d0) ' d^V d^V The expression for -— -^ may be deduced from that of -— -^ TT by putting for 0. We then get (fV ,^d2^ sin^^e^F sin^^dF do?^ dr^ r' d0^ r dr 2sin0cos0 / d'^V dT \drd0~~d0) r" Adding these together, d^F d^V _d''V 1 d^r idV _ 'd^'^'d^~'d?'^'?d¥'^r~d^~^' ... r„ . d'^V d^V d^V (6) Transform — -„ +— -x + ^-2 = ^ ^ dw^ dy^ d%^ into a function of r, 0, and 0, having given a? = r cos 0, y = r sin sin <^, % = r s\n0 cos 0. A slight artifice will enable us to do this with considerable! facility. Assume p = r sin 0, so that y = p sin (p, X = p cos 0, |0 = r sin 0, w ~ r cos 0. 40- CHANGE OF THE INDEPENDENT VARIABLE. Taking first the two variables y and ^, we find as in the preceding example d?V d'^V _(fV 1 ^V } dV dif^ dz^ ~ dp^ p"^ d(p>^ p dp ' In exactly the same way, the equations of condition being similar, we find d^ r d^V _d^V \ d?V idV d^'^'d7-~d?'^?d¥'^rd^' Also, as in the first part of the last example, 1 dV _ 1 dV cotOdV p dp r dr r^ dO Adding these three expressions, d^V d^V d^V dy^ d%^ da? _^y \ d^V \ d?V 2dV cot^ dV _ ^'d?'^V^d¥^p'd^' ^rl^ ^~7"de^^' By substituting for p its value, and making some obvious reductions, this becomes dJ'irV) 1 d'^V d f . ^^ dV \ dr^ sm^O dcff d.cos6\ d.cosOj This important equation is the basis of the Mathematical Theories of Attraction and Electricity. The artifice here used is given by Mr A. Smith in the Cambridge Mathe- matical Journal, Vol. i. p. 122. (7) Transform the double integral ffoj'^-iy^-^dyda! into one where u and v are the independent variables, w, y, w, V being connected by the equations ,v + y = u, y = uv. CHANGE OF THE INDEPENDENT VARIABLE. 41 . , Idx dy dw dy\ , , Here dy dw =^ [ — t^ - ^ —\du dv. \du dv dv dul doB dy dx Now — =1, -f-=u, — =0. du dv dv Therefore dy dx = u du dv, and ffw'"-'y"-'dydx = ffu"'^'"' (l -v)'"-'v''-'dudv. This transformation is given by Jacobi in Crelle's Journal, Vol. XI. p. 307 : it is of great use in the investigation of the values of definite integrals. (8) Transform the double integral jje'^'^y'dwdy into one where r and 9 are the independent variables, having given w — r cos 0, y = T sin 0, fje^'-'y'dxdy = - !je'\drde. (9) Having given \ X = r cos 9, y = r sin sin (p, ^ = r sin cos 0, transform the triple integral HfVdxdydz into a function of r, 0, and 0. Using the same artifice as in Ex. 6 we find fjjV dxdyd% = fffVr^ dr sin ddOdcj). This is a very important transformation, being that from rectangular to polar co-ordinates in space. If we suppose V = I, jjjdxdy d% is the expression for the volume of any solid referred to rectangular co-ordinates : and it becomes fffr^dr sin Odd dcj) when referred to polar co-ordinates. 4:2 CHANGB OF THE INDEPENDENT VARIABLE. (10) Having given z a function of x and y determined by the equation a?^ y^ %^ it is required to transform into a function of and when 0!= a sin 6 cos (p, y = b sinO sin (p, and consequently % = c cos 9. In this case dx dx —-= a cos 6 cos 0, — — = — a sin sin 0, acr a0 dy dy ^ . ^ -— = o cos 6 sm (b, -— = b sm cos 0, da ' 00 ' d% . ^ dz de = -"^'"^' d^ = ''- Hence - To eliminate a, substitute its value given by (2) in (l) ; dy then y^ = 2a?v 1- b. ^ ^ da To eliminate both a and b, differentiate (2) again ; then (Py (dy'^^ y^T^'^ m-"- (2) Eliminate a from the equation y = 00^ + ae^' ; dy my = (n — mx) af^~^. dw (3) Eliminate a from the equation m y = ax H — ; a the result is w f — J - « — + m = 0. \dw) dw (4-) Eliminate a and b from the equation y — ax^ — bx = ; the result is — 4 ^ + -^ = 0, a.r X dx x^ 44 ELIMINATION OF CONSTANTS AND FUNCTIONS. (5) Eliminate the constants m and a from y = m cos (jx + a). DiiFerentiating twice, -— = — r'^m cos {rx + a). Multiplying the former by r^ and adding, dF y (6) Eliminate m and « from the equation 'if' = m {c? — a?'^) ; , . ^y idyV dy the result is xy — — + x \—-\ - y —— = 0. dx^ \dxj dx (7) Eliminate e from the equation x - y = ce '-^ . Taking the logarithmic differential and eliminating, dy X - 2y + y — =0. dx (8) Eliminate a and /3 from the equation (x - af + iy- (if = r\ dy Differentiating, («" - «) + (?/ - p) ;p = 0- Differentiatmg agam, 1 + ( v^J + (^ -p)y-i = 0, \dxl \dx/ dy whence « - p = - ,., , x - a= ,<, -;— , d^y d-y dx daP dx^ Substituting these values of y - /3 and x - a, we have dc"^ = r ix" in which a and /3 no longer appear. ELIMINATION OP CONSTANTS AND FUNCTIONS. 45 This is the expression for the square of the radius of curvature of any curve. {9) Eliminate m from the equation (a + m/3) (v^ - my^) = my^ ; the result is (10) Eliminate a, 6, c from the equation z = ax + by + c, y being a function of x. Differentiating two and three times with respect to a?, d^z d?y d? z d? y = o— — -, and — ; = o . dx- dx^ dx^ dx'' Eliminating 6, we have d^ z 4py d^ % d?y dx^ dx'' dx^ dx^ This is the condition that a curve in three dimensions should be a plane curve. (11) Eliminate the exponentials from y „x „— * Multiply numerator and denominator by e*, then ^ ~ gS^ _ 1 ' whence e^' = , and 2x = los- . y-l' ^y-i' . dy and differentiating, — = 1 - y^. d X 46 ELIMINATION OP CONSTANTS AND FUNCTIONS. (12) Eliminate the power from the equation Taking the logarithmic differential we have dy m wy dee n a^ + ne^ (13) Eliminate the functions from y = sin (log od) ; , . ^d^y dy the result is x^ -— + .a? — - + ?/ = 0. doB^ dx (14) Eliminate the exponential and circular functions from y = ae"^" sin nx. Taking the logarithmic differential 1 dy - — = m + ncotnx, y da? Differentiating again and eliminating cotnx by the last equation, we have d^V dy dx dx (15) Eliminate the arbitrary function from the equation z=^xy(l)(y). Differentiating with respect to x only, — = 'U(h('u); and therefore x % = 0. dx ^^^^^' dx (16) Eliminate the function (p from the equation y — nz = (p {x — tnz). Differentiating with respect to x only, dz , ( d«\ - n —- ^ (b (x — mz) 1 - m -J— . dx ^ \ dx) ELIMINATION OF CONSTANTS AND FUNCTIONS. 47 Differentiating with respect to y only, d% dz I - n -— = - m m (x - m%) -— ^ dy ^ dy d% d% whence m —— +n-r- = l. dw dy This is the differential equation to cylindrical surfaces. « — 6 , ice — a\ (17) If <^[ , by the elimination of the function we find d% dz (x-a) -—+ {y-b) — =z -c. dw dy This is the differential equation to conical surfaces. (18) Eliminate (p and y\f from the equation Differentiating with respect to a?, Differentiating with respect to y, Multiply (1) by co^ (2) by y and add, , dz dz then a - — v y-— = n%. dx dy This is the differential equation to all homogeneous func- tions of n dimensions. It is to be observed that the two arbitrary functions are really equivalent to one only, for the original equation may be put under the form 48 ELIMINATION OF CONSTANTS AND FUNCTIONS. This is the reason why both functions disappear after one differentiation. If we proceeded to a second differentiation we should find or -p-g + 2a?2/ - — - — I- y 3— g =n{n - 1) x; dar dx dy dy for the third differentiation ^d^ % , d^% ,. d^z „d?z dcc^ dw^dy doedy^ dy^ and so on to any order. See p. 27. (19) Eliminate the functions from the equation z = (p(x + at) + v// (a? - at), w and t being variable, d' z ,. ,, -~=(f) {a) + at)+\lr (a; -at), d^ z — - = a^\f'{z)\ = (^{x). dw ELIMINATION OF CONSTANTS AND FUNCTION!?. Similarly, dz — {l - a; {ay + bx) + log \|/ (ay - hx), and as the functions are arbitrary their logarithms are also arbitrary functions, and we may replace them by the general characteristics F and/. Therefore, differentiating with respect to X and y successively, --— = hF' {ay + bx) -hf {ay -bx), z ax • 50 ELIMINATION OF CONSTANTS AND FUNCTIONS. - -—== a F (ay + ox) + af (ay - ox). % ay - ^(~)' = h' F" {ay + hx) + W f" {ay - hw\ Differentiating again, 1 d?z 1 fdzV \d'% 1 ld%\' zd^^-Ady) ="^ («2/ + 6a.) + aV {ay-hx). Multiplying by a^, 6^ and subtracting, we obtain as the result of the elimination of the functions [d'% \dx^ % \dx) J \dy'' %\dy] \ (23) Eliminate the arbitrary functions from (1) a?/(a) + 2/0(a) +^x|/(a) = 1, where a is a function of .r, «/, and % given by the equation (2) «?/'(a) + 2/0'(a) + ^v/.'(a) = 0; /'? 0'? '^' being the differential coefficients of /, d), -vl/^. Differentiating (l) with respect to x^ {^/'(a) + 2/ <^'(a)+i.>/.' (a)} ^ +/(«) + v/. (a) ^ = 0; which by the condition (2) is reduced to /(„) + ^(„)^ = 0. In the same way, differentiating with respect to y, we have ^(a) + >/,(a)^ = 0. Since from these two equations it appears that — - and dx d% — are both functions of a, the one may be supposed to dy be a function of the other, and we may wjite ELIMINATION OF CON&TANTS AND FUNCTIONS. 51 dx \dy Eliminating the function F from this equation there results /^\ fd^z\ _ I d^% y _ This is the differential equation to developable surfaces. (24) If u =-f{x,y) = Fir.z), and r = (p {aoe + ess) — -^ (ax - hy), 1 du 1 du 1 du then _ + _ + - =0. adw b dy cdz du du dr du d% dw dr da dss dx du du dr du dss dy dr dy dz dy -— = { a + c -—■] (h' (ax + cz) = ayl/' (ax - by)y dx \ dx) ^ dr d r -r- = C(t) (ax + cz). — = - bylr (ttX — by) : dz ^ ^ ^' dy ^ ^^ , , 1 dr I dr therefore, + - -— = o, a dx b dy _ 1 du 1 du du [1 dz 1 dz and 1- = — + a dx b dy dz \a dx b dy Also I « + c -J— ] (j) (ax + cz) = a-^' (ax — b y), \ CvX / J c -r-0' (ax + cz) = - 6\|r' (ax — by), 1 dz 1 dz 1 whence (- = ; a dx b dy c , , „ 1 du 1 du 1 du and therefore +7 — h = 0. a dx b dy c dz 4 — 2 CHAPTER V. APPLICATION OF THE DIFFERENTIAL CALCULUS TO THE DEVELOP- MENT OF FUNCTIONS. Sec. 1. Taylor's Theorem. This theorem, the most important in the Differential Cal- culus, and the foundation of the other theorems for the development of Functions, was first given by Brook Taylor in his Methodus Incrementorum, p. 23. He introduces it merely as a corollary to the corresponding theorem in Finite Differences, and makes no application of it, or remark on its importance. The following is the statement of the theorem : If u =f(,v) and a; receive an increment h, then du d^u h^ y— + &c. TT cos y Now tan'^a? = y, and a; = coty = — ; 2 smy therefore TT — = 2/ + sin ycosy + ^sm 2y (cos yY + ^ sin 3y (cosyY+kc. Again, let k = - I x + -] = -, ; then \ xj sm y cos y tan~^ (a? + ^) = tan~M 1 = - tan"' - = |-tan"'a?; \ wj OS 2 therefore TT sin?/ J sin 2?/ ^^ sin 3?/ ^ sin 4^ 2 cos 2^ ^(cos?/)^ ^ {cos yY ^ {cos yY Again, let A = - (l + a?^)^ = r-^— ; then ° sm^2/ tan-n.^-(l+^^)H=tan-^(^^f^)=-tan-'(tan|) = - |; therefore, as tan"' x = y, 2 - = - + sin 2/ + ^ sin 22/ + ^ sin 3?/ + &c. If we differentiate this series we find = 4 + cos y + cos 2?/ + cos 3t/ + &c. In these formulae ?/ lies between and ^w.. * Calc. Biff. p. 330. ^6 DEVELOPMENT OF FUNCTIONS. (7) Let u= cot"'i.2.3 &c. (14) If «*" log w = a{y), and if u =f(y), f and (p being any functions whatever, then u may be expanded in ascending powers of a? by the theorem This is Lagrange's Theorem. See Equations Nume- riques. Note XI ; Memoires de Berlin, 1768, p. 251. The Theorem of Laplace is an extension of the preceding made by assuming the given equation in y to be y ^ F\%-v op(p(y)\. Then if u=f(y), and if we put fF{%) = f(z), and i-fF{z)=f;{^), and <^F(^) = 0,(;^), V €i% V m J, Memoires de VAcademie des Sciences, 1777, p. 99. 64 DEVELOPMENT OF FUNCTIONS. In these theorems, if we make /{y) = y, we find and y = F{%)+(t>,{z)Fi^)- +_ [{0,(^)}"-r (;.)] — + &c. Ex. (l) Let y^ - ay + b = 0, or y = - + - y'^ ; a a Expand y in ascending powers of - . h Here f(y) = y, (y) = y\ x = -. a Therefore ^J0(.)}^ = 6(^)', ^. {0(-)P = 9.8. Q V &c ' T,r, b ^ h^ ¥ b^ 6' „ ^ Whence ?/ = (i +_+ 3 - + 12 - + 55 — + &c.) a a"* a" a a (2) Let a - y + by" = 0, or 2/ = a + &2/"- Expand y in terms of 6. Here f(y) = ?/, 0(?/) = y", % = a. Then 6^ b^ y = a {l+a"-'6 + 2w.a^"-^ + 3n(Sn-l) a'''-^ + &c.l. '^ 1.2 ^ 1.2.3 * (3) Let b —y + ta^ =0, ov y = b + ca}'. Expand y in terms of c Here / {y') = y, (p (y) = a-', % = b. Then c c c y = b + a^'.~ + 2 log a n^^ +3" (log afa^^ + &ec. 1 ^ 1.2 ^ ^ ^ 1,2.3 If ft = 1, or ?/ = 1 + cff/^, ^, ft c no « = 1 + a - + 2 log a h 3^ (loff «)'^ + &c. ^ 1 ^ 1.2 ^ ' 1.2.3 See Ex. 15 of the preceding Section. fore DEVELOPMENT OF FUNCTIONS. 65 (4) Let y = a + X log y. Expand y in terms of x. Here f{%) = z, f'(z) = 1, %= a,

»T? (5) Let a - y + 6 («/" + cy"") = 0, or y = a + b^y" + cy"^) : expand «/ in terms of b. Here 0(^) = z" + cz^, z = a; therefore 2/ = o+(o"+ca'')- + {2wa^"-* + 2c(»i+»')a"+''-^ + c^2ra^''-^} + &C. In the preceding examples it will be seen that the expan- sion of y in terms of b is the solution of an equation either algebraic or transcendental, and Lagrange has shown that the series always gives the least root of the equation. (6) Let y^ -ay + b = 0: 1 expand y" in terms of -. a b Here /(») = ^", 0(^) = ^^ z = - . it Whence b" , b^ 1 n(n + 5)b^ I n(n + 7)(n + 8) b^ 1 a"'- a^ a i .2 a* a- 1.2.3 a" a^ w(n + 9)(w + 10)(w + ll) 6M . 1.2.3.4 a^ a* ^ 5 66 DEVELOPMENT OP FUNCTIONS. (7) Let I - y + ay^ = : expand y" in terms of a. n n(n + 2r-l) , n(n + 3r-l)(n + 3r-9^)a^ «"=!+-« + — ^^ a^ + — ^^ — ^ 1 1.2 1.2.3 w(/i + 4r- 1) (»^ + 4r-2) (n + 4r-3) ^ 1.2.3.4 (8) Let 1 - 2/ + ae^' = : expand ?/" in terms of a, w(w+l) „ „ n{n^ + 3n-\-5) ^ 1.2 1.2.3 Lagrange has shown* that if by his theorem we develop the /I* negative power of the root of the equation y = % + i€^{y), and if we only retain the terms involving negative powers of %, the result gives us the sum of the 'nP^ negative powers of the roots ; while, as has just been stated, the whole series gives the ^th negative power of the least root. (9) If the equation be cy^ — by + a = Oy of which the two roots are a, j3, then ^■^/3»"W ^^~T'b'^ 1.2 b^'\b) n(n-4s){n- 5) c^ (a^ ^ 1.2.3 the series only continuing so long as there are positive powers b . „ a of - , that is, negative powers of - or %. * Equations Numiriques, p. 225. DEVELOPMENT OP FUNCTIONS. 67 (10) Let a-hy + cif = 0. Then if we represent the sum of the inverse v^^ powers of the roots by 2 (a~"), we have ^/ «x i^W (ay-' c n(n-2r + l) fa\^''-^c^ 2(a-)=(-)fl-„(-) .- + T?— (j) ^ n (n - 3r + 1) (n - 3r + 2) fay"'^ c 1.2.3 the series being continued only so long as it involves positive powers oi -. a If in these equations we substitute - for y, and then find the sum of the inverse w*^ powers of the roots of the transformed equation, we obtain a series for the direct w* powers of the roots of the original equation. (11) If we thus transform the equation in Ex, 10 it becomes c — by + ay^ = ; and if a, /3 be the same quantities as before, continued so long as there are positive powers of - . (12) Let u = m + esinu. Expand u and sin it in terms of e. The expression for u is u = m ■\- sin m . - + sin 2m ^ + - (3 sin 3 m - sin m) I 1.24' 1.2.3 + (8 sin 4m - 4 sin 2m) 1- &c. ^1.2.3.4 5 — 2 68 DEVELOPMENT OF FUNCTIONS. The expression for sin u is sin 2 m e 3 sin 3 wj — sin m e' sm M = sin m H + 2 1 4 1.2 + (2 sin 4m — sin 2m) v &c. 1 .2 .3 (13) We might employ Lagrange's Theorem to express h in terms of u from the equation du , d~ u h^ d^u h^ u + -—h + -— + — + &c. = 0; dx dso^ 1 .2 doo^ 1.2.3 but the following method is more convenient, as it gives at once the law of the series. It is easily seen that the series is the development of some function of w + h, which when A = becomes u. Let u =/(,a?), then /(a? + A) = 0. But since u =f(jv), w = f~^ (u), and if we call k the increment of u due to the increment h of w, x + h =f~^ (u +k), or, expanding by Taylor's Theorem, da? d^off k^ ^x k^ X ■ith = x+ -—k + -— — - + -—z + &c. du du^ 1 .2 du^l .2.3 But from the given equation we have u + k = 0, or k = — u, and therefore dx d^ X u^ d^ X u^ k = - —— u + — — 7-3 — - — -{- &c. d u du^ 1 . 2 du^ 1.2.3 dx . d^ X dv If we put — — = - V, then -— - = -u -— , du dw dx ^ X d ( dv\ f dy — - —_«-—« — ] = — [v -—] V, and so on. du dx \ dx) \ dx) d u^ / d \2 u^ Hence, h = vu + v-—v. 1- v — — v . f- &c. dx 1.2 \ dx) 1.2.3 This is the form of the expression which is given by Paoli, Elementi d^ Algebra, Vol, 11. p. 40. DEVELOPMENT OP FUNCTIONS. 69 (14) As an example of Laplace's Theorem, let us take y = log (z + a? sin y), and expand e^ in terms of x. Here f{y) = e^ F(z) = logz, fF(z)=f^(%)=!,, (p (y) = sin p, ^F («) = 0^ (z) = sin (log z). Therefore // («) = '^i -^{- 3sin(log2r)r r . „ x-»„ . „ . x, x-i T~i50/(^)i = V^[2-3{sm(log^)}^-sm(log;s)cos(log«)J = ^ { 8 - 9 sin (log z^-9. sin (log ^^) + 3 sin (log «^) } . ,,ri . „ V "^ sin (log «^) .a?^ Whence e^ = ^ + sin (loff ^) - + ^ — h ^ *= ^ 1 z 1.2 3sin(log^) , . ,, . ^, o . ,1 ,xi "^^ + ^^ — 1 8 - 9 sm (log ^) - 2 sm (log «^) + 3 sm (log zr) \ + &c. (15) Again, let z, = e'+*<:°sy : expand y in terms of oc. Here F {z) = e'', 0^ (2?) = cos e"", F' {z) = e". Therefore y = e'' + e' cos (e') - + e'' cos (e") {cos (e'') - 2 sin e^(e'')| -^ + e'' cos (e'') {(cos e^)" - Qe" cos (e") sin (e"^) + 9 e'== (sin ^f - Se'^ + &c. ^ M . 2.3 Sec. 4. Expansion of Functions by particular methods. The preceding Theorems sometimes fail from the function which is to be expanded becoming infinite or indeterminate 70 DEVELOPMENT OF FUNCTIONS. for particular values of the variable, and, more frequently, they become inapplicable from the complication of the pro- cesses necessary for determining the successive differential coefficients. Recourse must then be had to particular ar- tifices depending on the nature of the function which is given. One of the most useful methods is to assume a series with indeterminate coefficients, and then to compare the differential of the function with that of the assumed series ; so that by equating the coefficients of like powers of the variables conditions are found for determining the assumed coefficients. This method has the advantage of furnishing the law of dependence of any coefficient on those which precede it. Ex. (1) Let u = e^. (l) Assume u = tto + a^x + a^a/^ + &c. + a„ a?" + &c. (2) Differentiating (l) we have du ^ Differentiating (2) we have du 1 r X „ , X — = «! + 202^? + &c. +wa„a?"~^ + {n + l)a„+i<2?"+ &c. (4) da? Now e'' =1 + x + - — + &c. + + &c., 1.2 1.2...n and substituting in (3) for e^' the assumed series, it be- comes dtL — = jflo + «i ,2? + cfg 07^ + &c. + a„a?" + &c.( (5) dx X ^ 1 + a? + + &c. + + kc.\. '- 1.2 1 .2 ...W ' Comparing now the coefficients of a?" in (4) and (5) we find a„4.i= <«»+ «n-i + + + Sec + >: "* n+l\ 1.2 1.2.3 1.2...WJ DEVELOPMENT OP FUNCTIONS. 7l whence any coefficient is determined by means of those which precede it, except the first or agj the value of which is easily found by putting oe = in the original equation, in which case Uq = e^" = e. Therefore, forming the successive coefficients from this first one, r 2a?2 5a?^ 15 a?'' 52a?^ ^ 1.2 1.2.3 1.2.3.4! 1.2.3.4.5 * (2) Let u = e"^"*. Then by the same process as before we find the co- efficient of the general term to be given by the equation If ^n - S ®ra - 4 ^« - 6 o 1 7i + 1 ^ " 1.2 1.2.3.4 1.2.3.4.5.6 ^ There remains to be determined a^, which is easily seen to be equal to 1. Hence we find 6®"'^ = l+a7 + + &c. 1.2 1.2.3.4 1.2.3.4.5 1.2,3.4.5.6 (3) Let u =e Then 5^= ' _ _sin~' ar dx (1 - a}')h 13 1.3.5 and (1 - a?2)-2 = 1 +^a)^ + -^- a?* + — — '-—at^ + &c. ^ ^ 2 2.4 2.4.6 Therefore, assuming a series as in the preceding examples, we find for determining the coefficient of the general term, «« + != : l«n + ian-2 + ^r—r^n-i + &c.} : n + 1 2.4 ■' Also, it is easily seen, that Oq = 1j therefore ■„_, d?^ 2x^ 5 a?* 20«J?5 1.2 1.2.3 1.2.3.4 1.2.3.4.5 (4) Let w = («(, + «i a? + a2 -^J^ + &c. + a„ a?" + fee.)". Assume this to be equal to Aq + J^x + J^ar + &c. + ^„a?" + &c. 72 DEVELOPMENT OF FUNCTIONS. and take the logarithmic differentials of both expressions: equating these we have m {«! + 'ia^oo + SffgO?^ + &c. + (»^ + l) a„+ia?" + &c.| Oo + «ir ... + (^ + 1) (P + 2) «p+2 (cos ,2?)^+ ... } (sin wy. Putting 1 — (cos xy for (sin wy, and taking the coefficient of (cos wy we find it to be pap + p{p-l)ap-{p-{-l){p + 2) a^+g; and this must be equal to the coefficient of (cos wy in the original series multiplied by n^: equating these we have the condition (n^ - p^) (^p + 2 — ~ ip + l)(p + 2) ^' by means of which any coefficient is given in terms of that two places below it. There remain to be determined by other means the first two coefficients a^ and «i. For this purpose make w = (2r + 1) — in the original equation, r being any integer. Every term on the second side vanishes except the first, and there remains Gq = cosw(2r + 1) — . To find «!, make w = (2r+l) — in the second equation, when we obtain 74 DEVELOPMENT OP FUNCTIONS. sm w (2 r + 1) — ay = n — ^^^— — — = ^ cos (w-l)(2r+l)— . sm(2r + 1) - 2 Starting from these values and giving p successively all integer values from upwards, we find TT 7^ w^(w^ — 2^) coswr a^oo^ — a^oo^ + &c.; e —1 and subtracting the latter from the former, oa(l-e^) , , . — - — — = - a? = ^\aiOO + a^oif + &c.}, and comparing the coefficients of like powers of oc, «! = - ^, a^= 0, a^ = 0, &c. Also it is easy to see that a^=l', we may therefore assume 00 OB ^ 00^ ^ X'^ =1-- + B, ^3 + &c. e'^-l 2 M.2 M.2.3.4 i5j, ^3, Sec being coefficients to be determined. The coefficient of o?^" in the expansion of is there- n fore (-)"'*'^ '^ — - , and consequently the corresponding coefficient in Jf- is (-)-^ ' ^^"- e^ -1 2^" 1.2.3...(2?i) If therefore Cg„ be the coefficient of a?^" in -~ — , we have the equation DEVELOPMENT OP FUNCTIONS. 77 (-)'-(,4-)t 2.3... (272) — ^2n. But Can = -7 7 7 (-;-) f-;^ l^ when a? = 0, 22"i.2...(2w- 1) V<^W W+i; and if in Ex. (27) of Chap. II. Sec. 1, we make r = 2w - 1, 0? = 0, /_d_y"-l / 1 \ ^ /XSn-l j_ (-jSn-l _ fp2»-l _ ^ ,2«-l> r o , 2w „ , 2w(2w-l) „ ,^ * 1 1.2 'J Substituting these values, we find B, {-y'^n ^^3„_, , ^„_, 2n '— ^ o2m /'o2w 2-« (2« rj2n-l _ ig^n-l _ ZH. l2n-l> + jgSn-l 22n-l ^ V J j2„_n _ ^^. 1 * 1 1.2 ^ -' These coefficients 5i, Bs...B2n-ij are of great use in the expansion of series, and bear the name of Bernoulli's numbers, having been first noticed by James Bernoulli in his posthu- mous work the A7's Conjectandi, p. 97 ; but the complete investigation of the law of their formation is due to Euler, Calc. Diff. Part ii. Cap. V. (10) To expand tan 9 by means of the numbers of Bernoulli 1 e(-)^2e _ 1 ^ 1 r 2 . *^^ ^ - (T)! e(-)^2e + 1 - (Zji |l - 1 + ,(-).2a) • The coefficient of 0^""^ in the expansion of this function will be the same as that of a?^" in the development of e + 1 multiplied by 2^"(-)". By what has preceded it appears, therefore, to be equal to 2^" (s'-^"- 1) 1.2... (2w) ^'"-'' 78 DEVELOPMENT OF FUNCTIONS. Hence, giving n successive values, we find ^ 4.3 „ ^ 16.15 _ ^, 64.63 tan = — B^e + ^30^ + B,e' + &c. 1.2 1.2.3.4 1.2.3.4.5.6 (11) To expand cot 6 by means of Bernoulli's numbers, cot e = (-)i -p^,^,-^ = (-). |i + ;hW3-J . Now the coefficient of ^^""^ in this expression is the same as that of a?^" in the expansion multiplied by (-)"2^": € ■" -I it is therefore equal to (-)"2^"^,„_, Hence we have cot = i BiO ^30^ - &c. e 1.2 1.2.3.4 CHAPTER VI. EVALUATION OP FUNCTIONS WHICH FOR CERTAIN VALUES OP THE VARIABLE BECOME INDETERMINATE. If w be a function of x of the form — , and if for the value x = a, P and Q both vanish ; u, taking the form -, is indeterminate and its true value will be found by dijfferentiating the numerator and denominator separately and taking the quotient of these differentials : that is, using La- grange's notation, the real value of u will be u = ■—,, when x = a. But if the same value (x = a) which makes P and Q vanish also make P' = 0, and Q' = 0, we must differentiate again, and so on in succession, as long as the numerator and the denominator both vanish when w is put equal to a. There- fore we may say generally that the true value of u when a? = a is p(») P'"* and Q^"* being the first differential coefficients of P and Q which do not vanish simultaneously when w is put equal to a. This theory of the evaluation of indeterminate functions was first given by John Bernoulli, Acta Eruditorum, 1704, p. 375. Ex. (1) 1-0?" , u = when \ - a; a? = 1. Here P = 1 - a?" ; Q = 1 - .0? ; P' = - n.v'-^ ; Q' = - 1 ; 80 EVALUATION OF INDETEEMINATE FUNCTIONS. and therefore when a? = 1, u n. 1 — + X'' 9 + X TTX — 1 TT IS u= — + ix" X (e^'^^' - 1) find its value when a? = 0. (7ra7-l)(6'^^- 1) + 2 7r/i? , Here u = ^^ -, ; = - , when a? = 0, Differentiating three times, we find the real value to be EVALUATION OF INDETERMINATE FUNCTIONS. 89 which is the sum of the reciprocals of the squares of the natural numbers. (35) The sum of the series 1 1 I + -r + -:; ^ + &C. to CC. 1^ + w^ 3^ + 00^ 5'' + of IT TT IS «^ = -—: r = CO — CO , when x = 0: 4>W 207(6'^^+ 1) TT 6^^-l , u — — — ; = - , when x = 0. 4 X (e^* +1) Differentiating once, we find u — which is therefore the sum of the squares of the reciprocals of the odd numbers. The reader will find other examples of a similar kind relating to the summation of series in Euler's Cole. Diff. p. 760, seq. Sometimes the value of an indeterminate function may be most readily found by throwing it into a form in which its real nature is more easily seen. (SQ) If «^ = 2* sin - ; find its value when a? = co , . a sm — 2* This function may be put under the form a , a 2* , . , .„ o ^ , sin which, if — = 0, becomes a —-— . When a? = cc, = 0, 2' e 5 ' J sin n , „ and — — — = 1, and therefore u ^ a. u (37) Let u = (a^ - x^)^ cot {- I | > = 0.co, when [2 \a + xj J iV = a. 90 EVALUATION OF INDETERMINATE FUNCTIONS. This may be (a'- put u inder the form IT [a - //(,37). See his Memoires de Mathematique et de Physique^ Vol. i. p. 44, and Crelle's Journal, Vol. x. p. 303. In the same Journal^ Vol. xii. p. 134 and p. 292, the reader will find some discussion on the real value of this indeterminate ex- pression. CHAPTER VII. MAXIMA AND MINIMA. Sect. 1. Explicit Functions of One Variable. If uhe an explicit function of cc, then the real roots of — - = are the values of w which render u a maximum or doD a minimum. If a root oi -— - = substituted m give a negfative dx dx^ * * result, the corresponding value of w is a maximum. If a du . d-w . . , root of -— = substituted in -— - give a positive result, dx dx' ' the corresponding value of «^ is a minimum. du ^u If a root of -— = also make -— ^ = 0, the corresponding dx dx~ ^ ^ ^u value of w is neither a maximum nor a minimum, unless dx^ also vanish for the same value ; and generally, u is neither a maximum nor a minimum for a given value of <2?, unless the first differential coefficient which does not vanish for that value of ob be of an even order. If, when 0? + /i is substituted for x in w, the expansion in terms of h assume the form PA« + &c., where a is less than unity, the values of x which make u a maximum or minimum are to be sought in the roots of the equation 1 du dx 94 MAXIMA AND MINIMA. P If u be of the form — , the equation for determining w dP dQ IS Q— P —-— = 0, unless the same value of oo make Q = 0, ax doo in which case -— = - , and its real value must he found by dw ^ the usual method. The corresponding value of u will be a maximum or a minimum according as the value of x when substituted in Q -— — P ——^ gives a negative or a positive Qj 00 Qj CG result. The values of x which make Q = make m = co, unless P = at the same time ; and this value of u will sometimes be a maximum, sometimes not. If the same value of X which makes w = co make — a minimum, then w = co u is a maximum ; — otherwise not. The investigation of the maximum and minimum values of u is sometimes facilitated by the following considerations. If ?^ be a maximum or minimum, and a be a positive quantity, au is also a maximum or minimum. When w is a maximum or minimum, ©m^""*"^ is so also; , a . . , but IS mverselv a minimum or maximum. If M be a positive maximum or minimum, aw^" is also a maximum or minimum. If ?^ be a negative maximum or minimum, «w^" will be a minimum or maximum. The same remarks apply to fractional powers of the function w, except that when the denominator of the fraction is even and the value of u negative, the power of u is impossible. When w is a maximum or minimum, log u is generally a maximum or minimum, except when the same value of the variable makes both u and — vanish. This preparation of dx the function is frequently made when the function u consists of products or quotients of roots and powers, as the dif- ferentiation is thus facilitated. Other transformations of u are sometimes useful, but as these depend on particular forms which but rarely occur, they MAXIMA AND MINIMA. 95 may be left to the ingenuity of the student who desires to simplify the solution of the proposed problem. In testing by the sign of whether the value of u be a maximum or a minimum, the following consideration will sometimes shorten the process. If — - be of the form xi^.v^. v^...Vj^^ and y = ^a determine the cone of greatest surface. (30) To inscribe the greatest cone in a given sphere. Let the radius of the sphere be r, and the distance of the base of the cone from the centre be w. Then .t = 1 r gives the maximum cone. MAXIMA AND MINIMA. 103 »^ (31) To find the point in the line joining the centres of two spheres, from which the greatest portion of spherical surface is visible. If r, / be the radii of the spheres, a the distance of the centres, and a? the distance of the required point from the centre of the sphere whose radius is r. a Then x = -^ t^ . ra + r (32) A regular hexagonal prism is regularly terminated by a trihedral solid angle formed by planes each passing through two angles of the prism ; find the inclination of these planes to the axis of the prism in order that for a given content the total surface may be the least possible. Let ABC ah c (fig. 6) be the base of the prism, PQRS one of the faces of the terminating solid angle passing through the angles P, R. Let S be the vertex of the pyramid. Draw SO perpendicular to the upper surface of the prism. Join OM, RP, SQ intersecting each other in N. Then it is easy to see that 3IN = NO and consequently SO = QN, and as the triangles POR, PMR are equal, the pyramids PSRO and PMRQ are equal, so that, whatever be the in- clination of SQ to OM, the part cut off from the prism is equal to the part included in the pyramid SPR, and the con- tent of the whole therefore remains constant. We have then to determine the angle ONS or OSN so that the total surface shallbe a minimum. Let AB, the side of the hexagon, = a, AP, the height of the prism, =6, OSN = 9. Then ON = NM =\a, and SN = \a cosec 0, and QM = ^a cot 6. The surface of ABPQ = ^a(2b-^acot0). The surface of PQRS = PR . SN = cosec 0. 2 Whence the total surface of the solid is a ^ 3-?a"^ 3a (2h cot 6) + cosec 0, 2 2 which is to be a minimum. Differentiating we have 1 , COS0 , , f n ^ - 3 — 14 a- 6^ + 6* > 0, or a > 6 {2 + S^J; that is, the angle of the cone must be less than 15°. When this is not the case, the ellipse increases continually till it coincides with the base. It may happen that the maximum value of the section is less than the base of the cone; and this will be the case unless the vertical angle of the cone be less than 11°. 5?'. Sect. 2. Implicit Functions of Tivo Variables. If u be an implicit function of two variables cV and y, — = will determine the values of x for which v is a dx ^ maximum or minimum, y will be a maximum for a given d'u value of ^ - 6.r^ +1 = 0; one root of this equation is x = — \, which gives y = 1 ; dru dx'' - 6x - 2y and as = 5— = + 4, « = 1 is a maximum. du 2y - x' ^ dy (2) y^ + 2yx^ + 4 I -I — r— must also hold, m order that d x' dy"^ \dxdyj the function may not change its character from a maximum to a minimum, or vice versa, in the course of the changes of the variables. From this condition it appears that n 71 fj 11 — - and -—7 must have the same sign ; and the function dar dy^ will be a maximum if that sign be negative, and a minimum if it be positive. , du du If the values of w ana y which make — — = 0, — = 0, also dx dy make the second differentials vanish, there will be no maximum or minimum unless the third differentials also vanish, while the fourth neither vanish nor change their sign. If u be a function of three variables x y z, the con- ditions for a maximum or minimum are du du du dx dy ^ dx- and Lagrange's condition becomes in this case \d'u d'u / d^u yW 0. MAXIMA AND MINIMA. Ill Therefore <2? = 0, y = gives neither a maximum nor a minimum, and x = a, y = a gives u = — a^, a minimum when a is positive, and a maximum when a is negative. da; dy Eliminating y between these equations, we find X' {{^'~-iy-i} = o. whence oa = 0, a? = ± 25, y = 0, y=^<2^: a? = 0, y = give u = 0, a maximum; a? = ± 23, ?/ = r^ 22 give w = — 8, a minimum. (S) %i = of y"^ {a - X - y), a a , d' , X = - . V = - give u = — a maximum, 2 3 *= 432 a? = 0, y = 0, give «* = neither a maximum nor a minimum. (4) W = ^ (a + x) (x + y){y + %) {^ + b)' Taking the logarithmic differential, since log w is a maximum when u is so. Then • 1 1 = 0, 1 du 1 — — = — u dx X 1 du 1 u dy y 1 du 1 u d% % a + X X + y 1 1 X + y y -T % 1 1 = 0, = 0. y -^ % % + b or ay - x^ = 0, x% - y^ = 0, by - z^ = 0; . a X y z whence -=- = - = -, X y % b or a, a?, 2/, ^, b are in geometric progression. Let each of these ratios be equal to — . Then, multi- n plying them together, 7= — , on n= (-] . b w* \a) 112 MAXIMA AND MINIMA. Let \ogu = v, then proceeding to the second differentials we get, on substituting for x, y, z the values na, n'a, n^a, d'^v 2 d-v 2 div^ d'n (l + riy d-if d^n^ (l + n)' ' d^v 2 d~v 1 d%^ d'iv' {\ + ny^ dwdy a^n^{\ + ny d'v 1 d^v dyd% d~n^{l + n)' dwdz so that Lagrange's condition becomes 0; 12 4 a^-n}° {I -^ ny a^w'" (1 + n)'*' and the corresponding value ii = : — j j— is a maximum. (a* + hiy k (5) Let - = nv'^ + 2xy + ty^ ; P X and y being connected by the equation 1 = (l + p^) oe" + 2pqwy + (1 + q') y*. Differentiating these two equations, = {roG -f sy) dx + {sw + ty) dy, = {(1 +p^) X +pqy] dx + \pqx + (l + q[')y\ dy. Multiply the second of these equations by an indetermi- nate quantity X, add it to the first and equate to zero the coefficients of the differentials : this gives X j (l + p^) w + pqy} + rx + sy == 0, X {pqx + (1 + q^)y] + sx + ty= 0. Multiply these equations by x, y respectively and add, then, by the original equations X + - =0. Substituting this value of X in the preceding equations, and grouping together the terms multiplied by the same variable. MAXIMA AND MINIMA. 11« |- (1 + />-) - A w = - I- pq -sj y I- (1 + q") -t\y ^ ~ [- pq- s\ ic \P ] ^p Multiplying these together so as to eliminate x and i/, we find k^ k -(1 +p- + q^) -- {(1 +q^)r -2pqs + (1 + p^~)t] +rt - s" = 0, P P a quadratic equation in -, whence a maximum and a mini- P mum value may be found. This is the equation for determining the radii of maximum and minimum curvature in a curved surface. (6) Let u = ay (c — is) =h!S (a — .v) = ex (b — y). Then a? = ^ a, y = ^b, ^ = ^ c, give u = ^ahc, a maximum. (7) Let U = acos-w + h cos^y ; a? and y being subject to the condition ?/ — , and the value of the minimum sum may be found. For from the geometry of the figure we have the equations V' + viv + w^ = d\ ti^ + uiv + w~ = b^, u^ + n v + v^ = c^, and 7iv + uw + vw = — ?-, si w?- being the area of the triangle. Adding the sum of the first three equations to three times the last, 2{u + V + wY = a~ + 6- + c^ + 4 . 3^ m^ ; whence u +v + w = \^(b there is a loop in the inferior conchoid at O as in the figure ; when a = b the loop degenerates into a cusp ; and when a ') = fia;) f{w) - Fi,oc) F(./), cF{x + w') = F{w) f{w') + f{w) F (c^0, cF{w - w') = F{w)f(.v) -f{w) F{w'). It is obvious that the preceding formulae are analogous to those connecting sines and cosines of circular arcs. For these and other properties of the catenary connected with the involute of the parabola, see a paper by Professor Wallace in the Edinburgh Transactions, Vol. xiv. p. 6^5. (7) The Quadratrix of Dinostratus. If the radius CQ of the circle ABD (fig. 18) revolve uniformly round C from A to B, while the ordinate NM also moves uniformly parallel to itself from A to C, the locus of their intersection will be the quadratrix of Dinos- tratus. To find its equation, let AM = x, PM = y, AC = a. Then from the uniformity of the motion of CQ and MN^ we have ACQ : ACB = AM : AC; IT a, &c. from the origin. The farther these points are removed from the origin the more nearly is the curve perpendicular to the axis of cV, the value of — at the intersection being i(27i-l)-, 2na being the abscissa of the point where the, curve cuts the axis of x. Rolling Curves. (8) The Cycloid. This curve is generated by a point P in the circumference of a circle hPc (fig. 19), which rolls along a line AA'. To find its equation put IS 4 GENERATION OF CURVES. 0'b = a, PO'b==0, AM = X, PM = y. Then AM = Ab - Mb, or cV = a (0 - sin 0), PM = O'b + O'd, or y = a {\ - cos 0). These two equations taken simultaneously represent the curve, or, if we eliminate between them, we obtain as its equation x = a vers" ' i^ciy — y^)^- a If we take C, the highest point of the curve as our origin, and put CN=.v, PN = y, and cO'p = (p, we should find cV = a (l — cos (p), y = a {(p + sin (p) ; - whence y = a vers" ^ - + (2a,v — w^)i. a It is easy to see both from geometrical and analytical considerations that the cycloid is not limited to the space between A and A', but that it consists of an infinite number of portions equal and similar to AC A' and touching each other in cusps as in the figure. After the Conic Sections there is no curve in geometry which has more exercised the ingenuity of mathematicians than the cycloid, and their labours have been rewarded by the discovery of a multitude of interesting properties, im- portant both in geometry and in dynamics. The inventi&n of this curve is usually ascribed to Galileo, but Wallis in a letter to Leibnitz* says, that it is mentioned by Cardinal de Cusa in a work published in 1510, and that in the MSS. the date of which is about 1454, it is "pulchre delineatam", therein differing from the printed copies. Ro- berval proved that the whole area of the cycloid is three times that of the generating circle, and this discovery, which was the cause of many disputes between rival claimants to the honour of making it, drew the attention of mathematicians to the study of the properties of this new curve. Among others, Descartes occupied himself with the subject, and he * Le'ibn. Opera, Vol. ju. p. 95. GENERATION OP CURVES. 135 showed how to draw tangents to the curve, and proved that the tangent at any point P (fig. 19) is perpendicular to the corresponding chord BQ of the generating circle, and con- sequently that it is parallel to CQ: from this also it readily follows that if QR be a tangent to the generating circle at Q, QR = PQ = arc BQ. Wren was the first who rectified the cycloid, and he showed that the length of an arc measured from the vertex is equal to twice the chord of the generating circle which is parallel to the tangent at the extremity, so that the whole length of the curve is equal to four times the diameter of the generating circle. Pascal discovered the means of finding the area and the centre of gravity of any segment of the curve as well as the content and surface of the solids formed by the revolution of the segment round the axis of the curve, and the base of the segment, and to the solution of these problems he challenged all mathematicians in a letter which he circulated under the name of Dettonville, offering at the same time a prize of forty pistoles to the first and one of twenty pistoles to the second person who should solve them. Wallis and Lalouere appeared as candidates for the prize, but none was awarded. To Huyghens is due the discovery that the evolute of the cycloid is an equal cycloid in an inverted position, and that the radius of curvature is double of the chord of the generating circle which is perpendicular to the tangent. He also discovered the important dynamical property of the tautochronism of a cycloidal pendulum ; that is to say, that a body under the action of gravity falling down an inverted cycloid with its base horizontal, will reach the lowest point in the same time from whatever point it begins to fall. Two of the most remarkable properties of this curve were discovered by John Bernoulli: 1st, that it is the curve along which a body will, under the action of gravity, fall in the shortest time from one given point to another not in the same vertical : 2nd, that if any arc of a curve as AB (fig. 21), the tangents at the extremities of which are at right angles to each other, be evolved into a curve BA', beginning from B : and if the same operation be performed on A'B, beginning from A\ and so on in succession, the successive 136 GENERATION OF CURVES. involutes will continually approximate to a common cycloid, the axis of which is parallel to AC*- The preceding are only a few of the most important properties of this curve ; for a detailed account of all which the industry of mathe- maticians has discovered, the reader must be referred to the treatises on the cycloid which have been written by various authors. Such are the Histoire de la Roulette of Pascal ; the History of the Cycloid of Carlo Dati ; the Treatise de Cycloide of Wallis ; the Historia Cycloidis of Groningius in his Bihliotheca Universalis ; and the work of Lalouere called Geometria promota in VII de Cycloide lihris. (9) The Companion to the Cycloid. If the ordinate Q,N (fig. 20) of a semicircle be produced till it be equal to the arc CQ^ its extremity will lie in a curve which is called the companion to the cycloid. The co-ordinates of a point in this curve are, putting CO = a, CN^w, CN=y, COQ = e, .V = a (l — cos 0), y — aO. It has points of contrary flexure at the extremities D and d of an ordinate passing through the centre of the gene- rating circle. The space COD is equal to the square of the radius; the whole area ACa is equal to twice that of the generating circle, and if the line AC be drawn, the area AMD is equal to the area CLD. (10) If instead of supposing the point P to be in the circumference of the generating circle we suppose it to be either within the area of the circle or without it, the curve traced out is called a Trochoid. The equations to such a curve are ,v = a (9 — n sin 0), y = a (l — n cosO), where n is the ratio of the distance of the tracing point from the centre of the generating circle to the radius of that circle. * John Bernoulli, Opera, Vol. iv. p. 98. Euler, Commen. Petrop. 1766. Ivegendre, Exercices du Calcul Integral, Tom. ii. p. 491. GENERATION OF CURVES. 137 (ll) Epitrochoids and Hypotrochoids. When the generating circle rolls, not on a straight line, but on the circumference of another circle, the curve generated is called an Epitrochoid or a Hypotrochoid, according as the curve rolls on the exterior or interior of the fixed circle. Let (fig. 22) be the centre of the fixed circle, C that of the generating circle, a, h their radii. Let A and Q be the points originally in contact, P the tracing point. Then if we make CP = A, CN^ X, PN = 2/, AOB = 0, so that QCB = ^ 0, o we find 'a + h x = OH+ HN =(a + b)cose- h cos | — — ) 0, = CH - CK =^ {a -^ b) sin e - h sin (^^) 9. If we suppose the generating circle to roll in the inside of the fixed circle as in fig. 23, we should find (a -h\ ^ o) = (a - b) cos Q + h cos ( — - — 0, , (a - b\ . y = {a, - b) smO - h sm I — - — j Q When h = b these become the equations to the Epicycloid and Hypocycloid respectively. When a and b are commen- surable the curve will re-enter after a number of revolutions of the generating circle equal to the least common multiple of a and 6 : in such cases the curve is expressible by an alge- braical equation between c^ and y. When a and b are in- commensurable the curve will never re-enter, and is ex- pressible only by some transcendental equation between as and y. \i h = b and 6 = a the equations to the epicycloid are X = a {2 cos Q — cos 2 0), 2/ = a (2 sin - sin 2 9), or .17 = a { 1 + 2 cos (l - cos 0) } , y = 2 a sin (1 - cos G). 138 GENERATION OF CURVES. Whence, squaring and adding, w' + y^ = a^ { 1 + 4 (1 - cos 0) } . But we have also (x - ay + 2/^ = 4a^ (1 - cos Oy. Therefore ( . The perpendicular from the origin on the tangent is du du ydoB — rudy dw dy nc ^ " (da^~ + dy')h " ( /duV' (duY\^ " {(duy JJHyTl ' \[d-J "■ WJ I Ifej -^ [dyj ] if w be a homogeneous function of n dimensions in cV and y. The portion of the tangent intercepted between the point of contact and the perpendicular on it from the origin is du du w y — 00 dw -^ ydy dy doc (d.v^ + dy '■)i Uduy ,duy\-^ The portions of the axes cut off between the origin and the tangent, or the intercepts of the tangent, are y — CO ~— along the axis of y^ dco . 00 — y -— along the axis of oc. dy ^ These I shall call y^, oo^ respectively. Ex. (1). The equation to the hyperbola referred to its asymptotes is coy = m^. , du du , , . , 1 nen —- = y, -—■ = oo, and the equation to the tangent is doo dy y {og' -a!)+x(y' -y) =0; or yoff' + xy = 9.xy = 2m^, 144 TANGENTS TO CURVES. dy m^ dx x^y Since -r- = r » the subtanffent = « -— = j = - x\ dx x^ dy m as xy = m^. ,. , 2m^ The perpendicular on the tangent p = — -^ g-^ . {x + y )a Also yQ = y + — = 2«, and Xq= x + — = 9.x. X y Hence the product of the intercepts of the tangent = a?ot/o = 4a;2/ = 4m^ is constant ; and the triangle contained between the axes and the tangent, being proportional to this product, is also constant. (2) The equation to the parabola referred to two tangents as axes is Hence the equation to the tangent is y {ax)h iby)h The intercepts are x^ = {ax)i, y^ = (J)y)h ; or and is possible, and therefore the axis of y is an asymptote: this is one of the ordinary kind. But if we put the equation under the form it is easily seen that when x = QO , y = c. On the other hand, if a? > a, y is impossible. Hence the line whose equa- tion is 2/ = c is an asymptote to an impossible branch of the curve ; that is to say, a branch of the curve leaves the plane of reference when a? = =t a, but tends to return to it again when .t? = ± CO , coinciding then with the line whose equation is y = c. The form of the curve is given in fig. 27, where the dotted curve represents the impossible branches of the curve lying in a plane at right angles to the plane of the paper. On the subject of asymptotes to curves, the reader may consult in addition to the work of Stirling before referred to, Newton's Enumeratio Linearum Tertii ordiniSf and Cramer's Analyse des Lignes Courbes, Chap. viii. * Cnmhridfje Mathematical Journal, Vol. ii, p. 23(). 156 TANGENTS TO CURVES. Sect. 2. Polar Co-ordinates. If the curve be expressed by a relation between r and d, then the tangent of the angle (0) between the radius vector dO and the tangent to the curve is r — . The subtangent, which is the portion of a perpendicular to the radius vec^ tor at the origin intercepted by the tangent, is r^ — ; and dr the perpendicular from the origin on the tangent is r' l-QT If the curve be expressed by a relation between ii and G where w = - , the subtangent and perpendicular are equal to r — r- and .,, 1 respectively. du { , (du\ni ^ ^ Asymptotes to spirals are determined by finding what value of d makes r infinite; and if the same value of 9 dG make r^ — either finite or equal to zero, a line drawn through dr the extremity of the subtangent parallel to r is an asymp- tote to the curve. Spirals may have asymptotic circles : these are found by the condition that an infinite value of G gives a finite value for r. Ex. 1. The equation to the spiral of Archimedes is r = aG. The angle between the radius and tangent is . dG d) = tan V — - = tan ' G. ^ dr ,,2 The subtangent = — . a TANGENTS TO CURVES. 157 The equation to the locus of the extremity of the subtan- gent is evidently / = - = ae% a being measured from a line 90" distant from the original axis as r is at right angles to r. If in a similar way we find the locus of the extremity of the subtangent of the curve r' — aO^, and so on in succession, we shall have a series of spirals, the equations to which are 1.2' 1.2.3 1 .2 ... (w - 1)' the angle 6 in each case being measured from a line 90° distant from that in the preceding curve. (2) The equation to the hyperbolic spiral is a r = - , or ?^ = - ; B a d9 therefore the subtangent =-—=«. ^ du The locus of the extremity of the subtangent is evidently a circle, the radius of which is a : and as 6 = makes r = co while the subtangent remains finite and equal to a, it appears that a line drawn parallel to the axis at a distance a is an asymptote. (3) The equation to the lituus is a 0i r = -I , or 7/ = — ; 02 a then = tan~^ (- 20), subtangent = 2 a^a; and as = makes r = co and the subtangent = 0, it appears that the line from which 6 is measured is an asymptote to the curve. Also since r^O = a^ it appears that if a circle be described with radius r, the sector between the axis and the radius r is of constant area. (4) The equation to the Lemniscate is r" = a^ cos 2O. 158 TANGENTS TO CURVES. The perpendicular on the tangent is — : (h = tan-'r — = tan"' (- cot 2 0) = 20- -. (5) The equation to the logarithmic spiral is e r = c 6 « . Then d) = tan~^a, and is therefore constant; ^ = r sin (tan ^ a) = ^ y The subtangent = ra. The locus of the extremity of the subtangent is the involute of the curve, the equation to it bping e r^=^ ar = acef^ , and therefore a similar spiral. Also if ra be the subnormal, that is, the portion of a perpendicular to the radius vector at the origin cut off by the normal, the locus of the extremity of r.^ is the evolute of the spiral, its equation being r c - r2 = - = - e« . a a (6) The equation to the Cardioid is r = a (l — cos &). If / be a radius in the direction of r produced backwards, r' = a J 1 - cos (0 + tt) 5 = a (l + cos Q). Therefore r + r' = 2 a, or the chords passing through the pole are of constant length. tan = tan 10; therefore cp = ^6. (7) Let the equation to the spiral be r" = a" sin nO. Then tan

for a true double pomt, \dxdyl \dx') \dy^) ^ 164 SINGULAR POINTS OF CURVES. / dru \' [d^u\ ld~u\ „ . ,. , ■ I - — 7, -r-T = for a point oi osculation, { d^u \^ (d^u\ fd^u\ — — :r\ < lor a coniua-ate point. Ua^dy) Kdco'j \dyV i & v If the point be more than double, it is necessary to pro- ceed to higher differentiations, but the formulae become too complicated to be of much use. The second of the preceding conditions furnishes an easy demonstration of the following general property of curves of the third order. " The three asymptotes of a curve of the third order being given, the locus of the points of osculation is the maximum ellipse which can be inscribed in the triangle formed by the asymptotes : the locus of the conjugate points is within, and of the double points without this ellipse." If we refer a curve of the third order to two of its asymptotes as axes, their intersection being the origin, its equation must evidently be of the form, ax^y + 9.hwy ->r caoy"^ =:h. ^^ du , o Hence — = ^axy + 2b y -\- cy-j dx — = ax^ + 2hx + 2cxy, dy d"u d'^u d^u . -T-rj = ^ay, -r^ = 2cx, ~~~ = 2 (ax + h + cy). dx dy^ dxdy Therefore by the condition for a point of osculation {ax + 6 +cyY - acxy = 0, or a^x" + acxy + c^y"^ + 2ahx + 2hcy + b^ = 0, which is the equation to an ellipse. That this ellipse is the maximum ellipse inscribed in the triangle formed by the asymptotes is easily shown. The equations to the three asymptotes are X = 0, y = 0, and ox + cy -\- 2b = 0. SINGULAR POINTS OF CURVES. 165 From the last it appears that the intercepts of the axes 26 2c cut off by the third asymptote are — — and . Also from the equation to the ellipse it appears that it touches he the axes at distances and from the origin, or that the points of contact bisect these two sides of the triangle. If in the value of — derived from the equation to the dco b h ellipse we substitute the values - - and for cc and y, a c we find — = — - J which is the same as that derived from dx c the equation to the third asymptote, and as these values of X and y satisfy both the equation to the ellipse and that to the asymptote, it appears that the ellipse touches all the three sides of the triangle in their middle points, which by Chap. vii. Ex. 19, is the property of the maximum el- lipse. The latter part of the theorem is too obvious to need demonstration. This proposition is due to Plucker, Journal de Mathtmatiques, {Liouville) Vol. 11. p. 11. Points of Contrary Flexure or of Inflexion, Ex. 1. The equation to the Witch of Agnesi is xy = 2a (2ax - x-)^ ; whence we find d'y 2a^{3a-2x) dx^ X (2 act? - ,i?^)i ' d^y . Sa . 2a 3a ^ — — - = gives X = — and « = ± — ,- , and as [■ h and dx^ ^ 2 ^3^ 2 3 a . d^y h, when substituted for x, make -— change sign, there are two points of contrary flexure corresponding to these values of x and y. 166 SINGULAR POINTS OF CURVES. — T and — both become infinite when co = and when dx^ aw w = 2a, but neither of those values gives a point of in- flexion, since y is impossible when x is negative or greater than 2 a. (2) The curve whose equation is w^ — 3b w'^ + a^y = has a point of inflexion the co-ordinates of which are 'V=b, y = — a^ (3) Let the equation to the curve be aar' + by^ - c'^ = o. There are two points of inflexion, the co-ordinates of the one being cV = 0, ?/ = c I - j , those of the other x = c l-\ , y = 0. (4) Let the equation to the curve be a?* — a^x^ -f a^y = 0. There are two points of inflexion corresponding to a 5a 65' -^ 36 (5) Let the equation to the curve be y = b + {x — a)^ , where m and n are both odd. If — > 1, X = a gives a point of inflexion, the tangent n being parallel to the axis of x. If — < 1, X = a gives a point of inflexion corresponding n to — - = CO, the tangent being perpendicular to the axis of x. dx' SINGULAR POINTS OP CURVES. 167 (6) In the curve of sines the equation to which is . OP y = c sin - , a there is a point of inflexion wherever the curve cuts the axis of X. In polar curves points of inflexion are found by the conditions that at such points — - = 0, and changes sign in passing through zero. (7) In the lituus r^ = — ; whence we find u When r = ± a 22 or = i, — = 0, and changes sign on either side of the point corresponding to these values : the point is therefore one of inflexion. (8) In the Lemniscate of Bernoulli r^ = a^ cos 2 6, r^ dp Sr' and ^ = - — = — ^ . Hence the origin is a point of inflexion for two branches of the curve. (9) The equations to the Trochoid are OB — a {0 — esmQ)^ y = a {1 - e cos 6), whence we find (Py e (cos d — e) dx^ (l -ecosOy therefore when cos Q =e and y = a (l - e^) there is a point of inflexion. The preceding examples are taken chiefly from Cramer, Analyse des Lignes Courbes, Chap, xi. 168 SINGULAK POINTS OF CURVES. Multiple Points. Among these I include all those points for which we find —- = - 5 including points where several branches inter- sect, or nodes, points of osculation, cusps, and conjugate points. (1) Let the equation to the curve be ajf — a^ — bx^ = 0. dy ScD^ + 2bw , ^ , Here — = = - when co = 0, y - 0. By the dw 2ay if J usual method of evaluating vanishing fractions we find dy ()oc -r 2b dx dy 2a — d,v (dy\^ 6a' + 26 b whence -7- = = -, when x - 0. \dt)o) 9,a a 1 7,1 Therefore — — = ± - 1 , indicating; that at the origin d,v \aj ^ * two branches of the curve intersect, making with the axis /6\5 (b\^ of w angles the tangents of which are I - ) and — I - 1 respec- tively. (2) The curve x^ — ayx^ + by^ = has at the origin a triple point formed by the intersection of dy three branches of the curve. The values of — — at the point dx fb\i are =*= - and 0. See fig. 29. (3) The curve x^ — 2ax'^y — 2x'^y^ + ay^ + y^ = has at the origin a triple point, the values of — being ± ^ and 0. The form of the curve is given is fig. 30. SINGULAR POINTS OF CURVES. 169 (4) Let the curve be /»■* - 2 ay^ - 3d~y~ - 2a^iv'^ + a* = 0. Here — - = 4 (a?^ — o-a') , __ = — b{ay + a y)- ax ay Both of these vanish when y = and w = ^ a, and when y = — a and <2? = 0. There are three double points corre- sponding to these values of oc and y. dy ■ /4\2 For !, = 0, ..= +a, ^ = =ty , rfy / 4 \ '^ w = 0, tl? = - «, T~ = =*= I For the form of the curve see fig. 31. (5) In the curve x^ + x"y'^ — 6ax"y + a'y~ = we find ( 1— I = at the origin, or two branches there touch \CvlV J each other as in fig. 32. (6) In the curve w^ + 6.2?* — a^ y'^ = we find at the origin rd'u\ (d'u\ I d^uV (6fu\ (d'u\ / d'u y \dx^) \dy^J \dwdyi y^l \dwdy, d ?/ Avhich indicates a point of osculation, and as — = at the dw origin, the two branches touch the axis of a?. See fig. S3. (7) The curve (hy — cxY = {,v — ay has a cusp of the first species when x = a; the common tan- gent is parallel to the axis of x. See fig. 34. 170 SINGULAR POINTS OF CURVES. (8) The curve 0?* - ax^y — aocy^ + o^y^ =■ has at the origin a ramphoid cusp, the axis of co being the common tangent. See fig. 35. (9) The curve 2/* - axy^ + a?* = has at the origin a ceratoid cusp touching the axis of x^ and also a branch touching the axis of y. See fig. 36. (10) The curve ay^ — a?^ + hx"^ = has a conjugate point at the origin, since x = 0, y = satisfy the equation, but x = ^ h when h is small make y impossible. At the origin — takes the form - , and its true value is ^ dx h\^ , . . ) 5 which indicates that there are two impossible branches a) passing through the plane of the axes at the origin. (11) The curve whose equation is {c^y - x^y = (x - by (x — ay, a< b, has a conjugate point whose co-ordinates are a y = -72 but the differential coefficients are possible till we come to the third, showing that the impossible branch has a con- tact of the second order with the plane of the axes. (12) In the curve a^y^ — 2abx'^y — x^ = there is a point of osculation at the origin, and one of the branches experiences an inflexion. Such a point is called one of oscul-inflexion. Sec fig. 37. . SINGULAB. POINTS OF CURVES. 171 (is) The curve y^ + ax^ — Jfwy'^ = has a ceratoid cusp at the origin and an inflexion in another branch at the same point. The cusp has the axis of x as tangent, and the inflected branch touches the axis of y. The form of the curve is that of the letter R. See fia;. 38. (14) The curve {y — c)' = (cP - ay (a; - b), a > 6, has an oval between oo = a and w — b. When a; = a and y = c there is a point of osculation, the common tangent being parallel to the axis of x. See fig. 39- (15) The curve {x^' + y^y = 4>a^x^y'^ has at the origin a quadruple point, a pair of branches touching both the axes. The form of the curve is best seen by transferring the equation to polar co-ordinates, when it becomes r = a sm2d. The greater number of the preceding examples are taken from Cramer"'s work, Chap. x. and Chap. xiii. CHAPTER XI. ON THE TRACING OF CURVES FROM THEIR EQUATIONS. Sect. 1. Curves referred to Rectangular Co-ordinates. Before proceeding to give examples of the application of analysis to determine the form of curves when their equations are given, I shall say a few words on the principles of the interpretation of symbols in analytical geometry, as a know- ledge of these is requisite for the understanding of the views which I have adopted both in the preceding and in the following pages. By the principles of the Geometry of Descartes, the position of a point in a plane is known when its distances from two axes Ox, Oy intersecting each other at right angles are known : and a curve is defined as a series of points for which there exists the same relation between the ordinate y and the abscissa a'. This relation is expressed by means of an equation f{-v, y) = between x, y and constants, which is called the equation to the curve. If we assign a series of values to one of the two variables a? and y, the corre- sponding values of the other can be found by means of the equation f(x, y) = 0: now so long as we consider this only as an arithmetical equation, the only values of a; and y which we can use are positive numbers. If we agree that the values of x are to represent lines measured from O (fig. 40) along Ox, and values of y lines measured from O along Oy, we can by means of the arithmetical values alone of X and y determine the positions of all points within the angle xOy. But the equation f(x, y) = for any value of one variable will frequently give an expression for the other variable which is not arithmetical, such as - a or (- a^)i, or more generally (+«")". Now there is no necessity for in- terpreting these expressions which are uninterpretable in TRACING OF CURVES. l7o arithmetic ; but it is clear that we shall gain an advantage in the generalization of our results if we are able to interpret these expressions in any way consistent with the original definition of the symbols employed. It was soon seen by the early cultivators of this geometry that the first of these expressions (— a) could receive the geometrical interpretation that, if a represented a line measured in one direction, (-a) represented the same length of line measured in the opposite direction. This extension of the interpretation of the sym- bols is of great importance, since it enables us to express by the one equation, f(x, y) = 0, the position of a point in all parts of the plane in which the axes OiV and Oy lie ; and no curve is considered to be completely traced unless the negative, as well as the positive, values of the variables be taken into account. This however is merely a matter of convention, and we might, if it were thought proper, restrict ourselves to the positive values of the variables and confine the curve to the angle wOy. If instead of inter- preting (—a) to mean the measuring of the length a in a direction opposite to that originally taken, we use the more general definition that — a means that the line a is to be turned round through two right angles, we are led to the general interpretation of such an expression as (+a")", viz. that the line a is to be turned round through the n^^ part of four right angles. This gives us a farther extension of the use of the equation /(-3?, y) = 0; for, as the turning of a line through a given angle is not confined to any one plane, we are enabled to express by the equation to the curve the position of a point situate in any part of space. To explain this, let us suppose that for a value x = a, we obtain a value m ?/ = (+)"6; this implies that the length h is to be measured not along the axis of y, but along a line inclined to it m at an angle -27r: but as the axes are supposed to ^remain n ^ '■ perpendicular to each other, this angle must be taken in a plane perpendicular to that of the original axes. Hence, if there be a series of values of y all affected by the same 174 TRACING OF CURVES. quantity (+)", they will give rise to a branch of the curve lying in a plane inclined at an angle — 2 tt to the plane of the original axes. If for different values of x the index of + change its value, the branch does not lie in one plane, but is a curve of double curvature. 1^ This use of the interpretation of the symbol (+ a")" has not been generally adopted, but it is quite as legitimate an extension as that of the negative values of the variables, and for the thorough understanding of the course of a curve it is quite as necessary. For all the ordinary pur- poses however of the equations to curves it is sufficient to use only the positive and negative values of the variables, and to these I shall restrict myself, only observing, that when such an expression as (— a?)^ occurs, it is not to be called imaginary, nor is the curve to be said therefore to have no existence for that value ; but it is to be interpreted as indicating that the curve there leaves the plane of the axes, which for convenience I shall call the plane of reference. The student who wishes for more information regarding the general interpretation of formulae in Analytical Geometry is referred to a paper by the Abbe Buee in the Philosophical Titans actions for 1806, to Mr Warren's Tract on the Geometrical Interpretation of Imaginary Quantities^ and to the Cambridge Mathematical Journal, Vol. i. p. 259, and Vol. II. p. 103 and p. 155: the last two papers being by Mr V^alton. When we proceed to trace a curve from its equation it is advisable in the first place to solve the equation with respect to one or other of the variables, if the solution be in a form which enables us to determine readily its value for different values of the other variable. After that we may proceed in the following way. 1. If .V be the variable which is expressed in terms of CO, assign to w all positive values from to co, marking those which make ?/ = 0, ?/ = co, or ?/ impossible. The first gives the points where the curve cuts the axis of x, the TRACING OP CURVES. 175 second gives the infinite branches, and the third, showing where the curve quits the plane of reference, gives the limits of the curve in that plane. 2. Assign to on all negative values from to co, pro- ceeding as in the case of the positive values of x. In both cases attend to both the positive and negative values of y, so as to obtain the branches on both sides of the line of abscissas. 3. Find whether the curve have asymptotes, and deter- mine them if they exist. d tj 4. Find the value of — , and thence deduce the maxi- dx mum and minimum points of the curve, and the angles at which the curve cuts the axes. d" 7/ 5. Find the value of — ^ and thence deduce the nature of the curvature of the diiferent branches, and the points of contrary flexure if such exist. 6. Determine the existence and nature of the singular points by the usual rules. Ex. 1. Let the equation to be discussed be cv' - a' r = r- From its form we see at once that there are always for each value of x two values of y equal but of opposite signs; hence the curve is symmetrical with regard to the axis of cV. Let a; be positive ; when w is between and «, y is impossible, and the curve does not exist in the plane of reference: when a; = a, y = 0: when x > a, y is possible, and increases without limit as x so increases. Let X be negative; when x is between and b, y is impossible, and there is no branch in the plane of refer- ence: when X = b, y is infinite: when x >b, y increases without limit as x so increases. Hence it appears that the curve has six infinite branches. Since x = — b makes y infinite, the ordinate at that point is an asymptote. Also since 176 TRACING OP CURVES. 2/ = ± \ — i- = ± ,r 1 - - 1 + - ; ^ (a? + 6)2 V a?V V W on expanding, and neglecting negative powers of a?, we find 2/ = ± (cT - 16) as the equation to two asymptotes inclined at angles + 45° and — 45° to the axis of x. On combining the equation of this asymptote with that of the curve, we find that there is a value of a? correspond- ing to an intersection of the curve with the asymptote. Difi*erentiating the equation to the curve, we find dy 2x^ + 3hx^ + a? 2-^ dx {x - a^Y {x + 6)t This equated to zero gives a cubic equation, which must have one real root negative, since all the terms of the numerator are positive : this indicates a minimum ordi- nate. The course of the curve shows that the other two roots of the cubic must be impossible. du When cT = a, — is infinite, or the curve cuts the axis dx at right angles. dx^ cave to the axis of x when x is positive, and convex when it is negative. The form of the curve is given in fig. 43, where OA = a, OB — b ; ON is the abscissa corresponding to the intersection of the curve with the asymptote; and OM is the abscissa of the minimum ordinate. (2) Let the equation to the curve be The value of -^— ^ shows that the curve is always con- 9.x — a This curve, see fig. 44, has four infinite branches, and the equations to its asymptotes are a 1 / a = -T M? + - 2H 6 TRACING OP €UK^'^S. 177 The curve cuts the axis of x at right angles at tlie origin, and at distances + a and - a from the origin : at the latter two points there are points of contrary flexure, while the origin is a cusp. There is a maximum value of y cor- responding to a value of oe between and — a. (3) a-y^ + 2a^y -,v^ = 0. Solving the equation with respect to y, we find y = =*= \'^ + -z ,v \ x~ When a; = 0, y = 0, and ^ = — co . This will be readily seen by putting the original equation under the form f 2«^ y \y+ — \ -w^ = o, which when x = gives y = 0, and y -\ =0 or y = — co • To determine the eiFect of increasing a? positively, let us consider the two values of y separately. Taking the upper sign and expanding the radical in ascending powers of X-, we have la?* 1 . 1 a?^ \ -, + &c.j, 1 x^ 1 . 1 0?'^ ^^ y = - ~^ — 7. + &C' ^ 2 a^ 2l 1 . 2 a« Now when x is small, the first term gives the sign to the series, and y is therefore positive ; and as no value of X can make «/ = 0, this branch of the curve lies always in the first quadrant, and extends to infinity, since y = co, when X — OS' Taking the lower sign and expanding the radical in descending powers of a?, we have a^ /, 1 a^ y= .^,l+&c. X \ 2 x^ which when a? = » is negative and infinite: expanding in ascending powers of x, we have 2a^ j\ x^ 1,1 X' y = -— - -o -, &c. X \2 a~ 2~. 1 . 2 a'' 12 178 TRACING OF CURVES. which when x = is negative and infinite ; hence this brancli lies wholly in the fourth quadrant. For the negative values of x it is sufficient to observe that as the original equation remains unchanged when — m and — y are substituted for + x and + y, it follows that the opposite quadrants are symmetrical, and we need therefore only investigate the form of the curve in the first and fourth quadrants. To determine the asymptotes : since y = — co when x = 0, the axis of y is an asymptote to the branch in the fourth quadrant : also by expanding the value of y in descending powers of a? we have, neglecting the terms involving nega- tive powers of a?, as the equations to two other asymptotes. Differentiating the value of y, we find that at the origin d 77 — = 0, and therefore that the curve then touches the axis of w. dw We also find a minimum value for y when a? = ± S^a. This minimum value of y belongs only to the branches in the second and fourth quadrants, and not to the branches in the first and third quadrants. d^y Without proceeding; to find the value of — 5, it is not ^ * dw^ difficult to see that at the origin there is a point of con- trary flexure, since the curve there both touches and cuts the axis of x. The form of the curve is given in Fig. (45). When the equation cannot be solved with respect to one or other of the variables, it is necessary to have recourse to particular artifices suited to the case under consideration. (4) Let the equation to be discussed be .71^ — Sawy + y^ = 0. When a? = 0, t/^ = : the multiplicity of values of y shows that there is a multiple point at the origin. Differentiating, we have dy ay — x" ^ -^ = — = - when X = 0, ?/ = 0. dx y' — ax TRACING OP CURVES. 179 To find the true value of this fraction, differentiate its numerator and denominator ; then ay ax dx dii a, is given in (fig. 47), where OB = 6, OA = a. The reader will find a great variety of curves discussed in the work of Cramer, hefore referred to. For lines of the third order he may consult Newton''s Enumeratio Linearum Tertii Ordinis, and Stirling''s Commentary on that work. Sect. 2. Curves referred to Polar Co-ordinates. When the equation to a curve is given by an equation r=fiO), a fixed point is to be taken as origin, and a fixed line passing- through it as the axis from which 9 is to be measured. The values of 9 which make f(9) = are then to be found ; these give the angles at which the branches of the curve which pass through the origin cut the axis. By giving to 9 the values and »Z7r we find the values of r when the curve cuts the axis; and by giving to 9 the value ^{2n+ I^tt we find tbe values of r when the radius is perpendicular to the axis. dr By making v^ = ^ we find the values of 9, for which r is d u a maximum or minimum. After determining these points in the curve, the asymptotes, both rectilinear and circular, are to be sought out; and when these are known there will generally be little difficulty in finding the form of the curve, except when singular points occur ; and these are to be investigated by the usual process. It is to be observed that in all cases we must substitute both positive and negative values of 9, and that when the re- sult gives a negative value for r, it is to be measured along TRACING OP CURVES. 181 the radius vector produced backwards : if this be not attended to, the curve will want branches or spires, and will appear to l>e discontinuous. Some authors neglect the negative values of r, and trace the spiral only with the positive values of the radius vector ; that this is an incomplete mode of tracing the curve may easily be seen by transferring the equation from polar to rectilinear co-ordinates, when it will be found that, according to the principles of interpretation used for the latter, the tracing of the curve from its rectilinear equation will give more branches than that from the polar equation. The re- mark which was made regarding the interpretation of the symbols in rectilinear co-ordinates applies equally to polar : there is no necessity for interpreting all the symbols which arise in our operations, but we gain much in the generality of our formulae when we do interpret them, and we should sacrifice many advantages by not doing so *. Ex. (l) Let the equation to the curve be r = a cos 9 + b, a > b. When 0-0, r = a + b, a. maximum. From 6 = to = cos~^ ( I , which is an ansle in \ aj . * the second quadrant, r is positive and continually diminishing till when = cos"' ( j it is equal to 0, and therefore the curve passes through the pole cutting the axis at an angle whose cosine is . O; From = cos"' ( ~ ~) to = tt, r is negative and in- creasing, and being measured on the radius vector produced backwards it traces out the portion OEB (fig. 42) of the curve ; and when ^ = tt, r = — (a - b) = OB. * It has been usual among writers on this subject to neglect the negative values of r and so to deprive the curves of their due allowance of branches: a marked instance of this may be seen in the spiral of Archimedes, which, as usually traced, appears shorn of one half of its length. Professor De Morgan is, so far as I know, the only writer who has insisted on the interpretation of negative values of r. See his Diff. Cnk. p. U2. 182 TRACING OP CURVES. From = IT to 6 = cos~^ ( ~ ~) ^" *^^® third quadrant r is still negative and diminishing, and traces out the portion BFO of the curve. When 9 = cos"' [ — -), r = 0, and the curve passes again through the pole, cutting the axis at the same angle as before, but measured in the opposite direction. From = cos"' ( — -) in the third quadrant to = 2x, r is positive and increasing, till it again reaches the maximum value a + 6 or OA, after tracing the portion OGHA of the curve. On increasing the values of Q the same values of r recur, showing that the curve is complete ; and it is obviously unnecessary to give to negative values, since these will give the same values for r as the positive values Stt — have done. When a = h the smaller oval OEBF vanishes, and the point O is a cusp ; the curve then becomes the common car- dioid. (2) Let r = a sin 3 be the equation to the curve. r = when 36 = nir; that is for = 0, 6 = - , 6 = — , 3 3 3 3 When = 2 TT or upwards the same series of values again recur. The curve therefore passes six times through the pole, and as r never becomes infinite, it must consist of six equal loops arranged symmetrically round that point. A little con- sideration will show that the form of the curve is that given in fig. 49- This curve belongs to a class represented by the general equation r = a sinm^, the properties of which have been very elaborately treated of by the Abbe Grandi, in a paper in the Philosophical Transactions for 1723, and in a book called rather quaintly Flores Geometrici. From a fanciful notion that thebe curves resembled the petals of roses, he gave them THACING OF CURVES. 183 the name of " Rhodoneae," and endeavoured to trace analogies between them and the flowers after which he had named them. The first paragraph of his paper in the Philosophical Trans- actions will give an idea of his way of treating the subject : " Suos Geometria hortos habet in quibus, aemula (an potius magistra ?) naturae, ludere solet, sua ipsius manu flores elegan- tissimos serens irrigans enutriens ; quorum contemplatione cultores suos quandoque recreat ac sumraa voluptate per- fundit." (3) Let the curve be r = a (sin 20 - sin 0) = a sin Q (2 cos 9 — l), r is equal to when sin = and cos = ^? or when = 0, = 7r, 9 = — i 9 = . o o O S The values of r recur when = 2 tt ; and as r never becomes infinite, it appears that there are four loops arranged round the pole, one pair being smaller than the other. From 9 = to 9 = ^tt, r is positive. From 9 = ^77 to 9 = TT, r is negative as 2 cos 0—1 is nega- tive, and sin 9 is positive. Btt From 9 = TT to 9 = — , r is positive, since both factors are negative. From 9=-^ to 0=27r, r is negative. The form of the curve is easily seen to be that in (fig. 50). (4) Let the equation to the curve be r = a (tan 9-1) (fig. 48). When = 0, 7' = - a — OB if OA be measured in the positive direction. From = to = ^tt, r is negative and decreasing, and traces out the portion BDO of the curve. When = ^TT, r = 0, and the curve passes through the pole, cutting the axis at an angle of 45°. From = Itt to 0= Itt, r is positive and increasing, and traces out the portion OEL. 184 TRACING OF CURVES. When 9 = ^TT, r = co. To see whether this corresponds dG to an asymptote we must find r^ — . dr Now ^a(l -{- tan-O) ; h f r^ 1^ - ^' ^*^" 9 -if a (sin - cos 9)' _ dr ~ a(l + tan^0) "~ (sin-0 + cos"^^) ~ ' when 9 = ^TT. Therefore AL drawn perpendicular to the axi« at a distance OA = a is an asymptote to the curve. From 9 = — to 9 = — , r is negative and diminishing, and it traces out the portion KHACO: the prolongation AK of AL being an asymptote to this branch. When 9 = — , r = 0, and the curve again passes through 4 the pole, cutting the axis at an angle of 45°. Sir z^^"". From 9 = — to 9 = — , r is positive, and traces out the portion OFN ; and when 9 = ^tt, r= co, and it is seen as before that a line BN perpendicular to the axis is an asymp- tote. From 9 = — to = 2 TT, r is negative and diminishing, and traces out the portion MGB. When 9 = '2,'K the curve joins on to the first portion, and is therefore complete. It is obviously unnecessary to consider negative values of 9 as they are included in what has already been done. (5) Let the equation to the curve be , „ sin S9 r - a — . COS0 The form of this curve is given in fig. 46. CHAPTER XII. ON THE CURVATURE OF CURVED LINES. Sect. 1. Radius of Curvature. When the curve is referred to rectangular co-ordinates, if a be the radius of curvature MS)T P = ITT- dsG^ m being made the independent variable ; and dy' y being made the independent variable ; and 1 fd^co d^y\ 3 ~p ^ \d? "*" d^l ' the arc being made the independent variable. Ex. (l) In the parabola, the equation to which is y" = ^mx, 2 (m + w)i P = m^ Ex. (2) In the ellipse ^, + -, = l, a~ b' p = ; , where e ~ . ab a 186 ON THE CURVATURE OP CURVED LINES. (3) In the rectangular hyperbola referred to its asymp- totes xy = m^y and P = -2m^ (4) In all the curves of the second order the radius of curvature varies as the cube of the normal. If N be the length of the normal, N=y\l + (-^J J ; and therefore P = ' - if — - All the curves of the second order are included in the equation y- = 9,px + qoo" -^ dy therefore y-i-=p -^ q^x, CLOG d?y (dy' ^ d'y (dyy and 2/7-2+3- = ^' ^ dx^ \dxl Therefore p — —z- . p (5) In the cubical parabola Sa^y = ob^, (a* + w'f P ~ ~ 2a^off (6) In the semi-cubical parabola Say"^ = 2,v^, (2 a + 3,37)^072 P= - 3^. a / X T .1, 1 -A ^y (2ay-y'y^ (7) In the cycloid -— = ; ^ ^ ' dx y (dyV' 2ff d^y a ^ .„ xi \dwj y dar y ON THE CURVATURE OP CURVED LINES. 187 C ~ -- (8) In the catenary y = -{e^-\-e "^ ), 2 dw 2^^ ^ ^' dw^ (?' ^ 2c' (9) In the tractory y ■\- {a? — y^)^ — — = 0. Taking the expression for p in which y is the independent variable we find, (10) In the hypocycloid x^ + yi = as, p = — 3 {awy)^. If the curve be referred to polar co-ordinates r and 0, then hST . fdrV'' d^r ' r^ + 2 -— —r- \ddj dO' or, if it be expressed by the relation between r and the per- pendicular on the tangent (p), dr dp (11) In the cardioid r = a (l - cos0), (8ar)^ (12) In the lemniscate of Bernoulli r' = a^ cos2d, dr _ (a'-r*)i (dr'v'^a' ^ __ _ , ^— j=__r, rf^r a* d^r a* + r* = r - r, r dO' r' ' df?2 r' ' a"' 188 ON THE CURVATURE OF CURVED LINES. (13) In the spiral of Archimedes r = a9^ (a' + r'O^ P = — z ~- ' 2a~ + r' (14) In the hyperbolic spiral r = -^ r (a' + r^)i P^ a^ • (15) The equation to the lituus being r'-=— , r (4 a* + r*)t P ~ 2o-(4a*-r')" (16) The equation to the trisectrix being r= «(2cos0 =t 1 ), (5 ± 4 cos 9)^ p = a — . ^ 3 (3 =t 2 cos 6) (17) In the logarithmic spiral when referred' to p and r, r p p = mr, p = — = — : . mm (18) In the involute of the circle p~ = r^ - a^, and P =P' br (19) The equation to Cotes' spirals is p = — ^ ^i r r {a? + r^)i P^ ~^b ■ ^20) In the epicycloid 3 c' (r^ - a') P Therefore p = p c — a'^ c^-a^ (c2 _ d')^ (r^ - a,')^ Sect. 2. Evolutes of Curves. When a curve is referred to rectangular co-ordinates, the co-ordinates (o, /3) of its centre of curvature are given by the equations ON THE CURVATURE OF CURVED LINES. 189 \dxl dy „ \dxl d'^y doD d^y dw^ dso^ To determine the equation to the evolute it is necessary to eliminate x and y between these equations and that of the given curve ; but the complication of the formulae renders this elimination always very troublesome, and most frequently impracticable. The few cases in which it can be effected we shall give. (l) In the parabola y^ = 4^aoo, whence a = Sw + 2a, B = - ^^, ; ^ 4-a' ft ^ Q, n. therefore oc = , y = (^— 4a^j3)i. 3 Substituting these values in the equation to the parabola, we find (4a^/3)3 = — (a — 2a), • or 27«/3^ = 4(a - 2a)% the equation to the semi-cubical parabola. (2) In the rectangular hyperbola referred to its asymp- totes wy = m^, 'J p 77" lV whence 2a = Sx + '— , 2l^ = Sy -\ . X y Adding 2(a + /3) = 3 (a? + 7/) + ^ = ^ ^—^ , xy m or 2w^ (a + /3) = (a? + t/)^ or x + y = (2w^)i (a + /3)i Similarly, subtracting 2m~ (« - /3) = {x - yf, or x - y ^ (2nr)^ (a - fi)'. 190 ON THE CURVATURE OF CURVED LINES. Adding and subtracting, 2<2? = (2m^)^ {{a + /3)^ + (a - (^)^, 2y = (2m'ys {(a + /3)^ - (a - fi)H. Multiplying these together, and observing that xy = m^, the equation to the evolute is found to be (a + /3)3 - (a - /3)^ = (4 (dy\^ 2a d'y dw'- a r whence y - fi =t 2y, or ?/ = -i8- Substituting this value in da dw dy' we have d{-^) (- -/3) da 52«(-^)-(-/3y^5r which is the equation to an equal and similar cycloid, but in an inverted position. (7) The equation to the tractory is dy^ y d X (a^ — y^)^ whence /B = — , y = --:, y ' P o. 1 • • 1 • 1 • d^ dw ^ , substitutmg this value in — = we find da dy da a which is the differential equation to the catenary. (8) The equation to the catenary is dy^ ^ c dw (2c w + w^)^ ' GN THE CURVATUKE OP CURVED LINES. 193 whence x — a= — (c + w) and x = , , ., . . dQ dx substituting in —— = , da dy da which is therefore the differential equation to the evolute. (9) The equation to the logarithmic curve is t/ = «e" ; ^, oi y /3±(/3^-8a^)^ whence ii — (i = ^ , or v - — y -^ — = 0. ^ '^ y ^ 2^ 2 From this jr = and 4a-^ + /3± (j3^-8a^)^=0 is the equation to the evolute. In curves referred to polar co-ordinates the most con- venient mode of finding the equation to the evolute is by the relation between p and r. If p and r be the co-ordinates of the curve, p^ and r^ evolute, p be the radius of curyature ; then p =f(r) being the equation to the curve, rf = r + p^ -%pp, 2 2 2 ^** P!=r -p\ P-r — . Between these four equations we can eliminate p, r, p, and so find a relation between p and r , which is the equation to the evolute. (10) Let p'^^r^- a\ Then p = p^ rf = r" ->r p^ — 9.p^ O O Q = r- — p^ = a\ and. p"^ = r^ — p^ =z a?. 13 194 ON THE CURVATURE OF CURVED LINES. Hence p^ and r^ being both constants, the evolute is a circle. (11) In the logarithmic spiral p = mr, V whence p = —^ p^ = r^ {l — m") , r; = r + — ^ - 2r^ = r" , and p^ = mr^, the equation to a similar logarithmic spiral. The logarithmic spiral may even be its own evolute ; that is, one convolution of the curve may be the evolute of another convolution. To find the condition that this should be the e case, let r = e"' be the equation to the curve. Let P (fig. 52) be a point in the curve, PN the normal at that point touching a point Q in the convolution which is the evolute of the convolution AP. Then since the curve makes a constant angle with its radius vector, the angle SPT must be equal to the angle SQP; that is, PSQ must be a right angle. Hence the radius SQ is separated from the radius SP by some whole number of circumferences together with three right angles, or if JSP = e, ASQ = e - (4r + 3) - . If SP = r, and SQ = r , But Q being a point in the evolute, r = ar^, so that £ _ 4r+3 ■TT e« = ae« a 2 . 4r + 3Tr whence a = e « ^ , or a" = e(4'+3) I ^ ON THE CURVATURE OF CURVED LINES. 195 which is the condition that the parameter a must satisfy in e order that the spiral whose equation is r = e" may be its own evolute. (12) In the Epicycloid p^ = — \ g-^, rf = r + --L p2 _ 2 __ p^ = r% fi^^ .)(.-., = - (a^ + c^ - r% Substituting for c^ — r^ its value in terms of p,, ^ a* + (c' - a^) »•; = — pi ;>/' a' and p/ r.3 I «, 2 _ c" — a' which is also the equation to an epicycloid. 13—2 CHAPTER XIII. APPLICATIONS OF THE DIPl'ERENTIAL <3ALCULUS TO GEOMETRY OF THREE DIMENSIONS. Sect. 1. Tangencies. If F{w,y,%) = be the equation to a curved surface, the equation to the tangent plane at a point x, y, % is dF , , dF ^ , dF (x -w)-— +{y -y) — + {%-%) — =0, ^ ^ doe dy dz where x, y\ % are the current co-ordinates of the tangent plane, <», i/, % those of the point of contact. If the equation to the surface consist of a function homogeneous of n dimensions in w^ y, % equated to a con- stant, the equation to the tangent plane becomes ,dF ,dF ,dF doB dy dz F {w, y, %) = c being the equation to the surface. If p be the perpendicular from the origin on the tangent plane, dF dF dF dx dy dz \[d^l "^ \d^) "^ \dij j and if the function be homogeneous of n dimensions, nc ^^TJdFy TdF? TIFvP' APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 197 The equations to a normal at a point a?, y, z are cc - X y' — y s;' — % dF " dF " Ilf" ' dx dy d% Ex. (l) The equation to the Ellipsoid being 00^ y^ %^ - + r- + ^ = 1' a^ 0- c that to the tangent plane is xoe' yy %%' a' h' & The perpendicular on the tangent plane from the origin IS given by the equation 1 _ Ix^ f %\^ p^ W "^ 6* "^ ?/ * If we wish to find the locus of the intersection of the tangent plane with the perpendicular on it from the centre, we have to combine the equation to the tangent plane, xx' yy %%' 2 ^ 1.9 ^ 2 ' with the equations of a line perpendicular to it, and passing through the origin w'x b^y &% X y % These last may be put under the form ax by' ex' . 2 '2 , a2 '2 '^ '2v^ x y % a b c x^ y^ %^ since -; + 7^ + ^ = 1- Multiplying each term of the equation to the tangent plane by the corresponding member in these last expressions, x^ y, % are eliminated, and we have for the locus of the intersections x'~ + y"' + z''-^ = (arx" + h'y'- + c-z'''-)--^. 198 APPLICATION TO GEOMETRY OF THREE DIMENSIONS. This is the equation to the surface of elasticity in the wave Theory of Light. (2) Let the equation to the surface be oi)y% = im". The equation to the tangent plane is / ' / OB It Z - + ^ + - =3. X y z The intercepts on the tangents are and the volume of the pyramid included between the tangent 1 ^ *T, ^- * T • 9*2/^ 9o? plane and the co-ordmate planes is = — . ^ ^22 The volume of this pyramid is smaller than that of any other pyramid formed with the co-ordinate planes by a plane passing through the point w, y, %. The length of the perpendicular from the origin is given by 1 1/1 1 1 \ 2 p 3 \oe^ y^ ^ (3) The equation to the Cono-Cuneus of Wallis is {a" — w^) y^ - c'z^ = 0, and the equation to the tangent plane is therefore y'^wx — {a" - aP^) yy + ^-^J^i '•> where v == x~ + y. APPLICATION TO GEOMETRY OP THREE DIMENSIONS. 199 (5) The equation of the heli^oide developpable is ,v sin^ -^ —> + vcos{ } = a- \ h a j ^ \ h a j The cosine of the angle which the tangent plane makes with the plane of xy is dF d% dFY idFy /ZFvP" djo ) \dy I ■*OT Let ^^ —^ = 0, then h a dF . ^ x(w cos^ — «/ sinO) = sinO ^ ^ ^ dx a {oB^ + y" - a?y^ dF n y {^ ^os — 2/ sin 6) -— = COS0 — : ^ dy a{cV + y^ - a )2 dF 2'7r — — = — (x cosO — y siny). dz h 'i\\ ' Now {cG COS0 — y sin^)^ = x^ cos^O + y^ sin^0 — 2xy sin0 cos0, and from the equation to the surface 2xy sin0 cos0 = a^ — os^ sin^0 - y^ cos'^0 ; therefore (<2?cos0 - ^sinO)- = x" + y^ — a?. Hence dF . ^ X dF ^ y -— = sm0 , — — = cos0--, dx a dy a, —-= — {x^ + y' - a~)5 . dz h From these expressions the cosine of the inclination of the tangent plane to the plane of xy is found to be 27ra The inclination is therefore constant, and equal to that of the helix, which is the directrix of the surface. 200 APPUCATION TO GEOMETRY OF THHEE DIMENSIONS. (6) Let the surface be FresnePs surface of elasticity,, the eqaation to which is The equation to the tangent plane is (2r^ — cP) OCX 4- (2r^ - lf)yy' + (2r^ - c") %%' ^ r% where r^ = .2?^ + ^^ + «^. The perpendicular from the centre on the tangent plane is {w^ -T y^ + z^Y When a curved line in space is given by tlie equations of two of its projections^ w = 0(sf), y = >//(s?>, the equations to a tangent at the point w, y, % are dx , , . , ^y t ^ X —a! = --{% - z), y - y = -—{%- z}. dz dz The direction cosines of the tangent are da dy dz ds^ ds^ ds The equation to the normal plane is (a?' — 0?) dx + (y — y) dy + (z' — z) dz = 0. The equation to the osculating plane is {x' — x) {dyd'z — dzd^y) + (y' - y) (dzd^x — dxd^z) + {z' — z) (dxd-y — dyd^x) = 0. (7) Let the given curve be the helix, the equations to which are z z X = a cos — , y = asm — . h' ^ h The equations to the tangent are h ix' - 0?) + 2/ («' - ^) = Oj h (y ~ y) - x (z - z) = 0. APPLICATION TO GEOMETEY OF THREE DIMENSIONS. 201 If 6 be the angle which the tangent makes with the plane of xy, tan = - , and is therefore constant. a The equation to the normal plane is a;y' — ycc -v h(%' — %) = 0. In finding the equation to the osculating plane we may for simplicity assume ^"z = 0, that is, make z the indepen- dent variable. This assumption readily gives us as the equation to the osculating plane, h {xy' — yw) + a^ {%' — %) = 0. In both of these equations if we make w' = 0, y = 0, we find % = % ; that is, both planes cut the axis of % at the same point, which is the corresponding co-ordinate of the point in the curve. (8) Let a curve of double curvature be formed by the intersection of two cylinders, the axes of which cut each other at right angles. The equations to the curve are ^ . ^ Q 9 9 7 9 01- + %- = a , y^ jf %'' = b-, the point of intersection of the axes of the cylinders being taken as origin, and the axes as the axes of x and y. The equations to the tangent are XX + »%' — a^, yy + %%' = &". The equation to the normal plane is x' y' % X y "% The equation to the osculating plane is, making % the independent variable, and therefore d'^z = 0, When a curved line in space is not given by the equa- tions to its projections, but by the equations to any two surfaces, F{x, y, z)^^, F,{x, y, z) = 0, 202 APPLICATION TO GEOMETRY OP THREE DIMENSIONS. we have dF ^ dF ^ dF ^ —— dx + — - dy + — — d% = 0, dx dy d% dF, dF. dF. ^ — — dx H — ; — dy + - — d% = 0, dx dy dz Y. , . , . ■, . dx dy . irom which equations we can determine — , — in terms d% d% of x, y, % : and these values are then to be substituted in the equations to the tangent, and to the normal and osculat- ing planes. (9) Let the curve be that formed by the intersection of a sphere and an ellipsoid. It is determined by the equations x^ y^ z^ ? + ? + ? = '' ^'^f^"'"^- From these we find dx c? }?-(?% dy b- (? - a? % d% (? d^ - b^ X dz c'^ or — b^ y therefore the equations to the tangent are X {x - x) ^ y (y - y) ^ z (z - z) a' (6' - c') b' {c' - d'') & {d - h') The equation to the normal plane is X y z This curve is the spherical ellipse ; that is, it is a curve described on the surface of a sphere such that the sum of the arcs of great circles drawn from any point in the curve to two fixed points on the surface of the sphere is constant. (10) Let the curve of double curvature be the equable spherical spiral. This is formed by the intersection of a sphere with a right cylinder the radius of whose base is one half of that of the sphere, and which passes through the centre of the sphere. The equations to the curve are therefore 3r + y- + ;y^ 2a 2a In order that these equations may hold we must have either cV = 0, or y = 0. Taking the former we find 2y^ a a it'/ 'm ^ j « - «' V"=2"2' o^' 2/ = 41« («-«)}^' and^ = ^-. Now if a>a the value of y is possible, and there are two umbilici, the co-ordinates of which are el? = 0, ^ = ± 1 Ja' (a - «') }i, % = —— . If a (8) In the surface, the equation to which is wyz = m^ there is an umbilicus at the point w = m, y =■ m^ % = m. CHAPTER XIV. ENVELOPS TO LINES AND SURFACES. The earliest questions the solutions of which involved the Theory of Envelops or Ultimate Intersections were those which related to evolutes of curves, investigated by Huyghens,* and those relating to Caustics, a subject introduced by Tschirnhau- sen;-|- but these authors did not follow any general analytical method for the solution of such problems. Leibnitz was the first who considered the general theory of questions of this kind, so well adapted for exemplifying the utility of his Cal- culus; and in two memoirs in the Acta Eruditorum^X he gave a general process for the solution of all problems which depended on the successive intersections of lines whether straight or curved, the position or magnitude of which were changed ac- cording to some law. This method is the same as that usually employed, no important modification having been subsequently introduced, and may be stated in the following manner. If u = f {Xi y, %, a, b, c . . .) = be the equation to a surface, a, 6, c ... being parameters determining its position and magnitude, the envelop of all the surfaces formed by the variation of o, 6, c ... is found by eliminating these quantities between the equations du du du da do dc When, as is often the case, there are one or more equa- tions of condition between the parameters, the method of indeterminate multipliers may frequently be conveniently employed. The same method of course applies to lines in two dimensions. * Opera, Vol. I. p. 89. t Acta Eruditofum, 1682. i 1692, p. 168, and 1694, p. 311. ENVELOPS TO LINES AND SURFACES. 221 Ex. (l) Find the equation to the curve which touches all the straight lines determined by the equation m y = aOD + - , a where a is supposed to vary. m Here u = y — aoo =0, a du m — = — ,r + — = D, da a^ whence a" = — , and substituting this value we have ?/ = 2 (mct7)2, or y~ = 4ma?, the equation to a parabola. (2) Find the equation to the curve which touches all the lines determined by the equation y = ax + r (l H- a"'^)^, when a is supposed to vary. Here u = y — aoo — r (l ■\- a~)^, du i ra \ Multiply by a and add to the original equation. Then y = rU\+a')^ ^^1 = y— ^ ' therefore y^ = ; also x^ = ^ . ^ \ + a" 1 + a"^ Adding, we have x^ + y~ - r^ •, the equation to a circle. (3) Find the envelop of the series of parabolas whose equation is y^ = a {x — a), a being the variable parameter. .^ du . w Here — = gives 2r/ - x = 0, or a = - ; 222 ENVELOPS TO LINES AND SURFACES. whence ?/'-=_, or « = ± - , 4 2 the equations to two straight lines. (4) To find the envelop of the series of ellipses defined by the equation 00^ if — + — = 1. a^ {k - af du . aP' y^ Here _ = gives -5 - ^^^—^ = ; kcD^ ky^ whence a = -^_ -^ ^ k — a = a?3 + ys got — yt and on substituting these values in the original equation we find as the equation to the envelop (5) The straight line PQ (fig. 41) slides between the rectangular axes Aog^ -^y '•> find the locus of its ultimate in- tersections. Let AP = a, AQ = b, PQ = c; then the equation to PQ is x y -+f = l, a «, h being subject to the condition (f -\. W = c\ Differentiating with respect to a and 6, xda ydh ^ ^ , , „ — - + - = (1), ada + bdh = (2\ X (l) — (2) = gives on equating to zero the coefficients of each diff^erential. CO V , Multiply by a, 6, respectively, and add ; then by the first two equations, \a b ENVELOPS TO LINES AND SURFACES. 223 therefore a^ ^ c^x^ b^ = c^y, and, substituting these values of a and h in the equation of condition, we obtain % . 2 2. as the locus of the ultimate intersections of PQ. (6) If the equation to a straight line be a a and b being subject to the condition a h -+ - = 1, m n the locus of its ultimate intersections is ce\l (y -J ^[i which is the equation to a parabola referred to two tan- gents as axes. (7) Find the envelop to the series of parabolas de- termined by the equation y = ax-(l + a) —, 4>c where a is the variable parameter. The result is a parabola, the equation to which is o?^ = 4c (c — y). This is the equation to the curve touched by the parabolas described by projectiles discharged from a given point with a constant velocity, but at different inclinations to the horizon. The problem was proposed by Fatio to John Bernoulli, who solved it, but not by any general method : it was the first case which was brought forward of the locus of the ultimate inter- sections of curved lines. — Commercium Epistolicum of Leib- nitz and Bernoulli, Vol. i. p. 17. (8) Find the curve which is constantly touched by the circles determined by the equation (.^? - ay + y^ = b^. 224 ENVELOPS TO LINES AND SURFACES. a and b being the co-ordinates of a parabola, so that 6" = 4w2a. The resulting equation is y~ = 4.,m (x + m), which is the equation to an equal parabola, the vertex of which is shifted through a distance — m. (9) Find the envelop of the series of ellipses defined by the equations 9 O O 7 9 -i + -^ = 1, - + -=!• a"^ b mr n The resulting equation is CD y m n The equations to four straight lines in the space contained by which all the ellipses lie. (10) Find the locus of the ultimate intersections of chords joining the extremities of conjugate diameters of an ellipse the axes of which are a and b. If ,x\ y be the co-ordinates of the extremity of a di- ameter, - ?/', oc' are the co-ordinates of the extremity of b a its conjugate. Hence the equation to the line joining their extremities is & X , f «A X (y w) —y .r + - M/ + ao = 0, a " V b) x and y being connected by the equation to the ellipse '2 '■? The resulting equation of the locus of the ultimate inter- sections is -^+ "^=^' a, ¥ the equation to an ellipse, the axes of which are -j, — r. ENVELOPS TO LINES AND SURFACES. 225 (ll) If from every point in a curve of the second order pairs of tangents be drawn to another curve of the second order, find the* curve which is constantly touched by the chord of contact. Let US suppose for simplicity that the second curve is an ellipse referred to its centre, its equation being - + I = 1' I- Let the co-ordinates of a point from which a pair of tangents to (I) is drawn be a, /3, then the equation to the chord of contact is ^+^ = 1: IL a, /3 are supposed to be the co-ordinates of a point which is always in a curve of the second order : they are therefore connected by the equation Aa' + 25a/3 + C/32 + 2Z>a + 2^/3 + 1 = 0, III. Now to find the curve which is constantly touched by (II) differentiate (II) and (III) with respect to a and /3. seda yd3 -^ + ^ = 1' (1) (Aa +B(i + D)da + (Ba +C(i + E)=0: (2) X (l) + (2) = gives us X^+ Aa + B3+ D = 0, \l + Ba+Cfi + E=:0. Multiply by a, (i and add, then by (II) and (III) Substituting in the preceding equations we have (2>a + ^/3 + 1) ^ + Ja + 5/3 + D = 0, (3) {Da + Eft+1)^, + Ba + C^ + E = 0. (4) 15 226 ENVELOPS TO LINES AND SURFACES. Between (II), (3), and (4) we can eliminate a, j8, and we obtain the final equation 2 2 (C - £"-) -, -2{B-DE)^^ + {A- D') I , a a~b 0* + 2 (CD- BE) -, + 2 (AE - BD) ^^ + AC-B' = 0. IV. This being of the second order, it appears that the locus of the ultimate intersections of (I) is a conic section. This is a case of the general problem of reciprocal polars. The curve (I) is called the directrix, the point a, j8 its pole ; and the line (II) the polar with reference to (a, j8.) The curves (III), (IV) are the reciprocal polars, and possess a great number of corresponding properties of considerable in- terest, but the nature of this work precludes us from entering on that subject. The reader who is curious in such matters will find memoirs on these related curves by Poncelet, in the Annales de Gergonne, Vol. viii. p. 201, and Bobillier, Tb. Vol. XIX. p. 106, and p. 302. He will also find these questions along with others of a similar kind very ingeni- ously treated, in a short tract on '' Tangential Co-ordinates," by J. Booth of Trinity College^ Dublin. The method em- ployed by that author does not come within the scope of the present work, but it merits attention, as affording a ready solution of many curious problems which yield with difficulty to the power of ordinary analysis. (12) A plane whose equation is w y z a b c a, &, c being subject to the condition abc = m^, will always touch the surface whose equation is ^ 27 (13) To find the envelop of the system of spheres determined by the equations {x — a)'^ + {y - by + %^ = r\ a^ + b^ = c". ENVELOPS TO LINES AND SURFACES. 227 Differentiating with respect to a and 6, {oe - d)da ^- {y -h)dh = Q (l), ada + 6d6 = (2) ; X (2) + (1) = gives on equating to zero the coefficients of each diiferential. \a + (a? - a) = (3), X6 + (?/ - 6) = (4). whe dip) -^h (4) gives whence ay = hcc^ or - = - , a h Xc^ + ax + by - c^ = o. -r. "^^ ?/ liut as - = -, a b ax + by aw + by {x^ + y^)^ _____ = _ = ± ____ , C ± (,»^ + ?/^)5 whence X = — — . c Substituting this value of X in (3) and (4), squaring and adding, |c =t (x~ + y'^)^^ = {x - ay + (y - by = r~ - z\ by the original equation ; and this is the equation to the envelop. (14) To find the surface always touched by a plane which cuts off from a right cone an oblique cone of constant volume. Taking the vertex of the cone as origin, and its axis as the axis of %, the equation to the cone is at^ -\. y^ = (^%^ ^1) where c is the tangent of the half angle of the cone. The equation to the cutting plane is Ix + my + nx = vi^ (2) /, m, n being the cosines of the angles which it makes with the co-ordinate planes, so that V' + m- + »^'- = 1, (3) and V) being the perpendicular from the origin on the plane. 15—2 228 ENVELOPS TO LINES AND SURFACES. Extracting the square root of (l) and substituting in it the value of z from (2), we have (w^ + yy^ = 1 ^, n n which is the equation to the projection on {ocy) of the section of (l) by (2) ; and as the radius vector is a rational function of X and «/, the origin, that is, the vertex of the cone, must be the focus of the projection. Comparing it with the general equation to a conic section referred to, its focus a (I- en 1 + e cos (0 - a) ' or {iV^ + y-)2 Ts a (i _ e^) — e cos ax — e sin a y, we find , „. cv & (P + m?) « (1 - e') = — , e- = ^ ; n n whence the area of the projection is and the area of the section is therefore TTC'V The volume of the oblique cone cut off is which is to be constant. Neglecting the constant multiplier and extracting the cube root, we may put {n^-c^{l^+m^)]h = ^' ^' v = a{rr- c^ {l^ + m^)]^ (4) We therefore have the equation Ix + my + n% = a {n^ - c^ (I" + m^)\i, Z, m, n being connected by the equation l^ + m'^ + nr = 1. ENVELOPS TO LINES AND SURFACES. 229 The result of the elimination of /, m, n is or \i?%^ - (.t?2 + 2/2) j i [ac + {c-%'' - (.t?' + y")}^ = 0^ The factor ac + [c^is''^ — (x^ + y^)\h = o, is the equation to the required envelop. Transposing and squaring, this becomes c^^^ — (<27^ + y^) = a^c^, the equation to a hyperboloid of revolution of two sheets, the possible axis of which coincides with the axis of z. If the theory of reciprocal polars given in Ex. 11, be ap- plied to the surfaces of the second order, it will be found that the reciprocal polar of a surface of the second order is also a surface of the second order ; and that when the one surface can be generated by the motion of a straight line, the other can be so generated also. For the properties of reciprocal polars in surfaces the reader may consult the memoirs indi- cated in Ex. 11, and also one by Brianchon, Journal de VEcole Polytechnique, Vol. vi. p. 308. (15) Find the surface traced out by the ultimate inter- sections of the planes which touch the ellipsoid 0?^ y"^ %^ a^ \f c^ along the curve made by its intersection with the plane Ix + my + nz = S. If tV, y\ %' be the current co-ordinates of the tangent plane, its equation is XX yy zz a' + h' + c^ — 1, where x^ y, z are supposed to vary subject to the previous conditions. Differentiating we have xdx ydy zd Idx + mdy + rtdz - 0, (2) 230 ENVELOPS TO LINES AND SURFACES. xdx ii' dii % d% a^ If c-" X (l) + M (3) + (2) = gives, on equating to zero the coeffi- cients of each differential, w 'XI ^ ^ y y X— +/U-2 + /=0, X-2 + iUl-- + m = 0, -^ + /ti-2 +W = 0. Multiply by >«?, ?/, %, and add, then by the equations of condition, X + /ix + ^ = 0. Substituting for X in the preceding equations they become lj.{x-a!') = a?l-hw, iuL{y-y)=:b^m-Sy, iJi{%-z)=^c'^n-l%, whence CO — OB y ~ y % — % Now multiplying numerator and denominator of these / / / cc li z fractions by -,, , — , — respectively, and adding together the numerators and the denominators, , /2 /2 '" Ix + my + n% — d But on multiplying the numerator and denominator of these fractions by /, m, n respectively, and adding the numerators and the denominators, we also have — (/<2? + my + n% ) Therefore equating the two values of p we have oo''^ y'"^ %'~ {^ ~ C^"^' + *^2/' + *»^')}^ a^ W c^ dV' + 6~m~ + c~n' - V' ' as the required equation to the surface. (16) Find the equation to the surface which is constantly touched by the plane /*' + my + n% = v, ENVELOPS TO LINES AND SURFACES. 231 Z, m, w, V being connected by the equations p j^ m^ + -n? = 1, v^ — a^ v~ — b^ v'^ — c~ Differentiating with respect to /, m, n, v we have, (1) oedl + ydm + %dn = dv. (2) Idl + mdm + ndn = 0. Idl mdm ndn { l^ m^ ' n^ \ X(l) = ix{2) + (3) gives, on equating the coefficients of each differential, / (4) Xa;= ijlI +— -, m (5) \y-fxm^ ^^_^,. (6) n v~ - c" (7) \ 11 ' A j + ^ - ^ ^ /..2 ..N2 + / n I [y - c ) J Z (4) + m (5) + n (6) gives by the conditions, (8) \v = ^, X (4) + y (5) + ^ (6) gives \r = ^u + ^ + ^ — -, + Iw my n% whence (9) X (r*^ - v') = :, + -^ — — + —^ ^ (4)- + (5)' + (6f gives , „ , Z' w*' w^ ^"'' = ^" + ^^ ¥:z + (v' - f/f (v' - by {v^ - c'f ' 232 ENVELOPS TO LINES AND SURFACES. whence (lO) X^ (r^ -«') = - by (7) and (8) ; and therefore \ = — 7-^ , and /u = — ~ V (r — V'') r'^ — V Substituting these values in (4) we have w { 1 1 V (r^ — v^) OS vl whence 2_ ^2- y vm 7-2 _ 52 1)2 _ ^2 ' % vn ^2 _ „2 „,2 -8 Similarly and Multiply by <», y, % and add, then by (9) and (10) p 2 2 X V % + — -^ 1 = 1 . r^ — (f r^ — b^ T^ — (? This is the equation to the surface of a wave of light propagated through a crystalline medium. See Fresnel, Memoires de V Institute Vol. vii. p. 136; Ampere Annates de Chimie et de Physique, Vol. xxxix. p. 113; and Smith, Cambridge Transactions, Vol. vi. p. 85. If from the above equation we subtract 9 22 x'^ + y + % 2 — ^> and reduce, we find a^x^ b^V^ (?%^ r^ — c^ r" — b^ ' r'^ — & which is the form of the equation given by Fresnel. CHAPTER XV. GENERAL THEOREMS IN THE DIFFERENTIAL CALCULUS. In this chapter I shall collect those Theorems in the Differential Calculus which, d ependin g only op the laws of combination of the symbols of diff erentiation, an j no tion the functions which are operated on by these symbols, may be proved by the method of the separation of the symbols : but as the principles of this method have not as yet found a place in the elementary works on the Calculus, I shall first stater" briefly the theory on which it is founded. There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for sym- bols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are there- fore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u^ v two subjects on which they operate, then the laws are (1) ab (u) = ba (m), (2) a(u + v) = a (u) + a (v), (3) a'\aJ'.u = a"'^'\u. The first of these laws is called the commutative law, and symbols which are subject to it are called commutative symbols. The second law is called distributive, and the symbols subject to it distributive symbols. The third law is not so much a law of combination of the operation denoted by n, but rather of the operation performed on a, which is 234 GENERAL THEOREMS IN THE DIFFERENTIAL CALCULUS. indicated by the index affixed to a. It may be conveniently called the law of repetition, since the most obvious and im- portant case of it is that in which m and n are integers, and a"^ therefore indicates the repetition m times of the operation a. That these are the laws employed in the demonstration of the principal theorems in Algebra, a slight examination of the processes will easily shew ; but they are not confined to symbols of numbers ; they apply also to the I symbol used to denote differentiation. For if w be a func- tion of two variables w and «/, we have by known theorems in the Differential Calculus, d d d d — . — - (u) = — . -rr— (u) ; dx dy dy dw Also considering u and v as functions of x only. and besides dy ( dy Id y« + » The principal theorems in Algebra which depend on these laws, and which have therefore analogues in the Differential Calculus, are the Binomial Theorem with the great number of theorems — Exponential, Logarithmic, and others, which are derived from it ; and the theorem of the decomposition of a multinomial of any order into simple factors with the various consequences which are deduced from it. It is to be observed that in all the applications of this method to the Differential Calculus, a ^constant has the same laws of combination with the diffei^entials that they have with each other, and therefore the theorems are true for complex i symbols involving constants and symbols of differentiation. Also, there are two ways in which symbols of differentiation may differ from each other, either by having reference to different variables in the same function, or by having re- ference to different functions of the same variable, and this difference gives rise to two totally distinct series of theorems as will be seen in the following examples.. GENERAL THEOREMS IN THE DIFFERENTIAL CALCULUS. 235 It is worthy of remark, that the indices in the greater number of these theorems may be any whatever : I shall not however make any use of the interpretation of the for- mulas when the indices of differentiation are fractional. It is easy to see that when they are negative they are equivalent to integrals of a corresponding positive degree : for by the law of indices, \dx) \dx) \dwj . , . d\'^ ( d Also and therefore ["die" { — u= { — \dxj \dx \dwj this interpretation I shall frequently have occasion to use. The principle of the method of the separation of sym- bols of operation from their subjects was first correctly given by Servois, in the Annates des Mathematiques, Vol. v. p. gs. Some very valuable researches on this subject by Mr Murphy will be found in the Philosophical Transactions for 1837. (1) Taylor's Theorem. This theorem may be reduced into a very convenient shape by the separation of the sym- bols: for as k d h^ ( d\^ '' ^ y./vy Ida? ^ 1.2 \dxj •' ^ ^ h^ ( d Y „, ^ 1.2.3 \dail '' ^ ' we have, by placing the function outside, f A ), which is the index law. We may, therefore, put f{w + h) = EKf(.v), and throughout our operations consider h as an index. (2) Binomial Theorem for differentials with respect to different variables. If M be a function of two variables x and y, we have du du d (u) = -— dx + —- dy; dx dy or, separating the symbol of operation from the subject, d d Id d, \ d (u) = [-r-dx + -—dy] \dx dy J Affixing the general symbol n as an index to the operations on both sides of the equation, we have / d d , Y d" (u) ^ \-r- dw + -;- a V u. ^ ' \dx dy I GENERAL THEOREMS IN THE DIFFERENTIAL CALCULUS. 237 Expanding the operation on the second side by the Bino- mial Theorem, since the demonstration of that Theorem supposes only that the symbols are subject to the laws of combination before laid down, there results d"u d"u , , , d" (u) = -— dcT" + n- — - dx""-^ dy (n - l) d"u , 3 , „ + w ^ T-r--> daf"-^ dy^ + &c. 1.2 dx"-^dy' ^ (3) In the same way, by means of the Multinomial I Theorem, we may shew that if u be a function of any num- I ber of variables x, y, z... d"u dx^'dy^disy... ^ ^ dx^'dyl^dzy... 1.2...a.l.2...(i.l.2...y... where a + y3 + 7 + Sec. = n. (4) By the Theory of equations it is shewn that the \ expression X" + Jj 0?"-^ + Ja'^p""^ + &c. + A„_iX + A„ is equivalent to (x-a^) (x -Oa) ...(x-a„); G,, flgj ••• ttn being the roots of the expression equated to zero. It follows therefore that d^u d^u . d^u ^ d^u —, H Ai T—, — J- A2- „ ,^„ + &c. + An— — da?" dx^-'dy dx^-^dy^ dy"" is equal to ( d d\ I d d\ f d d\ \dx dy) \dx dyj \dx "dyj «!, ^2 ••• <^n having the same meanings as before. In this Theorem it is necessary that none of the quantities ] Ai ... An should contain u, x or y. (5) If w be a function of one variable x only, the pre- ceding Theorem becomes d"w d^-^u ^ d'^-'^u -JZn + ^1 ,7...-. + ^2 -1-^:^2-+ &c. + Au d.a?" da?"-' " dx d dx -N(^.-''=){5s-"') (i-"-) 238 GENERAL THEOREMS IN THE DIFFERENTIAL CALCULUS. , (6) By the theory of the decomposition of rational frac- tions, we know that Iv" + A^x''-' + &c. + J l-'= -— N, N, N, N„ + + — + + / x — a^ X — a2 00 — o^ w — a„ when «!, ttg, cfg .,, a„ are the roots (supposed all unequal) of x" + Jj cj?"-^ + &c. + 4„ = 0, and ' N-, = — ;; , — ^777*^ (a, - ag) (a, - a^) ... {a^ - a„) y (^ij with similar expressions for iVg, iVg, &c. ... N„. It follows therefore that lu) +^-y ^^'U ^""■^"'i " Or if M be a function of two variables, x and ?/, f/dy d" d" MVl" z* \dy) \dx ^dyl \dy) \dx ^dyl I dy^'^-'^f d d\-' ^^-y [drv-'^Ty) ''' If we suppose r of the quantities a to be equal to each other, they will give rise to a series of p terms of the form f d d\-P , . „ . I ^ M„ a — u where p receives all integer values from ^ \dx dyl 1 to r. The value of the coefficient M p is easily found. For if we put ) 1 ( d y-'i' ,^ / L ^ ^^' l.2...(.- p)Uj -^H-)- when . = «,. (^^1 GENERAL THEOREMS IN THE DIFFERENTIAL CALCULUS. 239 The results contained in the preceding four Examples are of great use in the Integration of Linear Differential Equa- tions, and in the sequel I shall have frequent occasion to employ them. The theorem in Ex. 6 was first given by Mr George Boole of Lincoln, in the Cambridge Mathematical Journal, Vol. ii. p. 114. (7) Binomial Theorem for differentials with respect to diflFerent functions. If w and V be two functions of cV, then d du dv — - (uv) = V —- + U-j- . dw dcV dx Now if we accentuate the symbol of differentiation which applies to v to distinguish it from that which applies to w, we may write dw \dw dw) Affixing the index n to the symbols of operation on both sides, dy . _ M d'v" \dwj \dx dw) or expanding the binomial on the second side by the theorem of Newton, we have fdy^ ^ d^u dvd"^^u (n-l)d"v d'-^u -— (uv) = V ~ — + n—- —-—r + n -— -— T, + &c. \dwj ^ ^ dw" dw dw"-^ 1.2 dw'dw''-^ This is the theorem of Leibnitz who arrived at it by in- duction for integer indices ; but it is true whether n be integer or fractional, positive or negative. (8) This theorem may be extended to the product of any number of functions by means of the multinomial theorem, so that we have 1/ rf \" / d\l^ I d\y — — — 1 .2. ..a. 1.2... Q.l .'k /3.1.2...7 ] /here a + ^ + y + ... = n. 240 GENERAL THEOREMS IN THE DIFFERENTIAL CALCULUS, (9) If n be negative in the theorem of Leibnitz, I d\-" 1 — 1 (uv) = f do)" (uv), and therefore f"dcv''{uv)=vf"dw''u-n — ["-^^dw^'^u dcV n(n+l) d^v . „ , ^., + ^ f''^^dx"^'u - &c. which is the general form ula for integ ration b^^ (10) In the last expression let m = 1 ; then ["dafu = ; and rdx^{vi) = -« — + &c. ; l.2...n-l\n n+\ doo 1.2 w+2 e^.^^ / or if w = 1, ^ , , . or dv oc^ d^v I dx iv) = xv + — - - &c. which is the series of Bernoulli. (11) In the theorem of Leibnitz let v = e°% then as dv ^^ d' — - = «e = av, we have —— = a, and therefore dx da) whence («+-—) ?^=e-"''(-— ) e'^^'w. V a<2?/ \da}) This result is of great use in the Integration of Linear Differential Equations. (12) If we assume as before A. we have E^'^fiw) =f(x + nh). Now £''/'- l=--^_-(£^-l)=-^^_-(e*^--l); i GENERAL THEOREMS IN THE DIFFERENTIAL CALCULUS. 241 or expanding the exponential ^^j^ E'^'^-l ,d h' f d\' h^ f dy E^ - 1 '^ dcv 1.2 \dwj 1.2.3 \d.vj * Apply these equivalent operations to f(w), and indicate the successive differentials by accents affixed to the f; then f{^v + nh)-f{.v)=^^{hf%v)+~f'Xa^)+ j|/'"(^)+ &c.} But j-^ =^-(«-l)A + £(«-2)/.^^(«-3)A^ &c. + 1. E'' — l Therefore, writing these in an inverse order and effecting the operations indicated, we find f{w + nh)-f(a>) = h[f(w)+f\.v + h) + kc.+f{a^+(n-l)h]] + &c. + &c. (13) Since we have jpnh -t d l+i:^+^2V&c. + £<'^-^>*=— ^ =(jE''*- l)(e*^-l)-S E — 1 d we may expand the factor (e'^^- 1)"^ by means of Ber- noulli's Numbers; (See Chap. V. Sec. iv. Ex. 9) when it becomes h\dx) 2 1.2 dw 1.2.3.4 \dx) d Applying these equivalent operations to -^f(^) or f (jv), multiplying by h and transposing, we have (^"'' - l)/(a?) = A 5J + jE^ + &c. + E('^-i)^' + l£«^'|/ {x) - — A^ (£«* - 1) f" (x) + — ¥ (E""^ - 1)/'" U) - &c. 1.2 JJ \ J 1.2.3.4 That is / (^ + nil) -f{x) = h [1/' (a?) +/ (^ + /O + &c. + f{oG^{n- \)h\ +i/'(.r+wA)] + 16 • 242 GENERAL THEOREMS IN THE DIFFERENTIAL CALCULUS. - &c. &c. The results in the two preceding examples are of great use in the approximate evaluation of definite integrals. Poisson, Memoires de V Institute 1823. (14) Having given the transcendental equation we can expand oo in terms of c by means of the logarith- mic method of solving equations : for the root of the pre- ceding equation is the coefficient of - in the expansion of 1 1 . This is easily found to be c + — c^ + -^ — ^—& + — ^^ — - — e + &c. 1.2 1.2.3 1 .2.3.4 Instead of co substitute -r- ; then e dx = E^ and c = ^- -E~*. doe doa Hence we have _d_ d ^_,_ 2^ /^^^^_2A. W (d doB doa 1.2 \dx) 1.2.3 \dwj Applying these equivalent operations to jdwf{ai) we find f{w) =/(^ -h)+ ^^f {^ - 2h) + ^^f" {.V -3h) + &c. This very remarkable theorem is given by Mr Murphy in the Philosophical Transactions. (15) In a similar manner we may prove the more general theorem, f(os) =f(jv - nh) + nhf {w - {n + l) h^ h^ h^ + n{n + 2)—f" {a)-(n + 2)h\ +n(n + 3y f"{.v-{n + 3)h} + kc. GENERAL THEOREMS IN THE DIFFERENTIAL CALCULUS. 243 (16) We know by the Calculus of angular functions that TT 11 -G = sin 6 - — sin 3 G + — sia 5 $ - &CC. 4 S- 5- Putting for the sines their exponential values and replacing (-)^d by (h-—] we have — n — = e a* — e dx ( e doe — e dx) + &c. 2 da? 3^ ^ Applying these equivalent operations to (p (w)^ we find TT , d 2 dx — h — (p {os) = (p(a) + h) — (p{w - h) --_{ m ■' in (I) fdx (sec xY - tan x. 246 INTEGRATION OF FUNCTIONS OF ONE VARIABLE. By simple algebraic transformations we may frequently put an integral into a shape in which one or other of the preceding formulae is at once applicable. (1) / =_ / = --log (a + 6a?"). ^^ J a + hw'' nbJ a + bo)" nb ^^ r da) r doc ^ r d{a-x) ^^' J{2aw-x')^^ J {o'-ia-wfY^^' J {ar-{a-wyY^ a — CO x = cos ~ ^ = vers ~ ^ - . a a dcV , i(x^ + 2ax)^ + cV + a] (^> i(^^T^i = ^^q— ^^ }• ^ ^ J (a* - x')'^ " 2 j {«* - (a^yyi 2 sin ^. ^^ Ja' + x'~ ^J a^ + {wy ~ 2a^ W) ' /' dx X r d {x^ — a^)\ y"^ J {(x^ - a') {W -x')Y^ J {¥ -a^- {x" - a^)\l J a + ox + ex c J = sm dx \ "^2c/ b\^ 40, or < 0. Hence we have r dx 2 /2c^? + 1\ (7) / 5 = -tan-M ^- . r dx _ 1 / 2x - 1 + 5h ^ J 1 +x -x^ ~ 5I °^ Uo? - 1 - 5y ' r ^ r dx , ^ (9) / -= tan- 1(207 - 1). ( \ f ^^ - 1 /2^^ + l \ ^ ' Jl +3a? + 20?^ ~ °^|2(a7 + 1)J ' INTEGEATION OF FUNCTIONS OF ONE VARIABLE. 247 The integral /- — —, is reduced to dcV I r dx 1 r dx I p c^ Jn by 4ac-6~ ' °'' ^° ?JUac+b' / b yW according as the upper or lower sign of c is taken ; and these are of the forms (/) or (e) respectively. Hence -y = log {2.2? + 1 + 2 (1 + ct? + ^-)H. (1 + 07 + w^y^ ■" r dx , , /a? — l\ (12) \-r ^. = sin" -\. ^ ^ J (1 + 2cJ? - 0(f-)\ V 25 ; ^^^^ i{x^-l-\)l = ^°^ ^^''^ - 1 + 2 G^.^ - /r - l)i}. ^^*^ J (1 - 07 - ^^)^ = ""^ = sin The integral / — may be split into J x"^ + nx + Q + px + q dx a r(2x + p) dx f ap\ r dx a r{9. \ 2 } J a^ + px + q 2 J x' x^ + px + q 2 J x'^ -j- px + q the first of which is integrable by (c) and the second by (b). Hence (15) \^„ = log (a- + 2bx + x'Y J a" + 2bx + x^ ^^ ^ ^ tan-^l-^±^l ^ ^v r(^^v-l)dx - ,, , 3 ,07+1 (1^) / 2 ^ o = log (07V 207 + 3) - -^ tan-^ ~- . ^ -^ J x^ + 2x + 3 ° 2^ 22 /,«\ r(l-xcose)dx . ,/o7-cos0\ (17) -^ - = sin0tan-M — r— r- ^^ 1 - 2o7cos6' + 0?"^ V sinO / — cos 6 log (1 - 2 ti? cos 9 + x^)k In this example the numerator may be readily split by ob- serving that 1 = cos^ + sin- 0. 248 INTEGRATION OF FUNCTIONS OP ONE VARIABLE. ^=T^tan-'--^- log (1-^ + ^0^. ■27 + ,3?" 2.3a 3S (.s) /^ By multiplying the numerator and denominator of a frac- tion by the same quantity it may frequently be split into in- tegrable parts or reduced to an integrable shape. (20) fdo) — = (ai- - a?)^ - a sec"* - . J so a (21) fda^ (f!_+^ ^ (.^e ^ ^,^^ ^ 1 L^-^^ri 1- , ^ r d,» (a? + a)^ , w , Tci? + ice^ - a^)^) a)§ dw (23) /"t rr-7 rrj = -7 {(^ + «)t - (^ + 6)^. ^ ' J(a? + a)2+(a?+6)i 3(a-6) *^ * /• d,a7 /- x'^ dw X ^^^^ i (1 - = tan X — cot x = — 2 cot 211?. [(cos xy (sin a?) J (44) / = - / — l^-L = _ tan hv. J a{l + cosci?) a J (cos-^xy a -^ ^ ^ r dxsinx 1 1 , (45) / - — = log ia+b coso?). J tt + 6 cos cT 6 The integral / ; may be reduced to the form J a + b cos X (c) ; for as cos X = (cos ^x)' - (sin ^ ^a + 6cosa? (a' - &')i \\a + bj ^ J 1 ^ /b + a cos ,v\ {a^ — b^)^ \a + b cos x) ' or r dx 1 f (& + a)^ + tan|- {b - d)^ a)\ J a + bcos X " (6'' - a')5 °^ [(& + a)2 - tan^ (6 - a)^^ ' according as a > or when a > 6, 1 , [a tan let? + 6 - (6' - a^)2] , . gr-i log {— f 7 7U ^f ^^*^^" ^<^- - a'^)2 ° ^atan-Jci? + 6 + (6^ - a'^)2j and = (h'' _ «2\* ° l/y fan _, , ^ r d<» _ , /Stanlci? + 4\ (48) / : = |tan-M ^ . , . r doe 1 , /S tan \x + 1\ (49) / , — = i log f . ^4+5sin.?7 ^ ^V2 tan 1^7 + 4/ .^qx /- da; ^ r dx (secwf J a (cos xy + b (sin xY J a + b (tan o?)^ 252 INTEGRATION OF FUNCTIONS OP ONE VARIABLE. , - r dw 1 , /tan ai\ (51) / -„=— tan-M — ^ . ^ ^ 1 + (cos wf 22 V 2 W rdx ^mw {co?,wY 1 rdxsmw\l-\-a^{co?,xy — l^ ^ ^ J -[ + d^ (cos wf ~ a^ J 1 + a^ (cos xf I . . I r dw sin X = — Jdx sm a? :, / — — ^ a^ aJl+a^ (cos xy H — - tan ~ ^ (a cos .5?). (S3) /--^^—^ f J a + o tan . j |6-(6_a)(cosa?)^5^ {b-af \\ b ) J By means of the formulae for expressing the products of sines and cosines of angles, in terms of sums and differences of sines and cosines of angles, we easily find f cos (rn + n)x cos (jn — 7i)x\ (55\ (dxsmmxcosnx=—^{ 1 >. ^ ^ J I m + »^ m — n J fsin(/» + w)<2? ^\xv{m — 7i)x\ (b^'^ \dx^\wmx^\mnx ^ -k\ K ^ ' J ^ [ m-vn m — n J \%va.{m-^n)x sin(m-w).2?l (57) /rf.i?cosm<»?coswa?= k\ ■ + >. ^ ' ^ J '^\ m+n m — n J (cos4(J7 cos6t'^l cos2a? + — ^^— J. INTEGRATION OF FUNCTIONS OF ONE VARIABLE, 253 , . -_ , fsin6 2 J ,- - ^, . . - f sinScT sin 4.37 sin6<3?l (60) Jdx cosa? sin2xsin3tV = -^ix ■{ >. Integration by Parts. Integration by parts often decomposes a function into an integrated part and one easily integrated. The general for- mula is -, dv . du •' dx ^ d.v (1) fdx cT?e«* = 6«* (- - -^) . \a (Tj (2) ^dCG logcV = CO (logcT' — 1). (3) ^dcB ocXo^x = — Qogx — 1). , . r. , a7" + W, 1 \ (4) Jdx x^nogx — ( log a? 1 . (5) fd,v sin~\v = crsin~'cr + (l - x^)^. dw . , a?sin~^a? . r aw . a,-sin w , rdwwsm-^x . oM . , (8) /da?sin-M^ — I =(a7 + a)sin-- ] - (ax)i, \a + xj \a + xj . ^ ri ' ,1 l^a-x\i x^ . ,, f2a-x\^ , /- a?^rf<^' (9) fd...n-'i{—.) =-s.„-i(— ) Hj^-i^^r^s = — sin~^ I + — sin - ' (4ff/ - x^y. 2 \ a J 2 2a 8^ 254 INTEGRATION OP FUNCTIONS OP ONE VARIABLE. (11) / ^ tan~\77 = (c?? — 1 tan Kv)tsin~^ a^ + n^ On comparing these expressions with the formulae in Ex. (lO) of Chap. II. Sec. 1, of the Diff. Cal. it will be seen that they may be deduced from the latter by making r= — 1. Rational Fractions. If — be a rational fraction, in which the numerator is of lower dimensions than the denominator, it may always be decomposed into a sum of simpler fractions differing according to the form of F. V may consist of factors of the forms I.. X — a, II. (x — ay, III. x^ + ax + b^ IV. (.1?- + ax + by. I. To every factor of the form x — a corresponds a M partial fraction of the form , where * X — a M = --— when x = a. dV dx INTEGRATION OP FUNCTIONS OP ONE VAEIABLE. 255 II. To every factor of the form (x — a)" corresponds a series of partial fractions of the form 7 ~T„ + -? ^ r + &c. H , (x-ay (x-ay-' x-a Any one of the coefficients as Mp is given by the equation Mp = (-- ] —) when x= a, where Q (x - ay III. To every factor of the form x'^ + ox + h corre- ]\£x -\- N spends a fraction . To determine the constants x'^ + ax + h M and N, the expression dV (2x + a) - (Mx + N) — = dx is reduced by successive substitutions of — (ax + b) for x- to the form Ax + B = 0, and from the conditions A=0, B = 0, M and N are found. IV. To every factor of the form {.x^ + am + 6)" cor- responds a series of fractions of the form Mx + N MiX + N, „ M„_,x + N„_i {x^ + ax + by (x^ + ax + 6)"~ ^ c^'^ + ax + b To determine M and A"" let V= Q(x~ + ax + b)'^; then if by the successive substitutions of — {ax + b) for x^ the equation U - (Mx + N)Q = be reduced to the form Ax + B = 0, the equations A = 0, 5 = are conditions for finding M and A^, If now we put U- (Mx + N)Q X'^ -f ax + b 256 INTEGRATION OF FUNCTIONS OF ONE VAEIABLE. where U^ is necessarily an integral function, we can, from the equation U, - (M,a,' ^N,)Q = 0, determine M^ and iVj as before, and so in succession for all the other partial fractions. The fraction having been thus, by one or other of these methods, decomposed into a sum of simpler fractions, each of them may be integrated separately by known processes, and so the whole integral is found. M If the partial fraction be of the form , we have lV — a M f — ~ = M log (a? - a) = log {co - a)^. M If the partial fraction be of the form , we have ^^ r dx M 1 M I = • . J {co - ay {r - 1) {w - ay-^ MiV + N If the partial fraction be of the form , we have rda;{Mw + N) ^^, ,^ ^„ _.^, , Ma + N .(a!-a.\ ■ Mw + N If the partial fraction be of the form {{a^-ay + ^^Y r dx (Mx + N) _ M 1 J l(^a;-ay + (^'\'- " 2(r-l) \{x - af + (3'^-' dx U "^:i T^^r The expression for the last integral will be found in the following chapter on formulae of reduction. (1) Let - = V x^ + — . W If ~ = V x^ + 5x^' [• dx 1 ^ (<2?^ -X + S){x+ If^ 1 a;'^o 3 a? 13 ,2 r dx 1 b (a -{■ bx^\ J x" {a + bx^) ~ Sax^ Sd'' ^^ \ x^ ) 1 r ux 1 1 /a + bx^ a + bx^y 3a (a + bx^) 3d 1 /•-ii_=-.iog(^ J 1 + x" 25 ^ Vl 1 + 22c'J? + a?'^\ 1 , + -g tan~^ 22a?+<2?' 9hx 1 - a?' r dx , /I + -^^A* 1 ^ _i dx - a ^x" dx rx~ ax , /I + a?N4 il^'=^°nr^' -*ta„-',r. r dx J x(a + (a + 6 a?*) dx 1 , /a + 6a?* /-da? 1 l+,r/l + a7 + x''\ 2 J 1 - .17^ ~ 6 ^ 1 - a? Vl - ^ + -a^V ■\ 1 tan" 2.35 35,2? 1 - x^ INTEGRATION OF FUNCTIONS OF ONE VARIABLE. 261 .. r 00' dx 1 , (ccy Rationalization. p Integrals of the form [dx x^{a + hx'^y can be rationalized, when IS an integer, by assuming a + ox = %^, and, , m + 1 p . n n ' when h - is an integer, by assuming a + bx = x ss'^. (1 (2 (3 (4 (6: (7 (8 (9 xd,v 2 r x^dx . .1 ((a;'- if S . ., \ J (x-l)i ^ 1 7 5 ^ i r dx 1 r(a + 6^)5 - a^] J X (a + bx)2 as ° [(« + 6^7)2 + aaj /-— ^=_tan-M =-cos-M— . J xibx — ayi a^ \ a ) a^ \oxJ / -T-7 T = (^ - 1)^ 5- + - COS-^ - . J x^{x-\^l ^ M 4. r dx X 2 f 2a + & . jdx x{a + x)^ = 3 (a + ,a?)s /4<2? — 3 /eZcT A^^ {a + a?)^ = 3 (a + <»)M 4 V 7 5 \{a + c»)^ da{a + xf 14 11 3a^ {a + x) a -il d^ x^ r ax x" bx^ — 2a . , „ , / dx 3 b' 2x^ - 1 - (1 + a,^)^. a?^ (1 + x^)^ 3x^ r dx x^ .^ 1 fa?^ Sa?"* 5.3. /(! + .)(! ^^¥ °^in TT^ /• f . r dx _ S /(^ + '^')' + 2iT| ^^^^ J (1 - x') (1 + .t7^)i " ii ^^ 1 (1 - x-y^ ] ' 264 INTEGRATION OF FUNCTIONS OF ONE VARIABLE, r dx 1 ((x^ - l)i + 2^a}\ , , r dbc J (c + ew^) {a + hw^p \ac + aex^l 1 . c{a + bw^)i + w (be" — ace)i log J ^-^ , ae "^ ^ (1 + c-j?) (1 + a? - x^)i [2(1 + X - x^)h] ' , ^ r d^ , [S+ X -9,(l-x-x^)^ <^^> / o^.)(i-.-^)r '°sl hlo -]■ Various functions can be rationalized by assumptions for which no general rule can be given : familiarity with the transformations to which different substitutions lead is the best way of acquiring a knowledge of the most convenient assumption in particular cases. \w + (1 + K 1 + a?* we «have r dx{l + a?')^ 1 r dz I f r dz r dz \ J 1 -w^ " 2lJr^^ " il" U 1 +z^ '^ J r^zv (40) By the same assumption we find that r dx x^ J_ ((l+x*)i + 2ix] ]_ . /2^x\ 266 INTEGRATION OF FUNCTIONS OF ONE VARIABLE. (41) If du = — - — -, , assume ,2? = ;& (2 5 . 4< . 9.7.5 9.7.5.3 A 9.7.5.3 ,x + ■ a'^x + a H vers " - , 5.4.3.2 5.4.3.2.1 / 5.4.3.2 a ] (3) Let the function be -—z r— . The formula of reduction is r dx 1 X 2n — 3 1 r dx J (a' + x^y " 2W-2 a2(a- + a?')"-^ "*" 2w-2 a:' J (a^ + x^y- By this the integral is reduced to J ^ dx 1 , <» -= - tan"^ - + x' a a Let w = 4 ; = — /dl when n is odd, ^ cr (1 + m^f^ ^\ X ] . r dx (1 + a?^)§ , and / — — — i = — when n is even. J or {\ + CB^p w Let w = 6 ; dx 1 (1 + 00^)^ 4 (1 +x^)i 4.2 (1 + x')h Jx'(l + x^)i 5 x^ 5.8 x'^ 5.3 X (8) Let the function be ; — -7 ; ^ (a + bxp r x^dx 2x^{a + hx)^ 9>m a r x^~^ dx J (a + hx)^ (2w+l)6 2m + Ih J {a + hx)^ Let 7W = 3 ; r x^dx /ct?'^ 6 ax^ 6.4 a^x 6.4.2 a^\ Let w = 4, 6 = ] 8.6 „ 8.6.4 + — ' — - a-x^ a^x r x^dx ^ . Ja^xl Sa'-^J x{a + bxy^ (10) Let the function be (a + 6c'p)3 i ' r x"dx Sx" (a + bw)i 3n a r x'"'^ dx J (a + bx)i~ {3n + 2)b 3n + 2I) J (a + bx)^' If n=l; r wdx 3 (a + bx)^ f a\ J (a + bx)3 " 5^~~~ V * ~ 2/ ■ If w = 2 ; /- x^dx 3(a + bx)i {(a+bxf 2 , . . «''| / i = — ^^ z — ~ { a (a + bx) + — > Ji,a + bxf b^ \ 8 5 ^ 2/ (11) Let the function be {or + a^y dx 1 X 71 - 3 r dx /ax L W //. — ^ r uaj T~^ ~Ji ~ (n - 2) a^ 71 2a|-i ^ (n-2)a^ J , . „ J (a'^ + af)2 ^ ' (a^ + a?^)2 v / (a^ + x~y n = 5; , r dx /I 2\ 0? *^ (a^ + ci?')i U' + ^' «V 3a^(a2 + .^^^^ • w = 7; /. rf.r _ I 1 4 1 1 ^p i (a^ + a?^)a ~ \(a^^+~xy ■'■ 3^^^^^ "^ i^j 5a'(a' + x-)l^ If w = 7 ; 272 INTEGRATION BY SUCCESSIVE REDUCTION. a?" (12) Let the function be -. ; s;t; ^ ^ (a + 6a? + cxY r x^dai a?"-^(a + 6a?+ca?2)5 n - 1 a r J?""^ 2n -1 b r af'^dx In c J {a + ba} + c ^ ' ^ ^ fdx (sin x)"' (cos a?)" (sin a?)'"^' (cos a?)""' w-1.,,. . ,„ = 't i !^ i — + fdx (sin xy"(,cosxY-^ ... (4) m + n m + n fdx (sin a?)" (cos x)" (sin .!p)"''^'(cosa?)""''' m + n + 2 m+1 m+1 fdx (sin a?)™ (cos x)" (sina7)"'+'(cosa')" + ' m + n + 2 n+l n+l 18 /da7(sinc'}?)™ + ~(cosa?)"...(5) /da?(sin *')™(cosa?)" + " . . . ((>) 274) INTEGRATION BY SUCCESSIVE REDUCTION. COS 00 fdx (sin cvy = {(sin H by (3) /sin 4a? . \ 3(27 = 4- - sm 2a;- + — . ^ V 8 / 8 fdoff (cos xy = ^ (sin x cos tV + j?) by (4) sin 2 ,2? tl'H- 2 n / .^ sina?r^ .. 4 ^ x9 ^1 1 ^ X j a/i? (cos it')'' = < (cos a?)* + - (cos a?) + - > by (4) -. f sin 5 a? 5 sin 3 a? . 1 = + { + ;; + 5 sm a?>. ^ [ 10 6 j Jdx (sin a;)^ (cos a?) ^ = ^ |(cos oof + -\ by (4), 1 /I . 1 . — - sin 5a? + - sin 3a7 - 2 sin oa 16 \5 3 r 1 y • i , v5 (sin a?)^ f ■> 2] ^doc (sin a?)* (cos a?)"' = < (cos a?)'^ + -> . {dw (sin xf (cos a?)^ = 1 — cos 1 0,2? cos 6x -v 5 cos 2 a? 1 J ^ ' ^ ' 2^10 6 / ^doB (sin a?)^ (cos a?)^ 1 /I 5 = -5 - cos 9 ^ J?. 2- / ; — = — : ] (cos xy^ - 3 cos X J - J (sin a/y 2 sin x rdx (cos xY 1 f(cos ci?)* 1 , .. X / ^^ i- = \ ^ ~ - 1 > - 2 loff (sin x). J (sinxy (sincT?)~\ 2 j & v y /•da? (sin ^)^ ( 2l 1 / — = { (sin a?)'^ > . J (cos a?)' \^ 3j(cosa?)' rdx (sin xY 1 ( . 2) / ^^ = — ;:^Sincl? — > . J (cos ■3?)*' 5(cosa7)' [ 7j r dx 1 , / X / -; : = h log (tan x). J sin X (cos xy 2 (cos a')"^ r dx 1 8 / 7-r- — 7 = — ; -, r cot 2x. J (sin xy^ (cos xy 3 sin d? (cos o?)^ 3 r dx 8 cos 2 a? r ] 2 1 ^ (sin<27)' (cosa?)* 3 \ (sin 2 a?)' sin 2 a? j r dx 1 1 , , X / ^^ r — = X J-. 1- loff (tan X). J (sin xy cos X 4 (sm xy 2 (sin cv)^ (17) If the function be (tan .r)" the formula of reduc- tion is /■ , , . (tan wY ' r , . . o /rfcr (tan xy = ^ ~ fdx (tan c-r)'' " \ 18—2 276 INTEGRATION BY SUCCESSIVE REDUCTION. If the function be the formula of reduction is (tan wY r dec 1 1 ^ , ' / -: = -^ ; - \di (tan ccf (m - 1) (tan a?)" " ^ ^ (tan a?)"-^ * ^dx (tan ouf = - (tan a?)^ - tan cc -^^ x. fdx (tan wy = - (tana?)® (tan xy + - (tana?)^ + log(cosa?). /d 00 = - 1 (cot xy + 1 (cot xy + log (sin x). (tan xy "^ (18) If the function be a?" cos,2?, the formula of reduc- tion is ^ dx .a?" cos ,2? = a?" sin a? + w .a?" ~ * cos .2? — w (w — 1) /da? .j?""^ coso?. jdx x^ cos 0? = /J7^ sin .2? + !^ a? cos ci? — 2 sin x. fdx x^ cos X = a?^ sin a {a^ + 4) . . ,„(^sin<*'-3cosa?) 6e"'^(asina7-cos<2?) fdx e"' (sm xy = e^'i^mxy -^ +— -^ — , . , ,, . -' ^ ^ a' + 9 (0^ + 1) (a^ + 9) e"^ fa sin 7.2? - 7 cos 7 <27 /d^ e" (sin ^) ■ (cos at = - | ^.^,3 3(a sin5<2?-5cos5a,') a sinS.r- 3 cosSa? 5 (a sin .27 — cos <2?)] aN-25 ~ a' + 9 «' + fdxe'^^icosxy'^; - cos <2?) 1 ^ J" INTEGRATION BY SUCCESSIVE REDUCTION. 277 (20) If the function be -; the formula of (a + b cos «)" reduction is r doD - 6 sin <» J (a + b cosivy (w - 1) (a^ - b^) {a + b cos wy~^ (2n — 3)a r dx (n - 2) ■ r dw "*" (w-l)(a'-6^) Jia+bcosaiy-'" {n-\){a?-b^) J{a+b cosxy-'' Let w = 2, then C dx J (a + b cos oay 1 r -6sind7 2a -i\(^~^\^ '''^iT ^ a^ -h' Va+b cos w "*" («^ - 6')^ *^" \U + 6/ *^" ijJ* Hence also we find /<^r 2 X (15) Let a (xdy + 2y dx) = xy dy. Dividing by xy we have [dy 2dx\ \y X ) = dy\ y x'y= Ce''. whence (16) Let dx + (a dx + 2by dy) (1 + x')k The integrating factor is (l + x'^) ~ §, and the integral is a? -1- (1 +a;^)h = (7e-(«^+6/). (17) Let (2x - y)dy + {2a - y)dx = 0. The factor is (2a - y)~^ ; multiplying by it we have (2x—y)dy + (2a-y)dx (2a-y) (dx — dy) +2(o. + ,v-y)dy INTEGRATION OP FUNCTIONS OF TWO OR MORE VARIABLES. 283 Integrating this we find a + tv - y = C (2a - y)^. (18) Let y dx — w dy = xd,v + y dy. The factor is {w^ + y")~^i ^^^ ^^^ integral is tan~^ - = log {od^ -f- y^)^ + C. . s -^ , (dti y dcV\ (19) Let ydy-wdx-h\~-- — - = 0, - The integrating factor is (^/"^ - r^%) + dz {x + ?/) = 0. Making y constant, and therefore dy = 0, we have 2 dx dz + - = 0; X + y y + ^ whence 2 log {x + y) + log [y + z) = (p (y). T^m • • 2(dx + dy) dy + dz ,,. . , Dmerentiating + — = («) ay, X + y ^ + ^ or, 2 dx {y ^ z) + dy (x + 3y + 2z) + dz {x + y) = we have '' \dxl \dx I \dx 7 Kdx j 288 DIFFERENTIAL EQUATIONS. The result of this transformation is different according to the nature of the roots of (3). 1st. Let all the roots be unequal; then by the theo- rem given in Ex. 6. Chap. xv. of the Differential Calculus, the equation (4) becomes + N,(^^-a.y'x (5) where iVj = , («i - as) («! -as) ... {ui - a„) and similarly for the other coefficients. But by the theorem in Ex. 11. of the same chapter, A similar transformation being made of the other terms, we find y = N^e"^'' Jdjee-^^^X + N^e"^"" fdx€-''^''X + 'hc. + N^e'"'^^ /dxe-'^'^'-X (6). It is to be observed that each of the signs of integration would give rise to an arbitrary constant ; and that this must be added in each of the terms when the integrations are effected. The value of y would then appear under the form y = Ni e"^^ (fdxe- '^^^X + C,) + N^ e'^^^ (fd.ve- "^'^ X + Cg) + &c. -^ N,, e""' {jdcce-^^X + C„) (7). Cj, C^...Cn being the arbitrary constants. The functions Ce"' which arise in the integration are called complementary functions. 2nd. Let r of the roots of the equation (4) be equal to a. Then by the Theory of the decomposition of partial DIFFERENTIAL EQUATIONS. 289 fractions we know that the factor ( a\ will give rise \dx to a series of r terms in (5) of the form the coeificient Mp being equal to 1 Id 1.2... (i^^\\ \d% d y^^'^ {z — ay 4j i.2...(^--i) W; f(z) d \-P when % = a. Now {-r--a\ ^= e"^ Pda?? (e'"^^) \dai j or, introducing the arbitrary constants which arise from the integration, d ( a] ^ X= €"' fP dxP (6-«-^X) \dx J + 6«^ (C'o + C\.v + &c. + C'p_,xP-'). Therefore the complete value of y is y = 6"^ {Mrf' dx' {e-^^X) + il/,._i /'- * dx'-^ (e'^^X) + &c. + Mjdx{e-''^X)\ + A^jg"'^ Idx^E-^^^X) + N.^e''^'' fdx (e~«2*X) + &c. + iV„_,6°»-'* fdx {€-"'•■-'' X) + 6«^(C'o + C\x + C\x^ + &c. + C',_i.t?'-') + Ci6«i^ + Cse"^^ + &c. + C„_,e"-^. (8) There are in all exactly n arbitrary constants as there ought to be. 3rd. Let there be a pair of impossible roots, which must be of the form a+(-)^/3 and a-(-)^/3; then the coefficients of the corresponding terms in (6) are of the forms 19 290 DIFFERENTIAL EQUATIONS. And as 6i" + (-)*^^^' = e"^' {cos/3 \ 1.2.3 1.2.3.4.5 1.2.3.4.5.6.7 J C = C COS wa? 4 sin T^a? = C cos w<2? + Cf sin w.r, w as the constant is arbitrary. As operating factors of the form -— + n'^ very fre- \CliV J quently occur in differential equations, it is convenient to keep in mind that the complementary function due to it is of the form C cosnx + C' sinnx. (8) Let — -^ + nry = cos ma?. Then «/ = < ( — i + w^/ Qo%mx -\- Cno%nx -v C 'sva.nx ^ \\dai] J cos mx , . = — r + C cos nx + C sm tIcJ?. w — m" (9) Let + 5 h 6« = sm wa?. therefore sinwc?? ^ , 1 . ^ ^1 w = — + C cos (22 a? + a) + Ci cos (32 a? + 18). nr - 5mr + 6 ■ d^y (10) Let _4 + 2/ = cr". I Then y = li + f — j > a?" + C cos a? + Cj sin a? r/ = cr''-w(w-l)a?"-2+^(w-l)(w-2)(w-3).i?"-*-&c. + 6^cos.i?+ Ci sin a?. 296 DIFFERENTIAL EQUATIONS. (11) Let j^-«'2/=*''- The roots of is* — a* = o, are « = + «, % = — a, %= + {-)^a, ^=-(-)2a; therefore y=- — + Ce''' + C^ e""^ + C^ cos {aw + /3). a (12) Let -4 + 2a^— 4 + «"*« = cos a?. This is equivalent to I© +^^' = 003.. From which we find y = — — + (C + Cisin{axsin \ \ n n) \ n nj] , acos^^f^ / • 27r 2,r\ / . Stt SttNI + e n ^l C5COS a.3? sm h — + C^ sm aci? sin y — > \ \ n n I \ n n ) ) + &c. + &c. This is evidently the solution of the equation d-y ,2» - - a"'v = 0. Euler, Calc. Integ. Vol. 11. Sect. 2, Cap. iv. (16) Let -— - aPy = cos mx. + 6 DIFFERENTIAL EQUATIONS. 299 . COS mx ■, ^ „„ then y=- + Ce''''' + C^e*^* «^|c3Cos («*• ^ + I) + C.sin (^aa; ~ + |]| C5C0S lax — + — j + Ctsin lax — h — > . (17) Let al'y + 710!"-^ -^ + — ^^ ~ a""' -^ + Sec. = X; ^ da;' 1.2 dcP^ /z being a positive integer. Here the operating function is I « h 1 , which is com- posed of n equal factors ; consequently = 6-'^V"^^''e''"X + 6-""(Co+ Clc^? + C2c'»^ + &c. + C„_la?"-'). The term e''^" f" dx" e"'" JC may either be integrated by successive steps, or by the general formula for integration by parts ; or what will generally be more convenient, the function I a + — - ] may be expanded in ascending powers of — - . If n were negative or fractional, the first term would retain the same form, but the form of the complementary function would be different from the difference between the roots of (x + ay = 0, when n is integer and when it is fractional or negative. I cannot however here enter into a discussion of the difficulties of this subject, which is closely connected with that of General Differentiation. Euler, Calc. Integ. lb. (18) Let -r^^-. ^ + &c. + -^+«=X - dx" dx""-' dx -^ 300 DIFFERENTIAL EQUATIONS. This may be put under the form — y = X. d dee d N***^ - 1 Kdw) Now the factors of — ^^-^— are the same as those d doB ( d \"+^ d of ( — I - 1, omittinff 1 ; therefore if w + 1 be odd, \dx) dx the integral of the preceding equation consists of a number of terms of the form 4 sini0e^^°'^cos^(30 + 2a?sin 0)/d^6-*'*^°'^^sin(a7 sin^) 4 sinl0e^*='''^sinl(30+2*sin0)/da?e-^^°=®Xcos(.^4.i... Stt dwj \ Sir dx Hence decomposing -jcos (;r^) [ into partial fractions, find ^ V2 dw) \2 dx) \ 2 dx) — + — -^ + &c. 2 dx and therefore y= - e^"" fdx e'^"" X + e"^"" fdx e^"" X 3 + €^ fdxe '^ X- e~ ^'fdxe^' X 5ir Sir Stt Sir -62 'fdxe~ -^ 'X + e^^'/dxe^'X + &c. - &c. IT 3ir Stt irx Sir Stt + C/e" 2 + C,'€~^" + C,'e~^"+kc. Euler, Calc. Integ. lb. (20) liet the equation be n{n - I) d'^y n (n - 1) {n — 2) (n - 3) d^y y + — ^ + — — — — -r + &c. = X. ^ 1.2 dx^ 1.2.3.4 dw^ The factors of the operating function in this case are the sam^ as those of the algebraical function The quadratic factors of this expression are given by the formula (1 + ^)-- 2 (1 - %^) cos 20 + (1 - %f, , . (2r+l)7r where w = . 2n 302 DIFFERENTIAL EQUATIONS. From this we easily find the simple factors of the ope- rating function to be -— ± (-)5 tan0; ax Therefore decomposing it into partial fractions, as in the previous Examples, we find that y consists of a number of terms of the form 2 (cos 0)"~ ' ( sin {oG tan 9) fd.v X cos {po tan 0) n isin {cc tan &) Jd.v X cos {po tan y) 1 - cos {po tan 0) jdoo X sin {x tan 0) j ' Q receiving the values — , — , — , &c., so long as they ^ '2n 2n 2n o . IT are less than — . Euler, Calc. Integ. lb. It sometimes happens that the inverse processes, such as { d \-' , I a \ X^ fail, from the coefficients becoming infinite, \dx } in the same way as the formula for integrating x" fails when n = — \. Thus for instance, a] 6 = = 00 when m = a. dx I m — a The method to be adopted in such cases is the same in principle as that used for determining the value of / — . It is this : since the function becomes infinite in these cases, we so assume the arbitrary constant in the complementary function as to make the formula assume the indeterminate form -, the true value of which may be easily determined by the ordinary rules. The assumption made with respect to the arbitrary constant is that it shall be negative and infinite, so that the difference of the two infinite quantities may be finite. (21) Let the equation be ay=e . dx DIFFERENTIAL EQUATIONS. SOS The solution of this by the usual formula would be mx ay = € , ax y = + Ce . a — a To determine the real value of this, let us take the equation dy^ da? the integral of which is y = Jl — + Ce'"'. m — a Now C being an arbitrary constant, we may assume it to be equal to 1 m — a mx ax SO that y = + C, e"*. m — a When m = a, the first term of this becomes - ; and its true value is easily seen, by differentiating numerator and denominator with respect to vw, to be ose"^ when m = a. Therefore „ ax , /^ ax y = We + C\ e is the solution of the equation dy ay = e"-". dec ^ (22) Let the equation be d -—-ay = e-. dw J The solution of this by the usual method would be y = r^^r + e""(Co + C,x + &c. + c,._X"')» which is a nugatory result. Proceeding by the same method as in the last example, we find the function whose true value is to be determined to be 304 DIFFERENTIAL EQUATIONS. when m ^ a. {m — ay By Ex. 15. of Chap. vi. of the Dif. Calc. we find the true value of this to be 1.2.3 ... (r - l)r' therefore w'e' is the solution of the given equation. (23) Let the equation be d-y 2 — - + n^y = cosnx. dor The solution of this by the usual rule would be cos wa? y = — + CcoBniK + Ci sin n^ — V? nx. If we assume C = C' ^ we have to find the true value of the function cos ma? — cos>4<:p , when m = n. This is easily seen to be x sin no! 2n so that the solution of the given equation is xsinnx , y = 1- C cosnx + C, sin nx. ^ 2n This example is one of great importance, for in the ap- plication of analysis to physics, equations of this form fre- quently occur; and as the value of y is not simply periodic, but admits of indefinite increase, it indicates a change in the physical circumstances of the problem. Cases of this kind occur in the theory of the disturbed motions of pendulums and of the Lunar perturbations. DTFPEEENTIAL EQUATIONS. 305 (24) If the equation be — + n'^i V = cosn.t?, Kdce) J ^ we shall find by means of Ex. 21, of Chap. vi. of the DifF. Calc. that the integral is (— )'' of' cos {na; + r—\ (2w)''1.2.3 ... r 2/ = Sect. 2. Equations in tvhich the coefficients are func- tions of the independent variable. Equations of this class cannot be generally integrated by one method, but a considerable number may be reduced to the class discussed in the preceding section. I. In the first place, all equations of the first order may be reduced to equations with constant coefficients by a change of the independent variable, or by some equivalent process. The general form of a linear equation of the first order is dx P and Jl being functions of a?. Assume dt = Pdx, so that t = fP dx ; then the equation becomes dy X dt ^ P the integral of which is by the preceding section, y = e-'{dtye'+ Ce'S or putting for t its value y = e-fP'^'fdxXefP'^' + Ce-^^^'; which is the complete solution of the equation, (l) Let the equation be (1 — w^) [. ,xy = ax ; dx 20 806 DIFFERENTIAL EQUATIONS. w d D Here Pdx = — and jPda = - log (l - w'^y^. Therefore y = a + C {l — ary^. (2) Let (l + w")l ~ + ny = a (I + cv")K dx Here Pdx = . , (Pdx = nlog; Ix + (1 + x^)H, (1 + x'y •' "" ^ V -r y p and the solution is 2(n + 1) ^ ^ ^ 2(n - 1) ^^ ^ •> + C \(1 +x')i-x]\ (3) Let -^ + ^ rfc^ 1 — x^ (l — a?)^ The integral of this is 1 fi + a!V ^ /I -''» a + 4* \1 — xj \1 + '3? (4) Let (l — x^)i — nv ■= X (\ — x-^. The integral of this is \nx (l - x^y + 1 - 9,x'\ „ ■ _i n^ + 4 \ II. Equations of all orders of the form (a + &cr)" ^ + Jj (a + 6.^)»-i — ^ + &c. + A,y=: X, where A^, Jo, A„ are constants, can always be integrated by a change of the independent variable. In the first place, if we assume a + bx = h%, ' ■ the equation evidently takes the form 6";.» ^ + J,6-^ ^»-> ^ + &c. + i„2/ = Z ; d% dz ^ DIFFERENTIAL EQUATIONS. 807 where Z is what X becomes, when we substitute m it % — - for w. As ?/, 6""', &c, are constants, this equation may, by dividing by 6", be put under the form ^" -4 + A z--' -—4 + &c. + A:y = Z', dz"' dz '■ where J/ = ~ , A^ = — % &c. and Z' = - . b b' If dz In this equation make — = dt, or ^ = e*. Then by Ex. 6. of Chap. iii. of the DifF. Calc. we have a;'- ^^ = — ( ■ 1 1 ( 2 I ... ( — - r + 1 ) , \dt J \dt j \dt ) d'y d f d dz^ dt so that the substitution of t for % will give rise to an equa- tion of the form + 5i 7 + Bo f + &c. +B„y = T. -r^ + Bi 7 + Bo-— ^ df" dt"-^ " dt" where T is what Z' becomes when we substitute in it e for %. The coefficients ^i, B.^^ &c. are constant, so that this equation is integrable by the method given in the last sec- tion. This transformation was first given by Legendre, Memoires de VAcademie, 1787, p. S36. o d'V dy (5) Let ,/-4+ cT— -?/ = cT''^ dx dx Making w = e*, the transformed equation is d^y the integral of which is mt m — I or y = —o h CcV + — . m" — 1 x 20 — 2 308 DIFFERENTIAL EQUATIONS. (6) Let a?^_^^ + 3,^-1 + 2/ = d<.2?' dA^ " (1 - xf Changing the independent variable from x to t, and making oc = e*, this becomes (d V 1 the integral of which is (a? \ i- 1 )'' + - (C + Cilog*). (7) Let Let = dt^ and therefore 1 + a? = e*. The trans- 1 + ^ formed equation is dhj d'^y dy i -I the integral of which is 8 1 8 2/ = — — -, (1 + icVj + C{l+ xy ^ 85 (1 + a?)2 51 ^ ^ ^ ^ + Ci cos log (1 + cvY + C2 sin log (l + x)^. ' (8) Let ^^,-3.^^H-7.^-8, = X dx" dx dx When the independent variable is changed, the operating function is found to contain three equal factors, hence the integral is 2 rdx rdx rdx X ^ ,„ _ , ^ ,, ^„, y = x^ — — +0!^ [Co+Cilogw+C^ (log^)n. J X J X J X X ° ' y In other cases the reduction may be made by artifices suggested by the form of the equation. (9) Let ^ + _ ^ _ o2 o_ dx'^ X dx DEFERENTIAL EQUATIONS. 309 dx^ doc^ ~ dx Now : f ^ = A' T-^, + 2 Therefore -^ + - -^ = - ; ;^ . dx^ w dx X dx^ The given equation may therefore be put under the form d~ {xy) -^^-a»(,.y)=0; which is a linear equation with constant coefficients. The integral of this is evidently and therefore y =^~ {Ce"' + C^e''"') X is the integral of the given equation. /XT d'y 2 dy ( o -\ This may be put under the form d I d Integrating with respect to - — r(r-l) „ d'-'u „ 1 d'^-^^z n(n-l) ^ dx'-" ' ' "-* The (?• + l)*** term of this transformed equation will dis- appear if d^u r d^'^u r{r — l) d^~^u dx^ n dx^~^ n{n — l) dx^~^ ' ' which is a linear equation of the Z*^ order. Since du ^ dx can always be solved, it appears that we can always make the second term of a linear equation disappear. Some equations which are not linear may be reduced to that form by a change of the variable. Let dy + Pydx = Xy^ + ^dx; assume «" = — , when the equation becomes u du — nPudx = — nXdxy which is linear with respect to u. See Jac. Bernoulli, Opera, pp. 663 and 731. 314 PIPPERBNTIAL EQUATIONS. (11) Let dy + ydai = wif div. The equation in u is die — 2u dcV = — 2x dx\ from which u - —=x + ^ + Ce^''. y" soy doe i (12) Let dy + ^ = coy^ dw. '■ 1 — w The integral is (13) Let ay dy -hy" dx = cocdoG. Assuming y^ = u this becomes adu — 2bud,v = 2coedoe, Avhich is linear in u. The integral is u = y = - - 00 -T + 6 6 " . (14) Let oiiy^dy + y^doo = . By assuming y^ = u, we find 3a^ C u = y' = + — ,2 ^ ay (15) Let 2/^2/ T "''*^ ~ ~3 The integral found by assuming y~ = u is y^=e ^ 1 5". •^ act? 2a^ (16) Let dx - oeydy = oo^y^dy. 1 , • ■■ Puttuig ci? = - , this becomes dv + yv = - y% dy DIFFERENTIAL EQUATIONS. 315 which is linear with respect to v. The integral is 1 X 2 -y^ + Ce~"'2 . Sect. 3. Equations integrahle by separating the variables. T. Homogeneous equations of the first order and degree can always be integrated by means of the separation of the variables. If the two variables be w and y, assume y X - = », or - = - %, CO y and by means of one of these equations and its differential eliminate one of the variables and its differential from the given equation. The resulting equation involving % and the other variable always admits of the variables being separated. This method of integrating homogeneous differential equa- tions of the first order was first given by John Bernoulli. See the Comm. Epis. of Leibnitz and Bernoulli, Vol. i. p. 7. Ex. (l) Let the equation be xdcV + ydy = mydoo. y Assuming ~ = z, the transformed equation is dx zd% — + z = 0, X 1 — ^nz + z~ dx 1 2z — 7ndz 1 mdz or — + - -f = ; X 2 1 — mz -r ^ 2 1 — mz + z the integral of which is 1 1 / ,. m '^ dz locr X ->, — loo; (1 — mz + zn -] / = C. * 2 * ^ ^ 2 J 1 - mz + z' If m > 2, the denominator of the part under the sign of integration is of the form (z - a) iz j , and therefore / mdz a~ + 1 (z — a^ loff \ - mz -Y z" a- - 1 ° I ] \ a^ 316 DIFFERENTIAL EQUATIONS. 1 1 a^ + 1 /% - a\ and loa; x + -\o2 (l - nns + ss^) + ; log / \ = C Substituting; for % its value - , we have 0? , a^ + I fay — a^cV\ loff (x^ - mxy + r)2 + —7-0 r log = C. ^ ^ ^ ^ ^ 2 (a^ - 1) ^ \ ay -w ) Let ?w < 2, so that we may assume w = 2 cos a. Then /d% 1 , / ^ sin a \ ; = tan-i ; l-2cosa.«+« sma Vl-«cosa/ and therefore the integral of the equation is oM 1 / 2/ sin a \ ^ loff (x^ - mwy + y)'^ + cot a tan M = C. ^ ^ " " \x —yco?, a) Let W2 = 2, or 1 — m% + z^ — (l - z)-. Then the inte- gral of the equation becomes logG^-2/) = C- ^, or X - y = Ce •^~J' . (2) Let *'c?2/ - y^^ = C-^" + y'^)^dx. Making y = 12?^, this becomes dx d% whence x = C \% + {l + ^^)^|j from which or = 2 Cy + C*. (3) Let {x^y + y^)dx = Sxy'^dy. Assuming y = xz, we find dcV Szdz X 1 — 2^" an equation which is easily integrable, since the second side is a rational fraction. The final integral may be put under the form {x^ - 2y~f = Cx\ DIFFERENTIAL EQUATIONS. 317 (4) Let y^doo + {xy + sc^) dy = 0. Assume w = yz^ when the transformed equation becomes dy dz — + T S J y 2% + z^ icy ni J. rjfjK \ and the final integral is y = C [ J . (5) Let cedx + ydy = m {wdy — ydw). Assuming y = w%, we find the integral to be log {or + 2/^)i = m tan"^ - + C. CG (6) Let y^dx ■{■ ardy = xydy. Assuming cc = y%^ we find as the integral y = C^" (7) Let y'^dy + Sy'^wdx + Qx^da; = 0. The integral of this is y" + 2x^ = C (x^ + y~)l (8) Let x^ydo! — y^dy = x^dy. The integral is y = Ce ^-^ . y (9) Let xydy — y"dx = {x + yY e~x dx. The transformed equation is dx ze'^dz X (1 + z}-^ ' and the integral is X y {x + 2/) log - = xe'' . (10) Let the equation be x^dy — x'^ydx + y'^dx — xy'^dy = 0. In this case the transformed equation is reduced to X {I — z^) dz = 0; which may be satisfied by a? = 0, I — z' = 0, or dz = 0. S18 DIFFERENTIAL EQUATIONS. This last is the only differential equation, and therefore is the solution of the equation. It gives as the integral ^ = c, ox y = ex. The other two solutions correspond to particular values of the arbitrary constant. The first or x = gives c = co, the second or %- = 1 oives C = ± 1. II. Equations in which the variables can be separated by particular assumptions. (11) Let {mx + ny + 'p) dx + {ax + by + c) dy = 0. Assume ax + by + c = ^, mx + ny + p = u; whence adx + bdy = d%, mdx + ndy = du, and therefore 'md% — adu bdu — nd% dy = 7 ' «^^ = 7 J mo — na mo — na by means of which the proposed equation becomes {m% — nu) d!^ + {bic — a!s) du = 0, which is a homogeneous equation integrable by the usual assumption. If — = - this method fails, but the given equation is n a then easily integrable : for eliminating m it becomes b (cdy + pdx) + (ax + by) (bdy + ndx) = ; and by assuming ax + by = % whence bdy = d% — adx, the equation becomes {ac — bp + (a — n) %] dx = (c + !^) d%, in which the variables are separated. Euler, Calc. Integ. Vol. i. p. 26l. (12) Let dy = {a + bx + cy) dx. By assuming bx + cy = z we find the integral to be 6 + c (a + 6a? + c?/) = Ce\ Euler, lb, p. 262. DIFFERENTIAL EQUATIONS. S19 (13) Let dy + hhfdx = a^ x'^ d x. Assume y = ;^', by which the equation becomes r%'''~^d% + h^%-'''dx = c^w"^dw. In order that this may be homogeneous we must have r — 1 = 2r = TO ; whence r— — \^ m= -2, so that the transformed equation is d% ,„ dx a^dx \-lr — = , z^ z x'^ a homogeneous equation in which the variables are separable. This equation was first considered by R,iccati in the Acta Eriidiforum, Sup. viii. p. 66, and it usually bears his name. It may be converted into a linear equation by assuming 1 dz b^z' dx^ when it becomes -— ,- a%"x"'z = 0. dx" (14) If in the equation of Riccati m = 0, the variables are immediately separable. It becomes then dy + b-y~ dx = a^dx or dx = — — r , a'-^ - b~y^ the integral of which is a -by The assumption y = z^' is not the only one which renders the equation of Riccati integrable. If we assume y = AxP + x'^z, the equation becomes x''dz + {qx'i-' + 2b'AxP+^ + b\v'9)zdx+ (pAx^-y+bU^ay^Pydx = a^ x'"' dx. This will be reduced to an equation of three terms, if we have p -1 = 2p, q ^i =,p + q^ pA + b^A^ = Of q + 2b''A=0. 320 DIFFERENTIAL EQUATIONS. The first and second conditions agree in giving jo = — 1, and from the second and third we find , 1 so that the assumption is 1 % and the equation is then reduced to or (15) If W2 = — 4, the equation becomes dz + }r%^ —- = — ;-, in which the variables are separated. The integral is ah — 00 + h^x^y If in the equation = Ce ^ . dx 1 d% + h^%~ — - = a^x'^'^^dx we assume ^ = — , x^ u . 7 o o _i-o h^dx we have du + a^u^x'^^^dx = — r- ; x^ and in this equation making (in ■{■ S^ x'^'^^dx = dv, and for shortness putting o' o, ^^ 2^ + 4 = P'j = a , = - 7Z, TO + 3 m + 3 m + 3 it is reduced to du + /3^ M^ dv = a^ v" dv, which is similar to the proposed equation, and is therefore g integrable if w^ = - 4, or w = — . If tz be not equal to 3 — 4, we may transform this equation by the same assump- tions as before, when we shall obtain an equation of the form du' + (i"u"dv' = a"v'"'dv', DIFFERENTIAL EQUATIONS. ^ 321 which is integrable if oi = — 4, furnishing a corresponding value for m. In this way we may proceed, continually transforming the equation and finding values of m which render Riccati's equation integrable. It will be found that these values are included in the formula 4r m. = — 2r - 1 T being an integer. Another series of values for m may be found by making y = — in the original equation, when it becomes du + a'u^x'"dx = Ifdx; and this being transformed by the assumptions a^ ^o ^-'^ o ^'^ m + 1 m + 1 m + 1 we find die + (3^u'dv = orv^dv, which is similar to the proposed equation and integrable if n be of the form , that is, if 2r - 1 m 4r 4r , or m = — m + 1 2r - I 2r + i Hence all the values of m are included in the formula 4r 2r ± 1 (16) Let dy + y-dw = —^ . Then !^i4±^f^=Ce'.*. y {xi — 3c^'3) + 3 (17) Let dy-y'dw = ^. Then ^j-~ = tan + C 3.23a?3(l + *'?/) \ wi (18) The equation dy + ay'^w'^dx + bx'"y'i dx -=-- 21 822 DIFFERENTIAL EQUATIONS. can be made homogeneous if (p + 1) (1 - g) = (m + 1) (1 - n), by the assumption y = %^-''' or = ^^~^. There is an exception to this \i n = 1 and g = 1 ; but in this case the equation becomes dy + y {awP -{- hx'^) dw = 0, in which the variables are ah'eady separated. (19) Let aydcc -vhwdy + ai'"^y'" {cydoD + ewdy) = 0-, dividing by ocy we have dx dy ( dx dy\ a— + h — ^x'^if \c~ + e— =0. X y \ X y j From this it appears that the assumptions x"^y^ — u<, x'^y^ = V will simplify the equation. It becomes after these substi- tutions du r,dv — + u'^vP — = 0, U V me — nc ^ na—mb where a = — , p = ; — . ae — be ae ~ be The integral of this is evidently + 75-= c. a p . .„ ra a e . li a = and p = 0, i. e. if — = - = - , the inte- n b e gral takes the form log u + log V = C •■, or iiv = C ; or x^^Uf^^ = C. (20) Let (x -f y)- dy = a^ dx. DIFFERENTIAL EQUATIONS. 323 Assume x +y = u, a^ du whence ay = ^ , a^ + IT in which the variables are separated. The integral is 2/ + c y -\. ce = a tan . a -^(21) Let {y - a?) (l + a?~)§ dy = n {l + y")^ dw. To separate the variables assume w — u y= , , 1 + xu when the" equation becomes dw u du \ ■\- OD^ (l + U^') |w + ^(l + V?^i\ * To integrate this put 1 + «^° = ^^, which gives dx dt TV^^ t \nt^ {f-\)\Y 1 + s^ and again putting t = , we find t4i s dx 2ds 2nds 1 + .a?'-^ ~ 1 + s^ (n + 1) + {n - 1) s' ' which is easily integrable. Euler, Calc. Integ. Vol. i. p. 270. Sect. 4. Equations which involve y and its differen- tials in powers and products. I. Equations of the form are to be resolved (when possible) into the simple factors I'^.Al'l-u), (^-^0=0; \dx V \dx 'J \dx V and each of these is to be integrated separately. Any one of these integrals, or the product of any number of tliem, will be an integral of the proposed equation. 324 DIFFERENTIAL EQUATIONS. (1) Let f^V-«^ = 0- \dxl TT dy dy Here a=0, --+a = 0; ax aw therefore y = aa + c, y = — ax + Ci^ are both integrals : also (y — ax — c) (y + ax — Cj) = 0. If we suppose c and Ci to be the same, this may be put under the form {y — <^)^ = ^^ ^^• /dy\ 2 (2) Let y^ i^\ - 4o^ = 0. The integrals are y^ = 4a<2? + c, y^ = — 4ati? + Cj, and (^^ - 4>ax - c) (y^ + 4 ; I (1 -J- C'Y) the singular solution is t^s + «! = «!. 't) Sometimes an equation which is not of Clairaut's form may be reduced to it by being multiplied by a factor. (14) Let ay ( — ) -h (2 .j? -&) — -?/ = 0. \ Ct X I Cl X Multiply by 4^, and let y^' = u, and 2ydx = Then a (-—] + (4a? - 26) 4^^ = 0, \dxj aw 328 DIFFERENTIAL EQUATIONS. du (b die a (d 0\' 11 = 00 — - . , , , dx 2 dw 4 \d 31' v/hich is of Clairaut's form. The general integral is u=y'= Cx - l-C- ~a\ . The singular solution is ^ay'^ + (2a/ — 6) = 0. (15) Let «'^2/ 1~) + (hoe" — ay^ — ah) ba;y==0. \dxj dw On multiplying by ^wy, and taking or and y^ as the new ^ variables, the equation becomes of Clairaut's form, and the integral is ^ b + aC The singular solution is ay^ + b (cV — a^y = - I If y==P.v+ Q, where P and Q are both functions of p, we have by dif- ferentiation / dP dQ\ dy = pdw — Pdw + cV — y — dp^ \ dp dp) ( dP dQ\ ^ whence {p - P) dw = [w ! dp, V dp dp] which being a linear equation in w may be integrated, so that Ave have w expressed in terms of p, and as y = fpdw, we can eliminate p and so obtain a relation between w and y. (16) Let y = wp^ + p^\ The integral is y^ = (w + l)^ + C. (17) Let 2/ = (l + p) <3? + p'- Then y = 2 (1 - p) + Ce"^. Substituting in this the value of p derived from the equation, we have the required integral. (18) Let y —2pw = a{l + p')k We find 2^0^ = -~[p(^+ pV - log {p + (l + /)^}] + C. DIFFERENTIAL EQUATIONS. 329 By eliminating p between this and the given equation, the integral is determined. (19) Let y = w{p-{\+p')^. In this case Q = 0, and we have to integrate doo _{i + p'y^ - P ■, and then to eliminate p. The result is IV. Homogeneous equations of the second order. . . dy d~y If an equation involve x, v? -7- ? -rr~„-, and if we assume ax dx^ dy X and y to be both of one dimension, -— will be of di- ■^ dx d" 1/ mensions, and — - will be of — 1 dimensions. The equation dx'^ then is said to be homogeneous when, adopting this scale, the sum of the indices in each term is the same. To integrate an equation or this lorm, let —-= p, — ^ = q ; CtcU CvtV V then by assuming y = ux, q = -^ the quantity x can be ' eliminated so as to give a relation between u, v, and p. But as dy = pdx = udx + xdii, we have dx du X p — u and as dp = qdx, we have also vdx = xdp. Whence v du = (p — u) dp. From this v may be eliminated by means of the given equation, and we have a differential equation of the first order between p and u : by integrating this we obtain p in terms of u, and then x in terms of u from dx du X p — II in which the variables are separated. SoO DIFFERENTIAL EQUATIONS. (20) Let ^^''j4= [y-''^] ' da; \ dec) put y = uw^ q = - i then V = (u — pY, and dp = (j) — u) du. This being a linear equation is easily integrated, and we find p = zc + 1 + Ce". da? du e~"du Then .2? 1 + Ce" C + C' C — Cai and log — = log {C + e~") or e~" = — : whence y = x log which is the required integral. „dy d^y dy (21) Let o!^ -^ --4 - iJ? -^ + 2/ = 0. £?d7 dw da; The integral is , p (C-l)^i+{2y+(C^-l>?^ (C+l)^i- {2y + (C'-l)a;}-^' There is also a singular solution y = Ca'. Sometimes an equation may be considered homogeneous by reckoning a; as of one dimension, y of 7i dimensions, and dy . . d^y consequently — of (n — I) dimensions, and of (n — 2) ^ -^ da; da;- ^ dimensions. In such cases assume y^afu, p = x"-~'^t, q = co^~'^v \ then by steps similar to those in the last case we arrive at a differential equation of the first order, between t and u, which being integrated will enable us to determine the relation between a; and y. d'^y dy (22) Let w' -^^ = (^^ + 2 ivy) £ - A>y\ Assume y = x~Ui p = wt, q = v. Then -u = ^(1 + 2w) - 4iU'. DIFFERENTIAL EQUATIONS. S31 But we have du (v - t) = dt (t — 2ii), and therefore 2icdu(i -2u) = dt(t -2u). This is satisfied by 2u du = dt, or by * t — 9,u = 0. The first gives u^ + C = t, and therefore diV du cV u'^ — 2u + C When (7 = 1, this gives cV^ = (or — y) log - . a When C = I — n^^ this gives lI?^" = a is"- (n + l) x^ — y (71 — 1) w^ + y When C = I + n~, this gives y = a;^ h + 71 tan ( n log — The other factor t - 2u = 0, gives y = Ccv\ as a singular solution. If X be reckoned of dimensions so that ?/, — , — ~ dw dx^ are of the same dimensions, a homogeneous equation may be integrated by assuming p = uy, q = vy ; whence as dy = uy doa and ic dy + y du = vy dx, — — udo3 and du + u^'dx = vdx. y From this last if we eliminate v by means of the given equation, we have to find u in terms of x, by integrating an 332 DIFFERENTIAL EQUATIONS. equation of the first order, and then by means of — = u dx, we can determine the relation between a? and y. dy From this we find u du dx V = u^ + -— —J and — = {a' + a?-)^ u {a' + x')i ' therefore u = C \x + (a^ + x-)^ ; whence log(C'y) = Canog{w+{a'+w')i]+C.v{x+(a^+a;'y^\. r . ^ ^^y ^y idy\" \dx) (2*) Let .,_ = ,_ + . y H-T^rrTp- The integral is (a^-x^y^ ^^ c \nb + (a' - x")H = 61og— i '-, n ny b and c being arbitrary constants. V. Equations of the second order in which one or other of the variables is wanting. If the deficient variable be the dependent variable y^ by d^y dp . ' n T n ^ putting — - = -^ we have an equation oi the nrst order between p and x, by the integration of which we obtain p in terms of x^ or x in terms of p ; and then by means of the equation y = fpdx = xp — fxdp, we can find the relation between iv and y. ^dy^^]^ a^ d-y 2x dx" ' (^^) ^^' {'*©T dy 1 d^y dp puttins; — = p and — - = -— this becomes ^ ° dx dx^ dx ^ 2x dx DIFFERENTIAL EQUATIONS. ' 333 , „ dp 9.xdx therefore ^ = , (1 + p')^ a^ ' p af ^ a;^ + ab whence r- = — + C = , (1 + p')i a' a' ' This is the equation to the elastic curve. Jac. Bernoulli, Opera, p. 576. (26) Let (i+^^)^+i + (^y = 0. The integral is C'y = (1 + O) log (1 + Cx) - Cx + C. The equation between p and x is . _^i dp dp p which is integrable when divided by (l + p^)K The complete integral is y = (a^ + h^ — x^p - b log ^ ; ; — , ^ ^ ^ '^ c(x - a) where 6 and c are the arbitrary constants. (28) Let a'^(a' + ai'y^ + a'~ = x\ daf dx The integral is „ ^ m^ ^a?x 2 {a? + x^)i ^93 9 ^-C.(.^ + .>^-^a^Clog^:ti^^^ c (29) Let (.r + a)— | + .r -^ = do? ^ 334 DIFFERENTIAL EQUATIONS. The equation r x^P s Lv + a) p = ~ .vp''. dcV becomes linear by assuming p = -, and the complete integral is y + c == log (w~ - c'.) - - log , c \x — cj c and c being arbitrary constants. If the independent variable {x) be wanting, we put d'^y dy dp dp . — - = -7- --r- = V • -V i ^^^ then we have an equation be- doG^ dw dy dy tween p and y from which by integration we find p in terms of y^ or y in terms of jo, and then w is known from the equation dy = pdw. d'^-y , dy , dy { (dy\H dp From this 3— (|?2/ + 1) = 1 + F? dy and dy ^ y dp ^ ^ , ^ 1 + / -^ ^ 1 +p'' a linear equation in y, which being integrated gives y = p + C(l+p')K w = f— = logp + C\og{p + {l+p')i}-i-C, J p whence by eliminating p we obtain a relation between x and y. The integral is ,i7 = C+H^^ +cos-i^^^ , \ a J y where C and a are arbitrary constants. DIFFEBENTIAL EQUATIONS. S3 5 (S*) Let y;^3+y =1, o,-,p^ + j,'=,. d~y _ jdyY , _ dp dy Putting this under the form f dp . dy multiplying by y and integrating, we have {pyf = a^ + y^; whence w -{■ c = {or + y~)i. ^ ^ \dw) -^ doir \\dcc] Kda/V j . d^y dp , . , Putting --— = p — , this becomes ^ dx" ^ dy' which is of Clairaut''s form. The general integral is there- fore p^Cy-^n{l +a^O% whence Coo -^ C = log [Cy + n{l -^ a-C-)^}. The singular solution is p = whence y = na sin a C + .V a The examples in this section are taken chiefly from Euler, Calc. Integ, Vol. i. Sec. iii. and Vol. ii. Sec. i. Cap. 2 and 3. CHAPTER V. INTEGRATION OF DIFFERENTIAL EQUATIONS BY SERIES. The method employed for integrating Differential Equations by series, is to assume an expression for the dependent variable in terms of the independent variable with indeterminate coefficients and indices, and then to determine them by the condition of the given equation. (1) Let —^ + ax"y=0. Assume y = x'' (A + A^a;''+~ + A^a!^'' + '^ + ^30?^"+^ + &c.) Whence we find -^= a(a-l)Jc2?«-^ + (a + W + 2)(a + ?2+l)^c'»«+"+&c. a OS and aw"y = a Jci?"+" + «^i,a;"+2"+2 ^ ^^^ Substituting these values in the equation, and equating to zero the coefficients of the powers of x, we have a(a -1) A = 0, {a + n + 2) (a + n + 1) Ai + aA = 0, {a + 2n + 4) (a + 2n + 3) A^ + aA^ = 0, &c. The first of these is satisfied either by a = or a = 1. Taking a = and substituting it in the other equations, we find . aA , c?A ^1 = - 7—7^, ir» A = {n+l)(n + 2)' ^ 1 .2{n + l){2n + 3){n + 2y^ a^A A-i=- — &c. &c. 1 . 2 , S (w + \){2n + 3){Sn + 5){n + 2)^ so that ^~ ^ ~(w+l)(w+2) 1.2(71 + 1)(2 w + 3)(ra + 2)2 ~ ^'^ INTEGRATION OF EQUATIONS BY SERIES. S3 7 JBut as this contains only one arbitrary constant A, it is not the complete solution. Let us take a = 1 and call A', A/, Aq, &c. the corresponding coefficients ; we then find in the same way as before y = A' ]a; 1 &c. I ^ (w + 3)(n + 2) 1 .2{n + 3){2n + 5){n + 2y ^ which is another incomplete integral with one arbitrary con- stant. The sum of these two series is tlie complete integral of the equation. When n = — Q both the series fail, as the denominators are then infinite : but the true integral is easily found. „ ._ d^ij ay For if _^ + -^ = 0, and we assume y = AcV^, we have a (a - 1) + a = 0. This is a quadratic equation, which gives two values for a. If these be aj, a, the integral is y = A^x"-^ + AoX"-^' 2r - 1 The first of the preceding series will fail when oi = , (2r + l) and the second when n— — ■, r being any whole num- ber : the complete integral may however be found by the following process. Assume y = u + V log ex, where v is the particular integral furnished by the series which does not fail. On substituting this value of y in the original equation we obtain the system of equations d'V ——, + ax'^v = 0, dx d'u 2 dv V -r-„ + - + aux' =0; dx~ xdx x" the second of which serves to determine u. Euler, Calc. Integ. Vol. II. Chap. vii. 22 338 INTEGRATION OF EQUATIONS BY SERIES. d~y ail (2) Let -4 + -^' = 0. dx' X Then v = A \x + ■ -, — -, — • + &c.( ^ 1.2 1 . 2~ . 3 1 . 2' . 3- . 4 * The equation to determine it is d^7i 2 dv V au dx'^ oc dx x"^ X ' Assume ti = B + B-^x + B^x"^ + B-^x^ + &c. A Then we find B = ; J?, is left undetermined and a 3aA aBi r . 2^ 1 . 2 — 5a^A a" A a^B^ Bz = .. ,., _o ; ^ + l'.2-\3^ 1^.2^ 1.2^.3 &c. &c. But since we have introduced the arbitrary constant c in log ex, we may assume for B^ the value zero, and then we have , ,'1 3a ^ \W , 1 a 1^22 r.2^3^ «2 «S„3 . f aa? a-x' ax' „ . , ■\- A \x h ; — }- &c4 loi^c^. ^ 1 . 2 1 . 2^ 3 1 . 2- . 3^ 4 ^ "^ Euler, Ih. p. 156. . - -r d^ij ail (3) Let —^^+-4 = 0. d x' x^ Here -u = J ^a? - a;-^- + — x' x'^i-kcl 1.3 1.2.3.4 1 . 2 . 3~ . 4 . 5 ^ , . j 1 a?^ 8 .4.« ^ 100.l6.a^ „ 1 and ^ = - J <— ; + , — -x'^ + , „ , , «- - &c.> l4fr a l-.S" 1'. 3^2^4^ j . f 4a 3 l6a" „ 64a^ 5 + A\x- - — a?^ + a;- ^ a;'' + &c. ( log ca;. ^ 1.3 1.2.3.4 1.2.3^4.5 ^ "^ Euler, lb. p. 159. INTEaRATION OF EQUATIONS BY SERIES. SS9 (4) From the equation d'y dy ax" dx we easily obtain a particular integral. For if we differen- tiate the equation t times, we have X- — =^+ 0' + 1) + — ^ =0, and when x=Q d'+^y _ 1 d'y dx'''^^ r + 1 dx'' ' Thus any one of the coefficients in Maclaurin's Theorem is derived from the preceding one. Let the first coefficient, or the value of y when x = 0, be J, then we find as the particular integral V = A \l H h — ^ : - &C. \ Let this be put equal to -y : then assuming y = u + 11 log ex, and u = B + B^x ->r Box" + BoX^ + &c., we find by substitution in the given equation 3x~ Ux^ 5lx'^ , B U = 2J(X + -r-— ; - ~ ; + &C.) + -: V. 2^ 2^.3' S-'.S-'.l^" 1 Hence we have 3x'^ lla;^ 51 i»^ y = 2A(x + — — - - — — - — - + &c.) 2^ 2^ . 3^ 2^ . 3^ . 4^ o Q 4 /y* ryi"^ rp'^ fY> ^ 1' 1" . 2' 1- . 2'' .3- l^ 2" . 3^ 4" ^ "= B . . the constant — - being; included in c. A '^ Fourier, Traitc de la Chaleur, p. 372. (5) Let ,v' — ^ - c' ?/ = 0. 22 — 2 340 INTEGRATION OF EQUATIONS BY SERIES. The assumption y = ^ (^s-^'") gives {n {n — I) ... {11 - r + 1) - c'} «„ = 0. From this it appears that a„ = except for those values of n which cause the other factor to vanish. These values of n are r in number; let them be n^, n2...ny^ then, the corresponding values of «„ being indeterminate, we have y = Clc^'"l + Cgcv"^ + &c. + CrX\ (6) Let — — 7 = 0. Assume 2/ = 2 (a^a?") ; then -— , = "2 ln(n - 1) a„x''~~\, ax' ^ C^y Substituting these in the equation and equating to zero the coefficients of «''"", we have n{n- l)a„= c-a^^g. If n = 0, or 01=1, ffa and aj both vanish, and so con- sequently do all the superior coefficients. If n= — l, 1.2a_i = c^ai and a..i = 2.3a_o=C"«o ^i^d «^o = 1 .2 1 . 2 . < 7i = — 3, 3.4«_3=c-a_i and a_3 = C^ffli 1.2.3.4 w = - 4, 4.5a_. = c-a_o and a_4 = 1 .2.3.4. &c. &c H( snce we have y = a, (a; + 1 .2 + 1 + &c •) 1 . 2.3, .4 4. a, ,0 + --"- c^ + 1 c'' ^ 1.2.3a?- 1 . 2 . 3 . 4 . 5 £» «! and «() being two arbitrary constants 1 + &c.), INTEGRATION OF EQUATIONS BY SERIES. 341 This may obviously be put under the form y = X {Ae~'' + Be'"'), ( Euler, lb. p. l66. if A = i(a, + ^^, and 5 = J («, - ^ (7) Let ^+q^=o, y = X [ C cos — h Cj sin - ) . V X xj Euler, lb. p. 167. d " '?/ c (8) Let — •^- - _ ^ = 0. ax Xi The integral is cj \ 3i Euler, lb. p. l6S. y = [x^ - —] Je'^'^'"' + [x^ + — \ Be o C J V o C / (9) Let —^ -0^^-57/ = 0. dx^ The integral is Generally, the integral of ~ - c'x-'^^y = clx'^ r will be expressed in finite terms when X = 2r ± 1 Mr Leslie Ellis has given {Cambridge Mathematical Journal, Vol. 11. p. I69 and p. 193) some remarkable methods for reducing to finite functions the solutions in infinite series of certain classes of Differential Equations. Let the equation be of the form 342 INTEGRATION OF EQUATIONS BY SERIES. Then on assuming ?/ = 2 («,,«")» ^"^1 substituting in the given equation we obtain as the condition for determining the coefficients {71 {91- 1) -2^{p- 1)} a,, + g~a„_2'= (2). Now 71 (n - 1) - p {p - 1) = {n - ji) (n + ^j - l)...(3) ; therefore {n — p') (n + p — l) a„ + ^oa„_2 = 0. Assume (7^ + p - 1) a„ = {71 - jJ + ^) b„; 1 71 - p then a„ _ 2 = ^ o„ _ 2, 71 + p - 3 and (w - p + 2) (w, + ^ - S) &„ + (J-- &„ _ 2 = . . . (4). Again assume {71 + p — 3)b,^= (n — p + 4^) c„, and so on in succession. We shall thus obtain a series of equations of which the type is (w -p + ft) (72 + p - fji -I) In + q^l„-2'= (5)5 ju being an even number. , If p be even let ^j = ju, then ja — yu — 1 = — 1, If p be odd let p = fj. + 1, then ^j. — |j = — 1. In both cases the equation (5) becomes 7l(7^ -1)4 + g'4_2 = 0. This is the relation between the coefficients which we should obtain from the equation 5^ + " = " • (*')• Hence ^{l^x'') = C sin (qx + a) (7), that being the integral of equation (6). Now suppose (71-p + fi- 2) (72 + p- ^^ + 1) i^^ ^ q^i^^_^= 0, {n-p + ,j) (71 +p- ^-i)k^^ (f'K-i = 0, to be any two consecutive equations ; then {71 +p -,x-tl) i„ = (w - 2J + m) /<^« (8), but ?i - |j + /x = 7i + 2J - /^ + 1 - 2 (^ - ^) - 1 ; therefore L = k„ — INTEGRATION OF EQUATIONS BY SERIES. 843 n n 71 + p - /x + 1 k 1 and = 5 (n + 2 - ^ + m) k„.2 ; n + 'p - [x+ \ q^ therefore Now {n-p-{- ij) A;„A'"-" = xP-f'-'^ {n - p + /x) A;a?"-^+/^-^ da) \wP i^j therefore By the application of this formula y or 2(a„cr„) may be deduced by a series of regular operations from Csin {qx + a). If p. be even 2 (p — /x) + 1 gives the series 1 , 5, 9 If |3 be odd it gives the series S, 7, H (10) Let -^+qhj^^^, dx^ x-- where p = 2. The integral is y = C {sin {qx + a) H cos ((/*' + a)}. get? (11) Let —~, + q-y=—, dx X where p = 3. The integral is y = C < sin (qx + a) [ 1 7-; j + — cos (qx + a)> . This method may be successfully applied to reduce d'^-y , , 1 d'"-~y -^ + r7'«7/ = p (p - 1) - •^, dx'" ^ -^ ^ ^^ \v' dx"'-~ when p or p — 1 is divisible by w/. 344 INTEGRATION OF EQUATIONS BY SERIES. (12) Let -:^ + g3 =_Y^, dx'^ X- ax The complete integral is \ qoe) ^ {^\n\—qx -\- a] 1 , . o- / 0"-i + Cse^ "Isin ( ^ga? + a ) ( 1 ) + — cos [ — qx + a\\ . . ^ ^ d^y dy 2y (13) Let -^ + g-^=^. dx' dx x^ This equation presents a peculiarity, inasmuch as if we neglect a factor, which apparently disappears, we shall have a solution which is erroneous or incomplete. Assume w = 2 («„<:??"), then {n{n-\)-9.\a^^ (n - I) qa,,_^ = 0, or (ti -2) (n+ 1) o,j + (n - l)qa„_^ = (l). Let (n+l)a„= (n-l)K (2), then (w - 2) (n - 1) nh„ + (n - 2) (w - l) g&„_j =-. ... (3). The factor (n - 2) may be safely neglected, but (ti — l) must be retained, as it enters into the solution of the auxiliary equation d''% d% -— + g— =0. d.x dx From (2) we have 26 7^ + 1 and as, except when n = 1-, we have nh,, + (/6,,_i = 0, ^n 1 , = — -6„+j, except when ?2 = ; 7Z + 1 g 2 therefore cr„ = &„ + -&„+i (4). The solution of the auxiliary equation is INTKGRATION OF EQUATIONS BY SERIES. 345 and from (4) it appears that \ qx) \ qoo) This appears to be the solution of the equation, but it does not satisfy it unless C^ = 0, when it becomes \ qo}) which is only a particular integral, and therefore incomplete. This arises from our implying in the use of equation (4) that w6„ + Q'&H-i = is generally jtrue, whereas the equation (tz - l)(w6„ + ^6„_,) = 0, derived from the auxiliary equation d"% d% dw^ dw shews that 61 is not necessarily connected with &(,, since it may be satisfied by w = 1. To complete the solution, we have from (2) which is always true «o = - K and from (4) which is true for w = - 1, we have 2 a _ 1 = & _ 1 + - 6o» 2 2 or as 6 _ J = 0, a _ ^ = - 6 = a^. q q These quantities are independent of a^, ff^, &c,, therefore writing Ci for «„ as it is an arbitrary constant, \ qoo is a particular integral of the proposed equation, and the complete solution is 2/ = Ci ( 1 - — ) + C, f 1 + --") e-'?^ \ qxj \ qx) 846 INTEGRATION OP EQUATIONS BY SERIES. (14) To integrate cV'y pm d"'-Uj where p is an integer. Assume 2/ = 2 («„>'{?"), then if «„ = (n - pm + 1) ... (n - m + 1) 6„. But from the given equation 71 (n - 1) ... (71 -m + 2) {n-m(p+l) + 1} a„ + A;™ «„_„, = 0, from which 71 (n - 1) ... (n -m + 2) ... (w - w + l) &„ + /i;'"6„_m = 0- But this is the equation which wovild result from sub- stituting '^(b„w") in dw"" -^ therefore S(6„<.i?") is the solution of this last equation, and is therefore known. Calling it X, we have Let 771 = 2, ja = 2, then the integral of d~7/ 4 dti -^ -^ +k'y = dx^ 00 dcV /I dy~ (Jcx + a) IS y = X' I - ■ — C cos ; V'^' dwj cV or y = C \(3 - liTcv") cos {kx + a) + 3hx sin {kx + a)} . Ellis, Cam. Math. Jour. Vol. ii. p. 202. CHAPTER VL PARTIAL DIFFERENTIAL EQUATIONS. Sect. 1. Linear equations with constant coefficients. By the method of the separation of symbols the inte- gration of Linear Partial Differential Equations is reduced to the same processes as those for the integration of ordinary differential equations of the same class. Hence the theory which is given in the beginning of Chap. iv. is equally applicable to the present subject, and it is unnecessary to repeat it here ; I shall therefore content myself with referrino- to what has been previously said in the Chapter alluded to, adding that every differential equation of this class between two variables has an exact analogue among partial differential equations of the same class, and that the form of the solution of the latter is the same as that of the former. On this point one remark may be made which is of considerable importance in the interpretation of our results. As in the solution of ordinary differential equations we continually meet with ex- pressions of the form so in partial differential equations we shall find expressions of the form d in which the arbitrary function takes the place of the arbitrary constant. Now as the preceding formula is the symbolical expression for Taylor's Theorem, we know that d a -—.X 6 ^ (p(y) = (p(i/ + acv). Hence, in the solution of partial differential equations, arbi- trary functions of binomials play the same parts as arbitrary constants multiplied by exponentials do in equations between two variables. 348 PARTIAL DIFFERENTIAL EQUATIONS. (1) Let the equation be dx dz a~— + 0-— = c. dcV dy This may be put under the form d d\ a-— j^ h--] % = c; dx dyj ( ^ , dx-"- whence sr=a--+o-— c. V dx dy] Now supposing x to be the independent variable, and --— a constant, with respect to it, by the Theorem given in Ex. (11), Chap. XV. of the DifF. Calc. this is equivalent to J d h d a or, effecting the integration, and adding an arbitrary function of y, instead of an arbitrary constant, a a ^ Now by Taylor's Theorem or, as the form of cp is arbitrary, we may for - (b [y — x\ Avrite (p {ay — bw), so that !^ = }■ (b {ay — bcv). a ' It is obvious that if we had taken y for our independent variable, and considered — as a constant vdth respect to it, dx we should have had s; = y + {bx - ay). 349 PARTIAL DIFFERENTIAL EQUATIONS. d% dz (2) Let a — - = e""^' cos r ?/, dx dy ( d d\-^ ^ ss = \ a — ■ 6 cos rrj \dw dy) d d_ = r^y fdx e ""'"'^^ e'"* cos ry. But by Taylor's Theorem d ~ax-r— 6 ^ cos ry = cosr {y - ax) ; therefore d % = e""^'-^ fdx e'"'' cos r (y - ax) ; and, integrating with respect to x, "■•'"'H-;, \ m CO?, r (y — ax)— a r?,\n.r (y — a x)\ ax-- m'^ -V a^ r (m cosry — ar sin Q'y) or, sr = e''"' -^ f ^^ + (p(y-^ ax). The same method is applicable to any number of inde- j pendent variables. (3) Let the equation be du du du ~- + h -—+ c-— = xy%\ dx dy dz ( d d d\ or, \ j^ h -- + c --] u = xy % ; \dx dy d%) "whence [ d d d \-^ , „ . u ={-—■{■ — + c — xy% ■\- complementary function. \dx dy dzj If we expand the operating factor in ascending powers of b - — he — ) we shall have dy dzj fd\-U fd\-'f^ d d\ ^ f dy' d d\ \dx) { \dx) \ dy dz) \dx) dy dx] the other terms being neglected, because when the operations S50 PARTIAL DIFFERENTIAL EQUATIONS. are performed they vanish of themselves. The complementary function in this case is e . rfy '^~' (l){y,%) = (p{y -hcG, Si- CW); therefore, effecting the operations indicated, u = — y% — — {h% + ci/) -vhc ^ // (y), or ^ = e~^"^'''0 (y + aw) + >//(?/ - aw). Euler, Calc. Integ. Vol. in. p. 210. d'% dx d% (11) Let-; — — + a _- + &-— + a&;tr= r. dwdy dw dy Whence % = e'^"'-^'"''^ fdye"^ fdwe''-'V + e-"^ , and a-Jcos (-)2sin — >. Therefore _a£ [aS^co\ , _(j£ . [aS^w\ % = e"'" (2/) + e 2 cos I -g- j ^//i (2/) + e 2 sin ( -^ j f 2 C^)' (14) Let ^'^ . ,. ^^^ . , «. d^% „, f^^^ — - - (2a + 6) -— — + (2a& + a^) - a^h — - = 0. dx doD^dy dxdy dy In this case two of the operating factors are equal, and we find ^ = >// (3/ + 6 c^) +f{y + ax) + cr/i (;?/ + ax). (15) Let the equation contain three independent vari- ables, as in d^u d?u d^u d^u 2 , . „ -3 dx^dy dxdy"- dx^d% dxdz^ d^u d^u d^u - 2 , ., , + 6 + 7 = 0. dy^dsz dydz^ dxdydz PAKTIAL DIFFEREXTIAL EQUATIONS. 355 When decomposed into its factors this becomes d d\ f d d\ f d d\ dw dy) \dx d%] \dy d%j the integral of which is u =f(y + 2x, %) + (p (y, % - a;) + yp^ {x, z + 3y). (l6) Take the general equation with two independent variables d^z ^ d^z , d"z d'^z ^ - — + Ai -— - + As ^^-v + Sic. + A„ - — = F, dci?" dw^~^dy dos^~ dy~ dy"" where the index of differentiation is the same in every term, and the coefficients are constants, and F is a function of oo and y. When decomposed into factors it takes the form d d\ ( d d\ f d d\ 7i — ^ii~] b — ^'^i~] h — ^n-r) ^= ^i dx dy) \dai dy) \dai dy) where «i, a^ a„ are the roots of the equation ?*" + Ji^"-i + Jg?*"-" + &c. + J„ = 0. Now in decomposing the inverse operation into partial / (^ \ - (n - 1) fractions each of the coefficients involves f — — ) as a \dy) factor in the numerator, since the denominators consist of the products of {n - 1) factors of the form Hence giving to iVi, iVg, &c. the same meanings as in Ex. (6) of Chap. xv. of the DifF. Calc. z=nA- — a,— ] — V+nJ- — «o— — F \dx dy) \dy) \dx dy) \dy) ^. ( d d\-^ ( d\ -<"-" ^^ + &c. + iV^„ — -c?„— — V. \dx dy) \dy) f d \ ■~'"~^' If for shortness v/e represent ( -7— 1 F by Fj, and if we transform the operating factors by the formula d d \ ^ a,T— . -ax~ dx dy) 23—2 356 PARTIAL DirFERENTIAL EQUATIONS. we have d d a a d d c„x . —a„x- + &c. + &c. + N„ e '^'^ fdxe ^^ V^. The complementary functions are supposed to be included under the sio-ns of integration : if we wish to see their form we have merely to suppose Fj = in the above expressions, when after obvious transformations we find ^ =/i {y + ayw) ■Vfziy'r a.w) + &c. +fn(y+ a„w), d^% d~% d^% (17) Let -7-2+3 -— — + 2 --2 = a? + ?/. dw dwdy dy In this case ay - - 1, ag = - 2, iVj = 1, iVg = - 1. Also (-) (. + ,) = -^; therefore % = — — — - +fi(y - x) ^f^{y-2 00). Sect. 2. Equations in which the coefficients are functions of the independent variables. As in the case of the similar class of ordinary differential equations these equations may sometimes be reduced to forms in which the coefficients are constant. Thus, equations of the first decree of the form ■'O' — + jrr — = P;^ + Q, doc dy where JT is a function of a only, Y a function of y only, and P and Q functions of both x and y, may be reduced to the form d% ^, d% ^ dcV dy t ^y by assuming dy = -77 • This equation may be written dss ^ ^^ d ~ +(X-— ,-P)^=Q, aw dy PARTIAL DIFFERENTIAL EQUATIONS. 357 in which shape it is seen to be a differential equation of the first order with respect to x with coefficients which are func- tions of that variable ; it may therefore be integrated by the method of Chap. iv. Sect. 2. We may sometimes however reduce the equation at once to constant coefficients by changing both the independent variables at once. Greatheed, Philosophical Magazine^ Sept. 1837. _ . . ^ d% cl% JbiX. (I) L.et w- — vy~—=^n%. ax ay _ . dw dy ^ 1 . , By assuming — = cZw, — = dv, this becomes a; y f d d \-r- + n\ z ~0. \du dv Integrating with respect to u, we have / d \ d — Ml- HI »K —U^r- 5? = e ^'" '^{v) = e e '>(«); or % — e""^ {p — u). But u — log X, V = log y ; therefore t;-.. = log^|], and (^ (^ - ^0 = 01og (^) =/ (^) , so that ;y = ci?"/ (-) • If we had integrated with respect to -y, we should have found The interpretation of these results is that ^ is a homo- geneous function o^ n dimensions in x and y. This is obvious, as the differential equation is the condition of homogeneity of a function of two variables. d% d% x^ (2) Let w~-y-r =~ • ax ay y Changing the variables, as in the last example, we have d d .du dvj S58 PARTIAL DIFFEREXTIAL EQUATIONS. The integral of this is Su —v € 6 ^ = \- (p {u + v) ; or % = 1- \J> (wy). sy (3) Let = . d.v dy w + y Integrating with respect to w, we have / d \ . d d f> dss d d d r d,v rd.v '"'•'= e^y and e ^^e"''"-'^ = e ^ y - jj therefore % = e "'-^ {^^

(2/)} ; n therefore ^ = 'it ^{y — mx), 1 dz 1 d% % (5) Let - +_ = x dos y dy y^ Put xdx = duy ydy = dv; then the equation becomes dss d% r~ 1 = — . du dv 2v This is a particular case of the last example, and its integral is % = v^ (j) (v — u) ; and therefore ^ = y (p(y^ — ^^)- ,^. ~r dz d% (0) Let y — + X — - z. ^ ^ ^ dx dy Dividing both sides by xy and putting u = x^, v = y^, we have d% d% % du dv 2 (uv)^ Therefore, integrating with respect to w, d smce e {uvY = (uv + u^y. Hence % = {\{u -^-v) ->r {uv)^i(p{v - u) ; or, putting for ic and v their values, and omitting 2~2 as it may be included in the arbitrary function, we have % = (x + y) (p {if — X'). d% d% (7) Let sec <» — - + a -— = » cot v ; dx dy The integral is 1- % = (sin yY ^{y — a sin a). 1 60 PARTIAL DIFFERENTIAL EQUATIONS. (8) Let w' --+y--— = - . ^ dx ^ dy y d V dv By assuming — = du, — = dv, this becomes ^ ^ x' y' d% d% V du dv u^ Integrating with respect to u. V 1 _ _ jjf = — - + — + € ''dv(h (y). rw^i n "^^ ^^ (y ~ ^\ 1 herefore z = + (Z) 22/ 2 ^ \ wy j y ^ T d% . ,. , d% (9) Let.- + (l + y»>- = .y. — = u, — ^ ^ (1 + «/^)3 Assuming — = w, ~ -^^ = rf« we have d^ dz ^e;^ At + V whence %: = \- (b (v — u). Or, substituting for u and « their values in x and y, , . ^ (•!, *). (1 - a)x ^ \% z) Equations of the second and higher orders may sometimes be reduced by transformations similar to those employed in Chap. IV. Sect. 2. d"% d^z d^% (11) Let X- — - + 2xy-~-^ + y^——, = x'^y". dx^ dxdy dy PARTIAL DIFFEREXTIAL EQUATIONS. 361 „ . d.v _ dy , , , , rattinff — = ai.(, — = dv, this becomes ° cP y ( d d\\ {\du dv) \du dv) ] The integral of this is ^ = 7 ^7^ 7^ + ^"^ (u - w) + >i^ (u - u) or.= - ^ -.rfm,/(? (w + w) (m + 71 — 1) \x} \w (12) Let a?^ — — - y^ — — = a??/. ^ ^ dee" ^ dy' -^ By means of the same transformation as in the last ex- ample we find % == xyXagoe -ir wF y-\ ^■f{aJy), (13) Let (.^ + 2/) -a =0. XAj kAj \Aj II \Aj iKi If we put —- = v this becomes dv (a? + 2/) -1 av = 0, dy the integral of which is v = (x + yY(h (ct?), so that % = /c?,a7 (,i? + yy(p {cc) + v|/ (?/). (14) Integrate the equation d"% , d^s; n(n-\) d^s; d^z a^-—+nx''-^y -— + ~-^ + kc. + y" -— =0. dw" ^ da)"-^dy 1.2 dx"--dy' ^ dy" Assume dx = xdu, dy = ydv ; then by Ex. (6) of Chap. III. Sect. 1, of the Diff. Calc. we have generally '^^ dcc'dy'~ du\du~^] '"{du"^^ '^^j ^ Tv{Tv-')"-{tv-^'-'^} 362 PARTIAL DIFFERENTIAL EQUATIONS. Now if we put for shortness du \du J \du J Ldu-i the given equation takes the form \ LduJ ldu-i IdvJ 1.2 LdMJ ldv-1 But by a known theorem of Vandermonde if [^j?]'' = iv (a; — 1) ... (x - r + 1), 92- (fh — 1 I [wj + n[wy-' [y] + -Y7^ L'^]"-' [yj + &c. + W»= [a; + 2/]". Therefore, as the symbols of differentiation are subject to the same laws of combination as the algebraical symbols, the differential equation may be written r d dy Ldu dvJ l\(d d \ id d . ,1 vj \du dv J [du dv J d d \du d the integral of which is s; = "0„_i (v-u); /o, /i, Sec. being arbitrary functions. d"^ cZ;^ dig (15) Let a!y ; — h a^ V oy —- abz = V, dccdy dx dy (F being a function of cc and y). Putting as before — = du, — = dv, this becomes ° X y PARTIAL DIFFERENTIAL EQUATIONS. 363 the integral of which (see Ex. (11) of the preceding sec- tion) is ^ = e-(««+6«) fdv e"" fdu 6*" V + e-'' (p (ic) + e"*" >// (v) ; "or ^==^Jdyf-'fd^.v'-'V+^J(.v) + -^F(y). 00 y y CO , ^^ ^ „ d^% , d^% d% d% (16) Let or -— - - y" -— ; + so y ~- = o. ^ ^ dx^ ^ dy' dcv ^ dy By the same transformation as before we find ^ = f-j +^ i^y)' , , ^ d^% 2 d% ^d^% (17) Let -r, + - T- = «Vl- dx'' CO dco dy By the same process as in Ex. (9) of Chap. iv. Sect. 2, this may be put under the form d^(ao%) „ d^(cc%) d/v- dy^ Whence we find 1 % = - {(p(y + «'*?) + ^ (2/ - «^^)} , . ^ d^z „ [d^% 2 d% 2 (18) Let — - = aM -_ ^ a^/ Kdx-' cV dx x^ This may be put under the form d^% ^ d f d 2 dy^ dx \dx x and thence by the same process as in Ex. (lO) of Chap. iv. Sect. 2, we find dH ,^d^v , ( d\-'^ ~r-r, = a~ TT— ; , where v = x \ — z. dy^ dx- \dxj Integrating we have V =(l){x + ay) + \// {x - ay) ; r S64 PARTIAL DIFFERENTIAL EQUATIONS. and therefore This equation occurs in the Theory of Sound. See Airy's Tracts, p. 271. (19) Let —-, -- v-^=0. ^ dw X ay This equation is of the same form as that in Ex. (6) of Chap, v., and its integral will be found from that given there by putting a — for c, and changing the arbitrary con- stants into arbitrary functions of y. Hence we find -i'{y--yf[y-f\- (20) The integral of the equation may in the same way be deduced from that of Ex. (8) of the same Chapter : the result is Z=^x{F' {y+ Saw^) +f' {y - 3ad)} {F(y + Sad) -f{y - 3aa)^)\. 3a d^z ,^ fZ-^ 2z dx^ dy^ or ' The integral of this equation may be deduced from that d' in Ex. (10) of Chap. v. by putting - a' — for q^. This gives us %^ — {F{y- ax) -f(y + ax)\ + F\y - ax) +f(y + ax) ax d^z d'Z 2z ^ "^ dx" dx dy x'^ ' PARTIAL DII'FERENTIAL EQUATIONS. 36o The integral of this equation is deduced from that in Ex. (13) of Chap, v., by putting a — for q. This gives 2 2 a; = aF'iy) --F{y)+ af\y - ao!) + -f{y - aw). 00 w (23) The equation d^ ~ Iv' d.v'"-'^ ~ "' If ~ ' may be integrated by the same method as that in Ex. (14) of Chap. VI., by changing /r into -a^ — i"^"^^ putting arbitrary ay functions of y instead of the arbitrary constants. Thus if m = 2, we have X=F{y+ ax) +f(y - aw), so that the integral of "& d^is 2jo dis „ d~% dw^ w dw dy^ IS % '"''''' {10 ~v ^^^'^ "- ''''^ ^^^y ■- '''''^^' Hence if p = 2, the integral of d^is 4! d% „ d^z dw'^ w dw df is ^ = 3 1 7^(2/ +«<«?)+/(«/- a .2?)} —Saw {F' (y + aw) -f (y -aw)^ + o.V {F'\y + aw) + f" {y -aw)}. d^% 1 (d% dz\ 2 (24) Let — —r + 1- + -r - 7 j ^ = 0- dw dy w + y \dw dyj {w + y)- Assume yi-w = u, y - w = v, when the equation becomes d-!^ d-!s 2 d^ 2 dir du^ u du zi^' The integral of this by Ex. (l8) is s: =- \(p' (ic + v) + \^' (u - v) I :^ [(p (u + v) + yj/ (u - v)]. 366 PARTIAL DIFFERENTIAL EQUATIONS. Hence, Non-linear equations of the form P, Q, R being functions of w and ?/, may be transformed into linear equations by assuming dz d% wy (25) Let x—-\-y— = ^ dx ay % The transformed equation is, putting %d%= d% , ov s: = — , 2 d% d% w and the integral is dw dy ,"- = xy + (pQ „ dz „d% (l + «-)a (26) Let co^ — + r 3- = ■ • ^ ^ doG dy z By assuming 7 ^j7i= ^^' o^ ^' = (1 + ^")^5 this becomes -' (1 -r ^ )- dz d%' a;2 +2/2— - = 1, da? a^ and the integral is (1 + ^^)^^ = - - + (- - -) . X \y oc) (27) Let .i? — - H- 2/ — ='2xy (a' - »")l ft ci? ciy The integral is ^ = « sin \xy + f^ [|) | • PARTIAL DIFFERENTIAL EQUATIONS. 367 We might with advantage have applied the same trans- formation to the equations in examples (l), (3), and (4), as it is generally convenient to reduce the factor of % to two terms. Sect. 3. Equations involving the differential coefficients of % in powers and products. If the equation be of the first order make — = p, — = q. dec dy and from the given equation find q in terms of p, w, y, %, and substitute this value in the equation dp dq dp dq dy dx dz d^ which will then become an equation of the first order between four variables. The value of p found by integrating this, with the corresponding value of q will render d% = j)dw + qdy, (2) a complete differential, and this being integrated will give the value of %. The integral of the first equation will involve an arbitrary constant (a) ; and the integral of the second will introduce another (6), which is to be considered as an arbitrary function of (a) ; and we shall thus obtain an integral of the form f{oc,y,%,a) = (p{a), from which a is to be eliminated when a specific meaning is assigned to (b. Lagrange, Memoires de Berlin^ 1772, p. 353. (l) Let jr + (f =1, or g = (l - p-)i, dq p dp dq jj dp dx {l — p")^ dx d% (l — p^y^ dz' Substituting these values in equation (l) it becomes dp dp oxi^P d% dx dy This equation is integrable if we can integrate the system of equations dp = 0, pdz - dx = 0, (1 - p^y^ dz - dy = 0. S68 PARTIAL DIFFERENTIAL EQUATIONS. The first gives p = a, whence q = {l — «")^, and dz = adx + (1 - a^)i dy, so that z = ax + (1 — a^)^ y + 0(a). If we diiFerentiate this with respect to a we obtain the equation between which and the preceding we may eliminate a when (p is specified. (2) Let p^ = 1. The equation in p to be integrated is dp 1 dp 2 dp dy p' dw p dz whence dp = and p = a. The final integral is y p = ax + - + (j) (a). (3) Let % = pq. In this case we find p = y+ a, g = y + a therefore dz = (y + a) doo H dy^ dz zdy whence -; = dx ; y + a (y + ay z and therefore = a? + 0(a). y + a (4) Let p = (qy + zJK In this case we get fa qy- = a, and ^ = I - + / a\^ a whence dz = [z + - ] dx + — dy ; V yJ 2/ PARTIAL DIFFERENTIAL EQUATIONS. o69 or rf(. + 2)=(. + ?)"d^, whence | ar + -| {,v + (p(a)] +1 = 0. (5) Let q = p^z. Here p = - , g = — , and the integral is Z IS 2 — = ax + a^y + (p(a). (6) Let q = oap + p". The integral is . X -r o fdz\"' fd-z\" (7) Let a?«/«:>' — — =c". This may be put under the form / « ^JL.dzy f ^ -JL. dzy [jcmzm + n-—-] lyn zm + n - — =C. V d.vj V dy) By assuming c the equation becomes ■' = fx mdw, y = fy "-dy, z' = fzm+ndz ; dz'Y (d%'y dec ) \dy' j The integral of this found by the same method as in Ex. (2) is c z = a'^a;' + —y+^{a); and therefore m + n rn + n + y ^ m-a ^ j «--^ — Z m + n =. Cl'^x ni ^ y n + (h {a). 7/1 + n + y m — a n - (ia'" When m = a, x' = \ogcV, when n = (3, y' = ^ogy, when m + n + y = 0, z' = log z. 24 370 PARTIAL DIFFERENTIAL EQUATIONS. (8) This transformation fails when m + n = while y is not equal to 0. In this case the following method may be used. The equation may evidently be put under the form dzy fdi^y (dxj \dyj then considering a; as a function of % and y, doa dao ^ dx =■ —— d% + —- dvy d% dy dw and therefore whence d% 1 dx d a X dy then % = y^^ f (or — y'^). (6) Let {,v + ij)j- + (y- .r) ^ = ^• The auxiliary equations are, (x + y) dy - (y - x) dx = (l) (x + y) d% - %dx =0 (2) (y — x) dz - zdy =0 (3). Equation (l) may be put under the form xdy — ydx + xdx + ydy = 0; X whence tan~^ log (x~ + y~)i = a. Multiplying (2) by a?, (3) by y and adding, we have dz xdx + ydy z x^ + 2/^ ' whence -—^ — ^ = R ; and therefore '• . {x' + y-y . z={x^ ^ t/y^ / {tan- 1 - - log (c^'^ + y^y] is the required integral. ♦ (7) Let (x -2y) — + (2x - 3y) -— = z. dx dy The integral is (x-y)z = e^f(x -yf. This method may be extended to functions of more variables. Thus if dx dy dz and if from three equations such as 374 PARTIAL DIFFERENTIAL EQUATIONS. Pdy - Qdx = 0, Pd% - Rds) = 0, Pdu - Sdx = 0, we can obtain three integrals, U=a, V=b, W=c, the integral of tlie proposed equation is U==f(V,W), or (j) (U, V, W) = 0. (8) Let ^ du . . du ^ ^ du (u + y + z) -— - ■v\u + w-\-%) — - ->r iu + 00 ■{■ y) -— =-o(i-\-y-\'Z. doB ay a% The auxiliary equations are, {u + y + z) dy - {u + X + %) dx = 0, (u ■{■ y + %) d% - (u + X + y) dw = 0, {u ■{■ y + ?i) du - {oo ^^ y -^ %) dx = 0. Adding these three equations we have (u + y + z) (du + doo + dy + d%) - S {u ■{- oa -{■ y + %) doo = 0. Putting u + os + y->r% = 'o, this gives dx dv u + y + z 3v Subtracting the second equation from the first, we have (u + y + s;) {dy - dz) = {% - y) dw ; doo dy — dz or Therefore u + y + z y — dv dy — dz 3v y — ^ and V (y - zy = a. From the symmetry of the expressions it is obvious that we must have also V (x - zy = 6, V (u - zy = c. Therefore f{v(u- zy, V (x - zy^ v(y - !^y\ = 0. PARTIAL DIFFERENTIAL EQUATIONS. S75 /NT du ^ du ^ ^ du (9) -Let X -—+ hi j^ z) -— + hi + y) — = ?/ + x. a 01 ^ ^ dy d% Tlie integral of this is u + w + y + % = ff {so (u - y), x (y - .^)} . Mange's Method. Let the partial differential equation be of the form d^^' „ d^% d~% ' _ doo'^ dwdy dy where P, Q, R are functions of x, y, %, -— , -j-. dx dy Then if we form the system of equations dy — mdcc = (1) mdp + Qdq - Rmdx = J dy — m'dx = 1 •^ I (q\ mdp + Qdq - Rm'dx = J dz ds; where p = —- and g = — , and ?w, m are the roots of the dcV dy equation m^ - Pm + Q = ; and if from these two systems we can find two integrals U= a, V=b, then is the first integral of the proposed equation ; and the integral of this is the complete integral of the proposed equation. It is generally more convenient (when possible) to find another first integral, of the form and between these to eliminate p or g' so as to obtain an equation involving only one differential coefficient, and which is therefore easily integrable. Monge, Mcmoires de VAcademie des Sciences, 17S4, p. 118. , , ^ „ d"z d^z „ d~':s (10) Let q^ 2«g v + V ■ = 0. dar dxdy dy' 376 PARTIAL DIFFERENTIAL EQUATIONS. The auxiliary equations in this case are, q^m" + 2pqm + «^ = 0, whence m = , q qdy + pdx = 0, — pdp -\ dq = 0. Q Erom the second, since d% = pdx •{• qdy, we have dss = 0, or ^ = a. From the third we have pdq — qdp = ; . p whence - = dy (x) = c, 9 since % = constant, and therefore (p (^) = constant. From the equation p — cq = 0, we easily obtain ss=f{w + cy) =f{^v + y^(z)}, which is the required integral. d^ % d'z 4p dar dy^ <^ + 2/ The auxiliary equations are, dy — dx = 0, dp — dq -^ dx = (l) w + y 4 p dy + dcv = 0, dp + dq - — dx = (2) X +y From the first of (l) we find 4ipdcB y — X = c, and therefore dp — dq ^ = 0. 2y -a If we subtract from this last the equation 2p 2pdy 2dz 2qdy (dy - dx) = ---^ + 2y — a 2y — a 2y — a 2y — a (as pdx = dz — qdy) we have {2y - a) (dp - dq) + 2 {p - q) dy + 2d% = ; PARTIAL DIFFERENTIAL EQUATIONS. 377 the integral of which is (21/ -a)(p- g) + 2% = b =f{y - w), and therefore 9,% f (v — oc) p-q + ^-L^ i, (jG ■\- y cV + y From the first of (2) we find y + 0! = «!, and substituting this in the equation just found, it becomes d!s d% 2« f{y — w) dy dx «! Ui This is a linear equation and is therefore easily inte- grated. The result is %= -e^^y jdye ai±-^ ^ + 6'^+^ \// (.2? + ?/)? where o) + y is to be substituted for «i afier integration. (12) Let the equation be ,, d"% „ d'^ . „^ d^% If we put p -\- q = a, this takes the form , ^d"z ^ d-% d^% The equation for determining m is (l + qa) m^ - {q - p) «»* - (l + pa) = ; 1 . 1 . 1 + pa which mves wz = 1, m = — — . ^ 1 + qa We have therefore to integrate the two systems, dy - da) = 0; dp{l + qa) — dq (l + pa) = ... (l), dy (1 + qa) + dx (l + pa) = ; dp + dq = ... (2). TJie second equation of (2), gives p + q = b or a = b. The first equation, of (2) when put under the form dx + dy + a {pdco + qdy) = 0, S78 PARTIAL DIFFERENTIAL EQUATIONS. gives a? + y + (p + q) !S = a ; therefore ix + y + (p + q) x = (p (p + q). The first equation of (1) gives y — x = a, and putting p — q = (3, we liave p = l(a + /3), ^ = l(a-/3), and therefore the second equation of (l) may be put under the form dR ada , /-• , ^ qm — = 2 ; whence /3 = &i (2 + a')^ and therefore p-q = y\f{x-y){^ + {p-^ qy}h This first integral will enable us to determine the second integral. Putting p -^ q = a, P ~ 9 = f^^ we have dss = ^(a + I3)da) + ^{a- 13) dy = ^a(da} + dy) +^ (3(doo -dy); or, putting for (3 its value -v^ (a? - ?/) (2 + ar)h, dz = ^a {dos + dy) + J ((ia^ - dy) ^\r {x - y) {2 + a^)L This is integrable if we suppose a to be constant, and gives ^ + (^ (a) = 1 a (ci? + ?/) + >|/i C^' - 2/) (2 + a^)^' ; which, combined with represents the integral of the proposed equation. Poisson* has shewn how to obtain a particular integral of equations of the form p= (rt-syq (1) where P is a function of p, q, r, s, t, homogeneous with respect to the last three quantities, and Q is a function of x, y, z, and the differentials of ^, which does not become infinite when rt - s^ = 0. « Correspondance sur VEcole Polyteclinique, Vol. ii. p. 410. PAKTIAL DIFFERENTIAL EQUATIONS. 379 If we assume (l=f{p)i we have s = r/'(p), t=sf'{v)=^T{f'{p)Y; and therefore rt — s^ = (2) Hence the equation (l) is reduced to P=0; and on substituting in it the values of ^, s, and t, the quantity r will divide out, as P is homogeneous in r, s, and t, and the equation is reduced to the form F{p,f(p)f'(p)}='0, which is an ordinary differential equation, and being integrated determines the form of f(p) involving an arbitrary constant. The partial differential equation q = f(p) can always be integrated, and furnishes a value of % involving an arbitrary function and an arbitrary constant. This process comes to the same as finding what developable surfaces satisfy the equation (l). (IS) Let r^ -f^rt- s\ Assuming q = f(p} we find . r'{^-[f(p)Y}=0, whence /' (p) = ± 1 ; and therefore q = f(p) = i ^ + C, C being an arbitrary constant. On integrating this we find % = CcV ^ (p{y ^ w) as a particular integral of the given equation. (14) Let t + 2ps+{p^ - a^)r = (1) In this case Q = 0, and on putting q — f{p) we have, after dividing by r, {f{p)Y+^yf{v)+p'-a'=0', .(2) from which /' (^j) + p = ± a, and therefore g + 1^- i ap = C (3) 380 PARTIAL DIFFERENTIAL EQUATIONS. Now as every equation involving only p and q may be considered as representing a developable surface, it may be satisfied by the equation to a plane in which the arbitrary constants are afterwards supposed to vary. Hence assuming ss = ace + (5y + 7, we find p = a, Q* = i^j ^^^^ therefore so that a particular integral of (s) is % = ace + {C ^ aa - ^a) y + y- To deduce the general integral we must take for y an arbitrary function of a, and then join with the equation to the plane its differential with respect to a, so that the system of equations % = aoB + {C ^ aa — \ar) y + (p (a), = a; - (a i a) 2/ + ^' (a), is the general integral of (3), and a particular integral of (l). A different form of (p should be taken for each sign of «, so that this system is equivalent to two. The equation (15) (1 + q^) r - 2pqs + {1 + p^) t = 0, belongs to those surfaces in which the principal radii of curvature are equal but of opposite signs. On assuming q = f{p), we have 1 + \f{p)Y - ^pf(p) f (p) + (1 + p') [f (p)r= 0. The integral of this is q = ap + {- 1 - a^)^; from which we have % = (p{cv + ay) + y (— 1 - 0^)3 as the particular integral of the given equation. It is easy to see that this must represent a plane, as that is the only developable surface which has its principal radii of curvature equal and of opposite signs. PARTIAL DIFFERENTIAL EQUATIONS. S81 From the difficulties attending the integration of ordinary differential equations of a high order it will readily be under- stood that the integration of partial differential equations of the second and higher orders is a problem in the solution of which still less progress has been made. The subject has much occupied the attention of mathematicians, and pro- cesses have been given for integrating various classes of these equations, but they are unfortunately exceedingly long and complex, and the solutions are frequently given in a form which renders them practically useless. I shall therefore not give any examples of them here, but shall content myself with referring the reader to the original memoirs: such as those of Laplace, -M'emo^res de VAcademie, 1773; Legendre, Ih. 1787; Ampere, Journal Polytechnique, Cahiers xvii. et XVIII. ; and Cardinali, Sul Calcolo Integrals delV equazioni di difference partiali. ^ome examples of the application of Definite integrals to express the integrals of partial Differential equations will be found at the end of Chap. xii. CHAPTER VII. SIMULTANEOUS DIFFERENTIAL EQUATIONS. Sect. 1. Linear Differential Equations with constant Coefficients. The solution of any number of simultaneous equations of this class may always be reduced to the principles of the elimination of the same number of linear algebraical equations. For the symbol of differentiation may be treated exactly like any constant involved in the equation, and therefore the rules for eliminating, when the variables are involved along with constants, may be applied to equations in which they are in- volved, along with symbols of differentiation. Ex. (l) Let there be two simultaneous equations in- volving two variables, dx dy d To eliminate y, operate on the first equation with — and multiply the second by a ; we have then d^w dti dy u a-^ = 0. «-^ + ahw = 0. df dt dt Subtracting the second of these from the first, y disappears, and we have d^x — abx = 0. dt^ The integral of this is, making «6 = ra", From the first equation we find SIMULTANEOUS DIFFERENTIAL EQUATIONS. 383 It might at first appear that as we might obtain an equa- tion involving y alone, similar to the resulting one in w, there must be four arbitrary constants, and not two. But the second pair can always be determined in terms of the other two, and are therefore not arbitrary. This remark applies to such equations generally : and it is best to avoid the introduction of the superfluous constants by deducing (as we have done in this example) the other variables from the first without inte- gration. The real number of arbitrary constants is always equal to the sum of the highest indices of diiFerentiation in the different equations. (2) Let —- + ax + by = 0, CLv ~ + a,x + b^y = 0, be two simultaneous equations. Operate on the first with I — +61)5 and multiply the second by 6; then, on subtract- ing, y disappears and we have This may be put under the form dt j \dt ) where h and k are the roots of the equation z^ — {a + 5i) % + abi — a^h = 0. Integrating in the usual way, we find h — a „ n 1c — a ,. (3) Let -^ + 4)0? -i- 3y = t, €1 V ^y t -~ + 2x + 5y = e. 384 SIMULTANEOUS DIFFERENTIAL EQUATIONS. vEliminating ?/, we find 31 5 1 Whence ^ = ^ —t - -e'+ C,e-'' + Cge"''; 196 14 8 ^ 98 7 24, 3 ^ (4) Let -— + 5ci? + ?/ = 6*; ■J- + Sy - w = e^\ We find 3/ = ^e' + ^6"-(C,«+C„+C,)e-". (5) Let there be three simultaneous equations, doB — + hy + c% = 0, dy , , at dt ^ Operate on the first with I — ] -&"c', on the second with h"c-h—-, and on the third with he - c — . Then on dt dt adding, the terms involving y and % disappear of themselves, and there remains I ( — ) "C^^'^^ + ^'^^ + ^'O T" ■Vah"c'{-a"bc'\ w = 0. SIMULTANEOUS DIFFERENTIAL EQUATIONS. SC The integral of this is easily seen to be .V == C^e-^ + C,el^^ + Csey', a, /3, y being the roots of ^ _ (a'b + a"c + h"c) % + ab"c + a'hc = 0. The values of y and % are easily derived from that of a. If (6) Let -^ — ay — bio = c, -—; — ay ~- 00! = c . Eliminating y by operating on the first with I — 1 - a\ multiplying the second by a and adding ; there results This may be put under the form where h?, k^ are the roots of the equation ^^ + (a + b) z + a'b - ab'. Hence we find (to — /* ft w = — -. + Ci cos {hoc + a) + Ca cos {kcB + /3) ; ab — ab b'c-bc' h'+b ^ ^^ ^ k'+b ^ and y = -7, ,-, Ci cos {hx + a) C2 cos {k,v 4 /i). a b—ab a a d^x dx dti (1^ Let — - - 2 2 -^ + ci? = cos 2^, ^'^ df dt dt d^y dy dx ^ . ^ — - + 2 — ^ + h 6w + 5.37 = sin r. d/^2 dt dt ^ Eliminating y we obtain the equation J ( — - J + 5 (!-] + ei a? = 2 cos 2^ - 4 sin 2t -Qcost; 25 886 SIMULTANEOUS DIFFEREN*riAL EQUATIONS. or ■! f — I +2W[ — j + $> w = 2 cos2t - 4 -<^ = ; \dw ^ dy J \dw dy J the integral of which is there is no singular solution of the final integral. (12) Let dy or d'^-y fdii d'y\^ (d'y\^ y-a)~ + ± - l-± -o!^) -— ^ =0. dcV 2 dw^ \dcv dot- J \dx~) ^, ,.. dU . i he condition = gives us 4 A = fdw (2aa) - w^)^ = (2aa! — ai^)^ H vers"^ '-. '' ^ ^' 2 ^ ^2 fit (5) The equation to the ellipse being ' x^ if b I) w A = - fdcV (a^ — w^)^ = - . circ. area whose cosine is - + C a a a For the whole ellipse A = - ira^ = irah. a (6) The equation to the witch of Agnesi is xy" = 4>a^ (2a — x). (2 ax - os")^ + a vers"^ -/ + C. a] Taking this from x = 2a to x = 0, and doubling it on account of the symmetry on both sides of the axis of b. The form of this curve is given in fig. 42. If we wish to find the area included within ODCAHG, it is sufficient to integrate from 9 = to that value of 9 which causes r to vanish, and then to double the result. Let I I , then the area ODCAHG is equal to ^{{a' + 2b')a + 3b(a'-b-)i]; and the area OEBF is equal to ■I- 5(a- + ^W) (tt -a) -3b (a' - b')^. If 6 = a, the curve becomes the common cardioid, and . 3 Tra- its area is . 2 (15) The equation to the conchoid of Nicomedes when referred to polar co-ordinates is r = asec 9 + 6, and its area is l{aUan0 + 2a&logtan f- + _W b'^9] + C. (16) The curve whose equation is r = a sin 39 has six loops (see fig. 49), and it is sufficient to find the area ttq.'' inclosed by one of them. This is easily seen to be and therefore the sum of the areas of the six loops is ^Tra", or one half of the area of the circle which bounds them. (17) The equation to the spiral of Archimedes is r = a9. Hence the area = h C. 6 414 QUADRATURE OF AREAS. After n revolutions the analytical area swept out is n^ (2 irY c? ; but to obtain the geometrical area we must sub- tract from it the area corresponding to {n — l) revolutions, which gives us {^rC- — 37i + 1) as the required geo- o metrical area. In the same way we should obtain as the geometrical area corresponding to (n + l) revolutions, the o expression (Sn^ + 3n + 1) (Stt)^ — , and the difference between 6 these or the space between the arcs after (n + 1) and after n revolutions is n (27r)^a~, which is 7i times the space between the arcs after the first and second revolutions. (18) In the hyperbolic spiral 0'9 = a. The area swept out by the radius vector from to r is -^ar^ which is equal to the triangle formed by the radius, the tangent and the sub-tangent. If the equation to the spiral be given by a relation between p and r, we have j=l f ^^^^ ^ J (r^ - p~)^ ' (19) In the involute of the circle r'' — p~ = a\ Therefore J = — fdr r Cr^ - a^)^ = ^ + C. 2a'' ^ ^ 6a (20) In the epicycloid ^ c' (r^ - a^) P ~ 3 2 ' c" — a where c = a + 26, a and 6 being the radii of the fixed and generating circles respectively. Hence 2 a^ W- - rV ^ aJ jc- - a' - (r^ - a-)}^ - c (r^ - a-)i (c- - r-)^ c (c^ - a-) . , fr'^ - a^\ ^ = + sin"^ — s ' - 4a 4 / w'^ y^\ ^ Therefore V = c ffdx dy ll ^ ~ Va ) • Integrating with respect to y, we have 7= ^/ vr 3 2 16 and the whole solid is aP { — + 2 tt When a solid is generated by the motion of a plane area which moves parallel to itself, while its magnitude increases or decreases according to a given law, its volume is found by the formula V = cosa fvd!S ; V being the area, the axis of ^ being the direction of motion, and making a constant angle a with the normal to the plane. (9) Let the solid be the groin which is generated by a* square moving parallel to itself, its sides being the double 426 CUBATURE OP SOLIDS. ordinates of a circle of which 5^ is the abscissa. If y be the half length of a side, v = 4 \2ay-y')l CUBATURE OF SOLIDS. 429 This being integrated from 2/ = to y = 2a and doubled gives Sir^a^ as the volume of the whole solid. When the axis of the cycloid is taken as the axis of ,r, the equation to the curve is dy I'S.a — w\-^ but this is not a convenient form for finding the value of irfy^'dx. It is better to substitute for y and ,v their ex- pressions in terms of 6, i. e. y = a(0 + sin 6), x = a (l - cos 0); whence F = tt a^ fdO sin 9 (d + sin 0)'. The value of this taken from 6 = to = tt, is F= ira^ . V 2 3J (l^) The equation to the tractory is and the volume of the solid generated by its revolution round the axis of off, and taken from .a? = to ct? = oo is -i- Tra'^ (19) The equation to the Witch of Agnesi is cvy = 2a {2ay — y^)^. If it revolve round its asymptote which is taken as the axis of w, we have for the volume of the solid V = TT fy^ dw = Try' a; — Ztt fwydy = Try^Off — 4!7ra fdy (2 ay — y^)K The whole volume is 4S' = ^Trrw. This is the part of the surface included within the positive axes, and if we multiply it by 4 we have ^tttx as the surface of a zone of the sphere, the height of which is x : it is therefore equal to the corresponding zone of the circumscrib- QUADRATURE OF SURFACES. 431 ing cylinder. The whole surface of the sphere is 47rr^ or four times the area of a great circle. (2) The axes of two equal right circular cylinders in- tersect at right angles, find the area of the surface of the one which is intercepted by the other. The equations are o o o 090 w + js~ = a, w + 2/ = «" ; ana . = //...,{.. g)V(g]l T^ dz w d% Here = _ _ , -— = ; aw % ay therefore S ^ a f f^-^ = a f fy^^,. J J % J J (a' - wyi Integrating with respect to y from y = io y={a? — co')^-, S = a j^ dx = a^ ; and the whole surface, being eight times this, is Sa^. (3) Circumstances being the same as in Ex. (7) of the last section, to find the area of the intercepted surface of the sphere. The equations to the surfaces being 0?^ + 2/" + ^^ = a^5 x^ + y" = ^'^^j rrdwdy PC dccdy ^=''JJ-^'"'JJia'-^'-f)i- Transforming into polar co-ordinates r and 0, we have rdrdO the limits of r being and a cos 0, those of 9 being and ^ TT. Therefore ,S = a- /o5^ (1 - sin 9) = a- (Itt - 1). The area of the surface of the octant of the sphere is ^TTtt^; and therefore the area of the surface of the octant which is not included in the cylinder is equal to a\ or the square of the radius of the sphere. If the sphere be pierced by two equal and similar cylinders, the area of the rrran 432 QUADRATURE OP SURFACES. non-intercepted surface is 8a-, or twice the square of the diameter of the sphere. This is the celebrated Florentine enigma which was pro- posed by Vincent Viviani as a challenge to the mathematicians of his day in the following form : " Inter venerabilia olim Graecias monumenta extat adhuc, perpetuo quidem duraturum, Templum augustissimum ichno- graphia circulari Alm^ Geometri.e dicatum, quod testudine intus perfecte hemisphaerica operitur : sed in hac fenestrarum quatuor aequales areoe (circum ac supra basin hemisphaeras ipsius dispositarum) tali configuratione, amplitudine, tantaque industria, ac ingenii acumine sunt exstructae, ut his detractis superstes curva Testudinis superficies, pretioso opere musivo ornata, tetragonismi vere geometrici sit capax. Acta Eruditorum, 16.92. (4) Under the same circumstances to find the area of the intercepted surface of the cylinder. The element of the circumference of the base of the a dx cylmder being - ,, we have 9. {ax — w~Y %d,v a^ radx a ra %a,v a^ raaw 2 ^0 {ax — ti?^)3 2 Jo <2?2 and the whole area of the intercepted surface of the cylinder is 4a^, or equal to the square of the diameter of the sphere. If a solid be generated by the motion of a plane parallel to itself, the surface may be found by a method similar to that used for finding the volume. If li be the periphery of the generating plane, s the arc of the curve made by a plane perpendicular to the generating plane S = fuds. (5) Under the same circumstances as in Ex. (11) of the last section, to find the surface of the intercepted solid. If s be the arc of the circle passing through O and perpendicular to PQ, the area of the element PQpq is Ids, and the area of the surface AOB is fids. QUADRATURE OP SURFACES. 433 Now I = 2p cosec a = 2 (a^ - x~)^ cosec a, and as = r ; {a- - ;?~)^ therefore S = 2a cosec a f^'^ d% = 2a" cosec a ; and the whole surface is l6a"coseca. (6) Under the same circumstances as in Ex. (12) of the last section to find the area of the convex surface of the part of the cylinder cut off. s being the element of the circumference of the base, and *S' the element of the surface. xdx {a^ — ai~)i = 2 a tan a { C - (a^ - ar)^ \ . When .T = 0, ^S* = and C = a, therefore S = 2a tan a {a — (ar - ar)'i] ; and the ^vhole convex surface is 2 a" tan a. When a curve surface is formed by the revolution round the axis of ,v of a curve the equation to which is y=f(cv), the area of the surface is given by the integral, Kdii\ . r wax S =2 fxds = 2a tan a /— —, J (a'' — a")2 S=- 27r fd.vyh + f- (7) For the paraboloid of revolution we have y^ = 4>mcV ; therefore S = ^irm^i [dx (x 4. m)^^ 8 TT 1 ^ 3 = — m-^ (x + m)'s + C. 3 If the surface be measured from the origin, S = -^ irr-' [{x + m)^ — m^. (8) For the prolate spheroid we have 28 434 QUADRATURE OP SURFACES. 27r6 1 , a"-b~ and S = fdw (a^ - e'os^)K where e~ — s — . a -^ ^ ' or Integrating, we have „ 'n-ah { . pw BOB f e^x'XA ^ S^ y 2*' = ^; th. 2.1? 2a! dx + ydy = 0, whence. , by integration, a^ + y^ 2 b^ being an arbitrary constant. This is evidently the equa- tion to an ellipse. The equation to the lemniscate of Bernoulli is (or + y-y = c? {cG^ — y"). That to the orthogonal trajectory is (t'p^ + y"y = hwy \ which is the equation to a similar lemniscate, the axis of which is inclined at an angle of 45° to that of the former. If one of the variables be given as a function of the other and the parameter, as if y = f{x\ a), we cannot eliminate a so readily. But let dy = Pdoo' + Qda\ dy' then for one curve — = P, and the equation to the or- dx thogonal trajectories is dx H- Pdy = ; and as dy — Pdw + Qda, this becomes (1 + P^) dx + PQ,da = 0. As P and Q contain only x and a, this is an equation between two variables x and a, and to integrate it appro- priate methods must be employed. 444 GEOMETRICAL PBOBLEMS. (14) Let the curves be a series of ellipses expressed by the equation 2/ = - {c~ -x-)'^. a being the variable parameter. Here c [c' — 03 'y^ c and the equation for the orthogonal trajectory is a 00 (c^ - Olt) da = {c' + {a' - c") og'^\ doc. To integrate this put (c^ - aj 3 4 u and ■?/ = , whence, eliminating u and v with the assistance of the given equation, we find y- = ia (,v + 2a), the equation to a parabola. 446 GEOMETRICAL PROBLEMS. (16) If the equation to the evolute be v^ — u^ = — C3 ; and if the constant be determined by the condition that, when 4' u = c, a? = -^c, the equation to the involute is shewing that it is an equilateral hyperbola. If the trajectory is to cut the curves according to any other law than that of a constant angle a similar method is to be employed. (17) Let, for example, it be required to find the curve PP' P" (fig. 60.) which cuts a series of parabolas having the same axis and the same vertex so that the areas AMP, AM'P\ &c. are constant. The equation to the parabola being 'if' = 4) ax, the area APM = fo^yda; = 2f^''(aa;)i doe = W suppose. Differentiate considering cc and a as variables ; then 2 (aa))2 dcV + —^dw.da = 0, ■r €t Ct f* . -J or 2 (aa))-^ dco-\ [^{axp dx = 0. a But by the condition of the area being constant fff(ax)idx = b", so that the equation may be put under the form , , b~ da 9. w^doc + 5- = 0. 2 at Integrating, we have 3 a^ 2' Eliminating a by means of the equation to the parabola, 2 ^ 6V' _ C ■3 7/ ~ 4' GEOMETRICAL PROBLEMS. 447 or To determine the arbitrary constant, we observe that when X is indefinitely diminished, y must be indefinitely in- creased in order that the area may remain constant; this makes C = 0. Hence y = — , or 2xy = 3o; is the required equation, being that to an equilateral hyper- bola. (18) Find the curve which cuts a series of circles described round the same centre in such a way that the arcs intercepted between the intersections and the axis of x shall be equal. If x"^ +y' = ap-l y + B = (a sin G) — cos cb) = r . ^ 1 + a^ ^ ^ ^^ 1 +a- (1 +p')^ From which we have (y + B)(a + p) = (x + A) (ap - 1). Or putting x and y for x + A and y + B, which does not affect « = — this becomes dx a (xdy - ydx) = xdx + ydy, which is the differential equation to the logarithmic spiral. That curve therefore is the only one which has an evolute similar to itself. Euler in the memoir referred to above has considered the question much more generally, for he investigates the nature of the curve which has its n^^ evolute similar to itself, as well as the curve which has an evolute similar to itself but placed in an inverse position. This last is reduced to the previous case, for if the evolute be similar to the original curve but in an inverted position, the second evolute will 454 GEOMETRICAL PROBLEMS. also be similar to the original curve and in the same posi- tion, and its radii of curvature will diminish while those of the first evolute increase, as will be seen in (fig. 62). It is easy to see that this condition is expressed symbolically by affecting the coefiicient of similarity with a negative sign. The general equation for a curve which has its n^^ evolute directly similar to itself is that which has its n^^ evolute inversely similar is ■fin + d'^'p = 0. (24) Let us investigate a particular case of this last problem when n = \ and a^ = 1, which implies that the evo- lute is equal to the curve but in an inverted position. The equation then becomes 4^ + ^ = '- The integral of which is p = C cos

C V 2 ^ ) the equation to a cycloid referred to its vertex. If a^l, the curve is a hypocycloid. (25) In speaking of the cycloid I mentioned a property belonging to it which was discovered by John Bernoulli, viz., that if BC (Fig. 21) be any curve, the tangents at the extremities of which are at right angles to each other, and if this be developed, beginning from C, and if the involute CD be again developed, beginning from Z>, and so on in succession, the successive involutes approach continually nearer and nearer to the cycloid, and ultimately do not differ sensibly from that curve. The following demonstration of this re- markable proposition is taken from Legendre, Exercices de Calcul Integral, Vol. ii. p. 541. Draw the successive tangents MP, PN, NQ ... which will be alternately perpendicular and parallel to the first, from the nature of involutes. Let 6 be the angle which MP makes with the line AB, and put arc CM = w, arc CB = a, arc CP = «, arc CD = &, arc EN = iv\ arc ED = a', arc EQ = %', arc EF = b\ and so on in succession. Then if we were to draw tangents, making angles = dO with the other tangents, we should have dz dm dz MP~ ¥n~ qn But from the nature of involutes, MP=CM = cc, PN^DP = b-%, QN=:EN=x,kc. d% doG d% dm" Hence dQ = — = w h— % 00 h' — %' 456 GEOMETKICAL PROBLEMS. From the first we have ;;; = fcvdO, which ought to vanish when 6 = 0, and to become equal to b when 6 = -- The second equation gives dx =hdQ- %dQ = hdd-dQ jxde, and w =h6- pdO^x, TT which, when = 0, ought to vanish, and when 6 = — to be equal to a'. The third equation gives dii' = .v'dO = bOdO - d9 pdO'x, ■LQ2 and z'= — - pdd'x. 1.2 •' Proceeding in this way, we have 1.2.3 1.2.3.4.5 "^ ;j;('')=I --— — + &c. ±p«+id0=" + \r. 1.2 1.2.3.4 -^ Now the last terms in both of these expressions continually diminish, and if n be made sufficiently large they may be neglected. This may be seen by considering that since x = 6<") f-?^ ? + &c.'j = &(") (1 - cos 6). Vl .2 1.2.3.4 J ■ ^ Similarly .t?("' = 6^) sin 0. These equations belong to a cycloid, in which 1 6*"' is the radius of the generating circle. Thence follows the pro- position. (26) Find the surface, such that the intercept of the tangent plane on the axis of % is proportional to the distance from the origin. The intercept of the tangent plane on the axis of x is z -px -qy; 458 GEOMETRICAL PROBLEMS. hence we have % — p 00 — qy = n (jG^ + if •{■ z^)^. The integral of this equation is (27) Find the surface in which the co-ordinates of the point where the normal meets the plane of ooy are propor- tional to the corresponding co-ordinates of the surface. The equations to the normal being x — X + p {z - %) = 0^ y' — y + q {% - %), we have when ^' = 0, x =x -^ p%, y =y ^ q^i therefore x + p% — mx, y + qz = ny. Substituting these values in dz = pdx + qdy, and integrating, we find z^= {m- 1) x"" + (n - 1) y'' + C, which is the equation to a surface of the second order. (28) To find the equation to the surface at every point of which the radii of curvature are equal and of the same sign. The conditions that this should be the case are or 1 + p^ pq 1 + q^ = = 7 > r s t p dp i dq q dq 1 dp 1 + p'^ dx q dx 1 + q^ dy p dy Integrating these as ordinary equations and replacing the arbitrary constants, in the first equation by an arbitrary func- tion (F) of y, in the second by an arbitrary function (JT) of x, we find I + p" = Yq\ \ -^ (f = Xp^. GEOMETEICAL PROBLEMS. 459 From these we find But p and q ought by their nature to satisfy the equation dp dq ^.^ . ^ -— = -— , which in the present case is dy doc dx dy Now whatever be the form of the functions X and F, this equation is of the form (pi^i) = ^{y), and it can there- fore subsist only when each side is equal to a constant. Let 2 this constant be represented by -; then , 3 dX 2 .dY 2 (1 + X)-t-- = - , (1 + Y)-^^~ = -, dx r ^ dy r whence on integration we obtain r r , a and h being arbitrary constants. If from these we take the values of X and F and substitute them in those of p and q we have {a - x) b -y ^ " {r'-{a-ay-{b-yf\h' ^ ^ [,^ - {a - wf'-^h - y)"\^ ' Putting these values into the formula d% = pdx + qdy, and integrating, we have (x - ay + (y - by + (^ - cy = r^, which is the equation to a sphere. Monge, Analyse JppUquee. V CHAPTER XI. EVALUATION OF DEFINITE INTEGRALS. When we are able to effect the integration of any function, the determination of its value between certain limits of the independent variable offers in general no difficulty, as we have merely to subtract its value at one limit from its value at another. There are however many functions, the Definite Integrals of which we are able to find, although the in- definite integral cannot be expressed in finite terms. The evaluation of these integrals has become one of the most important branches of the Integral Calculus, in consequence of the numerous applications which are made of them both in pure mathematics and in physics : it is to functions of this kind that the examples in the following paper refer. The methods for evaluating those definite integrals whose general values cannot be found are very various, but they can generally be classed under the following heads. (1) Expansion of the function into series, integration of each term separately, and summation of the result. (2) Differentiation and integration with respect to some quantity not affected by the original sign of integration. (S) Integration by parts of a known definite integral, so as to obtain a relation between it and an unknown one. (4) Multiplication of several definite integrals togethei, so as to obtain a multiple integral, and, by a change of the variables in this, converting it into another multiple integral, coinciding with the first at the limits, and admitting of integration. By this means a relation is found between the definite integrals multiplied together, which frequently enables us to discover their values. (5) Conversion of the function by means of impossible quantities into a form admitting of integration. DEFINITE INTEGRALS. 461 These different methods will be best understood by their application to the following examples. We shall betjin with the function known as the Second Eulerian Integral, because, though its exact value cannot be found generally, its properties have been much studied, and to it a number of other integrals are reduced. 1. Second Eulerian Integral. The definite integral j^ dx e~'''tv'^~^i when 7^ is a whole number, is easily seen by the method of reduction in Ex. (13), Chap. II. of the Integ. Calc. to be w(w - 1) ... 3.2. 1. When, however, n is a fraction, its value can be found only in certain cases, but it possesses many remarkable pro- perties which render it of the greatest importance in the Theory of Definite Integrals. It was first studied by Euler, who seems at an early period to have seen its importance, and has devoted several memoirs to the investigation of its pro- perties ; on this account Legendre has named it after him, at once for the purposes of characterizing the function and honouring that great mathematician. To distinguish it from another integral with which also Euler had much occupied himself, and of which we shall afterwards treat, it is usually called the " Second Eulerian Integral," and Legendre has affixed to it the characteristic symbol F, applied to the index, so that he writes jrdxe-'w--^^V{n), which notation we shall adopt. Throughout the following investigations n is supposed to be greater than 0. In the first place we remark that by a change of the independent variable this integral may be put under other forms which are sometimes more convenient in practice than that which we have used. Thus if we put e"*' = y, the corresponding limits are .2? = 0, 2/ = 1 ; c2? = CO , y = 0. 462 DEFINITE INTEGRALS. and the integral takes the form (a) r in) = f,'d.v (loglj \ This is the shape under which the integral has been usually treated both by Euler and Lagrange, but it is scarcely so convenient as the preceding. Again, if we put at'"- = %, the limits remain the same as before, and we have (6) rin) = -J,''d^e-'". This last form is the most convenient for determining the value of the integral in one remarkable case when it can be found in finite terms. If n = ^ Let k = j^ d% e~^^ '. then as the value of the definite integral is independent of the variable, we have also k = j^dy e-y\ and therefore multiplying these together, k^ = f-d^e-^' . !,-dy e-y = T/o^cZ^/ d^e-^y'^^') ; since y and % are independent. Now assume % = T COS 0, y ='i" sin 9, then %'^ + y~ = r^, dy dz = r dr dd. To determine the limits we observe that y and % never TT become negative, and therefore 6 must vary from to — , while r varies from to oo, so that we have F = /,=* /o i ^ dr tZa r e-'*' = ^ tt ; whence (c) k = lir^^ and T (J) = Tri We shall now demonstrate the more important proper- ties of the function T {n) referring the reader who wishes, for a more detailed exposition of them to Legendre, Eocer- cices de Calcul Integral, Tom. i. and ii. DEFINITE INTEGRALS. 463 If we integrate by parts the expression jdw e'" ce^ we have Jdw e-'^.i?" = - e-' a?" + n fdw e"" a'""^ The integrated part vanishes at both limits, so that (d) r (n + 1) = nT (n). This may be looked on as a characteristic property of the function T, and is of the greatest importance, as by means of it we can reduce the calculation of T (n) from the case when n> 1 to that when it is < 1, and we have therefore to occupy ourselves only with the values of n which lie between and 1. If w be a proper fraction, r{n)=f^'=d.ve-''w^-'; T (1 - w) = /o'^e'^r"; and therefore r(n)T{l-n) = Jo'^ dw e-'' cv""-' fo"^ dy 6-^ y-" = rrd^dye-^^'^y^a^-'y-\ To reduce this, we shall use the transformation of Jacobin given in Chap. iii. Sec. 2, Ex. (7) of the Diff^. Calc. Assume x + y = u, y = uv, so that dx dy = ududv: the limits of u and v corresponding to those of x and ?/, are u = 0^ u = m = -, n = -, &c. pqr Hence if we know the integral U= fdx fdy fd% ... a'^-^ ?/™-^^"-^ ... , Avith the previous condition for determining the limits, we can find V. When the variables are two in number, it is easy to see that the integral is identical with that called the first Eulerian. Let us suppose therefore that there are three variables. Then U = fldw.x^-'fy^dyy—' f:^dzz^-\ where y^^^ I — w^ z^= \ — w -y. Liouville's Journal, Vol. iv, p. ICS. DEFINITE INTEGRALS. 469 Assume % = vx^, y = uy^, the limits of u and v are then and 1, and U takes the form But as 2/i = 1 - ct', and ^1 = 2/1- «^.Vi = (1 - P r(i) •'° Vi + ^ — v^ whence lY —— ( !1L _ i ) . 4 \2 (4) M. Catalan* has shewn how to evaluate a definite multiple integral which depends on those which have just been considered. It is ^^ da>idx2 ... da^„_i y == JJ ... /(«i<»i + «2<^2 + ••• + anX,J, (1), in which x^^ = 1 - x^^ - a?/' - ... - wl_i, and the limiting * Liouville's Journal, Vol. vi. p. 81. DEFINITE INTEGRALS. 471 values of the n — 1 independent variables are given by the condition cVj^ + cV2^ + ... + a!l_i < 1. Assume tVi = p^Ui + qiU2 + ... + r^ Un a>2 = P'^u^ + q^u^ + ... + r^Ur, I w,„=p,,u^ + q^u^ + ... + r„w„ j where the coefficients of u^ are «! a Pi = 1^2 = and the other coefficients are subject to the conditions. ^1 &c.; qi +q-i + ... + 2^ A "^i +'>'i + . . . + r/ = 1 P\qi +P2q2+ ... +i?„g„ = o gir^ + ^oro + ... +g„r„ = (3). The number of constants in (2) is n (n — l), the num- ber of conditions in (s) is ^n{n + l), so that there are ^n(n —3) arbitrary coefficients. Adding the squares of (2) we have by the conditions (s) «^l" + Wg^ + ... + u,^ = Xi^ + •^' f:.?'- a?." - 1 - 1 = log- n. DEFINITE INTEGRALS. 475 In like manner we find ^^^ •^o'^'^ ' — W^; = log m - log w = log (^- j . If we multiply = 1 — x + w^ — x^ + &c. 1 + X ■a?'" ~ ^ — a?" ~ ^ ^y ^i 5 ^^^ integrate from <2? = to ti? = 1, we find log X .a?™ ~ ^ - ci?" " ^ m , w + 1 , w + 2 cid^ -r r^ = log loff — 1- log &c. ■' " {\ -v cc)\og CG ^ n ° ^ + 1 ° w + 2 m (7^ + 1) (wi + 2) (?^ + 3) ... ~ 7^ (/Ti + 1) (n + 2) (to + 3) ... ' and if w = 1 — m, a?*""^ - <2?~'" m(2 — to) (2 + W2)(4< — m)(4 + m)... ■^ " (1 + ct) log x~ ° (1 - to) (1 + to) (3 - to) (3 + to) ... Now by the formulas expressing the sine and cosine of an angle in products of factors, Ave have In this putting % = , and observing that by Wallis's theorem - = ^^-V( , , , . v3/ \b} V7 2\ 2 /4,\ 2 ,qk1 /gN a we see that (c) Indx- — i = log tan TO - . ^ ^ -^ ^ (1 + <2?) log cv ^ V 2 ; Kummer in Crelle's Journal^ Vol. xvii. p. 224. (8) By integration by parts it is found that acos,rw — r sinrcT jdx e '^^ cosr<2? = — e — ax a~ + r 476 DEFINITE INTEGRALS. a sin vw + r cosr.^ id /t/cr6-"'%inr.r= -e""'' Hence taking the integrals between and cc we have (a) r<^<'»6-"^cosra; = -7^, (b) rdct;6-"'sin r^ = -:^. If we differentiate these expressions (n — 1) times with respect to a we have by Ex. (20) and (21) of Chap. ii. Sect. 1 of the Diff. Calc. .rr, COS nO (c) f, dsj c-j?"-! 6-"" COS r .2? = 1 . 2 . 3 . . . (7Z - 1) , {a^ + rj Sin 71 (d) {^'^dww^-'e-'''''%mraj= 1.2.3 ... (w - l) ~, {a' + T'f r where 6 = tan " ^ - . a In these expressions n must be a positive integer; but if it be a positive fraction, the only difference is that instead of the continued product 1 . 2 . 3 ...(«— 1) we must substitute the definite integral T (n). If we integrate (a) with respect to r, we have (e) f. — e-""sinrA' = tan-M- , Jfi X \a) no constant being added, as the integral vanishes when r = 0. In this formula if we make a = 0, we have ,.. r==da; . tt ( / ) / — smrx = — . ^■^ ^ Jo w 2 If we make « = in the formulae (a) and (b) we have (g) Jq^ dx cos 7\i' = 0, (A) f^y'^dw sin ra-' = - . From the integral (/) it is easy to see that (k) / — sm x cos TcZ' = — , ^ -^ Jo w 2 DEFINITE INTEGRALS. 477 when r lies between - 1 and + 1, but that it vanishes for all other values of r. The results (g) and (h) are very remarkable as giving the real values of what are apparently indeterminate quantities, the sines and cosines of an infinite angle. For as f^'^dw co^ rat = - (sin co - sin 0) = by (^), it follows that sin co = 0, and as fa^da) sin rx = (cos oo — cos 0) = - by (h), it follows that cos co = ; so that both the sine and the cosine of an infinite angle are equal to zero. In the formulae (c) and (d) if we make a = 0, we find the two remarkable integrals n\ r= 7 n-1 1.2 ... {n - l) ir iL) L ax 00^ * cos rw = cos n - , -„ . 1 .2 ... (/Z - 1) . TT (m") L dx af~^ sm rx = — — — sin 7i - . ^ / JO „,n Q If the index n lie between and 1 the corresponding formulae may be deduced without the consideration of limits involved in making a = 0. Since l^ da a.-'' e-'''' = T (l -?z) .'»"-', on multiplying both sides of this equation by cos rx dx and integrating from to co, we have j^ dx co'&rx ^r^ da a~" e~"^ = F (l - Qi) /^^"dx x^~^ cosrx. But fo'^dx cos fxf^'^da a~" e'""'' = f^^da a~" l^^dx e~°'^ cosrc-p -^d a' a a' + r Hence f^'^dx x'" ' cos j'A' = ~ J(i^da -^ V{\-n)^ a- + r 478 DEFINITE INTEGRALS. By the formula (d) in Ex. (5), v/e have rd d^ + j'2 ^,n 2 sin 1 (2 - w) TT 2 r" sin (^ 72 tt) Therefore 1 TT I {n) ( ir\ L dx A'"~^ cos rx = — — — r - — --, = — — cos \n-\, ^° r"T (1 - n) sin (-| Wtt) »•" V 2 / as r (w) r(l -n) = >2 TT sm Wtt In like manner we should find ["'dcV x^~'^ sin rx = sin [n — \. Jo r"" \ 2} Thus if n = ^, we have, since F (J) = tt^, and sinivr = cos^TT = S'^, ^o=> doj /ttX^ r'x.dw . To these integrals may be reduced fQ°^dxx'"~^e~""'° sinrx when 72<1. For, on integration by parts, the integrated term vanishes at both limits, and we have f°^ dxx'"~^6~°'^ sin rx = /^^da?<37"~^e~"^sinra7 f^^ dxx""'^ e'^''' cos rx. n — \ n — 1 When a = 0, this gives Y 1 r (tC\ TV (^dxx''-^smrx = f^ dxx'^~'^ cosrx = — cos^i- . fo^dx . 1 (p) If »2 = ^, / — - Sin rx = {Zrirp. I If in formulae (l) and (m) we assume a? = «" , they become . -X nT(n) [ 7r\ /o ^^ cos {r%-) = ^^,^ cos [n -j , . . - wr(w) . / 7r\ /o d^ sm (risr") = —-;^ sin ^^?^ - j . DEFINITE INTEGRALS. 479 Hence if n = -J^, (?') io'^^^ cosr^^ = 1 I — I = j^ d% sin r%^. The formulge in this article are due principally to Euler, Calc. Integ. Vol. iv. p. 337, See also Mascheroni, Adnota- tiones, p. 53. Laplace, Jour, de VEcole Polyt. Cah. xv. p. 248, and Plana, Memoires de Bruxelles, Vol. x. (9) To find the value of u = j^ dx e \ ^v. Diiferentiating with regard to a we find da Jq w Put - = %, the corresponding limits being a' = o, ^=co; <2?=co, ^ = 0. Hence -— = - 2z* ; da this is a linear equation, the integral of which is To determine the arbitrary constant, make a = 0, when / "^\ J (a) Hence f(^dx6 \ *"/ = — e""". This integral was first given by Laplace, Memoires de rinstitut, 1810. From this may be deduced the following integrals : (&) j^ dx cos I «•" + — 2 ) = — cos f — V 2a\ = ^(-| (cos 2a - sin 2a). 480 DEFINITE INTEaRALS. (c) fo'^da! sin Lv^ + - J = — sin f - + 2a| ~ 2 \~] (cos 2a + sin 2a). (d) !r^^^ ^^ a-v'^"^ cosJfct?^+ — ] sin Oi 2/ = _e-2«cos0cos 2a sin 6^ + (e) jo a.2? 6 V *"/ sin N a?'^ + — sin y / 1 = ![:e-2acos0gin /^ . n ^ 2a sinO + - Q I 2 Cauchy, Memoires des Savans Etrangers, Vol. i. p. 638. (10) Find the value of j^ dw e~"'^^^ cos, 9,rw. Calling the definite integral u, and diiFerentiating with respect to r, we have -— = - 2 L dec xe'"'"^^ sin 9,rx. dr On integrating the second side with respect to w by parts, the equation becomes du 2r = u, dr a since the integrated part vanishes at both limits, and the unintegrated part when taken between the limits is equal to u. This equation on integration gives ?•" To determine the arbitrary constant, put t = 0, then C = fo^ dwe~"-^^^ = ^—-^ so that 2a (a) u = ['^dxe~"''^ co^9,rx = — e «"• •^ 2a Laplace, Memoires de V Institute 1810, p. 290. DEFINITE INTEGBALS. 481 From this may be deduced the following integrals. /7\ ros 7 o 7!"- ('^ *"\ (o) /n a J? COS a~cv cos 2/0? = — cos -:: • ^ •' 2a V4 aV (c) fn'^t^A' sin a^ar cos 27*0? = — sin ; . Fourier, To-aite de la Chaleur, p. 5S3. /•CO cos ax (11) To find the value of w= / dx ;. Differentiating twice with respect to a, we have d~u r^ _, x^ cos ax .rs , = - / dx = - L "'^'^ ^^^ ^"^ + ^^' d d~ Jo 1 + X' By formula (g) of Ex. (8) f^^'dx cos ax == 0; therefore d^u -— _ M = 0. da^ The intejjral of this is w = Ce^ + Cie-". To determine the constants, we observe that u cannot increase continually with a, and therefore the term involving e" must vanish, or C = 0. This being the case we have, when a = 0, •w dx Jo 1 + a;'' 2 u = /-ss (^,27 cos «.a? TT „ Therefore (a) / -— = _ e" . ^ ^ Jo 1 + ct?- 2 On differentiating this with respect to a, there results X sin ax tt + Integrating with respect to a and determining the constant so as to make the integral .vanish with a, we find •» dx sin ax tt 1 + ci?- Laplace, Memoires de V Acadtmie, 1782. .31 r^i cT sin a<2? TT (6) / d.2; = - ^ ^ Jo I + x' 2 r'^dx sin a.x- tt ^ ,,^ v'o 0/' 1 + ii? 2 482 DEFINITE INTEGRALS. It is to be observed that the fornnula (a) is discontinuous, as the integral is equal to -I^Tre"" when a is positive, and to \'n-e when a is negative. Libri* has accordingly expressed the value of the integral in the following manner : /-«> , cos aco TT / e" e"'' \ / dx , = f- . By a similar method we find r^ 7 COSa.2? TT _-l / « ' ^\ (a) / ax = — , e 2^ COS — + sin — , . ^ ^ Jo 1 + c^■* 2t V si 22/ Laplace, Memoires de V Institute 1810, p. 295. dx '— see Jour, de VEcole Polyt. 1 + c27^" ^ J/« C3 f*0^ ft K* dx J75- (1 + xy see Jour, de Mathematiques, Vol. v. p. llO (Catalan). ^ X m n 1 1 -1 P r'^ dxcosrx (12) To find the value of / — -, a 1. Integrating (6) by parts, we find ,_. /-TT da? ^ sin a? tt, , . tt . / A {a) / = - log (1 + a) or = - log 1 + - , ^ ^ ^0 1 -2acosci7 + a- a ^^ ^ a ^ \ a) according as a < or > 1. Integrating (c) by parts, we find TT dw sint-r sin roo = — a'" ~ ^ or Jo 1 — 2 a cos x + a^ 2 according as a< or >1. (13) To find the value of / . , a dx 1 (a) f Jo 1 + x^ 1 - 2a coi cos r 1, the only difference is that the preceding result is to be multiplied by a~'". Poisson, Jownal de I' Ecole Poly technique, Cah. xvii. p. 6l4. (18) To find the value of / ; -. Jo 1 -\- (cos cV) On expanding the denominator, we have a series consisting of the even powers of coSc^■. Take one of these as (cos as)"''' : then -, . . ^. w (cosoeY^'^^ 1 ., .„ ^, , ax x since (cosc'^r'" = ^^ -I (cosxY^'^^ dx, ■' ^ ^ 2r + 1 2r + 1 '' ^ by integration by parts. In taking the limits between and tt fo'^dx (cos 0?)^''+^ vanishes as 2r + 1 is odd; therefore ff^doD w sin CO (cos ccY''' = 2r + 1 A^ sin <» , 111 ^ TT^ + &c.) = - 5 7 4 Poisson, lb. p. 623. 05 dciff sin ax , r . A^ sin <» ^ 1 „ ^ .. and L-^dx = 7r(l--+ + &c.) = — . -'" 1 + (cos xy ^ 3 " ~ ^ ■ (19) To find the value of f — ~ Jo 1 + x^ + x'"^ sin bx There are here three cases to be considered — according as a is less than 6, equal to & or some multiple of b, and greater than b, not being a multiple of it. Let a v/o e"^^' - e-™ [r^ + (tt - a)^ r^ + (3 tt - a)" J + |.J 1. -+&c.>. Now if we decompose e'" + 2 cos a + e"** into its quadratic factors^ we have e'' + 2cosa + 6~'' = 4 fsin -j |l + ( )[x Taking the logarithmic differential of this with respect to r we have = 2 e'' + 2 cos a + e — r therefore Sin r^)(e-^ + 6— ■^) " ^ ~ 4^^- (22) Poisson* has demonstrated the following formulae : If tf, = cos X -f (-)- sin ,r, v — cos *• - (-)^ sin a', so that w" 4- u" = 2 cos w.r, ?<■," - v" = 2 (-)i' sin WiV ; * Journal de VEcole Polytechnique, Cab. xix. p. 482. DEFINITE INTEGRALS. 493 r^ dx (l — p cos v) , „ , . _.- , . , then / ^^^ ~ j F (a + u) + F (a + tt) ^0 1 — 2/j cos ,%• + p" = ^{F(a+p) + Fia)] (1), dx sin X r -{F{a + v)-F{a + u)} I —2p cos X + jf '^ -,{F(a + p)-F(a)] (2). These expressions may be easily proved by developing the functions and the denominator in terms of cosines and of sines of multiples of x, integrating each term separately, and observing^ that Jq'^ dx cos mw cos nx = 0, f^ dx sin mx sin nx = 0, j^^ dx (cos 71 x)' = - , Jq^ dx (sin nx^ In applying these formula it is to be observed, (l) that p must be less than 1 ; (2) that for the particular value assigned to a none of the differential coefficients of F{a) become in- finite; (3) that the sum and difference of F {a + v) and F{a-\-u) be expanded in converging series; (4) that the func- tion under the sign of integration should not for any value of X between and tt become infinite, while the corresponding series remains finite, or vice versa. From the equation (1) may be readily derived the fol- lowing : r^ dxF(a + v) + F(a + 7() 2 tt „, / ^ —, - = :,F{a + p) (3). Jg 1 - 2p cosx + p' I - p^ ^ -^ In equation (3) put F (a) = e^", c being a constant. Then F{a + v) = e''" e"'''-'' {cos (c sin , V 2) { 2 ^ ^ 2 \ ^x . . X X since 1 + cos x = 2 cos^ - , and sin cX- = 2 sin - cos - . 2 2 2 Hence, putting x for ^x and therefore ^tt for tt at the limit, A'^ dx (cosxy cosax ir /l+p\" Jq 1 -2pcos2x + p^~ 2(1 - p^) \ 2 J In this make p = 0; then TT 1 (h) /o'' '^ d a? (cos xycosax= - -. Developing both sides of {g) and equating the coefficients of like powers of p, we find TT a(a-l),..a-r+l 1 (k) fJ'^ dx(cosx) cosaxcos2rx = t,- ^ ^ •'° ^ ^ 4 1.2.3 ...r 2 DEFINITE INTEGRALS. 495 Differentiating (g) with respect to a and then making a = 0, we have A'^ da; log (cos ct) tt , f^ + P (I) / ^-^ i — = log ^ Jq 1 -2pcos2a; + p' 2(1- jr) \ Expanding this last, and equating the coefficients of like powers of p, TT 1 (m) j^'^ dw log (cos .2?) cos 2rw = (-)' "^ • (23) Swanberg* has proVed the following theorems more general than those of Poisson. If 2Ji'=/(a + aw\ & + /3M^...)+/(a + a?5S 6 + /3«'' ...)j u and u having the same signification as before ; then dx M / 1 O 2 = , i/(« + «^ ' ^ + ^P''"-), '^o 1 - 2^ cos *' + _p'^ 1 - p^ Jo l-2«cosa? + «'^ 2r"^^ ^ A-r >' 2r- ^ ^ Let f(a,b...) = afb"'..., a = a = b = (i ... = 1. Then, changing x into 2x, and therefore the limit tt into ^TT, we obtain from the first of these expressions •2'^ dx (cosXci?)^ (cos fjiw)"' ... cos (IX + nijUL + ...) x r Jo 1 — 2p cos 2ci? + jj" ^ + pW /I + P"^'" 21-p'\ 2 J \ 2 (24) The same writer (p. 233) has proved the following theorems. If P=f{a+au\ b + j3ici^ ...)+f(a + av^, b + jSvf"...), {-y^Q=f{a + au^, b + (iuf ...)-f{a + av\ b + l3vi^...), where u and v have the same signification as before, * Nova Acta Reg. Soc. Upsaliensis, Vol. x. p. 271. 496 DEFINITE INTEGRALS. (2) r^^, = 7rf{a+ae-^\ 6+/3e-''''...)-T/(«,6...)- These expressions are easily proved by expansion, with the assistance of the formula in Ex. (11) : they are evidently sub- ject to the same cases of exception as the formulae of Poisson. Let f(a,b...) = aKb'" and a = a = 6 = /3=l; then changing X into 2\, ju into 2^, &c., we have •c= dw(cosXa!y (cos/x.r)'"... cos (IX + rtifx + ...)•%' ' 2^ V 2 / V 2 j dx 00 {co?>\ ccy {cos, iulw)"^ ... sin(ZX. + mju + ...)cV (') X TT ,1 ^-6-2^^' fl +6-2/^^'" TT 1 5: e If in (a) m, Src. be made equal to zero, the expression becomes r^ dx (cos XwY cos IXCC -TT I'l + e~'^^^^\ ^ ^^^ Jo ff + X~ " Jli \ 2" I ' Let f{a, h) = a} . e"^, considering two terms only, and a = a = /3 = l, 6 = 0. Then as before changing X into 2 A, we find by formula (l), •~ dx (cosXxy. e'wcos^^pQg (/\^ ^ ^ sin jux) w I h'~ + .37- TT /I + e-2XAW -M/t = — — - I e"*^ 2h \ 2 J In this expression put X = 0, then r » c^i?? e"* '^"^ '^*' cos (w sin u a?) tt "i"'' (e) / \ —^ = — e*"^ . ^ ^ Jo h^ + x^ 2/t DEFINITE INTEGRALS. 497 The student may exercise himself in deducing other in- tegrals from the general formulas by assuming other forms for the functions, and other values for the constants a, b... a, /3... . (25) Jacobi* has proved the following remarkable trans- formation of a definite integral : J^^ doD p''\cosx){%mxf''=\.S.5.1 ... i2r-l) f^ doo f{co^x)cQsr x\ / d \^' where /*'^ (^) = (t~) /l^)? ^^^ ^11 ^^^ differential coeffi- cients up to the (r — l)*^ inclusive remain continuous from « = 1 to z=—'l, or from x = to <2? = tt. To demonstrate this formula we must premise the following. If ^ = cos iV, 2r-\ d'"~^(l - ^'") ^ , , , , ^ sinr (x + atsmusmv, i/ + atsmucosv, z+afcosu) rf Ttt ["^■^^udv sinw . ^ . v(/(.t'+a< sinw sinw, ?/ + af sinw cosw, z + at cosu). dtJo Jo This transformation is given by Poisson, Memoires de rinstitut, 1818. (c) The equation for determining the vibratory motion of a thin elastic lamina is d^% ,^d^% — + 62 — - = 0, df dw' the integral of which is / d^ \ / d^ \ Now /+^ dtj e-'"^ cos y~ = tv'^ cos la^ + -"l ; and therefore f^^ dy e''''^ cos 1^ - y'^\ = tt^ cos a-. d^ Hence putting ht-~ for a^ \ DEFINITE INTEGRALS. 501 Also sin bt = b at cos bt . V dx-J dx^-' \ dxV Therefore as F (cv) is an arbitrary function, and as we d~ may write F M for 6 F (x), we have dx'^ •n-iis = fdy cos 1^ - yA f[x-2y {bt)i\ + fdtfdy cos {^ - y'^ F {x - 2y (bty^]. Poisson, lb. CHAPTER XII. COMPARISON OF TRANSCENDENTS. The integration of differential expressions frequently leads to forms which are not expressible by any finite combination of algebraic, circular, and logarithmic functions. Such integrals are called transcendents, and the study of their properties be- comes of importance as affording the means of classifying and arranging them so as to reduce them to the smallest number of independent functions. The class of transcendents which has been most studied consists of those called elliptic, from their being in certain cases capable of representation by elliptic arcs. They thus appear to be functions little more complicated than those which are represented by circular arcs, and to be naturally pointed out as the next subject of investigation. The pro- perties of these functions which have been discovered, relating chiefly to sums and differences of connected transcendents are very numerous ; but in the following pages I shall confine myself to elementary illustrations of some of the principal theorems, making use chiefly of those examples which admit of a geometrical interpretation. All elliptic transcendents are included in the formula Pdw I: I {a + hoB + car + dw'' + ex'^y^ by proper substitutions A + B (sin (pY d(f) which can always by proper substitutions be reduced to others contained in C + n (sin (py { 1 - e^ (sin (pyjh' The fundamental relation connecting transcendents of this class was discovered by Euler*, and may be thus expressed. * JVovi Comm. Petrop. Tom. vi. and vii. ^ COMPARISON OF TRANSCENDENTS. 503 d

) tan^ = ^- ^^^-^— --^- ^/-^, (c') 1 =f tan tan >// A {(p) A (a/^) where A (0) = J 1 - e^ (sin 0)^(i The functions F are called elliptic functions of the first kind ; e is called the modulus, and (p the amplitude of the function. Hence it appears that we can determine algebra- ically the amplitude of a function which is equal to the sum or difference of two given functions, and therefore also of one which is a multiple or a part of a given function. Thus if 03 be the amplitude of a function which is double of that whose amplitude is (p, we have to put \^ = (p and /x = (p^ in the preceding expressions, when we find as an equation for determining 02> tan 102 = t^^n (p ^1 — e^ (sin ,f(!/ — sin sin \|r { 1 — ^ (sin /i)^}^ = cos ji. If ^t = -| TT the corresponding arc is the quadrant of the lemniscate, and if \^ = (^, the value of (p given by the equa- tion 2^ (sm d)f = ^ ^^ 1+2^ will give the amplitude corresponding to the middle point of the quadrant. Functions of the form {d(p {l -e^{%m(py\ =E((p) are called elliptic functions of the second kind, and are con- nected by the following equation. The same relation between the amplitudes which satisfies the equation F(cp) + F(^|.)-F(^) = will also satisfy the equation E((p) + E(^jy) - E(,i) = e^sin0 sini/, sin ^ (b). (2) This formula leads at once to the demonstration of several properties of elliptic arcs : for if s be the arc cor- COMPARISON OF TRANSCENDENTS. 505 responding to tlie abscissa w in an ellipse whose major-axis is 1 and eccentricity is e, (1 - e^a^^ (1 -oo-y If we assume w = cos dy this becomes s = fd(p { 1 - e^ (sin 0)^} i = jE: (^). Now let /i=-l7r, then E {^tt) represents the quadrantal arc of the ellipse, and equation (6') gives as the relation between (p and y^ (l - e^)2 tan

// = J/F = iVZ, or the portion of the tangent intercepted between the point of contact and the perpendicular on it from the centre. From equation (6) then it appears that BM - AN = MY = NZ ; or the difference between two arcs of an ellipse is equal to an assignable straight line. This remarkable Theorem was discovered by Fagnani*. (3) If yj/ = (p, in which case tan^ d) = , , the two points M and N coincide in A", and we have BK - AK = 1 -(1 -e-)i, or the difference of the two arcs is equal to the difference of the two axes of the ellipse. * Produeioni MatJiemaliche, Tcm. ii. 506 COMPARISON OP TRANSCENDENTS. (4) Again, let us suppose xp- = 0, while ^i retains a general value ; then we have 2E{^)- E (fi) = e^ (sin cpf sin ^ ; while equation (of) gives (cos (py^ - (sin 0)" {l - e' (sin/i)^} a = cos /u, the condition for doubling or bisecting elliptic arcs. For doubling, we have 2 sin (h cos (h A (0) , , , , sin JUL = or • .M ' °^' tan -| ^ = A (0) tan ; 1 — e'^(sm0)* -* r^ r and for bisecting / • ^\2 1 - cos M (sin ©) = T— . ^ ^^ 1 + A (m) From this it appears that we can always determine an arc of an ellipse which shall differ from the double or the half of another by an algebraical quantity. Under certain cir- cumstances, however, when the origin of the arc is arbi- trary, this difference may be made to disappear, as in the following problem. (5) Find an arc MN which shall be exactly half of the quadrant BA of the ellipse. Let (p be the amplitude of the point M, >// that of N, that of K, determined as before by the condition tan^0= -T, (1 - e'y so that E(9) = iE (^tt) + ^ [1 - {i - e'y^. Now if we have it follows that E((p) + E (0) - ^ (v|,) = e" sin sin x/r sin 0. Therefore ^(v|/) - jE:(0) = 1 AX^tt) + 1 5 1 - (1 -e-)^} - e2sin0 sinx/, sina COMPARISON OP TRANSCENDENTS. 507 Hence, in order that MN or -E(^//) — E{(p) may be equal to i£ ('s'^)' ^^^ must have 1 — (l — e")^ = 2 e^ sin c/) sin \p- sin 0, which gives 2 sin sin \|/ = sin 0. On the other hand, we have from (a) 2 cos cos -v^ = cos 6 ; from which equations and \\r may be found, and so the arc MN determined. Abe], in the third volume of Crelle's Journal, has given a remarkable theorem for finding the sums of transcendental integrals, the amplitudes of which are connected by certain conditions. The following is the enunciation of the theorem. Let (x) be an integral function of x which is decom- posable into two integral factors, so that Let fix) be another integral function of x^ and let where a is any constant quantity : also let 9i (x) = a.Q ~ aiX + a.^x" + &c. + a„ti7", 02 {v) = Cq + Cjci? + Ca.r^ + &c. + 6',„a;'", where «o ? ®i ? ^2 » &c., Cg , Cj , Co , &c. are any quantities whatsoever of which one at least is variable. Then if {e,{x)Y(p,i,x)-{e,{x)Ycl>,{x)=A{x-x,){x-x,)...{x-xX where A is independent of x, we shall have, /(«) i_ r^i(a)l0i(«)j^+ 02 (a) 50, (a) 5 i" /(«; r t^i(a)10i(«JP+(^.(«) )02(a)^n ere C is a constant, and development of the function + r + C, where C is a constant, and r is the coefficient of - in the X 608 COMPARISON OF TRANSCENDENTS. according to descending powers of x. The quantities ei, eo ••• e^;. ^^e equal to + 1 or to - 1, and depend on the values of cTj, cVg-.-o?^, being determined by the equation (6) Let the integral which is to be considered be h \2 (1 + <2?")^ ' In this case assume 1 + a?" = ^ {x), 1 + A'" = 01 (c^), 1 = (p., {x), f {w) =oa -a, dx {og) = 1, 62 {po) = 1 + Ci<» : then {e, Or) p . 91 {x) - {e, (x)Y^2 (^) = 1 + ^" - (1 + c,xy = c17 (-2?""^ — Ci'X — 2Ci). The roots of this being 0, c^i, .^2 ... .:p„_i, Ave have 71 -2 relations between them from the conditions implied in the equation t'p""' — Ci^lV — 2ci = : these are 1 \ (2? I <2?2 ... tC^j _ J 2 {cc,) = 0, 2 {x^x.^ = ... E (-) = ± the upper or lower sign of the last equation being taken accordine- as n is odd or even. 's> Since fix) = x — a, f(a) = 0, and as f{x) is of lower dimensions than {(p{x)\^ we have r = ; consequently /•■•'"i dx r^'^ dx /■■'■'"-i dx e, / r + eo / r+...+e„_i r = const. 'Jo (1+^")^ ~Jo il+a^')r ^ " Vo (1+^")^ Lubbock, Phil. Mag. N'eio Series.^ Vol. vi. p. 121. Fao;nani has availed himself of the relation which sub- sists between the integrals fyd,v and fxdy, COMPARISON OF TRANSCENDENTS. 509 to compare certain transcendents of considerable interest. Since Jxdy + jydw = xy + const., if a symmetrical equation subsist between x and y, so that X is the same function of y that y is of x^ or that when <^' =

"^ [ x' - 1 j '^ ^^ [ f-l j = ^'^'2/ + const. Mr Fox Talbot * has extended to any number of variables the principle made use of by Fagnani in the case of two, and he has arrived at the following Theorem. If there be n variables x, y, :s, &c. connected by (n — 1) symmetrical equations, so that they are all similar functions of each other, then if " Phil. Trans. 183C and 1837. 510 COMPARISON OP TRANSCENDENTS. wy% ... ocij% ... = 00^), = (pdy)^ &c. 00 ' y we shall have J^ - 1) (if - 1) (r % + &c. = pi = (p (v) suppose; therefore S . dx = dpi = d . = a.^, (3) then will V„ and V„ being values of V corresponding to the values r^ and r„ of r. 34 522 COMPARISON OF TRANSCENDENTS. From the given equation (l) we easily obtain therefore integrating with respect to a?, But from the condition (2) we find on taking the limit cr = <2?j, that dx doo Similarly we find from (s) that at the limit w^ = x^ the same relation holds; hence (n» - K) It' dxgV^ F„ = 0. supposed not to 1 As r^ and r„ are supposed not to be the same, it follows that Since we have r^-r^\ dx dw ) it appears that when m = n. fx"d^gVn^ = -; the real value is Ix'dxgV^ =-V„—-{k-~+h^ F„) when x = x^, as may be deduced by the usual method for evaluating in- determinate functions. It is to be observed that the equation (3) involves an equation to determine r, which equation may be written as F (r) = 0. Poisson* has shewn that this equation has an infinite number of real and unequal roots, for the demonstration of which proposition I must refer to the works cited below. * Bulletin de la Societt Philomatique, 1828. Theorie de la Chaleur, p. 178. COMPARISON OF TRANSCENDENTS. 523 The function V is of great importance in the theory of heat, and the investigation of its properties has formed the subject of several elaborate memoires by MM. Sturm and Liouville. See Journal de MatMmatiques, Tome i. pages 106, 253, 269, 373, and Tome ii. p. l6. (22) Let Y^, and Z„ be integral and rational functions of ^i, (1 - iuL^)K cosw and sin to determined by the equations — {l-f/)-JL+ , -^ + m(m + l)Y^= 0, a/u. cifx 1 ~ /A "(t) dfx dfjt, 1 - fx dw then will f^,'d^f,'^dcvY^Z^ = o, so long as m and n are different. Multiply both equations by (l — ^u^), and assume (l — fj.^) -r— = — , when they become dfx dt d^Y^ ^ ^ ^^(^ ^ i)(i _^3) j.^^ ... (1), df d to ££.,£|_.,.„(„H-i)(.-.')z. = o. ..(.). Multiply (l) by Z^dtdcD and (2) by Y^dtdoo, subtract (2) from (l) and integrate with respect to t and w. Then transposing, and observing that (l — /u^) dt — d/u, we have {m (m + 1) - n (n + 1)} fdfi fdeo F„ Z„ f^-Z ^\ do)^ " d(D^ I -ffdtda>i^Y^^"-Z^^yffdtda.i^Y^ Now if we effect the integration of the first term of the right hand side with respect to t, it becomes 524 COMPAEISON OF TRANSCENDENTS. In taking the limits from ;f= — ooto #= + co, or from fx.— — 1 to jui = H-l, the part under the sign of integration vanishes, in consequence of the factor 1 — /t^ ; hence on in- tegrating with respect to w from to 2 tt we find that the first term of the right hand side of the equation is equal to zero. In the same way, on effecting the integration with respect to a; of the second term of the right hand side of the equation, we find it to become /-{''"S-"^}' which vanishes on taking it between the limits w = and ft) = Stt, because Y^ and Z„ are supposed to be rational and integral functions of sin w and cos w. Hence on integrating with respect to t and taking it between the limits # = - co and # = +co, or;u.= -l and /j. = + 1, the second term of the right hand side also vanishes ; therefore {mim + 1) -n{n+ 1)] P^^diuifo^''d(o Y^ Z„ = 0. So long as m is different from n this involves the con- dition that /_Vrf/x/o'"tZft,F^Z„ = 0. The functions Y^ and Z„ are known by the name of Laplace"'s Functions, that mathematician having been the first who studied their properties and pointed out their utility in the calculation of attractions. For the investigation of other remarkable theorems relating to these functions the reader is referred to the Mecanique Celeste, Liv. iii., or to O'Erien's Mathematical Tracts. Mr Murphy has ap- plied to the treatment of these functions a new and very remarkable analysis, which will be found in the introduction to his Elementary Principles of the Theory of Electricity. THE END. CORRECTIONS. PAGE LINE 15 9 (last letter) jor X read x 27 for lines 5 and 6 read 1 . 2...m . S— j-^^ . o nl } « 1 . a ... a . 1 . 2 ... p . 1 . 2 ... y ... 31 5 for + read = 72 % 11, 12 for «q" read a^ no 9 add =0 228 8 dele comma after to 266 6 for {\ + x)i read (1+,^?*)^ 11 for Dift". read Int. 280 3 for 3 6 read b 290 13 for I3x + a read px + a' 310 15 for dependent read independent 313 10 for P„ read P„y 315 6 (from bottom) for 2z — mdz read (2«- 364 15 for Z read z 381 19 for XII read XI 386 6 . d^x , d^y for ^ read ^ 7 . d^y , d^x ^"' dt- ''""^ dfi ■ m)dz 411 2 (from bottom) for Divichlet read Dirichlet 463 3 for e-" read e— " J & C. Walker Snip' ACME BOQKRIND'MG CO.. m, NOV 21 1985 100 CA^/iBRIDGE STREET CHARLL,v:r:;':v -.-cq BOSTON COLLEGE 3 9031 01072577 8 DATE DUE T ggf i ^5 (ill . p AOO \ ^ ' )0c; CAYLORD PNINTCOINU.S.A.