H/nfl ftfi lO^ CORNELL UNIVERSITY LIBRARIES Mathematici Library White Hall DATE DUE ^ „ -iir.,Y'^ - HOV )0 *^' I^AN 1 2 1999 ' f EB 2 e 20oa ' p GAYLORD PRINTED IN U.S.A. Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059412902 Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39. 48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. Digital file copy- right by Cornell University Library 1991. ^^^%- / '^&M£:^' CORNELL UNIVERSITY LIBRARY GIFT OF J. E. Oliver through Lucien Augustus Wait MATHEMATICS 0£P,. ^^(HT, A TEEATISE '^^ DIFFERENTIAL EQUATIONS, AND Oy THE CALCULUS OF FINITE DIFFERENCES. By J. HYMERS, D.D. LATE FELLOW AND TUTOR OP ST JOHN'S COLLEGE, CAMBEIDaE. SECOND EDITION, ENLARGED. LONDON : LONGMAN, BROWN, GREEN, LONGMANS, AND ROBERTS. 1858. PEINTSS BT C. J. OLAT, M.A. AT THE UHIVEMITT PEESa. CONTENTS. DIFFERENTIAL EQUATIONS. SECTION I. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND DEGREE. ABT, PAGE 1 — 5. Natcre and Formation of Differential Equations ....: 1 6 — 10. Exact Differential Equations of the First Order 4 11, 12. Homi^neous Equations 8 13 — 15. linear Equations of the First Order 11 16 — 18. Biccati's Equation, and other Equations, in which the Variables are separable by particular substitutions 14 19, 20. Euler's Equation 19 21 — 27. Factors which render integrable a Differential Equation of the First Order 22 28. Geometrical Problems producing Differential Equations of the Krst Order 30 SECTION II. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE. 29. Case in which the Equation can be resolved into its Simple Factors 35 30—36. Cases in which this resolution is impossible. Qairant's Form... 37 SECTION III. SINGULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS OP THE FIRST ORDER. 37—^0. Nature of a Singular Solution ; mode of deducing it from the complete Integral; its geometrical signification 47 41 — 44. Test of a Griven Solution being a Singular Solution. Mode of obtaining the Singular Solutions directly from the Differential Equation 52 VI CONTENTS. SECTION IV. DIFFERENTIAL EQUATIONS OF THE SECOND ORDEK, AND OF HIGHER ORDERS. AET. PAGE 45 — 48. Cases in which only three of the four quantities x, y, -j- , -rj, are involved 5'.> 4'.) — 55. Linear Equations of the Second and Higher Orders 09 50 — 66. Integration of Linear Equations with Constant Coefficients by separation of Symbols 70 07. Method of Parameters 91 G8 — 70. Method by changing the Independent Variable 92 77 — 82. Simultaneous Equations. Jacobi's Equation ]07 83. Solutions expressed by Definite Integrals 115 84 — 8 9. Appro.\imate Solutions of Differential Equations 117 SECTION V. DIFFERENTIAL EQUATIONS INVOLVING TWO OR MORE INDEPENDENT VARIABLES. 90 — 92. Explanation of the Notation used 126 93 — 96. Integration of Differential Functions of two or more Variables... 128 97—102. Total Differential Equations 132 103 — 111. Partial Differential Equations of the First Order 138 112 — 117. Partial Differential Equations when linear solved by separation of symbols 147 118 — 120. Change of the independent variables 155 121. Partial Differential Equations of a higher degree than the first. . . 1 02 122 — 129. Partial Differential Equations of the Second and Higher Orders, not being Linear 104 130, 131. Integration by a Series. Simultaneous Equations 174 132, 133. Singular Solutions. Geometrical Problems 177 CONTENTS. Vll FINITE DIFFERENCES. SECTION I. DUtECT METHOD OF DIFFERENCES. ART. r.VGE 1 — 6. Definitions and Principles 1 7 — 16. Differences of Explicit Functions 4 17 — 25. Relations between the successive Values and the Differences of a Function 8 20 — 32. Generating Functions 14 33 — 39. Separation of the Symbols of Operation from those of Quantity 19 SECTION II. INVERSE METHOD OF DIFFERENCES. 40— C3. Integration of Explicit Functions 2C 64 — 71. Numbers of BemouiUi 41 SECTION III. EQUATIONS OF DIFFERENCES. 72, 73. Origin and Nature of Equations of Differences 50 74. Linear Equations of Differences of the First Order 5 ) 75. Indirect Integrals of Equations of Differences 53 76 — 85. Linear Equations of Differences of all Orders 55 86 — 89. Certain Equations reducible to Linear Equations. Simultaneous Equations 71 90 — 95. Equations of Partial Differences, and Mixed Differences 80 96 — 99. Problems on continued Fractions. Functional Equations 89 100. Geometrical Probleras involving Finite Differences 97 Vm CONTENTS. SECTION IV. SUMMATION OF SERIES. ART. PAGE 101, 102. By Integrating the General Term 100 103 — 108. Recurring Series 107 109 — 122. Application of the Integral Calculus to the Summation of Series 111 123 — 133. Convergency and Divergency of Series 124 134 — 142. Interpolation of Series 132 ERRATA. DIFFEBENIIAL EQUATIONS. 37, line . 16, forfa.% read/dxx'^. 38, 5, d a (1 +p")t (1 +/>»)♦■ 69, 2, Ex.5 Ex. 6. 80, 16, nn' lln". 96, 6, =*=i'.-iy 96, 11. ip. --I 115, 14. «!- a|". — 17, «2" Oj". 160, 9. m = -l m = l. FINITE DIFFEEENCEB, Page 49, line 8, /or Art. 72 read Art. 67. 98, 20, COS. COS 6. INTEGKATION OP DIFFERENTIAL EQUATIONS BETWEEN TWO OB MORE VAMABLES. SECTION I. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND DEGREE. 1. In that part of the Integral Calculiis which relates to the integration of explicit functions of one variable, we have to determine the relation between y and x from the equation in the present portion, we have to determine it from the equa- tion or to assign the relation between x, y, z (where a is a function of the independent variables x and y), or between a greater number of variables and their functions, from the equation fdz dz ^[ix^ ^'^'^'^)=^' or from other equations in which a greater number of variables and differential coefficients of higher orders are involved. 2. A differential equation is said to be of the w"" order, when the differential coefficient of the highest order which it involves is the «"■. A differential equation of any order is said, moreover, to be of the first, second, &c., degree, when the differential coefficient H. I. E. 1 ■which marks its order, is raised to the first, second, &c., power : or when it involves a product at most of m dimensions in differ- ential coefficients and their powers, it is said to be of the m"' degree. To integrate a differential equation of any order, is to pass to the primitive equation between the variables and the con- stants, from which the proposed may have been derived by the process of differentiation. ^^ ■ ' ' _. 3. We shall begin with the simplest case, viz. that of differential equations of the first order and degree, which will be of the form ax Jif and N being functions of x and y. Every differential equation of the first order and degree is either the direct derived equation of a primitive ; or it results from the combination of the derived equation with its primitive, so as to eliminate a constant which enters in each only to the first power ; the former sort are called exact, the latter inexact. 4. First, let u=f{x,y)=0 be an equation between x and y, by virtue of which y is a function of x; then, as is proved in the Differential Calculus, -^ is given by the equation du du dv 1 . — ^ = dx dy dx ' the partial differential coefficients -5- , -=-, being formed as if the variables x and y were independent of one another; or, since du du n. • ^ J ^ J are lunctions 01 x and y which we may represent by JVf and^, dx a differential equation of the first order and degree, of which /( "> I/)=0 is the primitive or integral. Now if / (x, y), besides other constants wliicli are affected with x and y> contain a term + G independent of x and y, this will not enter into M and N, having disappeared in differentiating; and if there be no such term, we may add it, and/ (x, y) + C=0 is still a relation between x and y which satisfies the equation under this form it is called the complete integral ; and the con- stant C, which does not appear in the differential equation, is called the arbitrary constant; if the integral did not contain such a term as + (7, it would not be sufficiently general, and would be only a particular case of the complete integral. We shall presently give the test which every equation of this sort must satisfy, and the mode of integrating it. It is evident that no equation of the first order which is not of the first degree can be exact. 5. Next, let C^ be another constant which enters to the first power in the equation fix,y) + C-^0, then C^ will be affected with x and y, and will consequently appear to the first power in ax and if a value of C, be obtained firom either of these equations and substituted in the other, the result will be an equation of the first order and degree, involving all the con- stants which enter into/(a;, y) + C— 0, except C^. Hence whilst the direct derived equation of f(x,y) + C = 0,yiz.M+N^ = 0, does not involve the term C which is independent of x and y, there will be as many other differential equations of the first order and degree that have f{x,y)->rC=(i for their primitive, as it has independent constants entering only in the first power ; if any constant enter in a dimension above the first, the diffe- rential equation obtained by eliminating it, will evidently not be of the first degree. There are two principal methods of integrating equations of this sort, which consist either in separating the variables, by substitution, or some algebraical process; or in restoring the factor which makes them exact. Exact Di£Ferentml Equations of the First Order. 6. Let ^ be a function of x determined by the equation u=f{x,y)=0; then the equation which gives the value of -^ is du du dy _ Ma-N^ — O- dx dy' dx ' dx ' the differential coefficients -7- , -3- being formed on the hypo- thesis that the variables x and y are independent of one another ; then, as proved in the Differential Calculus, dM^dN dy dx ' Conversely, an equation of the form M+ N-^ = being proposed in which M and N are functions of x and y, if the condition dM^dN dy dx (which is called the criterion of integrabUity) be satisfied, the equation results from the immediate differentiation of an equa- tion of the form ./(a, y) = 0; and to find its integral amounts to finding a function of two variables f {x, y) whose differential shall be Mdx + Ndy, and then to put f {x, y) equal to a con- stant; if the above condition be not satisfied, there exists no equation by the simple differentiation of which, the given equa- tion can be produced. 7. To integrate the exact differential equation ax Let the equation from which it is derived be ,, du ,_ du „ 1 dM dN thenT-=if, -j- = N, and -^ = -r-; ax ay ay ax .-. u=JdxM+Y, denoting hj Y a, ftmction of y which may have disappeared, since M is the differential coefficient of u relative to x, on the hypothesis that x and y are independent ; du d , ., ■,,, dY •»■. .-. ^= N- J- (JdxM), and Y=fdy {N- ^ {J + 2y + »:y' = 0, where2 = ^log/y/|. .-. tan"' (y aJ-J +Jdx'/^= C, or y — KJ- tan {C—jdic\/rp). d_ dx 19 71 '~~ 7th Thus if we take p = x'", r = x", then q = , and the solation of i + ^'" + izi^-^)y+^f=o, is y = x' tan-^G -y " \ WJ + W + 2J Euler's Equation. 19. To integrate the equation ^ Va + lx-{-ca?+ex^ +Jx* + \/a + by + cy" + ey' +fi/* = ; or, considering x and y as functions of a new variable t. Let the function of t which expresses x be determined by the equation -^ = "/X, and therefore that which expresses y by the equation -^=— VF; also let x+y=p, x—y = q, p and q being functions of t. Then since {-^\ = X ; dxcPx_dX ^x_.dX similaily ^=i-^- •'* tie' "* \dx dy) = i {2& + 2C {x+y) + 3e (a'+j^^) + ^/{x' + y")] = h-Yc(x + y) + ^-[[(x + yy+{x-yy\ + 2/(a;+3/) 20 and J .§=^- r=6 (a; -y) + c {a?-f) + e{x'-f) +/(x*-y*) or V2 dt\q dt) ~^ dt^ -^ dt' or '^X-'^Y={x-y)->/G+e{x + y)+f{x+yy, the integral required. The discovery of this integral, which is due to Euler, was of great importance, as being the first step towards the foundation of the Theory of Elliptic Functions. 20. To integrate the equation Vl-c'sin'^/r + Vl-c'sin^ ^=0, or considering /r^ + Vl-c'sin'<^^=0. Let ^ = Vl-c^8in^<^, and .-. ^= _ Vl -c''sin>; 21 Let p = ^ + ->fr, q = ^ — yfr, d'p , . -•- -^ = — csinpco3q, d^_ • c cos^ sin g. = — (cos 20 — cos 2^) = — c' sinp sin q ; ■■■(ir-S = S|S^ .•...g(|) = .og(Odn„; .•. -^=Cs\nq; similarly ^= Csin^; .-. Vl — c" sin* + Vl — c'sin'i^ = Csin (^ + i/r) is the integral of the proposed equation ; which is only Euler's equation under a different form. The equations Vl+y' + Vl+a;* ^ = 0, are immediately reducible to the above form, viz. v'l-isin'-«/r + Vl-isin'0^ = O; the former by making a- = tani0, y = tanj'^; the latter by making 'Jx = cos 0, Vy = cos ^. 22 On the Factors which render integrable a Differential Equation of the First Order. 21. The most natural way of obtaining the complete integral of a differential equation of the first order, is to prepare it so that its first member may become an exact differential coefficient ; for then we shall have only to integrate and add a constant. This preparation is always possible by means of a factor, when the equation is reduced to the form -^ + K= 0. For let an equation f{x, y, C)=0 be resolved with respect to C, so that .-. by differentiation, = P+ Q-f, or ■A + -q = 0- Now the equation M+ N-^^Q may be put under the form ~+K—0, which agrees with the preceding, and may conse- quently be supposed to have arisen from the elimination of a con- stant between the primitive f{x, y, G) = 0, and its immediately derived equation. On this supposition, therefore, •^+K=0 is identical with ^ + -^ = : ax Q dx dx -^4(^^«S)' or ^ "!> (a;. 3') = Q (^ + -^) , identicaUy. The second member therefore is an exact differential co- efficient, which proves that there always exists a factor proper to render the expression -^ + K integrable. 22. But although the existence of the factor in every case is thus established, the investigation of it is usually attended with greater difficulties than the solution of the original equation. 23 For let P+ Q-^ = be an exact differential equation ; and let z, a function of x and y, be a common factor of P and Q so that P= Mz, Q = Nz, by the removal of which, the equation is reduced to the inexact state then because P+ Q-r- = is exact, ax dP^dQ d {Mz) ^ d (Nz) dy dx' dy dx ' dM T,^dz dN „dz dy dy dx dx y,dz Tuf^_ / and the equation becomes 1 ^ _ J_ fdM_ dN\ z dx~ N\dy dxj' which, being integrated, gives z; for the hypothesis requires that the second member should be independent of y. Similarly, if the factor be a fonction of y only, it will result from the integration of 1 dz 1 /dN dM\ z <^~ M\dx dy J ' dy M\dx dy I of which the second member is independent of x. 24 r^y. Hence, if in any equation M+ N-^- = we find 1 fi^- ^\ - X N\dy dx)~ a fiinction of x only, or l_ fdN_ dM\ ^ Y M \dx dy}~ a function of y only ; the factors which make it integrable are respectively /^^, Z""^. Ex. 1. ^ + {Py— ^)=0) the linear equation of the first order. This compared with M+N ~ = 0, gives M=Py-Q, N=l; ' ' dy dx~ ' N\dy dx) ' a function of x only ; therefore the factor is e^"^"". Ex.2. 2^+(i_a;2,)g = 0, M=y', N=l-xy, dN dM_ r, 1 (dN dM\ /■ , 3 , 1 ••• ryM\di—d^r-]^y-y=^''^?' therefore the factor is -3 . y Ex.3. {y-x)^-j-=0. ^ ' dx 2ca; The factor by which this is made integrable is a function of both the variables - d^', as may be shewn by introducing it, and applying the criterion of integrability to the equation. 25 Ex.4. ^ + 1^_1^«= = 0, dx qdx p dx where ^ and q are any functions of x, is made integrable by the factor r^ , and its complete integral is a{p+9.y) vi^ 24. In the case of homogeneous equations, a factor proper to render them integrable, is readily discovered by means of the property that if m be a homogeneous function of n dimensions of the independent quantities t and z, then du du nu = t-j-+Sj-. at az For suppose V, a homogeneous function of x and y of to dimensions, to be a factor which makes M + N ^ an exact dif- ' dx ferential coefficient, M and ^ being homogeneous functions of x and y oi r dimensions ; then if U denote the primitive function, it will be homogeneous and of m + r + 1 dimensions, and we shall have ax dx hence since VM and VN are the partial differential coefficients of Uvnth respect to x and y respectively, xVM+y FiV= {m + r+l) U; __± = __L_ 1^ ■■ Mx + Ny m + r+1'Udx' and as the second member is an exact differential coefficient, it follows that the first is so likewise, and consequently, that M+ N-p- is made exact by means of the multiplier dx 1 Mx 4- Ny ' II. I. E. 4 26 25. The property of homogeneous functions assumed above is easily proved. Let m be a homogeneous function of the inde- pendent quantities t and z oi n dimensions ; then if we change t into t{l + h) and z into s (1 + h), u will become M (1 + hy = M + nuh + &c. But by Taylor's theorem, u will also become du T du , „ at dz therefore, equating the coefficients of h, du du dt dz And, generally, if m be a homogeneous function of n di- mensions of any number of independent quantities <, z, w, &c., and we change them into t{l+h), z{l + h), w{l+k), &c., the new value of u will be equally expressed by m (1 + h)" or by e^ *" '* u ; and equating the coefficients of A"" in these two identical expressions, we get, separating as above the sym- bols of operation from those of quantity, d d d + . .{n-l)...{n-r + 1)u = (t^^+z^^+w dw Ex. xy + f+{xy-x^)-£ = Q. The factor is {xy + y^ X -^ {xy - a?) y ifx' . xy + f . xy-x^ dy " 2/x "^ Hfx dso~ ' is an exact differential coefficient, and gives the primitive by Art. 7. ^ + ilog(a^) + (7=0. 27 26. Whenever the variables can be separated in an equa- tion, a factor which makes it integrable can also be found. For suppose tliat M+N-^ = 0, by the introduction of two other variables u and z, is transformed into Ii+ 8^- = 0, so dz that (tsc cLz and suppose F to be a function of u and 2, such that if we divide B+S-j- by it, the variables are separated, i.e. ^con- o tains z only, and t?- contains u only ; is an exact differential coefficient; and consequently p., which, upon restoring the values of u and z, becomes a function of x and y, is a factor which makes if + ^-/ = integrable. Ex.1. a + Ja;V + a;= j = 0. Assuming y = - , we find a + Jar'y' + a;*-/ = a + iM' + a;T — m, and dividing by a; (a — m + Jm"), we get a + bxy ^ar ^ J ^^ a; (a — tt + Sk") a; a — m + iw" ' 1 1 ■ ■ a; (a - M + Jm") ax — a;'y + Jaiy is a factor which makes the proposed equation integrable. 28 The integrating factor here is -rrz rj ; as resiilts CC (1 ^ 3^^) —" Cb from the substitution by which (Art. 16) the variables are separated. Ex.3. /+aa; + (l-a;y)^ = 0. By assuming y = - — ^ — , it may be shewn that a factor which makes the proposed integrable is 1 y + ^ayx — a + cira? Ex. 4. M+N-^ =0, a homogeneous equation. In this case we know, that making y = xz, we have M=xy{z), N=x'4,{z), and ilf + i^^ = x'f{z) + x^<^ {z) [^ + x^); consequently, dividing by ^'^' {/(2) + ^

, 6, c) =0 be the equation to the curve A A' tJff referred to polar co-ordinates, and let it give for p -5- the value yfr {p, 6) independent of c ; then considering p and 6 as co- ordinates of the point A in the curve AB, t&n8AT'=:yjr{p,0), tarn SAT = p^, .: a = ^ i+^{p,e)pj-^ which is the differential equation to the trajectory ; or if it be orthogonal, Ex. 2, Let the curves be a system of circles touching a straight line in the same point, then taking that point as the origin and measuring 6 from the line, their equation is p = c sin d, 1 cos 6 dO dO sin ^ , , „■. ... l + «^pf =0, orcos^ + 8in^p^=0, cos d'^ dp dp 32 or -J- I — ^ I = ; .'. p — C COS 0, dp Vcos 6J '^ the equation to the orthogonal trajectory, ■which represents a system of circles passing through the given point and having the given line for their diameter. We may generalize this problem, by finding the orthogonal trajectory of all circles described through two given points. II. To determine a curve such, that the locus of the ex- tremity of its polar subtangent shall be a straight line. The polar subtangent is a line drawn from the origin per- pendicular to the radius vector to meet the tangent. Let p, 0, be the polar co-ordinates of any point P in the curve sought (fig. 2) ; then those of the extremity Tof its polar subtangent will be p'-j- and 0--^, which must satisfy the equation to a straight line, viz. p =■ c sec {0' — a) ; ,dd fn 'n-\ c P - = c sec {0—a — dp \ 27 sin {0 - a) c . ,. .d0 , c -5 = sin (0 — a) J- . and - = cos (0- a.) + C; P ^ ' dp p C+cos {0-a.) the equation to curves of the second order, having the pole for one of their foci. III. To find a curve in which SG varies as 8P, PG being a normal at P, and 8G a fixed line through S, (fig. 2). Taking SG for the axis of x, the equation to the normal at Pis (r-,)|+A'-.. = 0; therefore making F= 0, X= SG = .r + ,y -^ , 33 and 8P=^/^^Tf, .: a;+3/^| = e V^T/, or Va^ + y = ea;+ C, the equation to a curve of the second order. IV. To find a curve which is always cut by its radius vector at an angle proportional to the corresponding angle of revolution ; that is, ^SPTx a ASP, (fig. 2). Let p, 6, be the co-ordinates of any point in the curve, then the angle at which the radius vector cuts the curve, has for its d0 tangent p j- ; dd , a 1 dp cosnd /pY . a .'. p -7- = tan no, or - ^ = -: z > •'• r?v = ^m nff. '^ dp p da sm no \ UJ V. To find the locus of the centre of an ellipse rolling along a straight line. Let 0^ be the line along which the ellipse rolls, and which touches the ellipse at P; ON=x, NG = y, the co-ordinates of its centre; then CP is a normal to the locus of C (fig. 7), and therefore -yy^ dy_ dx. but a' + h'-CP'^'''^^ ON'' the difFerential equation to the required curve, in which the variables are separated. Generally, if p =f{p) te the relation between the radius vector and perpendicular on the tangent in any curve, then the locus of the pole, when the curve rolls along a straight line, will have for its equation H. I. E. 5 34 Thus if the cuiTe be a parabola, to find the locus of its focus we have p = \ap ; .:y = ^a^\l + (-|)]^ or y^a ^1 + (^^)\ the equation to the common catenary. VI. To find a curve such that the intersection of the tan- gent at any point and a line drawn from the pole inclined at a constant angle to the radius vector of that point, shall trace out a given curve. Let zASP=d, SP=p (fig. 2) be the polar co-ordinates of any point P in the required curve, YQ a tangent at P inter- sected by a perpendicular upon it from S in Y, and by a line through S inclined at a constant angle PSQ = a to SP in Q. Also let /.ASQ = ff, SQ=p be the co-ordinates of Q, which is supposed to lie in a given curve whose equation is -r =./ {6') ; then ff=e + a.; and, calling ^PSY= (f>, 1 cos (a + rf>) cos a. sin a. dp . , 1 dp -J = J— = s— -js, since cot d>= ^k- p pcos'''= +ccoa6, another curve of the second p COS cc order when (7=0. The result seems to fail when a = ^ tt ; but in that case, first changing the constant into C ■+ £21^ . -^^ cos a find -+ G'-dcos^ = ccos0. P SECTION II. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER BUT NOT OF THE FIRST DEGREE. 29. When a differential equation of the first order is of a higher degree than the first, we know that it is not obtained by the direct differentiation of its primitive, but results from eliminating a constant, (which enters into the primitive in a dimension above the first,) between the primitive and its de- rived equation ; the degree of the differential equation and the dimension of the constant eliminated above the lowest dimen- sion in which it appears, being always the same. The. general form of such equations free from radicals, is the coefl5cients being functions of x and y. If this can be resolved with respect to -y- into its n simple factors, it will assume the form then each of these factors put equal to zero, will be an equation of the first order and degree, whose integral may be found by the methods of the preceding Section ; and any one of these integrals, as well as the continued product of any number of them, will evidently satisfy the proposed equation. If, there- fore, we integrate the n equations, and complete them all with the same constant C, as the pro- posed equation is of the first order, we shall obtain the required primitive involving C in the n*^ power, by equating their con- tinued product to zero. 36 Ex.1. (±]\?^^-i = 0; \dxj y ax dy dy X r x* " dx •■• J + - = ±\/ 1 + - . or , ^ = + 1 ; dx y -y f Vy + ar' .-. +V/ + a;= = a;+0, and - -ifT^ = x -V G ; .: {V^+^-C-x){'/f + ^'' + C+x) = 0, ... f + a?-{C+xy = 0, or f = 2Cx + a'. and -logg + V'^) = '°K\/^ + ^-S=" + ''= .•. Var* + y^ + y = cxe", changing the constant, dy , r^-, — 5 dx\x/ and Vaf ■>r')^ — y — cxe^ ; . (Var* + / - cxe'' + y) (Vx" + / - case' - y) = 0, dy or / + (a; - cf = «' + 2na:, the equation to a circle. 37 This ia the solution of the Problem to find a curve in which the square of the normal is always proportional to the sum of the abscissa and subnormal. 30. When the resolution of the proposed equation into its simple factors is impossible, there are still various forms for which the complete primitive can be determined, or its determina- tion made to depend on elimination ; this is done by means of substitution, or differentiation, or other analytical artifices, of which we shall now give some instances. Obs. The arbitrary constant in what follows is often reserved under sign of integration. 31. "When the equation contains only one of the variables, X suppose, and can be solved with respect to that variable, so that x=f\-^\; let-T^ be denoted by^, then x=f(j>); and inte- grating the equation ^ =py>J parts, we get y = xp-Sdx-£ =pf{p) - !dpf{p) ; between which and the equation x=f{p), eliminating p, we shall obtain the required integral. Similarly, if we have ^=/(^), since dx _dx dy 1 df{p) dp dy' dp p dp ' we shall have to eliminate ^p between y=f{p), andx=felfe). J p dp Ex.1. x + x(-^] = l, or a; 1+y '=rf?-*^°"^ + ^' 2/ = Va; (1 - a;) - tan" a/^-^ + C. 38 Ex.2. y = a^\ +p', x+ C = alog {'^y' -a' +y). This is the solution of the problem in which it is required to find a curve such that the perpendicular on the tangent from the foot of the ordinate shall be constant. dy d Ex.3. y^Jl+f = ap. Here ^ = ^— _^-,^, ; .-. x+C = ^dp—j-- — ^.= ^ + alog — fl^ — - , p or X + (7=V«-,^ + «log^^,=^^J. This is the solution of the problem to find a curve such that the tangent terminated at the axis of x shall be of a constant length. .32. An equation not coming immediately under this case, may sometimes be reduced to it by putting j? = xz, or p =yz. Ex. [-^ ) + «a; -^ + a;' = 0. Let p = xz, az then X (z' + 1) + as = 0, or a; = - —— 3 ; ^ 1 + 2 dy_dy dx _ aV(2g°-l) ■"■ dz~ dx' dz~ (z' + lf ' and z must be eliminated between the integral of this, and the az equation x = — 7— — 3 . ^ 1 + z 33. When the equation contains both the variables x and y, provided it be homogeneous with respect to them, we may assume - = z; then the equation will take the form, (which is not solvable with respect to p by supposition,) 39 Suppose this capable of being solved with respect to z, and let it ... . dz giye s = 9 (p) ; now y = xz gives p = z + x-y- , 1 1 dz or - = J- , X p— z ax substitute the above value of z, and integrate this equation ; then p must be eliminated between the result which will be of the form loga; = i^(^), and^ = a;^(^). Ex. 3/— i=«v^+(iy-' .*. ^ = J) + w Vl +f ; " X _reVi+/ «Vvi+/ " Vi+jj" /' .-. loga;= {log (^ + Vr+y) +w log VT+^I + log C; I between which equations ^ must be eliminated. This is the solution of the problem, to find a curve such that the perpendicular upon the tangent from the origin shall vary as the abscissa to the point of contact. Let ?i = 1. - = 1 +y +p Vl +/, 'i-=p+\/l +/; X X ...»!l±^=2!'=,>+/)(^+vrT?r.§', or y^ = 2Cx — a?, the equation to a circle. 34. Another integrable form is ^ = ^^+/(^)' which 40 is called Clairaut's form, after the Mathematician who first considered it. Substituting p iox -^ , and differentiating, we get successively, dx -P-P + '' dx^dpJ^P> ' dx' which resolves itself into the two The first of these gives p = ^{x) suppose ; this value sub- stituted for p in the proposed equation, furnishes a relation between x and y which satisfies the proposed equation, but which involves no arbitrary constant, and cannot therefore be the complete primitive. The other equation must therefore lead to the complete primitive ; but this gives p= G, and by substituting this value of p in the proposed we find y=Cx+f{C). Hence Clairaut's form has the property, that the complete primitive is obtained by substituting the arbitrary constant G for p, in that form. If we integrate ^ = (7, we find y= Cx+G'; but the condition of the proposed equation being satisfied gives C =f{G), the same result as before. We shall afterwards return to the consideration of the other solution, which is called the singular solution, and is not de- rivable firom the complete integral. It is evident that Clairaut's form may be put into the rather more general shape y={x-\-c)p+f{p). Ex.1. 3, = ^ + ^il±£!) P 41 Differentiating, we get^ = jj + a; -^ + a f 1 2] ;^ , .-. g = gives^=C, and3,= (7x + ^^i+ill, the complete integral; and " /^ ■ . / « * + "~;;;5 = o givesp = + A/— - — , P V a;+ a which, substituted in py = a + p' {x + a), gives the singular solution -^V^=°^"' or / = 4a(a; + a). Ex.2, y = a; (p - c) + Va'' - c' +J9V, y = x{C-c)+ ^a'-c'+ CV, the complete integral ; 1/ (x — cV -5 j 4 2 — = 1, the singular solution. The two foregoing examples are the solutions of the problems to find a curve for which the locus of the intersection of the tangent and perpendicular upon it from the origin shall be a straight line, and a circle, respectively. For the co-ordinates of the said point of intersection are p{y-px) ^_y_-px 1+/ ' 1+/' and these substituted in the equations to a straight line and circle, viz. X + a= 0, F' + (X — c)' = a", lead to the preceding equations. Ex. 3. To find a curve such that, a, /S, 7 being the angles which the sides of a triangle on a given base and with its vertex a point in the curve, form with the tangent at that point, we may always have tan" /S = tan a tan 7. IT. T. K. 6 42 Let the given base SC=c, SN=x, NP=y, the co-ordi- nates of a point P in the curve, PT a tangent, PG a normal at that point, .4P parallel to SC; ^APT=^a, SPT=^,CPT=y (fig. 8) ; then tan^yS = tan a tan 7 gives 1 5-!- = tan a tan (7 — a) ; cos a fSG\'_ GN PN _ GN ("+y|y y ^y . x' + y' x — c dx' or 1 r- — 5 — = or '' + y^-' (^ + ^^) x—c X +y dp '^ dx /dpV ,^. dp .,^=^ + (,_,)|_^^|; ,. | = 0,or^=C; .•. \a? +y' =[x —c) C + -p! the complete integral, representing a conic Section, the two fixed points being its focus and center, and its eccentricity = C; and (a; — 2c)' + y' = 0, is the singular solution, representing a point, viz. the other focus. Ex. 4. y = xp + VP + aY, y= Cx + ^6"+ CV, the complete integral, ay -1- Va? = c^V, the singular solution. This is the solution of the problem to find a curve, such that the product of the perpendiculars dropped from two given 43 points upon the tangent may be invariable; for taking the line joining the two given points (whose distance suppose = 2c) for the axis of x, and their middle point for origin, and x, y the co-ordinates of any point in the curve, the equation to the tan- gent at that point will be .^^ Y-y^'£iX-x), orr=|x+(y-.. and the lengths of the perpendiculars dropped upon this line from the points (c, o), (— c, o) will be -pc-(y-xp) pc-{y- xp) _ and the product of these = — — ^ ^ — ~ = V, suppose ; .*. y = xp+ ^b' + a'p', putting a' = b^ + c'. 35. A still more general case is the equation y = xf(p) + ^{p), which by differentiation is reduced to a linear equation of the first order in a;; for we get ■• dp f{p) -p f{p)-p' which gives x = F{p); then p must be eliminated between this and the proposed equation. o dp ^dp Ex.1. y=xp'' + 2p; ■•- F=P+^^f£ + ^-^' dx 2x 2 or J- + z — T = ~ zs dp p-1 P -P 44 which is made integrable by the factor (^-1)"; .-. x{p- 1)'= - jdp^^^ = -2p+ log/ + a But p = — "i'v/+~; therefore substituting this in the preceding, we obtain the complete primitive between x and y. Ex.2, y = xmp -hnjl •\-p^ ; 2m- 1 Ex.3. y+p{a — oc)=nfdx'^l+j)^; ^ 2c"(K + ])^2(w-l)(a-a;)"-'^ This is the solution of the problem of finding the path of a point P which moves uniformly towards another point Q also moving uniformly in a straight line. For taking A (fig. 3) for the origin, and A£, which is per- pendicular to Bi/ the line in which Q moves, for the axis of x, we have, supposing P and Q to start together from A and B, BQ^nAP, or if AN=x, NP=y, AB=a, y + {a — x)p = n^dx VI +p'. 36. In the following examples the method of substitution succeeds. Ex.1. {l-f)xy=p[x'-f-^), which expresses that the normal bisects the angle between the focal distances; 2c being the distance of the foci, the origin at the middle point between them, and the line joining them the axis of X. Let p= — , .'. if = ^z — c' ; ^ y " 1+2' therefore, difierentiating, dz (!+«)'] dx 45 this resolves itself into a; = + , which gives y^-^ [x — cy=0, the singular solution ; and -=- = 0, OT z= C, which gives y^^ Cla?—- — ^j, the com- plete integral. Cx If we integrate p = — - , we get y^ = Cx' + C, where C must be determined by the condition of the proposed equation being satisfied ; and by this condition, in general whenever the method of solution raises the order of the equation, must the number of constants be reduced. By the same substitution may be solved the more general form axyp^ +P {a? — ay' — h) —xy = 0. E... 4_,=AV(iy-f 2+.. or dx X x\ \dx x) a? ' dti dz y dz Let y = xz, then -^=z-ir x-j- = - +x^- ; ^ ' dx dx x dx •'■ ^\-z^dx~x^a?-X^' If X= 1, we have sin"' a =sec~'a;+ C, or sin~* ^ = sec'' X+ C. x Ex. 3 and 4. y — oip / tc \ 46 Introducing polar co-ordinates, we get for the first, P" r, s de f{p) ■-f{fi), or -j- = - — |y+^="''^'' '^P-PVPMTW? The second gives ^/C = /(co3 0), or — ^ ^■' «'pV^„^~''' " P^~ /(cost/} The former expresses that the perpendicular on the tangent from the origin is a given function of the radius vector; the latter that the sine of the angle at which the radius vector cuts the curve is a given function of the cosine of the angle at which it is inclined to the axis of x. SECTION III. ON THE SINGULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 37. From the complete integral of a differential equation we can deduce as many particular integrals as we please, by giving to the arbitrary constant particular values. But some differential equations are satisfied by a relation between x and y not containing an arbitrary constant, and not deducible from the complete integral; such a relation is called, as has been said, a singular solution of the differential equation. The existence of such solutions depends upon the fact, that when a solution of a differential equation has been obtained in any manner, it will still be a solution after a quantity of any kind has been introduced in any way, provided the same derived equation result. This is merely an extension of the principle on which the arbitrary constant is added. 38. Before entering upon the general theory, it may be useful to consider the following particular instance. Let the equation y-.t + a + aftV (D dx \dxj be proposed, which, since it falls under Clairaut's form, has for its complete integral y=(7a; + a+a(7'; (2) C being the arbitrary constant. If we now regard C, not as a constant, but as a function of x, and differentiate, we get dC |.(7.(...a0f; and if we eliminate C between this and y = Cx-ira + aC^, we shall obtain a differential equation, but not the proposed one. 48 for that arises by eliminating C by means of the equation -^ = C. But if C be so determined as to make the coefBcient ax dC X of -J- vanish, that is, if C = — — , then the derived equation will be ^ = C, and the result of the elimination of C will be ax the proposed equation. Substituting for C its value, we get 2 2 2 x' X X ^ la ia ia a relation which manifestly satisfies the proposed equation ; for it gives ;f- = — ^: ^^^ these values of y and -r^ , being sub- stituted in the proposed equation, make it identical. But this solution contains no arbitrary constant, and being the equation to a parabola it cannot, either by making (7=0, or any other constant quantity, arise from the complete integral which is the equation to a straight line ; it is consequently a singular solu- tion, and arises from the complete integral by changing C into a function of x so determined as to make the term involving -J— disappear from the value of -^ . Thus we see how the singular solution arises from the com- plete integral; next, let us consider its geometrical signification. The proposed differential equation expresses that the curves to which it belongs have the property, that the tangent at any point is intersected by a perpendicular upon it from a given point, in a given straight line. Take the given point S (fig. 4) for the origin, and T8, AS respectively parallel and perpendicular to the given line AC, for the axes of x and y. Let TC be the tangent at a point whose co-ordinates are x and y, then its equation is Y-, = '£iX-x), 49 and the equation to the perpendicular upon it from 8 is ax and for their point of intersection but this point is always in ^ C for which Y=A8 = a; dy /dyV dx \dx, the same as the proposed equation. The complete integral y = Cx + a{l + C'), which represents a series of straight lines, evidently satisfies the problem for all values of C; for let Tb be any one of these lines, then y = —Y, is the equation to a line through S perpendicular to it; and combining the equations to get the co-ordinates of their point of intersection, we have y = -C'y + a{l+C), ory = a; the intersection consequently falls in A C. And as a straight line is its own tangent at every point, the equation y = Ca3 + a (1 + C), for all values of C, represents a line such that the intersection of the tangent at any point, and a perpendicular upon it from ;S', falls in the given line A C. Now the curve which is generated by the perpetual intersections of these lines will also satisfy the problem; for each of the lines will be a tangent to it, and therefore per- pendiculars from ,8' upon its tangents will intersect them in the various points of A C. To get the equation to this curve we must, according to the usual method, differentiate with respect to the parameter C, which gives = x + 2Ca, and eliminate C between this, and the equation H. I. E. '^ nO which gives X* y = — -. — V a, or 4a (a — 3/) = 3^, the equation to a parabola, vertex A, focus S, of which curve it is a well-known property that the perpendicular from the focus intersects the tangent at any point, in the line touching the parabola at its vertex. This result being obtained by exactly the same process as the singular solution was obtained, of course coincides with it ; hence it appears that the singular solution belongs to the curve which touches the family of curves resulting from the complete integral by making the arbitrary constant assume all possible values. The conclusions arrived at in this particular instance, we shall now shew to hold generally. 39. Having given the complete integral of a differential equation, to find its singular solution. Let fix,y,^\ = Q be a proposed difierential equation, and suppose it to result from the elimination of the arbitrary con- stant c, between the equation F {x, y, c) = 0, and its imme- diately derived equation M+ N^ = 0. ax Now change c into c any function of x and y, then our equa- tion becomes F {x, y, c') = ; and its immediately derived equa- tion is ,r + .V'j4-C'£=0 (1), dx dx c entering into M' and N' just as c did into M and N, and dc -J- being formed in the usual way for a function of two variables X and y, the latter of which is dependent on the former. Now C is the diiferential coefficient of F {x, y, c) with respect to c, regarding x and y as constant, and vsdU therefore usually in- volve X, y, and c ; and if put equal to zero, will give such a value for c as makes the last term of equation (1) disappear; 51 and then the elimination of c' must evidently produce the pro- posed equation /fa;, y, -^ I = 0. Let this value of c be sub- stituted in F{x, y, c') = ; then this equation is changed into ^ (^j y) = 0, and furnishes a relation between x and y which satisfies the equation f(x,y,-j-\=0, but contains no arbitraiy constant, and is not deducible from the complete integral by giving a particular value to the constant ; since it results from the complete integral by substituting for c a variable value deduced from the equation -5- F {x, y, c) = 0. Consequently, the relation ^ [x, y) = is the singular solution required. 40. To explain the geometrical signification of the singular solution of a differential equation. Let F{x, y, c) = (1) be the complete integral of a difterential equation between two variables ; if we differentiate it with regard to c, we have -^ F{x, y,c) =0 (2) ; and if between these equa- tions we eliminate c, we get

1, this can easily be effected ; for we must have -3— = 0, or K = a, constant ; ax dA a+l = m, and -3— = MK"" ; and so on for the other terms. Con- sequently it will be possible to make the two members identical, and 7/ = u will be a particular integral. In the same way the identity may be established i{ m = l. But if m < 1, there is no term on the first side corresponding to MK"'a'^ ; and since K cannot be equal to zero, it is impossible to satisfy the identity ; and therefore y = v, is a singular solution. Hence to discover whether a given solution, y = u, of a difiierential equation is a singular solution or not ; we must write m + A for y in the value of ---, and if the expansion in ascending powers of A in- dx • volve a power of A, whose index is < 1, the solution in question is a singular solution ; otherwise it is a particular integral. 42. To deduce the singular solutions from the differential equation, without knowing its complete primitive. Let y = M be a singular solution of the equation then by the preceding article, substituting u + hfoi y, we get f{x, u + h) =f{x, u) + Mhr + Nhr + &c. 54 ■where m, n, &c. are proper fractions ; .-. ^f{x, u + h)=-^f{x,u + h) = mMhr-" + 7iNhr-' + «S:c. ; consequently, when h = 0, -5-/(0;, w) = 00 . But -J- fix, u) is what -j-f{x, y) becomes when y = u; and therefore, conversely, every value u of y, which satisfies ■—■ =f{x,y), and makes 7-/(«, y) = 00 , is a singular solution of the equation 43. It is not essential to give the equation the explicit form -r=f{x,y). Yovh,i-T- = p, and let F= be the given rela- tion between x, y and p ; then we may regard ^ as a function of X and y determined by the equation F= ; hence forming the dV dV differential coefficients -5— , -5- as if y and p were independent of one another, we get dV dV^^Q 01- ^ = - — ^^ dy dp dy ' dy dy dp' Hence the condition -j-f{x, y) =ao is equivalent to -j- = 0, provided -5— remains finite ; and therefore the singular solutions of the equation V= F{x,y,p) = 0, are determined by eliminat- dV ing p between V=0 and -5— = 0, provided always that these solutions satisfy the proposed equation and do not make dV -J- = 0. And conversely if the solution of an equation be given, and we deduce from it the value of -r- =jp, then if this value dV ... . dV make -5- vanish without making at the same time -3- = 0, it is 55 a slngiilar solution ; otherwise it is a particular integral. It is evident that if we consider y as the independent variable, and put the equation under the form dx\ ^'=4'3.'S)=«' dyl the same reasonings would shew that singular solutions may dx dV be obtained by eliminating ^' = -5- , between F= and j-7 = 0, provided that these solutions do not at the same time make — = dx Ex. 1. To find the singular solution of dy , y y — x-r- -^x — *!-=a. ^ dx dy dx Here -=-= — a;+^ = 0; .•. p' = ^, and the proposed becomes (x + y — a)p=y + xp^, or {x + y — a)p=2y, or {x + y — ay = 4ixy, which may be reduced to the form Vx + Vp = Va. x — 'M-\ = c^. iocy = a*. dx/ The three former examples determine respectively the curves which have the properties that 0T+ OT' is constant, that the area of the triangle TOT' is constant, and that TT is constant, TT' being the tangent at any point meeting the axes Ox, Oy in T and T, (fig. 5) ; the last determines a curve such tliat the 56 product of the portions of two fixed parallel straight lines inter- cepted between the tangent at any point and the axis of x, shall be invariable. Here -y- = a; — tan"' p — , ^ , = : dp -^ 1+p' ' 1/ = J— i^ , .: x = cos ' Vy + Vj/ -/, wliich represents a cycloid whose base coincides with the axis of X, the origin being in the centre of the base. This is the sohition of the problem to find a curve always touched by the same diameter of a circle rolling along a straight line. dx Va;'+/-a' ^ -a: \__y 1 .•. the relation x' + r/^ — d' = makes -^ infinite, and satisfies the proposed equation ; it is consequently a singular solution of the proposed. But if we suppose the solution given, we may find whether it is comprised in the complete integi'al or not, by Art. 41. For we have y^'Jd' — x"; therefore substituting wd' — x' + h for y in the value of -^ , we get X — X V2A when developed according to powers of h ; and as the index of h is a proper traction, x^+y'—d' = cannot be comprised in the complete integral, and is therefore a singular solution. 57 7. (a;— a) (-^j —y-^ + a=0, to find whether y^ = ^aix—a), y=x, are particular integrals or singular solutions. We get dV Hence the solution y = x which gives p=l, does not make ^— vanish, and is therefore a particular integral. In fact the pro- posed equation being , .dy a dx its complete integral is (Art. 34) y={x-a) 0+-^; and this becomes y = x, when C = \. But the solution y = 4a (a; — a) gives yp = 2a; and these values of y and p reduce -j- to zero without making -5- vanish ; consequently «' = 4a (a; — a) is a singular solution. dy «• 2/'(|) + 2^2 + '^ + *^ = '- ^= 2y ipy+x), -^ = 2p{py + x). dV X The value of ^ which makes -^ vanish is ^ = --, and the dV result of the elimination of ^ is aa; + Jy - a;" = ; but as -^ vanishes for the same value of p, this cannot be a singular solution. In fact, it does not satisfy the proposed equation, and is not a solution of any sort. H. I. E. ^ 58 fdvy dv , dV JF „ the value of ^ which makes -=- vanish, isp=ay, and this value dV does not make -y- vanish ; but the result of the elimination dy of p, ay = hx, as it does not satisfy the given equation, is not a singular solution. 44. Every factor proper to make a proposed differential equation integrable, is made infinite by the singular solution. Let j^ +y*(a;, y) = be the proposed equation, F{x, y) = c its complete integral, and z the factor which makes it integrable, 30 that ^{|+/(-,^) \=iF{x,y); also, let y =u be the singular solution ; then this is not deduci- ble from the complete integral, and therefore if u be written for y in F(x, y), the result will not be constant; if therefore we substitute u for y, in the preceding equation, since the second member will have a finite value, and the factor -^ +f{x, y) of the first member will be zero, the value of the other factor z corresponding to this substitution must be infinite. This property will sometimes lead to the discovery of the factor which makes an equation integrable ; as in the example (a* - a;«) ^ + a^ = a V^TF^. a singular solution of which is x* + ^' — o* = ; if we try a factor of the form {a?-aY{y'+x'-a')'', we arrive at 7w = — 1, n = —\; and the factor which makes the proposed integrable is SECTION IV. DIFFEKENTIAL EQUATIONS OF THE SECOND OEDEE, AND OP HIGHEE OKDEES. 45. Evert differential eqiiation of the w* order admits of a primitive with n arbitrary constants. Let f(x, y, c,, c,, ... cj =0 be an equation between the variables x and y, containing n constants c^, c^, ...c^. Let the first n derived equations be 7- la;, y, ^ , ^^, ... ^„ , c,, ... c,; -0. Between these n equations and the original, the n constants may be eliminated, and the result will be Fix y iy ^ iy\=o (1) a difierential equation in which none of the constants enter. Conversely, a differential equation of the n* order being proposed, it must admit of a primitive containing n arbitrary constants, because that number of constants, and no more, can be eliminated in its formation. Hence every differential equa- tion of the n* order admits of a primitive containing n arbitrary constants. 46. Again, between the original equation and its first n — 1 derived equations, » — 1 of the constants may be eliminated, 60 and a differential equation of the (n— 1)"" order with one constant will result. Every such differential equation, having the same primitive with equation (1), is a first integral of that equation; hence a differential equation of the w* order has n first integrals, each a differential equation of the {n — l)"" order, and containing one constant. Also, between the original equation and the first r of its derived equations, r of the constants may he eliminated, and a differential equation of the r**" order containing n — r constants will result, which is an integral of equation (1). Now r constants can be eliminated in a number of ways equal to the number of combinations of n things taken r together, or n{n—\) ... (« — r + 1) 1.2.3 ...r ■ This then is the number of integrals which equation (1) has of the (w — r)* order, each a differential equation of the r^ order and containing n — r constants. 47. Of the general equation of the second order W ' die ' 2'' *; - "' we shall first of all consider the following particular cases, in which -T^ is involved with only one, or two, of the other quantities a;, y, ^ ; and which admit of integration, or rather of reduction to forms of the first order. I. ^\^^ x\=0. Let this by resolution give g =/(-), .•.|=/^/(x) + (7; and integrating again and adding another constant, we obtain the complete integral. The same process applies to -r^ =/(«). 61 Also, if we have F {g^ , -^ = 0, and put ^ = u, vre get -f f ^ , M j = ; and if this can be integrated, and gives u =f{x), it is reduced to the case just noticed, ,-. j/ = alog- + C7x+ C ■■. y = \ (2c' + x») -JT:^ + \ &x sin-' - + (7a; + C". II. ^( j4 , 'T'] — ^- -L®* *^is ^y resolution give 3-/(1). "I-/W- p«'i"5j-^^ rfa; _ 1 _ . , 1 p must be eliminated between these two equations. Ex. ag=^; .:f=P-, .-. log^ = ? + (7, da? dx' dx a' ^^ a , dy dy dx a . „, •. - + C=log^ -c m. i?'fS,3^) = 0. Let this give g=/(y); ^'^y-"- -- — 6"- dor' 62 and this, treated as above, gives a; = <^ (m) ; and if x = ^ (m) can be solved with respect to u, it is brought under Case I. .-. asin"'-^ = a;+C". V(7 Ex.2. «/g-l = 0; i(.-cT=iy-|. This is the solution of the problem, in which a curve concave towards the axis of x is sought whose radius of curvature shall vary as the cube of its normal : for this requires that (!+/)»- g = ny(l+/)'. Ex. 3. Voy T^ = l. ^ = I {^/^+ cy - 2 0-17^70+ c: This becomes of the first order in p and x, by putting p for -p and -^ for ^^ ; let its integral be ^ {x, p, C) = 0. 63 If this by resolution give p or -^ =/(«)> then y=jdxf{x) ; among other cases, this will happen when the proposed is of the form -TT^ + P-r-— Q; for the latter can be solved as a linear oar ax equation of the first order. If it gives X =f{p), then y = fdxp=xp-fdxx-^ = xp-fdpf{p) ; and p must be eliminated between these equations. Ex 1 ^+1^ = 0- .: x^ + '^ = —(x^]=0- da? X dx ' dx' dx dx\ dxj ' Ex.2. (l-.^g+l+(|) = 0; " ^ l+cx c\l+cx /' .-. 2^=^log{l+ca;)-?+C'. Ex.3. x{a + hx)-^ + {c-iex)-£^=:0; ^ 1 dp c + ex _ ex" p dx X {a + hx) ax~^ + i a + 5x ' :. p = c'ic"^ (a + hxf''^ andy = O+c'/tfa; x"' {a + Jaj)'" ' Ex.4 dx' 64 This is the solution of the inverse problem of the radius of curvature, in which it is required to find a curve whose radius of currature shall be a given function of the abscissa. Es.5. g+(^_i)J..- y = .' + ft- + C'. ^- ^©. S. *)-»■ Putting -j-^p, we get d^y _dp dy dp da? dy' dx dy' and the substitution oi p-4- for -y4 , and of ^ for — , will make the proposed of the first order in p and y ; let its integral be <}> ip, y, C) = 0.. If this by resolution give P = -t- =f{y)i tben If it gives y=f{p), then ^=sdy'-=y^jdyy,f=y^sdp^-^, p p p "y p p aftd^ must be eliminated between these equations. 65 dp 'dy yp^+^p'+^ = ^> ilog (1 +/) + n logjf = ilog C; .-. ^ = VC^^"-1, &ndix=jdy This is the solution of the problem in which it is required to find a curve whose radius of curvature shall vary as its normal ; for this condition gives - dx — or + according as the curve is convex or concave to the axis of X. If n = 1 the curve is a circle, if n = 2 a cycloid, if n = — 1 a common catenary. Ex. - ■HI)'-^3--h(l)T dx _ y d^~V{Gy+lf-f' As there is no substitution by which this can be generally reduced to an equation of the first order between two variables, the artifice to be employed in any case will depend upon the H. I. E. 9 66 nature of the example proposed. Among other substitutions for -74 1 tlie two following may be noticed, 'da?' a,-^=i.U±(y\\ and x^ = ^^^-2^^ ' "■ dx\ dx \xj] ' dx^ da? dx Ex.1. x»g==23,; ^!i5yl_2^=?^ or <^'(a?y) ^2 d{xy) dx^ dx X ' da? x dx ' d , . ^ , , Co? C .:^{xy)=Cx; and 3,= — + -. Ex. 2. {a? + yj '^ + a'y = 0, let | = s, then d / 2 dz\ a" » _ dir&^)'^x'{l+zY~^' ^dz d / , dz\ c^z dz _ ■'• ^ di'd^Vdi) (l+sT ^ ~ ' •••(-|)'-r^=-. ,dz or a; T- dx "J'^*^-' 1 ^ , Vl + s" or = /as Ex.3. -^ = ax + ly; d? ^^d^^^^^y)=^^^+^)^ which becomes ^ = hz, putting oo; + Z>y = z, and so falls under Case III. 67 48. Of the general equation of the second order involving X, y, ^ , -j^ , the following particular cases may be noticed, in which P denotes a function of x. ^cPy \dP dy da? 2 dx' dx this when multiplied by 2 -^ may be written d_ dx Thusif(l-a;=)g-a!g + 2^j^ = 0; 1_ dy 1 1 . _. ffw which may be put under the form y= G^ cos (y sin"' a;) + CI, sin (j sin"' a;). Similarly, {ax + Ja;") T;^+(-a + Ja;j-^+cy = 0, leads to , „ -y- = , . i^C-ci^ax 'Jax-rha? .-. y=Cxjdx.^^e-J^''. 68 48*- When the equation is homogeneous, reckoning the dimensions of j) and -p to be and — 1 respectively ; it may be reduced to an equation of the first order by putting y = xz, andg = |. For each term, if r denote its dimensions, will consist of some function of ^ multiplied by a factor of the form m and n being any numbers from to oo ; therefore upon making the substitutions stated above, every term will be divisible by x", and the equation will assume the form and if this can be solved relative to q, we shall have q = {.,p); but2 = a,|=l^|=(^-.)|; dx let this give dz p = ^{z), then i/r (z) = s + a; -g- , which will give the required integral z = ~ =f{x). putting I =p, y=xz, I = |, we find {l+p'^i = q^l + z' 69 nn+p*)^ , .dp which is the same as Ex. 5, Art. 18. Linear Equations of the Second and Higher Ordets. 49. The linear equation of the n* order is all the coefficients being functions of x, and each term of the first member involving either y, or one of its differential co- efficients, in the first power. The first step towards the integration of this equation is the establishment of the following theorem. If there be n particular values m,, Mj, m, ...m„, fiinctions of X, which, when substituted for y, satisfy the equation dry tZ-'V d^^'^y its complete integral is y = a^u^-\- a^u^-\- a^u^ + ... + a^u^, a^, a^, ... a^ being arbitrary constants. For let this value of y be substituted in the expression d'y . d'-'y , and it becomes Jn-1 or, collecting the terms multiplied by the factors a^, a^,...an, /d'u, d'-\ . , \ /d'u, „ = 0, or ilf =/ (w) = 0, which is called the auxiliary equation of the proposed linear equation. Hence the n real or imaginary roots of' this equation, OTj, TWj, ?Mg ...7n„, provided they be all unequal, will give n different particular values of «/, 6"'% e"^, ...e""*, which satisfy the proposed equation; and therefore its complete integral is 71 51. But if any of the roots axe equal to one another, as, for instance, m^ = m^, the value oiy becomes y={a,+ a,) 6"'"=+ a/^ + ... a„e'^^ which contains only n — 1 arhitraiy constants (because a^ + a^ can be reckoned only as a single constant), and therefore cannot be the complete integral of the proposed equation. In this case, in order to discover the complete integral, first suppose the two roots ?»,, m^, to be only very nearly equal to one another, so that m^ = m^ + k, where A is a very small known quantity ; then the part of the value of y corresponding to these roots is a.e"'^ + a^e'^^ = e"'' (a. + a/') = e"'^ (a,+ a,+ «, y + "^ ]~2 + *^^-) = e"'"' (q + c^ + \cjix' + 1 cji'x^ + &c.) , replacing the constants a, + a, and aji, by c„ c^ respectively. Now let A = 0, then this becomes e"'''^ (c, + CjX) ; and the com- plete integral consequently is ^ = (c, + c^) e"'^ + a^e"'^ + . . . + a„e"-^ 52. Generally, if we suppose r roots of the auxiliary equa- tion to be nearly equal to one another, and therefore to be represented by where \,\,...hr are very small quantities, the complete in- tegral takes the form 2^ = e"'''(a/'"4-a/>^+ ... +a/'-'') + a,^ie'"'+'== + &c., or, expanding e*'% e"^, &c., or, replacing the constants 2(a), S(aA),.- |y_i ' 72 by Cj, c^...Cr, y = e""'"" (c, + CjjX + ... + cX~' + terms multiplied by \, \, &c.) + ar+i«^'^' + &c. Now let A, = = Aj= ... = A,, in which case the auxiliary equation has r roots each = m^ ; then the solution becomes y = e"''' (c, + c,a; + c,a^ + . . . + c^-^') + a^^e^'+i" + . . . + a„e'""% which contains n arbitrary constants, and is consequently tha complete integral. 53. Of the correctness of the above modification for the case of equal roots, we may assure ourselves by the following reverse process. Let y = e°"M, then since by Leibnitz's theorem d'^juv) d^v d^ du njn-l) dT^v dSi_ „ m). y = e^ (i= + l) (A -!)■■' = 0; .-. y = e'' (a, + «,«) + /3 cos (x + a). 8 ^*y J^-2n*^+n*v = a■ y = ^ + (a + flSjO;) cos nx + {b + \x) sin wa;. 76 y=e'", [{k + mf + nJ^O; •'■ y = / 2 — 572 + e~^ (a + o.a;) cos «a; + e'"" (h + b.x) sin luo. dSi . dS „ dSi , C',„e"''''+' + ... + (7„e"°", a single constant being substituted for the sum or product of several constants in forming the complementary function. 58. Again suppose flj = p (cos 6 + V^l sin 6) = pe"^'' to be an imaginary root of the auxiliary equation, then since -4, is a function of a, it will be of the form R (cos a + V^ sin a) = Ber-^^, and the term involving a^ in the value of y will consequently be l^gCf cos e g(o+xp sin 9) \'=i fgxe''''' ^os 9 g - a»> sm « V^ jf or iZe'''«'^» {cos (a + xp sin 6) + s/~\ sin (a + xp sin ^) } y.^dxe-^"'^^ X[coB [xp sin 0) - V^.sin [xp sin ^)}. Now the term in y involving the conjugate root to a^, will result from this by changing the sign of V— 1 ; and therefore the sum of the two terms introduced into the value of y by the pair of imaginary roots p (cos^+ V— Isin^), will equal twice the real part of the foregoing expression, that is, 2R^'^»cos{a. + xpsm.ff) {/cfee* «»"»=* cos (a:psin0)X+ C] + 2Be^'^^ sin (a + xp sin 0) [Jdxe-=^'^'^ sin {xp sin 0) X+ C] : where 2i?e^«'=* { C cos (a + xp sin ff) + C sin (a + xp sin 0)], are the terms introduced into the complementary function, and may be replaced by gsipcoss jg pQg (jj^p gjjj ^ + c' sin {xp sin ^)]. 79 59. Exactly in the same way, if the imaginary root a^ = p (cos 6 + \/—l sin 6) = pe*^ occur r terms in the auxiliary equation, it will produce in the value of y, r terms of the form or since A^ is a function of a^ and may be assumed = i?„. (cos a„ + V~l sin a„) = ii^e"^^, of the form i?^,e^':oseela„-favsm6)V~ (J^^^«'g-a-pcoseg-xpsm9N/=i_X-^ or J?„e=^'=<'^«{cos (a„ + xp sin Q) + V^sin («,„ + xp sin 0)} X (/ia;)'"e-»*'™'*{cos {xp sin 0) - V^ sin {xp sin 6)} A'. But the root conjugate to a, will produce a term precisely the same as this except with — V— 1 instead of + V— 1 ; conse- quently the sum of these terms will produce twice the real part of the foregoing expression, that is, 2i?,„e^»s»cos (a„ + a;p sin &) (Afoj^e-^^^^cos (a:psin 6) X + 2i?„.e*''»^«sin (a,„ + xp sin 6) {fdx)'"e-'^">^^3m {xp sin 0) X; and to get all the terms introduced into the value of y by tlie pair of imaginary roots p (cos 6 ± V— 1 sin 6) that occur r times in the auxiliary equation, m in the above formula must receive all values from 1 to r. Also we see that the part of the comple- mentary function introduced by these roots, by substituting a single constant for the sum or product of other arbitrary constants, will take the form {c, + c^x + ... Cr.ia:'"') e^'=°^*cos {xp sin 6) + {c,'+c^x + ... + c',_^x'-') e'o^^'sin (a;jo sin ^). 60. We shall now give an example of each of the cases that have been examined. It may be observed that for equations capable of being reduced to either of the forms 80 the process may be greatly simplified. For in the former case since -v- e"" = m^, the symbol -7- is equivalent to the factor m ; so that the solution of the equation f [j-] y=c""' ^^ y = -71 — T + complementary function. And in the latter case if lyf j-}r can be expanded in a series of powers ' dx ' V^^ then as every differential coefficient of X higher than the wi"" is zero, Also if X= cos (mxH- a) or sin {mx + a), since in either case -y-j- = — rr^X, if the proposed equation contain only differential coefficients of an even order, that is, be of the fonn then its solution is cos(7w.r+a) \ . e. .■ y — — -J-, j7 V complementary fiinction. Ex. 1 . 5^3 _ 6n ?f + »w» f^ - 6«'v = «'. dx dx dx " the roots of the auxiliary equations are w, 2w, 3?i; and «'"= e"'"*"; •■• y = 71 \T\ "\ N n r^ + V" + <^/""+ c/""- " (log a — n) (log a — 2w) (log a — 3n) ° ' ^ the roots of the auxiliary equation are 2 + 5V— 1, 3, 3; and /(I) = 104, •'■ y- ■; — + (« + «'«) e'^H- (h cos 5a; + J' sin 5x) e^. 81 = a (a;' - 6a^ + 18a; - 24) + (c + c'x)e-^. cos (mx + a) „ , /;t o% ^, , /- •■• •3'=:^?r5^^+^*=«^(^^2 + ^)+(7'cos(xV3 + 7)- Ex. 5. ^ - a«w = X dx ^ Let Z» = a (cos + V- 1 sin <^) = oc* "^^ be a root of the auxiliary equation e" — o" = 0, where (/> = , r being any in- teger, then if B {z— J)"' be the term corresponding to this root in the resolution of (s"— a")'' into partial fractions, Hence the corresponding term in the value of wa""' y is _ .arcos<)g(o*sin(j)-B^-l-\/-l J^ _ gfucos^ jcog (aa, sin <^ _ w0 + ^) + V— 1 sin (oo; sin ^ — w^ + ^) | x/ ;;rj- COS {ax Sin — n(j) + ij)) J dx cos [ax sin (j)) Xe-"^ '='"* 2gar cos «^ H s=j- sin (ax sin <^ — n^ + = — {2r + 1) v, where r is any integer ; and to get all the terms we must take r from to J (« — 1) or ^ w — ] according as n is odd or even ; only the term that results from taking r = ^{n — 1) must be divided by 2. d"'ii d"ii Ex.7. ^-2cosea''^+a^y=.X da^ djd Let & = a(cos <^ + V— isin^) =ae*^-* be a root of the auxiliary equation z'^" — 2 cos 6a''z''+ a^=0, where (j>= - (2r7r + 0) r being any integer. Then if B {z — 5)"' be the term corre- sponding to this root in the resolution of (z*"— 2 cos 0a"z"+ a**)"', into partial fractions, since 6°= a" (cos 6 + V— 1 sin 6), 1 1 g-(n-i),;.Vn ~ 2n {b" - a" cos d) 6""' ~ 2n V- 1 sin ^a'J""' ~ 2^^^ sin da^'' ' Hence the corresponding term in the value of n sin Bc^'^y is 1 _ _ or 7^=^ga*cos">' -IV" 83 °^ JvCl e""'* {eos (aa;sln0-w^+(/)) + V^ sin {axsm-n^+)} X f dxe-'""'^ * {cos {ax sin ) - V^^ sin (aa; sin ^)} X Hence taking double the real part of this expression we get for the general term of the value of n sin Od^'^y g<«co8« sin (^^ sin + <^) /tire-'"™^* gos (aa; sin ^) X _gM:cos« cos (ox sin ^ - w^ + ^) /Jxe" '«'=»'* sin (aa; sin ^) Z, when .^ + F,^+...+p.y-0, (1), and assume y=u Jdxz, z being a new variable, a function of X ; then, separating the symbols of operation from those of quantity, we have H. 1. E. VI 90 -J- being understood to affect u only, and -7- to affect jdxz only. Hence, substituting for the differential coefficients in the proposed, by this formula, and again separating the symbols of operation from those of quantity, we get successively /d d'Y ,j [d d'Y'\, But since m is a particular value of ^, /(t-)m = 0; hence reversing the order of the terms and observing that/"' (-=- ] de- notes the same function of the symbol ^- , that — ~^ does of v. •^ dx dv^ we get for the depressed equation Similarly, if we know another particular solution m, of equation (1), then -^ f — M will be a value of « in equation (2), and we may depress this equation to another of the same form of the (n — 2)*" order ; and if we know r particular solutions of equation (1), we may in this way depress it to an equation of the same form of the (re — r)"" order. 91 Method of Parameters. 67. There is also another mode of integrating linear equa- tions, which deserves to be mentioned, called the method of Parameters, which we shall now explain. It may be stated thus. The complete integral of the linear equation of the «"" order will be of the form y = V^U^ + VjM,+ ... + t)„M„, where u^,u^, ... m„ are the n particular integrals of the equation when the term independent of y becomes zero ; and u,, v^, ... v„ are functions of x, determined by equations of the form Let the proposed equation be and suppose y=v^u^ + v^u^ + ... + «„«„ = S {vu), and as we hare made only one assumption respecting the n independent quantities v^, v^ ... v., we may make w — 1 more ; putting for the first of our additional assumptions S ( m t- j = 0, (1) ; Substitute these values ioi y, ■£■, j4 . ••• -jS » ^ ^^^ P™" posed equation, and the first member becomes + &c. + «« (^+i'i-^;Fr+ - +i>n«») +-^. 92 which manifestly reduces itself to X, since each of the quan- tities within brackets is equal to zero ; hence if the parameters Vj, v^, ..- v„ be determined subject to the above n equations of condition, the assumed value of y will satisfy the equation ; and since there are n equations in which the n quantities dv^ dv^ j changing the independent variable; and this is the method which has been attended with the greatest success. Thus in the example if we make y the independent variable, and consequently for ^ ^ write 4- -^^(±X dx' cfe" ^^^ ^' df ■ \dxj' dy the equation is transformed into the integrable shape d^x dx , df dy 69. In the above example there is no difficulty in fixing upon the new independent variable. In other cases we must consider x and y as functions of a third variable t, and substi- tute the values of -f^, -t4, &c. corresponding to that sup- position. When the equation is thus generalized, we must assume for x ox y some known function of t, adapted to the particular form of the equation, so that there may arise for de- termining the function of t which expresses y oi x, a. differential equation simpler than the proposed one; between the integral of which and the assumed function, if we eliminate t, we obtain the required relation between x and y. The generalized equation is x" X Hnr* d X using x, x, &c. to denote -t- , -^ , &c. Let X = cos <, .*. x = — sin t, x' = — cos t, therefore by sub- : 0, which gives ^ = .4 cos n« + J5 sin n< ; stitution -^ + nV = Oj which gives 94 .•. y = A cos (n cos"' x) + B sin (m cos"* x). Ex.2. x*-^ + cy = Q. The generalized equation is I It t ri xy —yx . _ let x' = — x\ or a; = - , .*. x" = 2a;', t hence by substitution we get y" + ly' + cy = o, or ^|^ + c<^ = 0; .". ty = ^ cos (< Vc + o), or y=^x cos ( ha). n n If c = — w', the solution is y = oj [c/^ + c^e"). where m denotes a given function of x. This when generalized becomes I II I II a; (£+«)4+»^-^^ 1 . I ^ rj 1 du X leta;=M, ori=Jaa;-, .•. -?- = — r, M dx X hence by substitution we get the integrable form Ex 4 a?^4-x^-v--^^ Here the substitution x = c* gives 95 e'-l> .-. y = c/ + c.e-' + ia + ia (e"' - e') log {^^ . c, , ax" X w — 1 Ex.G. f^ + -^f+^-^,y = 0. aa; a + oa; oa; (a + oa;) "^ The generalized equation is a" ^a + Ja; a;'^(a+6a;)'2'-"• Let a;' = a + Ja;, or e'* = a + bx; .'. x" = lx' ; hence, substituting, the solution of which is y = c^e""' + c/"'' = c, (a + Ja;f + c, (a + Sa;) " , m and to' being the real roots of m' + (-4 — J) to + jB = 0. If the roots be impossible, and of the form to + n V— 1, the solution is m n y = /3e"" cos {nt + a) = )8 (a + Ja;) " cos {log [a + Ja;)" + a}. The same substitution of course succeeds for the equation 96 which, hy putting a + hx = s, may first of all be reduced to the form „ d'y „_, d''~^y „ dy ., . and then to a linear equation with constant coefficients, by the formula where z = c', and^,,^j, &c. are such that the roots of k' -pjc*-' ■Vp.]^-^ - &c. . . . + p^Jc = are 0, 1, 2, 3 ... (w — 1) ; so that the preceding formula may be written =S(l-)s-)•••(^-)}- 70. The result just noticed ^dTy d (d \ (d , \ where x = e', may be written "^ tie" ^»W^' using the symbol P„ to denote the product of the n factors formed by subtracting fi-om -^ the n numbers 0, 1, 2, ... w — 1. Hence multiplying by a;" = e**, we get But by Art. 53 we have 97 which shews that in the expression f{-r] ^y we may remove the power of e' from the operation of the symhol f{-j], and prefix it as a factor of the whole expression, provided we add the index of the power to -j ; and conversely in the expression e'°'/f J +OTJ y we may transfer the power of e* to he a factor of y, and so bring it under the symhol of operation, by subtract- ing the index of the power from -j-\-m. Hence ^ ■;^=^''lT,-'»*]«"> 71. Hence an expression such as (a + Jx + c:j^ + ... + &*■) x" -TT, ax! is transformed by the substitution x = e' into aP„ (d) y + bP„ {d - 1) e'y + cP^ {d-2) e^y + ... + IP„ {d - r) e"y, where for convenience d is written for -t, the differentials of t dt' being omitted ; and consequently an equation such as {a + bx + ex' + &c.) x" -j^ + (a + h'x + c'x^ + &c.) a;""' -j-^ + (a" + h'x + &c.) a!"-= ^+ . . . + (a'"' + V'x + &c.) y = X is transformed into aP„ {ct)y+ a'P^, {d)y + a"P^, (^ «/+••• + «'"V + bP„{d-l) e'y + h'P^, {d-\)e'y + l"P^, {d-l)e'y+...+ V^e'y + cP^ [d- 2) ^y + c'P^, {d-2) e^y+ c"P^, {d-2) e^y+ ...+^'e^y + &c, = T {a, function of e'), H. T. E, 13 98 which is evidently of the form (where /„(«?), /^ (<^), &c. denote rational and integral functions of d) /o (^ 2/ +/. {{d)e''f{d)v=T„ and in order that this may coincide with the auxiliary equation we must have the two conditions f{d)--^' f{d) the latter of which gives //^^-ii^//^_.^-iM) id-r)... _{d) ^^ '~-<^[dY^ '~ir[d)-y{r{d-r)...~ 'ylr {d) ' where P^ denotes the product of an infinite nimiher of factors formed from -j-Wv by replacing dhj d—r, d—2r, &c. Hence, f{d) being known, T^=^ {/{(i)]''' ^i can be computed; then v is known from v + 4^ (d) e"v = T^ an equation which is supposed to admit of being solved when T^ is any given function of e', and thenw = P ^-^v Ex. 1. '^%^^''+'^)'^-£ + ^^ +Va'') y = ^' .: d{d-l)y + ia+l)dy+{b + q^e'')y=T, or {d^ + ad + b)y + ^e"y=T; --•y-^dT^i^'y^'^- As the proposed equation can be solved when a = — 1, J = 0, let the auxiliary equation be Assume y =f(d)v ; 101 and such values must be given to the constants a and h as will produce a value oi f{d) that will allow y to he determined from the equation y =f{d) v =f{d) /3 cos {qx+ai) when X=Q. If (^ he preceded by a negative sign, the value to be used for V in this and the following examples will be ae'^ + ;8e~*". f{d)=P,-j^^ = d{d-2)...{d-2n + 2); y=d{d — 2) {d — 4:)...{d-2n + 2) yScos {qx + a) ; £ dt' but the symbol of operation -r. — r prefixed to any subject, is equivalent to rt<^-rt r d 1 ^, d \ dt ' dxaf dx x"' y = x— a;'— i a;'—- af'"'-^ -^ ^ dx' dx a?' dx X* '" dx ai"''^'' x^ \ dx) jy /3co3 (ga; + g) Let n = 2, then the equation is and its solution y = /3 {— + aj{d)f{d- l)f{d - 2) e^y + &c. = T; then since f{d — 1) e^y = e'f{d) e'y ; .: f{d)f{d- 1) e-y =f{d) e'f{d) e'y = {/(d) e'Yy, and generally f{d)f{d- l)f{d- 2) ...fid-n + 1) e'^y = {/(^) e'}^, consequently the proposed equation becomes, if we denote the symbol of operation f{d) e' by p, {1 + a,p + a,p^+ ... + a„p'^]y= T; .-. .y = (1 + a,p + a,p' + ... + a^p")- T 105 resolving the function of p with which T is affected into its partial fractions, or y = iVjMj + iVjMjj + ... +iV„M,„, where u^, u^, &c. are determined from the equations, M, - Sj{d) efu, = T„ u,- qj{d) e'u, = T, , &c. 76. J£ f{d) =d'^, the equation just treated of is the linear equation with constant coefficients : for tliis leads to P, (J) y +p,P^, id - 1) ^y +p,P^ [d - 2) e'V + &c. = 2^; El ^t„ J ^ ju^, _i_ ^„ i_ . [d)' °'-^+f^'^ + J(fc)^> + *'=-=^ which is of the assumed form. For all other values of f{d) the equation has variable coefficients. If the equation be fid) = ^ d-n+\ ' which leads to a linear equation of the first order for the deter- mination of Mj, Mj, &c. x" ^ i^^^^) = 9,an«. ; if T, = 0, this gives m, = Caj'-'e'.^ \if{d) =a + b {d — r)~^, the partifsular values of ^ are determined from linear equations of the first order, -J-{u^{l— q^ax) x^ — T^x-'] = qj)u^x~'. Ex. {l-irax + lx'^c^^-\2r + {r-2)ax-^hx^}x^ + {r (r + 1) - rax 4 ibx^] y = X. H. I. E. 14 106 Changing the independent variable into t = logx, this be- comes (1 + oe' + be") d{d-l)y - {2r + {r -2) ae' - iJe"} dy + {r{r + l)- roe' + 2he"} y = T, OT {d-r){d-r-l)y + ae'{d+l){d+l-r-l)y + be"{d+2){d+2-l)y=T, or {d-r) {d-r-l)y + ad{d-r-l)e'y + hd{d-l)e"y = T, d , T d d — 1 5, _, or V + a J ey + b -j . -^ e y = T., "^ d-r " d-r d-r — \ ^ ' or (1 + ap + Ip^) y = T^, putting t — e' = p, or (1 + ap + bp')y= (1 - ^.p) {l-q,p)y = T, ; ••• y = ir^^ + 7^^) ^. = -^A + -^.''.' Vi - g'lP 1 - q^pJ where w, is given by the equation (1 - iip) Wi = T„ or u,-q,^i—^e\ = T^, or (<^ — r) (m, — rj = q^diu^, Further applications of this method may be seen in the Philosophical Transactions, where Mr Boole has treated this subject with great generality. As a concluding application of the method by changing the iqiependent variable, let us solve the equation Taking z for independent variable where a + - = a;, we find (.-l)«=g + 2.»|-«(« + l)(.-l).y = Z. 107 Now assume z = e', then {e'-\)d{d-l)y+'ie'dy-n{n + \){e'-l)y=Z, or e'(rf+«+]) [d-n)y-{d-n-l) {d + n)y = Z, or {d + n){d-n-\){e'y-y)=Z; or (2n + 1) (z - 1) J, = z"*'/c^g z"""^ Z- g-'/c^z a""' Z If Z= 0, we have cz"*' + c'z"" y= z-1 ' and if for s we substitute its value \{x + Va;''-4), we shall find , /x + 2 { , , c?m) where m = a;" + a;""' - (w - 1) x""* - (n - 2) a;"-' + &c. the general term of the series for u being / iSr(»-»"-l)(w-»--2) ... (71-27-+1), „ (- ir-^ 1.2 ■ 3 ...r ^(w-2r+wa;-ra;) x"-^-'. Simultaneous Equations. In these equations, which sometimes occur in the higher parts of Dynamics, instead of one equation between x, y, and the differential coefficients of y with respect to x, being given to determine the relation between x and y ; we have two equations containing x, y, and t (of which x and y are considered as func- tions) and the differential coefficients of x and y relative to t, to find that relation. 77. To integrate the simultaneous equations of the first order, ax + by + -£=T, a'x + b'y+ -£= T', 108 T and T denoting functions of t. Multiplying the latter by an indeterminate quantity w, and adding it to the former, we get ax + hy-\-m (u'x + h'y) + -j.{^ + '^V) = T-'r mT', d , , , ,, , J + mb' > ™ rni or -J- {x + my) + (a + ma ) ix H -, y) = T+ml . Let , = m, which will give two values of m, m. and a + mu ^ ' m^; then the equation, under this condition, becomes a linear equation of the first order ; and we obtain by integration {x + my) e'"-^" = Jdt {T+ mT') e'"""^" ; and by substituting successively the two values of m, we obtain two primitive equations each containing an arbitrary constant, which will farnish values of x and y in terms of t, and the re- lation between x and y, if t be eliminated. If the two values of m are equal, we shall obtain only one equation between x, y, and t ; but if this can be solved with respect to x or y, and we substitute the value so found in one of the given equations, we shall obtain a second relation either between x and t, or between y and t ; and then t may be eliminated as before. Ex. 1. 5x - 2y + -^ = e', Gy - x + -^ = 6" ; -I dt' .•. {5 — m)x + {—2 + 6m) y+ -j-{x + my) =e' + m^, let —1 = m, or m^' + m — 2 = 0, or to = 1, or —2; b — m :^Ax+y) + 4:{x + y) = e' + e'; dt x + y = ^+^+Ce-*'; e' 2 similarly « — 2v=^— -«"+ C,e"", which determine x and y in terms of t. 109 Ex.2. ^+5x + y=:e% ^^ + 3y-x = e"; ■'■ dt ^^ "*" "*^^ + (5 - m) X + (1 + 37n) y = e' + me^ ; .-. -= = 7», or 1 - 2m + m" = (1 - mY = 0. 5 — m ^ Hence the values of m are each = 1, and integrating, we find x + y = C,e-^' + ~e' + ^e\ By means of this, eliminate y from the fii-st equation, and we get and y = Le'+Le-+ G,{l + t)e-^- C/"". The more general form may evidently be reduced to the above by successively elimi- dy dx 78. To integrate the simultaneous equations of the second order, d^x „ , ,, , d'y ax + bi/ + c + -^ = 0, ax+by + c+-j^ = 0. Multiplying the latter by an indeterminate quantity m, and adding it to the former, we get d' {a + ma) x+{b+ mb') y + c + mc + ^{x + my) =0, no or ^ (* + '«^y + Ci) + (a + »»a') (a; + wi^^ + c,) = (1), .J, J + mV . , c + 'wc' it , = m (2), r=c, : therefore, integrating equation (1), and substituting successively the two values of m given by equation (2), we obtain the two required primitives; or if the values of m be equal, we must proceed as in the former case. Ex.1. :^-(3x + 43,-3)=0, ^ + (aJ + «/+5)=0; Let = m, or m''' — 4»» + 4 = (w — 2)" = ; .-. ^(a;+23/)-(a; + 23/-13)=0; .-. a; + 2^^ - 13 = ce' + c'e"*, and eliminating a; from the latter of the given equations, we find ^-2^ + 18 + ce' + cV = ; and a;=-23+c (< + i)e'-c' («- 1) e"* - 2ae' - 2a'e"'. 79. If we have three variables a;, y, a, which are functions of t, and if ^ + aa; + % + c« = r, J + a,a; + % + c,3=2;, ^+aja; + J^ + c^=7;, Ill ''"'<'> iTy then if we multiply the second and third by indeterminate con- stants w, m, and add them to the first, and assume a + wia, + m'Oj = s, b + mb^ + m'b^ = ms, c + mc^ + m'c^ = ms, we get ■j- {x + mi/+m'z) + s (x + my + me) = T+mT^+m'T^. But the three preceding equations give s—a = ma^ + m'a,, m {s — b^) =m'b^+b, m' {s — c^) =mc^ + c; and if values of m and m' be obtained from the two latter and substituted in the former, we get to determine s tlie cubic equa- tion {s-a) (s-6,) (s-Cj)-J,c,(s-«)-a,c(«-J,) -ba^{s-c^)-afi^c-aj>c=0. Hence if «j,, th,'; jw,, m^; m^, m,'; be the values of m, m, corresponding to the three roots of this cubic, we have by solving the linear equations, X + mji + mlz = jF, (t), x+m^ + m^z = F^{t), x + m^y + m^z = F,{t), from which equations x, y, z may be found in terms of t. It may be observed that if b^ = c„ c = a^, and o, = b, so that the cubic equation is (s -a){s- b,) {s-c^) - c^ {s-a)- a/ (s- 5J - V {s-c^ - 2c^aJ!>=0, then all its roots are real {Theory of Algeb. Eq. Art. 58). 80. This method readily leads to Jacobi's solution of the equation {A + Ax + A'y){x^-y)-{B + Bx + B'y)^ + {C+C'x+C"y)=(i, which, upon introducing the independent variable t, becomes [{A + A'x -H A"y) x-{B + B-x + B"y)} § = [{A + A'x + A"y)y-{C+C'x+C"y)}^^, 112 and resolves itself into ^-{A + A'x+ A"y) x+{B + B'x + B"y) = 0, dy dt -{A + A'x + A"y)y+ {C+ C'x+ C"y)=0; dz or {putting -^ = — [A+ Ax + A"y) z] into the system, dz -j- + Az + A'xz + A"yz = 0, ^^+Bz + B'xz + B"yz = Q, ^^ +Cz+ C'xz + C'yz = 0. Multiply the second and third by m and m respectively, and add them to the first, and assume A+mB+m C=s, A'+mB'+m 0'=ms, A"+mB"+m'C"=m's (1); then -J- {z + mxs + m'yz) + s {z+ mxz + myz) = ; which gives 3(1+ mx + m'y) e" = C, or zue" = C But equations (1) lead successively to s-A=mB+m'C, m{s-B')=A'+m'C', m'(s- C") =A" + mB"; the two latter of which determine m and m from s by the equa- tions m{{s-B'){s-C")-B"C'} = A'{s-C")+C'A" m'{{s- C") {s-B')-B"G'}=A" {s-B')+A'B" ." (2), and the values thence obtained of m and m substituted in the former give for finding s the cubic equation {s-A){s-B'){s-C")-B"C'(s-A)-A"C{s-B') - A'B (s - C") + A"BG' + A'B"C = 0. Let the three values of s in this equation be 5,, s^, s,; and let the corresponding values of m and m, be m^ and m.\ , m^ and m'jj, m, and m', ; then corresponding to the three roots of the cubic we get the three integrals. 113 and if we raise these equations respectively to the powers s^ — s, , S3 — Sj , «! — Sg , and take their product, z and e' are eliminated, and we find the required solution or (1 + WjO; + m\y)'^'^ x (1 + m^a; + m\y)'^-'' X (1 + wjjo; + m\y)''-''' = C, where wi^, ot'^ are given in terms of s^, m^, m\ in terms of s^, and TWg, m\ in terms of s,, by equations (2). This includes as a particular case the equation (Ex. 7, Art. 18), in which Euler separated the variables, {a + bx + ex' + y) -^ — {n + ex) 1/ = 0, or ex (^|-^) + («+^-^ + ^)i-"-^ = "- 81. When the proposed equations are linear with constant coeiEcients, we may separate the symbols of operation from those of quantity and then obtain by the ordinary processes of elimina- tion an equation containing only one of the unknown quantities : for as the symbols of operation here employed combine according to the same laws as ordinary algebraical quantities, they may be treated precisely as if they were symbols of quantity. Thus the equations of Art. 77 (suppressing the dt in jA may be written {d + a)x + by=T, {d+b')y+a'x=T; the latter gives ;/ = — y — 77- ; and substituting this value in the former we get ld + a){d + h')x-dhx={d+h') T-bT, an ordinary linear equation with constant coefficients, from which the value of x may be obtained, containing two arbitrary con- stants ; and thence the value of y by the equation y=^-T-\{d + a)x. H. I. E. 15 114 Ex. -^ + 2cx-y = f, ^-2cy + 5c'x = e^; e"" — 5c^x the latter gives for y the value y = — -^ — - — , and sutstitutlng this in the former we get {d'+c'')x = e'"'+2{t-cf). e"" 2 4 .-. a;= a + -2 (<- c<') + -3 + /3 cos (c« + a) ; m + c c c and y = (^ + 2c) a; — <° 771 + 2c 10 = — J 2 e"" -5f+ -„- - |8c sin (ct + o) + 2y3c cos (c< + a). m +c c / <- \ / 82. Again, suppose the equations of Art. 78 to be {d' + ad+b)x={md + n)y+ T, {d'-ad+h')y={m'd+n')x+ T'; obtaining a value of y from the latter, substituting it in the former and reducing, we find {d* — {a^ — b—b' + mm) d' + {ah' —ah — mn — m'n) d+hV — nn] x = {d'-ad + h') T+{md + n) T ; from which the value of x may be obtained ; and if it be substi- tuted in one of the proposed equations, we may deduce the value of y. Ex.1. {d-\Yx = '2dy + 005,1 {d^ + 2d-\-(,)y = -{d + 5)x + smt; eliminating y, we find {d* + bd^ + %)x = l co&t-2 sin t, 7 — — .-. a; = - cos < — sin « + c cos {t V2 + a) + c' cos {t Vs + /3) ; and the value of y may be now obtained from the former of the proposed equations. 115 Ex.2. {d^ + {n+^d+n^]x==dy+T, eliminating y, Ave find [d* - (2 +m^) d'-\-\]x= \d^-{n + l\d + 11 T, or {d^'-md-l) {d^ + 'md-\)x=\d^ -{n+^d+^,- T, which gives x ; and then y can be obtained from the former of the given equations. In these examples d is used for -j . Solutions expressed by Definite Integrals. 83. Sometimes the complete integral, of a differential equa- tion may be expressed by a definite integral, as in the following instances. d^' = a:^ + »w. Let u = a, I dte'i'" . e"" , a^ being a constant quantity ; ' Jo then j^, V, = < j "dtf-'e^i'^'e-n = o^" (^ + ^t) ; ax Jo change a, into a^, Oj, &c., and let v^, v^, &c. be the correspond- ing values of Vj , '^"'ii>„ = a„"(l+xi;„). «ic' Now suppose Hj, a^, ... a, to be the n* roots of unity; then multiplying each of these equations by arbitrary constants 116 (?, , Oj , &c. sucli that Oj + Cj + . . . + c„ = OT, and adding them toge- ther, we find ^H=i (Cj^'i + c^v^ +■■■ + c^v„) = m + x{c^i\-\- CjWj + ... +c„v„). Hence y = c^v^ + c^v^ + . . . + e„v„ is the complete integral of the proposed equation, containing only n — 1 arbitrary constants by reason of the equation of condition ; or it may be written y=l dte'n [ce^^ + c,ae»'* + c,aV'<* + . . . + c^^ a""' e°°" '^j , JO a being a primitive root of A;" — 1 = 0. Let OT = 0, « = 3, then -^ = a;?/, the solution of which is y=i c? •'• j3 "" «^2a;" = 0, or putting c for - ah, Assume z = af {A + Boifi + Ga?^ + &c.) ; then a{a-\)Ax''-^ \ + (a+,S) (a+/3-l)5x"*''-'+(a+2y3) (a-f-2^-1) Ca;"«P^+&c.[ = 0. 118 Hence /3 must equal n+2; and then to determine a, A, B, &c., we have a{a-l)A = 0, (a+ n + 2){a+ n + l) B + cA = 0, {a + 2n + 4) (a + 2n+S) G+ cB= 0, (a + 3w + 6) (a + 3n + 5)Z>+c(7=0, Hence a may be either zero or unity, and A remains undeter- mined ; calling therefore A and A' the two values of A corre- sponding to a = 0, a = 1, we get ^"^ (n+l) (w+2) "^ (n + l) {n+2) (2w+3) (2n+4) ~ ^'^ ^^•^t^ («+2)(»i + 3)^(n+2)(n+3) (2«+4)(2»j+5) ^^ 1 «fe and substituting this value of a in the expression y =-7- -5- , we shall obtain the value of y, involving only one arbitrary constant —n. A As the terms of the above series have divisors of the forms (« + 2) i + 1, where i is an integer ; if w be such that (n + 2) t + 1 = 0, one or other of the series will be illusory, and we shall only obtain a particular value of z ; and if w + 2 = 0, both series become infinite, but in that case the equation may be exactly integrated by Art. 16. Ex.2. (;.^-4)g + a.^-«V = 0. dx^ dx Assume 3^ = a^ («„ + a^a^ + a^^ + agx"^ + &c.), .-. y = (a'-n") a,x-+ {(a+/3)''- Ji=} a^a;°-'^-4a (a- 1) a^a;""' + &c. + {(a + »-/3)'-«'}a,a;''-^^-4 (a-2r+2) (a-2r+l) a,_,x"-*+ &c.. 119 the first term shews that a may he either + ra or —n, a, being the arbitrary constant ; the next terms shew that /3 = — 2, i.e. the series is a descending one ; and that the coefficients are derived from one another by the law (taking a = n), _ (w-2r + 2) (w-2r + l) **'" r{n-r) '''-'' + a\ [x-^ + nx--^ + 'L^* + 3) ^-^ + &c.}. As the solution of the proposed equation is (Art. 48) x\ y = % cos I n cos -^ I + a„ sm \n cos -z] , if we make x = 2 cos 6, we get the well-known result, cos nO = (2 cos 6^ - n (2cos 6^^ + liilLzi) (2cos 6)"-* - &c. 85. The transformation of Art. 71 may be employed in the case of linear equations to find the solution in the form of a series : for suppose the transformed equation to consist of three terms fM)y+A{d)^y+fA'^e-y = o. Let y = M^a;"" + M„4..a;'"" + . . . + K_^^^ + v,_^x'~' + u^' + &c., where m, in the general term m^ is an invariable function of the index r, and x" is the lowest power of x that can enter into the development of y. Now substitute this value of y (having first replaced x by e') in the proposed equation : then since f{d)e"=f{r)e", the first member will become /o {m) M„e"" + {/„ {ra + 1) «„« +/, {m + l) «.} e""^"'+ . . . 120 which will vanish, provided /o W M.,. = 0, /„ (m + 1 ) u„^^ +/i («i + 1) «„, = 0, and if for eveiy value of r from m + 2 upwards, we have /o ('•) Wr +/i W t' - 2 (2« - 1) u^/"^" -3{2n- 2) u^e''''" + &c. + 2«M„e'»-'" + (2w - 2) M,^.e'"-"' + {2n - 4) M,^,e'""'" + &c. - 4 (n - 1) nu„e"-^" - 4 (« - 2) (w - 1) M„_.e'"-'" + &c. Hence, equating the coefficients of the powers of e' to zero, we get «,-, = «■„, «.-, = - («-!)«„, «„_, = -(«- 2) M„,&c.; and since m„ = c, an arbitrary constant, ^ = a;"+ a;""'- (n - 1) a-""'- (n - 2) x"' + ... (-1Y + -^^-j-^ (n-r-1) (n-»--2) . . . (n-2r+l) {n-2r+ {n-r) x} a;""''-' + &c. Similarly the series corresponding to m = — n — l may be found. 86. In the preceding instances we arrive immediately at a complete result ; but it often happens that the solution we obtain by the method of indeterminate coefficients involves no arbitrary constant. To supply this defect, we must introduce, instead of H. I. E. 16 122 the arbitrary constant, a value of y corresponding to a given value of X ; that is, supposing these to be h and a, we must sub- stitute in the given equation, y = h + u, x = a+ t; then determine m in a series all whose terms vanish when t = ; and replace u and t by their values y — h and x — a; in this way it is evident the arbitrary constant will be involved implicitly ; for, from the complete integral /(x,y, (7)=0, C may be expressed in terms of a and h. Ex. 1 + ^=^--, this becomes -j- + }> -V u = g [a + t)" ; assume u=if- {A-\- Bfi + Gp + &c.) ; .'. = a^i'-' + (a + /S) 5r+^-' + (a + 7) CV'^-' + &c. + h + Af- + BV'*^ +&c. - ga" - mgoT ' t \ ' goT f - &c. .-.« = ], /S=l, 7 = 2, &c. A = .9a"' - J, 25 = ^^ma*""' - (/a" + b, 6C = gm{m-l) oT^ - gmoT'^ + gd^ - h, &c. 87. The approximate solution of a differential equation may sometimes be obtained in the form of a continued fraction by assuming _A^B^ Cxi ^~TTTT 1 4- &C. ■ First, suppose x to be very small, and for y substitute Aa^ in the given equation; then, retaining only the terms of lowest dimensions in x, A and a become known by equating the coefficients and exponents of those terms. Next, write y = 123 in the proposed equation, and in the resiilt put z = Bafi, and determine B, )3, as before, by supposing x to be very small ; Bafi then in the transformed equation in z, put z = ; and so on for the rest. Ex. my + {\+x)-j- = 0. Let y = Aaf- ; .-. {m + a) Aaf- + Aaa;^ = 0, or Aaaf^^ = 0; .-. a = 0, and A remains undetermined. Next, put dz and we get successively m{l + z) = {l+x) ^ , m+BxP{m-^)=0Bx^-\ or m = B^x^'; .-. /3=1, B=m; similarly, putting z = , and t = Cx'', we determine C and 7 ; and thus we obtain m— 1 x m + l X »i — 2 x m + 2 x m — 3 x _ A mx \ '2 3 "2 3 '2 5 "2 5 2 ^~iT 1^ 1+ 1+ 1- 1- 1 + &C. ■ Since the proposed equation when integrated gives y = A{l + xr\ the above continued fraction is the development of ^ (1 + x)~^. 88. We may approximate to the integral of a differential equation by successive substitutions, in a manner similar to that employed by Newton for the solution of algebraical equations, as in the following instance. Ex. j^ + n^y + ai^ + a = 0, where a is a very small quan- tity. 124 We may assume y = u+ aw, + a^Wj + a'wj + &c. which gives + «lS + '*''^^+H V + a' (^ + n\ + 2mm,) + &c. = 0. Hence, equating the coefficient of each power of a to zero, we get d'u „ -5-^ + nu + a = 0, -^^ + n\ + u' = 0, -t4+7i\+2uu^ = 0, &c (3). The first give u = = + c cos rue + c sinnx ; and this value substituted in the second reduces it to the form the integral of which by Art. (67) may be shewn to be Mj = cos nx (c, JdxX^ sin nx) + sin nx {c\+-jdxX^ cos nx). Similarly, these values of u and u^ substituted in (3) reduce it to the form d^u. „ „ which may be in like manner integrated ; and in this way the coefficients of the powers of a may be deduced one from the other by a uniform process. 89. We have seen (Art. 63) that the solution of -j^ + n^y =A cos {mx + a) +B cos {nx + /8), 125 18 y = c^ COS nx + Cj sm nx A B + -: tCos {mx + a) + -- a sin inx + fi). rr — m ' 2n Hence, if from the proposed equation we had to determine y approximately, we could not neglect the tenn A cos {mx + a) even when A is exceedingly small, provided m and n are nearly equal to one another ; because in the value of y this term is divided by n' — m^ which is very small. With respect to the last term in the value of y, we remark, that it is not periodical, but may increase indefinitely, as x increases. The equations which present themselves for solution in physical Astronomy are usually of the above form ; and upon the peculiarities just no- ticed depend some of the most interesting results in that subject. The following example supplies an omission at the end of Art. 83. When m is an integer we have obtained (Art. 74, Ex. 4) the solution of the equation cfw , m (m + 1) When «t is a fraction, assume y = a;"'^' m, then ay ay ax ax ''f-«f-^f+<^ »• uz t*i& uu ^H ax ax dz az Hence, multiplying the first of these equations by B,, the second by P, and the third by Q, and taking their sum, the factor V disappears, and we find nf-f)-*(f-S-^(S-f)=» <^> the same equation of condition as in the preceding Article. 100. If this equation be satisfied, which will be the case only when the proposed admits a primitive of the form f{x,y,z, = 0, equations (1) afford a means of determining V. Then du = VPdx + VQdy + VRdz, whence u can be found by the method of Art. 95, and m + C = is the required relation between x, y, and s. 135 Or, without determining V, we may integrate considering one of the variables constant, and add an arbitrary function of that variable ; then differentiate the result with respect to that variable and compare it with the proposed equation, and so the function added will become known. In the majority of cases which present themselves, the factor V is capable of being determined by inspection. Ex. I. (ay — hz) dx + [cz — ax) dy + {hx — cy) dz = 0. Divide by [ay — hz) hx — cy), and the resulting equation dx [cz — ax) dy ^^ _ci hx — cy [ay — hz) [hx — cy) ay — hz satisfies the conditions, considering x, y, z as independent, dP^dQ^ dQ_dR dR_dP_ dy dx ^ dz dy ^ dx dz ' and therefore the first member may be regarded as the differential of some function, m, of x, y, z considered as independent ; ■'•i^h^- and« = ^log(Ja.-cy)+«.; 1 _ dw ' ' ay — hz dz ' cz — ax c 1 dw [ay — bz) [hx — cy) h'hx — cy dy ' dw a — 1 or ^- = T • , , 1 ady — hdz ; hence aw = — t ■ — - — r — 5 dy b'ay — hz' h'ay — hz .". w = —T\Qg[ay — hz) + C; ,. „=llogf^^)+C = 0, or*^=C. b ° \ay — hzj ay — bz Ex. 2. xdx+ydy^-[3?^-f)[\- z"') dz = 0, x' + / = Cz^'e'^. When in an equation of this sort, the differentials enter above the first degree, it is not integrable unless it can be resolved 136 into rational factors of the form Pdx + Qdy + Rdz ; for what- ever be the integral, it must upon differentiation produce a result of that form. 101. If the equation Pdx + Qdy + Rdz = he susceptible of a primitive of the form/(a!, y, e, C) = 0, and be homogeneous and of n dimensions with respect to x, y, z ; then, putting x = vz, y = wz, and dividing by z", it becomes 8 [vdz + zdv) + T [wdz + zdw) + Udz = 0, S, T, U being functions of v and lo ; dz Sdv + Tdw °^ z ~ Sv+Tio+ U' hence the second member is an exact differential since the first is so, and it may be sometimes integrated by inspection, or by the method of Art. 94. Ex. 1. {y + z) dx + {x + z) dy + {x + y) dz = ; put x = vz, y = wz, dz {lo + l)dv + {v + l)dw _ dz , d (v + w + vw) or \-l — = ; z V + w + vw .'. log z' (v + w+ vw) = log G, OT xz + yz + xy = C. Ex.2. {y' + yz + z')dx + {x'' + xz+z')dy + {x^ -i xy + /) dz = 0. Make x = vz, y = loz, dz {w'' + w + 1) dv + {v^ + v+l) dw _ _ ■ ■ 'z~ V [w'' + w + \) + w {v" + V + \) + v" + vw + w" ~ ' but of the latter fraction, the denominator = vw {w + V+ 1) +v {w + 1 +v) +w{v+l+w) — {vw + v + w) {v + w + 1), 137 and the numerator = {^+v + w)d{v + w + vw)-{v + w + vw)d{v + w); , ds djv_+je+^vw} _d(l + v+w) Z V + W + VW 1 +V +10 ~ ' >»^{^^?Tf^}-%C; .-. zx-\-zy + xy=G{z + x-\-y). Ex. 3. 2 {:y + z) dx+{x+^y Jr^z) dy + {x + y)dz=(i. By the same substitution this becomes ^dz _ ^dv + dw dw Ex. 4. (a? -f + z') dx- z'dy + {y -x) (z' + xy + x')- = ■ z putting X — vs, y = wz, this is reduced to dw= {l + v^ — w^) dv ; ' * " --fdve-^ + C, (Ex. 4. Art. 18) w — V or = jdxe z= + Cfe, y-x s being constant under the sign of integration. 102. We have seen that the equation Fdx + Qdy + Bdz = 0, when the equation of condition (Art. 98) is not satisfied, does not admit of being derived from a single primitive equation involving two independent variables. The integral in this case will be exhibited by a system of two equations ; and the proposed equa- tion cannot be regarded as the differential equation to a surface, but to a system of curves in space all endowed with some com- mon property. H. I. E. 18 138 Ex. 1. dz= aydx + hdy. Since the equation of condition in this case is not satisfied, x and y cannot be independent, and we may assume y —f(x) ; ,". dz = af{x) dx + h -j-fix) dx, .: z = aSda:f{x) + bf{x), which, with y=f{x), constitutes the integral of the proposed equation. In general, if Fbe a factor which makes Pdx + Qdy an exact differential considering z as constant, and we find / ( VPdx + VQdy) =w + {z), it is eyldent that w + ^ (z) = 0, together with -^+-j-^{z)-VB = 0, satisfy the proposed equation, where <^ (z) denotes any function of z. Ex. 2. zdx + xdy + ydz = 0, d y + z\ogx + (t>[z) = 0, \ogx + -j-^{z) x' Ex. 3. [x {x — a) + y {y — h)] dz = {z — c) {xdx + ydy), x^+y' + 2{z)=0, x{x-a)+y{y-b) + {z-c)-^(j){z)=0. a z Partial Differential Equations. Equations of the first order. 103. In partial differential equations of two independent dz variables, the differential coeflScients of the first order -^ , dx dz -=- , of the dependent variable z, are usually denoted by the symbols p and j ; and -^-^ , , , -^-^ , the differential co- efficients of the second order, by r, s, t, respectively. A partial differential equation is said to be of the n**" order, when it in- volves one or more of the partial differential coefficients of the dependent variable of the n"* order; but none of a superior 139 order. To be the general equation of the to* order, it ought to contain the independent variahlea, and the dependent variable together with all its partial diflferential coefficients from the first order to the n**" order inclusive. To integrate a partial differential equation, is to find for the dependent variable an expression between the differential coefficients of which that relation exists which is indicated by the proposed equation ; and under the most general form possible. 104. The complete integral oif{x, y, z, p, q) = 0, the gene- ral equation of the first order, will involve one arbitrary or general function. For let u=F[x, y, z,

(v) ; consequently, du du between the three equations m = 0, -5- = 0, -5- = 0, it will be possible to eliminate ^ {v) and ^'{v), and there will result a re- lation f{x, y, z, p, q) — 0, wholly independent of the form of the function ^ (v) ; and it is evident that in general more than one function cannot thus be eliminated. Conversely, an equation of the form f{x,y,z,p,q)=0 being proposed, its complete integral to have all the generality possible must be of the form F{x, y, z, {v)} = 0, where the form of {v) is arbitrary. For example, let u = z + mx + ny — tp {v) = 0, where v= {x-a)* + {y -hy + {z - cY ; then ^=p+m-'{v){2{x~a)+2{z-c)p}=0, '^= q + n-'{v) {2 {y -h) +2 (z-c) q] ==0. dy 140 Hence, transposing and dividing one result by the other to eliminate '{v), we find p + m _(x — a) + (g— c)p q + n [y — b) + {z — c) q' ox p{y —h —n{z — c)} — q[x — a — m{z — c)} = n {x — a) — m {y — h), the partial differential equation of which the complete integral is z + mx ■\- ny = ^ {{x — of -[■ {y — hY ■\- {z — cf}. 105. To integrate an equation in which only one of the differential coefficients of the first order enters with x, y, and z. Let the equation be Integrate it, considering y as a constant, and in place of the arbitrary constant G, add a function of y of arbitrary form. The resulting solution, containing one arbitrary function, will have all the generality that can be attained. Ex {x^+f) dz dx s' + y; 1 dz _ dx 1 = 0; ■ ^'+f x'+f -12 . .,x . • . tan"' tan"* — = tan"' ^ (^) > or z-x={y^ + xz)<^{y), ^ (y) being arbitrary in form. The equation f{x, y, z, q) =0 is similarly integrated, the correction in this case being an arbitrary function of x. T-i dz ^ Ex. xy -J- + m = 0, zY = if>{x). 141 106. To integrate the equation of the first order, Pp+Qq = B, P, Q, and R being functions of x, y, and z. Let the primitive be F {x, y, z) = 0; therefore, denoting ^ F {x, y, z) by F' (x), and so on for the other dififerential coefficients, we get F'{x)+F'{z).j, = 0, F{y)+F{z).q = 0; .'. PF'{x) + QF'iy) + RF'{z) = 0. But dF {x, y, z) = F'{x) dx + F' {y) dy + F{z) dz = 0, .-. PF'{x)dx + PF'{y)dy + PF\z)dz = Q; and PF'{x) dx + QF(^) dx + RF'{z) dx = 0; .'. F'{y) [Pdy - Qdx] + F'{z) [Pdz - Rdx} = 0, (1) which is satisfied by Pdy- Qdx = Q,\ Pdz -Rdx-. ;:::} «• Suppose that by integrating these equations either separately or conjointly, we obtain M— a, N= h, two relations between the three variables and the arbitrary constants a and h, which satisfy them. By means of the equations M= a, N= b, any two of the variables as y and z can be expressed in terms of a and b and the third variable x. The complete primitive then becomes F{x,y, z) = {x,a, 5)=0, and the differential of ^ {x, a, h) must by virtue of equations (2) be identically equal to zero, therefore ^ {x, a, b) cannot contain x, and consequently = <^ (a, J) = Jd'-z'dx=0, or -===+— =0; V a — z •'■ Z -IT sin" — 7 sin''- + loga; = logJ, or xe « = 6; 144 but o = Va^+y, therefore sin"' - = tan"' - ; and b=f{a), hence are**""' ^ =/(V?+p), is the integral of the proposed equation. Ex. 2. J) {x — a) + q {y — h) =z — c. Here the reducing equations are {x -a)dz — [z- c) dx =0, {y — h) dz — {z - c) dy = Q ; X — a -h z — c z — c y — h _ .f x — a \ ' ' z —c ~^ \z —cj ' Ex. 3. mp + nq = l; .'. mdz — dx=0, ndz—dy = 0; .'. x — mz = a, y — nz = P; .'. y — nz-=f{x — mz). Ex. 4. {x — mz) p-\- {y — nz)q = 0; .'. {x — mz)dz = 0; .'. z = a, [x — mz) dy — {y — nz) dx = 0, or (a; — ma) dy — (y — na) dx = ; x — ma , , , .,,. .fx — mz = 1; but a=/(6), .-, z=f y — na " " \y — nz. 5. px -\- qy = nz, z= a?f y x z' ^ "^ \xj 7. rx^^qf = z^ i=i+/g-i). 8. z —j>x —qy = n Vx'' + y" + z", a:-'(3 + V^?T7TP)=/(|). 145 9. p{y + x) + q{y-x)=z, tan-^ - log (x^ + /)*=/ y "' ^' •'Wx' + y ,. dz dz dz dz „ /./a; y u 111. The following examples require artifices in the com- bination of the reducing equations. Ex. 1. (2as + cx) p+ {2bz — cy)q= {ay + hx) c. The reducing equations are {ay + hx) cdx — (2a3 + ex) dz, {ay + hoc) cdy = (25s — cy) dz. Multiply the first by y, and the second by x, and add them together, then cydx 4- cxdy = 'izdz, or z^ — cxy = a. Multiply the first by — b, and the second by a, and add them together, then — hdx + ady = —dz, or z+ ay — bx = ^; .: z + ay —bx = ^ (2" — cocy). Ex. 2. {y — b — n{z — c)}p-[x—a — m{z — c)}q = n{x — a)—m{y — b), the equation to surfaces of revolution. Here the reducing equa- tions are {n{x-a)-m{y-h)]dx-{y-b-n{z-c)]dz=Q, (1). [y -b-n{z-c)] dy + [x-a-m{z- c)] dx = 0, (2). [x-a-m{z-c)}dz + {n{x-a)-m{y-b)}dy = 0, (3). Multiply (1) by a;- a, and (3) hy y-b, and add, then [n {x-a)-m {y-b)] {{x- a) dx^ {y - h) dy+{z-c) dz}=0; .: {x-a)dx+ {y -b)dy+ {z-c) dz = 0; or {x-a)'+{y-hy + {^-cY^ct. 11. T. E. ^^ 146 Multiply (1) by m, and (3) by n, and add, then {n{x—cL) —m{y — h)\ {mdx + ndy ■\-dz)=Q-^ .'. mdx + ndy + dz=0, or mx + ny + s=^ =/(<") ; .-. s + mx+ny=f{{x-aY-{- {y-by+ {z-cf}. Ex.3. (y + z)p+{x + z)q=x + y. The reducing equations are {y + z)dz = {x+ y) dx, {x + s) dz= {x+ y) dy ; .*. {y — x)dz = — (x+y) {dy — dx) {x + y) {dx + dy + dz) =2 {x+y + z)dz; dx + dy+dz_ 2dz 2 {dy — dx) _ x+y +z x+y y — ^ ' ••• {x + y + z) {y-xY = a. Similarly, {x+y +z) {y — zy = ^; .-. F{{x + y + z).(2/-x)% {x+y+z).{z-yy'} = Q. Ex.4. {x + y + z)^+{t + y+z)^ > dz + {t + x + z)-^ = t + x + y. The reducing equations are {x + y + z)dz = {t + x+y) dt, {x + y + z) dx= {t + y + z) dt, {x + y+z)dy={t + x + z)dt; .-. {x+y + z) {dz-dt) = {t-z)dt, {x + y + z) {dx — dt) = {t — x) dt, {x + y + z) {dy - dt) = {t-y) dt, {x + y + z){dx + dy + dz + dt)=3{t + x + y + e)dt; 147 , dz — dt dt dx + dy + dz + dt hence = = —:r-, — r-; t-s x + y+z 3(x+3f + z + «)' .*. {x + y + z -\- 1) . {z — ty = a; similarly {x + y-{-z + t).{x-ty = ^, {x+y + z + t).{y-ty = y; .-. {x + y + z + t).{z-ty = F{{x + y + s + t).{x-tY, {!^+!f + s + i)-{y-ty}- Method of solving linear Partial Differential Equations by separation of symbols. 112. As the symbols of operation here employed -7- , -r- are independent of one another, and as they are subject io their combinations to the same laws as algebraical quantities, we may by separating the sjmabols of operation from those of quantity readily effect the solution of many partial differential equations by the same processes as were employed for differential equations between two variables of the like class. In particular the whole class of Linear Partial Differential Equations bears a strict analogy to equations of the same sort between two variables, both in the mode of treatment which they admit, and in the form of the solution obtained. Great use will be here made of the symbolical expressions for Taylor's Theorem, applied to a function of one or more variables : viz. d f{x + h) = e •^f{x) , fix + h,y)=e ^/{x, y), f{x + h,y + h)=e^""^'f{x,y); and this use of the symbols e '^, e "", must be particularly attended to, as having an important bearing on the interpretation of the results obtained. For since, when only two independent variables x and y are involved, after every integration relative to a; an arbitrary function of y, ^(y), must be added instead of an arbitrary constant, if an operating factor such ^^{j^-^-^. 148 be applied to a subject /(a;,y), we shall have by Art. 53, (taking notice that in operating relative to x, we must treat y and the symbol -j- like simple constants ; and the same holds relative to X and -7- when integrating or differentiating relative to y), Tx-"" 1^)"-^^^' y^ = '"'^^ Udxe- (j,)] = e'^dy fdxf{x, y — ax) +{y + ax) , where we see that an arbitrary function of a "binomial, viz. ^{y + ax) here takes the place of the term Ce" consisting of an arbitrary constant multiplied by the exponential quantity e"^, which is always met with in the solutions of linear equations between two variables. In the following examples we shall often for the sake of convenience write d, d', d" instead of -7- , -J- , -J-, suppressing the differentials ; and arbitrary func- tions of x and y will sometimes be denoted by c,,, c„. 113. We shall begin with the linear equation of the first order having constant coefficients. dx or e° 02 = e~ « '^ ^'%^+3+*(*4') Suppose f{x, y) to be a rational integral algebraic function of n dimensions ; 149 then z = hl + l{ad + hdyrf{x,y) = -{l--{ad + hd')+-,{ad+bd'y-&c.±\{ad + bd')"}f{x,y) c c c c . , bx\ + e where we stop with the «"" term of the development of the operating function, because any power of cZ or d' higher than the w"", applied tof(x, y), produces zero. Hence iif{x,y) =g, a constant, 9 . -5J. J / bx\ z = ^^ + e.,j>[y--); imless c = 0, when s —^ + (y J; and if/(a;, y) =gxy, xy ay + hx 2qab _h , / bx\ -=3-j-ff^-^+y+e''4>{y--). Again, supposing f{x, y) to be equal to ye""*"", or to g cos [mx + ny], the values of « will be found to be, respectively. .=-^ am + bn + c ^ "P\^ a)' c COS (mx + ny) + (ma + nb) ain (mx + ny) , _2., , , s = q. ^^ ^-rj ; — iXa ^^ ^ + e <'d>{ay-bx). ^ c + {ma + nby ^^ " ' 114. K the proposed equation be then assuming 2 = e°, or v = log z, where v denotes a new fimction of X and y, the equation is reduced to the same form as the pre- ceding, viz. : 150 hence the solution is ae '■\osz = e-^fdxe^f(x,y + ^-^)+,f>(y-^-^y Thus let fix v)= "^"^^^ ^], a5 + /3« __g. •^ ^ ' ^' (ay— bx) y bay — hx by' .f bx\_ a.b + ^a I a^ 1 ••J\^,y ^J— ^ -y b ay + bx' , ah+0a X «/8 1 / \ , . / ix\ ... „iog,= _^.__^-_log(a3,) + .|,(^y--j is the solution (where c = 0) of dz ,dz _ay + ^x e dx dy ay — bx'y' A particular case of this is dz dz az bz -J TO-j- = h dx dy y mx + y ' a hx for which z=3r "" .^'*^ .^{y + mx). Again, let f{x,y) = ^+^ + ^, then Jdxf{x, y + -^j =a\ogx+~\og[y + -^j + 27 v^iog (^^^ + \/f + «=) ; .'. aloga = aloga; + ylogy + 27y'|log^Vx + y'^j is the solution (where c = 0) of dz ^ ,dz /« , /8 , 7 \ ria; rf!/ Va; y Nxyi 151 A particular case of this is dz dz /a yS Ixy dz dz _ /a p 7 \ dx ^ dj/~ [x y V^/' for which = a:;"^™U/a;+ a/^J '".(f>{7/ — thx). 115. Next let an equation involving three independent variables be proposed, dz dz J dz „ or {d+ad' + Id") z = ;^yt ; _n_ ad' + hd" 2abd'd" \ ~ \d d'^ + d' J ^•^'' these being the only terms necessary to be preserved in the de- velopment of the operating function, since d'^yt = ; therefore, effecting the operations indicated, e = -^x''yt- — {at + by)x* + —abx' + {y-ax, t-hx); the complementary function being ^-M^^^,^ (y, t) = ,i>{y-ax,t- hx). 116. We come next to the case of the complete linear equa- tion of the second order with constant coefficients [a^-ifhdd' + cd'^ + ad'+fid + y) z =f{x, y). This may be integrated whenever the operating function can be resolved into two factors rational with respect to d and d'; the condition for which may be shewn, by putting the operating function equal to zero and solving it relative to d, to be {V - 4ac) 7 = Ja/3 - aa? - c^. Thus, suppose 5 = 1, a = c = 0, then 7 = ay3, and {dd' + ad' + ^d+ a/3) z =f{x, y), or 152 d . \fd i^^M^^y-f^^^yy' ■ + ^)z = (£ + ayfix,y)=e-^Jdxe'-f{x,y)+e-"0,, d_ Hence, changing the arhitrary functions, we find z = e^^^ldyi" [Sdxe'^fix, y)] + e'^'c, + e^'c',, for the solution of d'^z dz nds . o j-i \ Hence if /(a;, y) =6"^"", since in this case -j- and -j- are equivalent respectively to factors m and n, (wi + a) (w 4- yS) " Again, let f{x, y) = e"" cos ny, then {dxe^fix, y) = ; ^; '. z=- m + a J dy^" cos ny e"" n sin ny + ^ cos ny _^ ^.^^ ^ ^_p^ m+a n' + ^ + e-"c, + e-V,. Similarly, if we suppose a = l, 6=7 = 0, then c = — ^, and the equation is { (y) + e-j'^ {y) = e""'.^ (y + ^) + >^ (3^ - ^) z for the solution of d^'~Wdy'^dy-'^- 153 If J" = 4ac, we must have 6a/3 = aa" + c/S", or l^ = 2ai; then the proposed equation becomes {(Va^+Vo 4a7. 117. Kext, let the equation be of the w"* order the index of differentiation being the same in every term ; and V denoting a function of x and ?/ ; then the equation may be written Let a be a root that occurs singly, and b a root that occurs m times, in the auxiliary equation v" +j?, v''''+py^+ . . . +i>„ = ; then by resolution into partial fractions we shall have Hence substituting -y, for v, we get (Ad , BJ- , B^,d-' B,d .oAj-^v .-. z = Ae'^'Jdxe-°'^'d"^*' V+ BJ^' {Jdx)'"e-'"""d'-'*^ V+ &c. Instead of reserving the complementary functions under the sign of integration, we may obtain them explicitly by supposing V=0; then since jdx = c„ Udx)'^o = c; + xc;' + x\' + ... + x^-'c;-"-", H.I.E. 20 154 the part of the complementary function due to the root a and the m roots h is, changing the arbitrary functions, ^c,^.+ B^ {c\^+ xc\^,+ x'c",^,+ ... + x^-'c';:;^) ; and similar terms will be introduced by the other roots of the aiixiliary equation. ••• s = j'Tn'd'^ =-d-^[l+ n' ^^ j + &c.} cxy = d^cxt/, all the other terms being neglected, because when the operations are performed they vanish of themselves ; _ c 3 , •'• ^ — 'n ^ 2/ "^ ^v+nx'i' (^ i/-nx- \dxj \dxj dy dx \dy) \dyj ' or {d-2d'y{d-dyz = e"^''; since d and d' operating upon e""^" are equivalent respectively to tlie factors m and n. If the second member of the equation be x^+y^, then ^~d^-i(fd ^t32Z'=^(i+4+^4>-'+^)' these being the only terms in the development of the operating function which it is necessary to retain ; _ a;y x*y x' „ -n „ '^''2 , ,, d'z ^ d^z ., Ex.3. ^,+ (a + &)^^ + a6^=/(x,3,), or {d-\-ad) {d+bd') z =f{x,y) ; 155 = ^ {e-'-''Jcbrf{x, y + hx)- e-^^'Jdxfix, y + cuc)}. Let f{x, y) = 0, then {d + ad') {d + Id') a = gives z = e--',/, (3^) + e--^> (y) = ,/, (3, - aa:) + 1 (2/ - hx), which shews the form of the complementary functions, whatever be the second member of the equation. H f(x, y) be a rational integral algebraic function of x and y, then z=\d-^-dr'[{a+h)dd-+abd'^}+&c.]f{x,y)+^{y-ax)+^{y-hx). Thus let f{x,y) = c-'tm{x + y)-\-nocy, then z=[d^-{a + h) d'^d'] {c + m{x + y)+ nxy] = -{c + my) x' + -{m+ ny) a;' - — (a + J) (4ot + nx) x' + ^ (3^ - aa;) + \|, (y - hx). Change of the Independent Variables. 118. The method by changing the independent variables is applicable to partial diflferential equations, and will generally lead to the solution of any equation of a form analogous to one of those for which the method succeeds in the case of differential equations between two variables. The change of the inde- pendent variables that has the most extensive application is to assume x = e', y = e', as we proceed to shew in the following instances, making use of the formulaj of Art. 70. _, dz , dz T7- Ex.1. ax -T- + by -J- + cz = V ; this becomes dz T dz _ _- ''dt+^dv + ''=^' the integral of which has already been found (Art. 113). Let V^x'-'y^ then U=e'^'"; mt+nv et / lx\ . -T +e"6v--), am + bn + c ' \ a J z=- 156 or, restoring the values of ( and v, and changing the arbitrary function. «"*^.'* z= ^ \-x 'dil^, am + bn + c ^ \x Ex.2. -=^+(a + J)-y^^ + ¥^= + -^ dz + cy-3 — CZ = 0. This, by the same substitutions as in the foregoing example, becomes ad{d-l)z + {a + i) dd'z + hd' {d' - 1) z + c {d + d' - I) z = 0, or {ad + hd' + c) {d + d' - 1) z = ; M-dTL = e~°^ [v J + e'l^ {v-t); hence, restoring the values of v and t, z = x~''i>{2/x'h + ^[l). Ex.3. -'£ + (2a + i).^yJ|^ .-. [ad {d- 1) («Z - 2) + (2a + b)d{d-l) d' + {a+ 2h) dd'{d' - 1) + bd'{d' -l){d'-2)}z = 0, or {ad+bd'){d+d'-l){d + d'-2)z = 0; = <^fr \-\-e'y^{v-t) +e''x{v-t) 157 Ex 4 a:""^'^ I nx-'i/ ^'^ I "^""^^ x"-^^' '^"^ I &c •*• "" ^x" + "^ ydx-'dy^ 1.2 ^ y dx^'-'df^^^- + y» j-5; = 0. Here we have (Art. 70) {d{d-l) ...{d-n + \) + nd{d-l) ...{d-n + 2)d' + "^—^ d{d-l) ...{d-n + Z)a {d' -\) + &c.} s = 0, or {d+d'){d + d'-l) ... (! 1 (a — 5) a;] .-. z = \a,n \. -^\a J ^ &a y-lf-mx ] -J-, 1 dz m dz _az hz cz let a!" = «, y'' = v; dz dz f a h c \ .-. -7, + w-T-= — + — I — 7=^ s ; at av \nt nv nStvl .-. z = e.v^-. (V< + y^i z,)""^,^ (t, - mt) ; or a 120. Since when operating relative to x we may treat y and the symbol -%— as simple constants, and vice, versa when operating 159 relative to y, we may often solve a partial differential equation by integrating it with respect to one of the independent variables, and introducing arbitrary functions of the other instead of con- stants ; and then efifecting the operations indicated relative to the second variable, and interpreting the result in the usual way, which will be attended with no diflBculty when only the expo- nential symbols e "^ or e '" are involved. Ex. 1 . -3-5 — a^ -3-2 = ; this may be written ^,- {ad-yz = 0, or (£-ad')(J^ + ad')z = 0; .-. z=e'^ ^{y) + e'""' ^ {y) =^{y + ax) +yk {y- ax). TT d'z ^dz . d'z Hence «.^, + 2^ = aa.^, which may be written J ^' - a — -^-^-^ = 0, gives xz — ^{y-\- ax) +^ {y — ax). -n « td^z „d^z ^ .d^z . j,,.j Ex.2, aj^-s — a^-r = 0, or x* -^, — (ad) z =0; dx drf dx .■."- = e^'{y)+e-"^U{y)=[y + l) + ^[y-i (Ex. 2, Art. 69.) 2 . -"'^'='^-^; cfa TO(m+l) J1.X. 5. ^^- — ^ - « ^2 = <>' ^, = (^3 1^)- <^(y + «")^+f(-V-«^) . (Ex. 3, Art. 74.) -r, „ „ d^z „ dz „ , ~ o^s ., Ex. 6. a;" -t-^- 2inx ^r + ^mz — oV -5-5 = 0, dx dx ^ dy ^■^,= (^^j,y >(y + y^)+^(y-g^) . (Ex. 2, Art. 74.) If m = — 1, in Ex. 4, we get the solution of d^{xz) 2xz ^ d^jxz) dx' x" ^ df ~ ' ^(Pz „ dz „ ^ «dz or x -j—„ + 2x-j 2z — q d^ T-g = 0) viz. dx dx dif x^z = qx [^' [y + qx)-^' {y-qx)]-<^{y + ' {t + v}- tyjr' {v- t) - <}> (t + v) -^k [v-t) ; .: z {x+yy= {x+y) \' {2y) -y}r' {- 2x)} - ,p {2y) -Vr(_2a;). 161 Ex.8. g + ^^_a'^ = 0. (Ex. 3. Art. 74.) Hence if ?i = 1, we get, as in Ex. 1, X3 = (y 4- ax) + V^ (y — cw) ; and if n = 2, we find x'z = ^ (y + ax) +yjr{y — ax) —ax'{i/+ ax) + axyfr' [y — ax). Ex. 9. a'" 2 + ^"^^ £ — ^^ " ^- ^^^- ^- ■^- '^^•) •JO g Ex.10. (.^+y|y.-2(.^4-y|). + 2. = a,V. ^ = (^ + »-lKl + .-2) + "'^(f)+"'-^(|)- Ex. 11. fix) . g + m^) • I - «' J = 0, (Ex. 1. Art.48) z = {y + ajdx Ux)-i} + ^ [«/ - a/^ir {/(x)}"*], where /(a;) denotes a given function of a:. Ex.12. ^-«'.= i^ = 0. z = X* {^'{y + 3axi) +ir'{y- Sax^)} - ^ {«/>(y + 3aa;*) - -^ (y - Sao:*)}. E-'3- S^/(')-^,-^|-ME....A«..8) z = x/- If from the proposed equation we determine ^ dg dq ^'dx' dz' equation (2) becomes an equation for the determination of p of the form dx dy dz the integration of which depends on the integration of one of the equations dp — Ldx = 0, dy — Mdx = 0, dz —Ndx — 0. Let p =f{x, y, z, a) be found from these equations, a being an arbitrary constant. This value of p and the corresponding value of q found from the proposed equation, being substituted in dz = pdx + qdy render it an exact differential ; and thus a value of z will be obtained involving two arbitrary constants a and b ; and this will consequently be the complete primitive. The general primitive may be obtained by putting b = ifr{a), differentiating the equation with regard to a, and eliminating a. The result containing one arbitrary function is as general as any solution which the equation admits. Ex. 1. / + 2= = 1, dq _ p dp dq _ p dp 2 — V 1 p , ^^ tJi_n^ dx' dz Vl — io' <^2 ' dy + ^^-P dz^Vr^" dx^ ->JT^p^ dz~^' 163 which is integrable if we can integi-ate the system of equations dp = 0, pdz — dx = 0, '^\—p^.dz — dy = 0. The fii-st gives p = a, whence g' = Vl - a", .•. dz = adx + 'J\—cfdy, or z = ax + '^l — a^y + 'ia); between which and the preceding we may eliminate a, when the form of the function (a) is assigned. This is a simple case of the Problem of finding surfaces of equivalent area to a given surface, that is, such that any cylindrical surface parallel to the axis of z, may always intercept equal areas in the required and given surface. If P, Q denote the values of -3- , -5- in the given surface, the equation of condition is / + 2= = P»+<2^; which, if the given surface be a plane, leads to the equation of this example. Ex.2. p'"q' = c''- The equation in p to be integrated is dx mc^ ay \7n j-^ dz .-. dp = («), 164 = «a"-'a; - -s+i ^ + ^' (a) ; and it remains to eliminate a between these equations when the form of {a) is assigned. If the proposed equation be and we take new independent variables x and y' such that dx' 1 dy' e ^- ©"(1)"-- dz' -Y Now assume s' such that -5- = a"*™ , then the equation becomes /&Y' (dz\_ [dx'J -{dy') ~^' and is reduced to the preceding. Ex. 3. z =pq ; here, when the form of ^ (a) is assigned, a must be eliminated between the equations, = x + 4,{a), /„!^x. = -'^'W- 2/ + a ^^ ^' (y + a)" Thus if «^(a) = - , then it will be found that z = {c + V^)'. Non-linear Ei^uations of the Second and Higher Orders. 122. Here, besides the coefficients p and q of the first order which may be involved together with x, y, and 2, the equation must contain one or more of the coefficients of the second order r, s, t; so that in its most general form it will be F {x, y, z, p, q, r, s, t)=0. In partial differential equations of the second order, we can- not be certain of the form of the solution, nor pronounce before- 165 hand how many arbitrary functions it ought to contain. For let be an equation containing two arbitrary functions of v and w two known functions of x, y and z ; then p and q will be given by the equations ^ = — = 0- dx ^ dy ' and r, s, and t by the equations d^u _ dd _ d'^u _ dx^ " ' dxdy ~ ' dy^ ~ These together with w = make six equations, into which the six quantities ^{v), -^{lo), <})'{v), f{w), <}>"{v), ylr"{w), may enter. Consequently, it will not generally be possible to eliminate these six quantities and obtain a relation between X, y, z, p, q, r, s, t, independent of the forms of the functions and ^fr ; although particular cases do occur in which this can be effected. In general, if m = contain n independent variables and m functions of the form m{r+l), the former quantity expressing the number of equations, and the latter the number of functions. 123. As an example of forming a partial differential equa- tion of the second order, and of the sort we are now considering, by the elimination of two arbitrary functions, we may take u = y — x^{z) — 1^(2) = ; ^= l-x(f>'{z)q--f{z)q = 0; 166 •■• i' + 2'^ («) = ; .-. r + s(f) [z) +pq{z)+q''(f}'{z) = 0; .-. qr -ps + {qs -pt) (f> (z) = 0, or q\ — Ipqs +pH = 0. 124. The eqxiations (Pz , ^dz ^ d'z —dz ^ „ where P and Q are functions of x, y and z, must be integrated as equations between two -variables, y being regarded as constant in the former, and x in the latter ; and arbitrary functions of those variables respectively, being introduced instead of con- stants. {l-x)z = x'^<^[y)-^>i^{y). 125. The equations dd ^dz ^ dxdy dx dd -ndz _ dxdy dy where P and Q do not contain z, are reducible to the case of Art. 105, by considering j- or -5- respectively as a simple quantity v. Tj, dd ,_E^—— dxdy 1— y '^ ' . dz . . This being a linear equation in -^ which is made integrable dx by the factor , , we have 167 ••• z^[^{x) + ^ {y)] Vnp - y (2 + /) (1 - f). 126. Similarly, equations of the forms x( dz d^z J. d''z\ ^ fix V z ^ ^ &c — Uo may be treated as if they contained only two variables ; arbi- trary fimctions of y instead of constants being introduced into the solution of the first, and arbitrary functions of x into the solution of the second. To this case may also be reduced the equation ■^r'^'^' d^' Wdf ' ' dxT'dyV ~ ' d"z for by putting -7-^ = v, it becomes ,/ dv d'v J. d^v\ ^ dx ' dx' ' ' dx^^J which will give a value of v containing m arbitrary functions d"z of y ; and then -j-^ = v will give z involving n arbitrary functions of x. 127. To integrate the equation of the second order, Itr+Ss+Tt= V, where B, S, T, V are functions of x, y, z, p, q. The following process first given by Monge, may be frequently applied. By means of the relations dp = rdx + sdy, dq = sdx + tdy, eliminating two of the three coefficients, r and t, from the pro- posed, we get Rdpdy + Tdqdx - Vdxdy = s [R {dyf 4- T{dxy - Sdxdy], 168 which is satisfied by Bdpdy+ Tdqdx— Vdxdy = i R {dyY + T{dxy- Sdxdy = J ^ Let M=a, N=b, be two relations between x, y, z, p, q, and the arbitrary constants a, b, which satisfy these equations ; then M= [N) satisfies the proposed equation. This will be shewn by proving that it can reproduce the proposed equation. Let dy = mdx ; .'. iJm" — 8m + T= 0. For each root of this equation, we have dy — mdx = 0, Rmdp + Tdq - Vmdx = ; .*. dy = mdx, y Vm , Bm, , , , di=^ax--jrdp, )■ (2) dz =pdx +qdy. Hence M=a gives on diiFerentiation ^ dM , dM , dM , , , , = -5— . ax + -J— . max + -5— [pdx + qmax) dM J dM /Vm , Bm , \ + df^^+d^-[-T^''—T'^^)' wherein all the known relations (2) having been introduced, dx and dp must be independent, „ dM , dM dM , , , VmdM dx dM_BmdM dp T dq' dM dx ( dM . .dM Vm dM\ dM _ Bm dM dp~ T dq' 169 ^ dN { dN , .dN VmdN) dN^BmdN dp~ T dq' By differentiating the assumed eqiiation M= ^ {N) we have dM= ' (N) . dN. Now ,,, ( dM , ,dM VmdM] ^^=-r-^+^p+"'i^-d^+-T^ , dM^ dx + ^dy dM , J - . Bm dM , dM , + —{pdx + ady) + ^-^.dp + -^.dg_ [dm dM\ , - J, s 1 dM , „ , „ , tt ■, s " Vd^'^^'d^j i^y-""^ +Y di ^^'^p+ ^^- V»^)y and a similar value exists far dN; (dM dM\,, ,, ldM,„ , ^, T^ , ^ .•• (-^ +q -j-j [dy—mdx) +-j,-y-{Bmdp + Tdq— Vmdx) + -y,-^ (^wic^ + Tdq - Vmdx) [ . which may be put under the form Rmdp + Tdq — Vmdx — w {dy— mdx), or Rm {rdx + sdy) + T{sdx + tdy) — Vmdx = a>{dy^ mdx), where dx and dy are independent ; .•. Bmr + Ts — Vm = —otm, Rms+Tt = (i>; .: Iir + Ss+ Tt= V- - {Em'- 8m+T) = V. m ^ Hence, M=^ (N) satisfies the proposed equation. H. I. E. 22 170 According as the roots of Rm^-Sm+ T=0, are unequal or equal, we are thus supplied with a total or partial differential equation for the determination of z. As the reducing equations (1) may contain and as these together with dz =pdx + qdy will generally lead to an equation containing three variables, which will not always admit of a single primitive (Art. 98), it may happen that the first integral of the proposed equation cannot be determined ; but we must not thence conclude that the proposed equation does not admit of being solved. 128. Hence, to integrate the equation of the second order Rr+8s+ Tt=V, the process is to obtain a value of m from the equation Em' - 8m +7 = 0, to substitute it in the system dy — mdx = 0, Rmdp + Tdq = Vmdx, (1) to satisfy these conjointly or separately, by two relations between ^, y. 2> p> !?> M= a, N= b, then to put M= (iV) which will be a first integral of the pro- posed, and to integrate this equation of the first order. But this determination of the second integral from the first will often be attended with great difficulty, on account of its involving an arbitrary function; and therefore when possible it is often more convenient to find from the second value of tn another first inte- gral of the form M' = a/t [N'), and between these to eliminate ja or q, so as to obtain an equation involving only one differential coefficient, and which is therefore easily integrated. 129. If R, S, The constant, and V a. ftmction of x and y only, then the values of m will be numerical, m and n suppose ; and the integTals of equations (1) will be y —mx = a, Rmp + Tq = mJdxV+ b. 171 where, previous to integration, mx + a is substituted for y in F, and after integration the value of a, viz. y — mx, is restored : con- sequently calling this value Fj, since Bmn= T, we have a first integral of the proposed, viz. Bp + Rnq =V^ + '{y - mx) + yS, nx + a. being substituted for y before the integration is performed, and afterwards the value of a, y — nx, restored ; this gives jdxV^— F,, suppose, and jdx<^' [y — mx) =Jdx(f>' [{n — m)x + a]= (^ — "*^) ; hence, including the constant multiplier under the sign of the function, we have the complete integral required, jRa = F, + (y — mx) + -^{y — nx) ; which agrees with the result obtained at once by separation of symbols from the proposed when put under the form _ , d'z ,d'z „ , Ex.1. -T^-<^-3-r, = 0, or r-c't = 0. dar dy The two systems of reducing equations here are dy + cdx = I dy — cdx =01 dp + cdq = ) ' dp-cdq = )' from the former we get y + cx = a, p+cq = b; .•. p+ cq = 2cq + f} r + {^ -p') s - {I +pg^+p')t=0, or (1 +qa)r-ir {q —p)as— (1 +pa) t = 0, putting p+q= a. The values of m are to be obtained from the equation (1 + qa) rt^ — {q —p) am — 1 —pa = 0, which gives l+poL , l + qa' and the two systems of auxiliary equations answering to these values of m, are dy — dx=Q, {l+qa)dp—{l-\-pa)dq = G, {\+q(i)dy-\-{\-\-pa)dx=Q, dp + dq=0. (1) The latter gives p + q = b, or a = b, x+y + hz=a; .: x + y + {p + q) z = {p + q). The former system gives y—x = a^; and if we assume p — q = ^, the second equation of this system becomes (a= + 2) JyS - a^da = 0, whence /3 = Z>, . (a" + 2)' ; .:p-q={{p + qy + 2Yf{y-x). 174 Next to integrate this equation of the first order, since we have dz = \a {dx + dy) + ^/S [dx — dy) -=\a.{dx + dy)+\{cbi-dy) {a? + 2y -^ (y-x), this is integrable if we suppose a to be constant, and gives z + {a)=^a.{x + y) + {«' + 2)» >/', (3? - a) ; which, combined with '{a.)=^{x + y)+ jL-^ .^^{y-x), Va^+ 2 represents the integral of the proposed equation. IntegTBtion by a Series. When other methods fail, partial differential equations, like equations between two variables, can be sometimes integrated by a series ; and we shall now give an example taken from Euler of the process. 130. To integrate by a series the equation hz + a{x + y) {p + g)+[x+ yf s = 0. Let z=A{x+y)'"{x) +B{x+yr*''{x)+ C {x+y)''*^"{x) + &c.; and as the integral must be symmetrical with respect to x and y, we must, after the determination of m, B, C, &c., replace {x), (f>'{x), &c. by (f> {x) + -^ (y) , i + 7t = 7i — z is a whole number, or when b= {a +i) (a — t — l). Thus let 7» = — a, and therefore J = a (a — 1) ; it is evident, from the formation of the series, that B= C= &c. = ; ••• a = (a; + 2^r{H+^(3^)}. Let 7M = — a — 2, and therefore & = (a + 2) (a — 3), then "- {x + y)"^ 2 {x + yY^' 12 {x+y)' ' If J = 0, the equation becomes a{p + q) + [x+y)s = Q; and we must have a = — i or = t + l , and therefore m = 0, or »» = — 2* — 1 ; in the former case z = (i>{x) + My) -hi^+y) Wi^) + ^'{y)] in the latter, we have {x + yy^^z equal to the same series. 176 To this fonn may likewise be reduced the equation (ana- logous to that of Riccati and integrable in the same cases) by introducing new independent variables, when it becomes {u + v) --j—i +w-T-+m-T-=0; and hence s may be found in a series. But when m is a positive integer, the solution may be rea- dily obtained by assuming x = t'*'"" ; thpn is transformed into d^ _ 2m dz _ ^ d'z W '^~r'dt~'^'df' Hence, for the upper sign, we get (Ex. 5. Art. 120), /„ dy' {y + ct)+^{y-ct) _ where, after performance of the operations indicated, we must replace t by a;'"**"; and for the lower sign the solution is (Art. 65) ^f^-'^ i^ (t^ ^V '/'(y + gO + ^(y-cO ] . "'^ -Jt\lVdi) f^' j' where, after performing the operations indicated, we must replace t by a;'"^'". The latter result is obtained from observing that, if M be a solution of -j^ f- — c'y = 0, then a;""" -7- is a aar x ax ^ dx , . „ d'y 2m dy , solution of -5^2 4 -^—cy = 0. 177 Simultaneous Equations. 131. A system of partial differential equations which in- volve two dependent variables z and u, provided they are linear and have constant coefficients, may be treated in the same manner as simultaneous differential equations of the same de- scription containing only one independent variable. For the two symbols of differentiation -7- and -5- , since they do not affect each other, and obey in their combinations the same laws as ordinary algebraical quantities, may be treated as two inde- pendent constants. -c , dz dz ^ du ^ Ex. 1. -j--\-a^ + bz + c-j- = 0, ax ay dx du du , ,dz„ -j- + a^- + l>u + c -y- = 0. dx dy dx Separating the symbols, and for convenience writing d and d' instead of -7- and -y- , these become dx dy {d->r oJ' + &) 3 + cdu = 0, ((f + (W^' + J) M + ddz = 0. Now substitute in the former the value of u obtained from the latter, regarding d and d' as ordinary constants, and we get {{d^■ad!^-})f-ccd''\^ = 0•, or, if we denote 1 — 4od by — and 1 + "^cc by - - {d+m [ad' + b)] [d + n {ad' + b)}z = 0, the integral of which is z = e"^ (^ {y - amx) + e^-^iy - anx), where m and n have the values above-written ; and from this u can be found, -^ J'z du „ d^u ,dz dxdy dy dxdy dx d'z d^u ,^ d'z ,dz „ + « 3— J- = 0, or -j-Tj — ao - dafdy dacdy ' doe'dy dx ' ••■ z=fiy)+x4>(y) + e'-'^f{x). H. I. E. 23 178 132. If U=f{x, y, z, p, q)=(i be a partial differential equation of the first order, the singular solution if there be one, as in the case of two variables, will be found by eliminating p and q between the three equations -». f=«. f =»■ Ex. (s —px — qyf— o^ (1 +^' + 2°) = 0, which expresses that the surface represented by the complete integral has the per- pendicular from the origin on the tangent plane of a constant length. Here -^ = -{z-px-qy)x-ap = Q, = -{g-px-qy)y-aq = (i; eliminating p and q we find 0!= + / + «= = a". 133. We shall terminate this part of the subject by the following geometrical problems ; the first of them being the well-known solution by Monge of the Problem — to find a sur- face at every point of which the radii of curvature are equal and of the same sign. The conditions for this are expressed by the equations : Ex.1. -P—^='^^- -J_^ = l^, 1+^" dx q dx'' \ + ^ dx p dy' Integrating these, and replacing the arbitrary constant by Y a function of ^ in the former, and by X a function of w in the latter, we find l+/ = 2»r, \ + ^=fX; But as the object is to find z a function of x and y which shall satisfy the two proposed equations, the quantities p and q must by their nature satisfy the condition ;^ = ^ > which becomes \ dX \ dY (1 + X)« dx (l + r)» dy' 179 and is of the form ^{x)—-sfr {y), whatever be the functions X and Y; it cannot therefore subsist for all values of x and y which are variables independent of one another, unless each member is reduced to the same arbitrary constant, ■—=; suppose; we then have 2<7 dX , 2C dY re by integratioi constants, (1+X)« dx ' (1+F)« dy which give by integration, a and h being two new arbitrary Vi + x ' Vi+r" Hence the quantities X and T" become known, and then j? and q may be expressed in terms of x and y from equations (1), and substituting these values of p and q va dz =pdx + qdy and integrating, we get (x~ay-v(2,-by+{^-cy=c\ Ex. 2. To find the surface which cuts at right angles the surfaces represented by the equation xy + xz+ys = a", when a assumes all values. ?/ ~4~ z cc "^ z Here p——- . q = , and the equation of con- ^ x+y ^ ^+y dition is de dz , .dz , .dz the integral of which is {x + y+z).{y-zy=(^ = b^ + c, .: z^ + x' + y^^bx + cy, which represents spheres passing through the given point, and having their centres in the given plane. By integrating the auxiliary equations xdy — ydx = 0, dx {r — z) + xdz = 0, we get x{r — z) =tf>(^]. And if we wish to determine the arbitrary function so that when y = aa^, z shall equal r + bx, then c e Jf » :.• CyE' I'l A TEEATISE CALCULUS OF FINITE DIFFERENCES. SECTION I. DIRECT METHOD OP DIFFERENCES. Definitions and Principles. Art. 1. The Difference of a Function of one or more vari- ables is the result obtained by subtracting from one another the two values of the function that arise from giving to the variables contained in it, different assigned values. Thus if M^ denote any fimction of a variable x, and u,^ the same function of x+h, and if the value of u^ be subtracted from that of u^^, the result is called the Difference of u^ ; and the quantity h is called the increment of the principal variable x. In the Differential Calculus it is the first term only of the series, arranged according to ascending powers of h, expressing u^^—u^ or rather the coefficient of A in that term, with which we are principally concerned, and which we usually write h -5-^ . But in the Calculus of Finite Differences, it is the whole of that series which forms the object of our investigations, and it is usually written Au„ so that 2. In deducing the Difference of any proposed function m^, the increment of the principal variable x is always supposed to be known and finite ; its value however is not commonly taken H. D. E. 1 to be an iindetermined quantity h, but to be unity ; both for the sake of simplicity, and because that is the value which the increment must necessarily have, in order to pass on to the succeeding term, when u^ is regarded as the general term of a Series ; and it is in that light that it is by far the most frequently regarded in Finite Differences ; so that the operation which the symbol A prefixed to u^ implies is, for the most part, A«, = M,^, — u,. "°x+i There are, however, in this Subject, several important theorems which it is advantageous to investigate on the hypo- thesis of an indeterminate increment h for the principal vari- able, instead of unity ; as the process is the same on either supposition, and the result one of greater generality. And in any case if it should be desirable to introduce the same hypo- thesis, the expression f{x) must be prepared by first writing hz instead of x ; then when z becomes z + 1, hz will become h (s+l) or hz + h, that is, the corresponding increment of x will be h ; and if the requisite analytical operations be now performed upon f{hz), on the usual supposition of A2= 1, we shall get the result expressed in terms of x, and adapted to the supposition of an indeterminate increment for x, by restoring j in the place of z. 3. By a Series is meant a regular progression of terms increasing or decreasing in magnitude according to a certain law; hence, when that law is given, and also the place of any term in the series, the magnitude of the term may be found, and thus the successive terms of the series may be produced in order. The place of any term in a series is assign- ed by giving the number of terms by which it is removed from some one which is considered as fixed. This number is called the Index of the term to which it belongs. Thus in the aeries 0, 1, 8, 27, 64, ... x\... taking the first term as the point of departure, we have the corresponding series of indices 0, 1, 2, 3, 4, ... ,r, ... If the series be continned backwards, the indices must be considered as negative ; thus the backward continuation of the above series gives the terms ...-x\ ...-27, -8, -1, with the corresponding indices 4. Since the magnitude of every term is determined solely by its index and by the law of the series, it follows that any term is a certain function of its index, the form of which does not alter in passing from one term to another, but remains the same throughout the whole series. Thus in the above series every term is the cube of its index. This function analytically expressed is called the general term of the series ; (in the above series the general term is a;' ;) and it is evident that all the terms of the series will be produced from it in order, by substituting successively for the index ar, the progression of natural numbers ...-2, -1, 0, 1, 2, ... The general term of a series is usually denoted by u^, where u^ is a certain function of x determined by the nature of the series. Thus, u^ denoting the general term, the series will be Any group of consecutive terms u^, u^^, u,^, &c. are called successive values of the function %. 5. The excess of any term m^, above that which imme- diately precedes it, or the function u^^ — u^ is called, as has been stated, the Difference of the function u^, and is denoted by Amj,. (In certain cases, which will however be expressly mentioned, we shall take Am^. to mean m^^ — m^.) Hence the characteristic or symbol of operation A, prefixed to a given function of x, denotes the series of operations of changing x into a; + 1, and of subtracting from the altered value of the function the proposed value. It is obvious that u^^^ — u^ is itself in general a certain function of x, the nature of which is entirely dependent on that of the original function u^ from which it is derived, and is susceptible of a difference. The diiference, consequently, of the function Am,, (which must be considered as having Am for its characteristic, in the same manner as u^ has u) is A (Am,) = Am^, - Am,, which is usually written A'm,. In like manner A(AV) = AX = AVi-AV, A(AX) = AX = AV.-AX, 6. Hence if in any function of x we change x into x+1, and from the result subtract the proposed function, we obtain the first difference of the proposed function ; and the second, third, &c. differences are formed, each from the preceding, by a similar operation. To determine these differences of given func- tions, and to investigate the relations which hold between dif- ferences of any orders and the functions from which they are derived, is the object of the direct method of Finite Differences. We shall now proceed to give instances of finding the dif- ferences of various functions, according to the above definition. Differences of Explicit Functions. 7. To find the difference of au^ + c, m, being any function of X, and a and c quantities independent of x. A (aw, + c) = aM,^, +c— [au^ + c)=a {u^^ — m,) = a Am, . Hence, making a = 0, Ac = 0. 8. To find the difference of the sum of any number of functions of x. A (m^ + V^ + W^) = M^^i + v^^ + w^, - (m^ + v^+ w J = Mx+1 - M:r + Vl - ■"x + W^, - W, = Am^ + Avj, + AlCj,. If some of the fanctions he preceded by negative signs, we shall find a similar result, viz. A (m^ -v^ — wJ) = Am^ — Av^ — Aw^. 9. To find the difference of the product of two functions. A (u^v^) = M^it;^„ - u^v^ = (u^ + AmJ {v^ + Av^) - u^v^ = M^Av, + v^ Am^ + Am^ . Av^ = Mj, Av^ + r^, Am^. 10. To find the difference of the quotient of two functions. A C^ = ^' _ ?^ = K + ^%) Vx - fa + AQ u, _ «^Am,-m^Ai), 11. To find the difference of the continued product of any number of successive values of a fanction. A ("x «:«+, • • • Mx+») = «XM Wx+2 • • • Mx^«+l - «x «»+, • • • Wx+n Hence in the particular case where u^ = a + hx, since m^»^., = a + 6 (o; + « + 1) = m^ + (« + 1) J, A (MxMx+, • • • Mx4.») = «x«Mx« • • • M,^.„ . (« + 1) &. 12. To find the difference of a fraction whose numerator and denominator are the continued products of any number of successive values of two functions m, and v, respectively. [^ ^l~ VV ...V ^^^^"^'^^ ^xV^v^J. V^x'^x+l '•• "z+m' ^x'^x+t "x+m+\ Hence A = = ^^s? — ?^^- , and in the particular case where v^ = a + bx, and therefore ^'x+m+i = a + i{x + m + l) = v^+{m + l)b, ^ 1 _ (m + l)5 13. To find the differences of any rational integral function, and to shew that the n^ difference of a rational integral function of the «"* degree, is constant. Let u^ = Ax'^ + Bx"'^ + ... + 7«' + Kx + i, be a rational in- tegral function ; then its first difference is Au^ = A{{x + lY-x''}+B{{x + l)''-'-x'"'} + ... + I{2x+l)+K = nAx"-' + B.x"-^ + ...+I^x + K^, which is a rational integral function, one degree lower than the original function. In like manner, for the difference of this, or the second difference of %, we have A'u, = n{n-1) Ax'^+ B.x'^-' + ... +1^, and so on ; and for the w"* difference, we have AX = w(w-l) ...3.2.1.x Hence the n* difference is constant, and the differences of all orders superior to the w**" vanish. Also A" (a;") = 1.2. 3. ..M. 14. To find the differences of a", Aa' =a^'-a'- = a^(a-l), AV = (a - 1) Aa" ^a'^ia- If, A»a^ = a'(a-1)^ Also Aa"' = a"-*-^"' - a"- = a'- (a^"" - I). 7 15. To find the difference of log v^ . V ( ^v \ A(logv,)=logv,^,-logt)^ = log^ = log 1+— ^ . 16. To find the differences of sin v^ and cos v^ . A sint>^ = sin (t)^ + AwJ — sin v^^ = 2 sm — ^ cos («x+-5"^) = 2 sin ^ sin {«, + i (tt + Ay;,)}. A cos tJ, = cos (Vj. + Avj.) — cos r, = — 2 sin — ^ sin f v, H AzJ = 2 sin -^ cos {v^ + i (tt + Av,)}. Hence making Vj. — xQ-\-a%o that Av^, = ^, we get A sin {xS + a) = 2 sin \9 sin {x5 + a + ^ (tt + 6)\, which shews that the difference of the sine of an angle a;5 + a is found by adding i('7r + 0) to the angle, and multiplying by 2 sin \Q. Hence, repeating the operation n times, we get A" sin {xQ + a) = (2 sin i^)" sin {a;0 + a + Jw (tt + &)]. Similarly, A" cos (x0 + o) = (2 sini5)''cos \xG-^a.-^\n (7r + 5)j. Also A" sin irx = 2" sin tt (x + w) , A" cos irx — 2" cos tt (a; + «). 16*. To find the differences of tan v^ and tan"* v^. sin i;„, cos v. — cos v^.. sin v^ AtantJ^ = tanv„,-tant),= ^ '- 2±i £ sin {v^^.^ — v^) _ sin At>, Hence A tan xd — 8 sin 5 cos (a+l) ^cosa;^' Also A tan"' r , = tan"' v^, — tan"' «, = tan ' r~^ = tan ' l+«x+,«x Hence A tan"' x6 = tan"' 1 + (x + 1) a;^ • Relations between the Successive Values and the DiflFerences of a Function. 17. The successive values of u^ any ftmction of x are, as has been stated, the values which arise from substituting x + 1, a; + 2, &c., or more generally x + h, x+2h, &c., for x in u^. These values are usually written u^^, u^+a, m,^^, «S;c. ; but it is further requisite that the operation of forming these values should be denoted by a prefixed symbol, in order that the no- tation for the successive values of u^ may be analogous to that for its successive differences, and in order that we may be able to avail ourselves of the advantages which, in these investiga- tions, the Method of Separation of Symbols offers. Suppose therefore 2) to be a symbol of operation implying the change of X into a; + A in any function of x to which it is prefixed, so that -Dm:, = m^; then which may be written ZJ'm,. = m^,^^ ; similarly I/u^ = u,^^, and generally ITu^^u^^. Hence since Au^ = u^^ — u^, we have Am^ = Bu^ — u^ , which, if we separate the symbol of operation I) from that of quantity M^, may be written Au^={D — l)u^, and expresses that the operation indicated by A is equivalent to that indicated by 3 — 1. Also since u^^ = u^ + Au^, we have JDu^ = u^+Au^, which, if we separate the symbols, may be written i)M,= (l+A)M,, and expresses that the operation indicated by D is equivalent to that denoted by 1 + A. 18. To express AX by u^ and its n successive values, ^ We here take h instead of unity for the increment of the principal variable, as the investigation is precisely the same on either supposition. Am^ =m,^ -m^, AX = u^^ - u,^ - {u^ - u^) = u^^ - 2m^ + u,, = «x+>» - 3m^« + 3w^^ - M^ . Now suppose this law of the coefficients, which as far as we have gone is the same as that of an expanded binomial whose index is the order of the difference, to hold for the w"' difference, so that AX = then A»+'m,= - (Mx+«» -Pl'>^xMn-l)H + ••• ±F2«:^a» + Fl'^x^ ± O = «x+,a4.l)» - (1 +I>l) M^»* + iPl +P^ Wx+^^X)» - • • • whidi is the same alteration with regard to the coefficients as occurs in passing from {z — 1)" to (s — 1)"*'. If therefore, for any value of n supposed a positive integer, the coefficients of the expansions of AX and {z — 1)" are the same, they will always be the same; but these coefficients are identical, as we have seen, when m= 1, 2, 3 ; therefore they are always the same ; H. D. E. 2 10 . An _ n (w — 1) + mWx+* ± ^x ; or, supposing A = 1, A« w (n — 1) _ A M^ = M^^^ - MU^^^, + '^ ^ W„,^2 - • • • + «Mx+i ± «x • 19. If for the descending values u^„ , M^+n-i i &c- we sub- stitute their equivalents (Art. 17), the second member becomes J)'u,-nD'"'u^ + ''^^~^^ i)»X-&c. ... + w., or, separating the symbols of operation from those of quantity, {D" - niy-' + " ^f~ ^^ J"^ - &c. ... ± 1) M„ .-. A''M,= (i?-l)X, each term of the development of (2) — 1)" being understood to be prefixed to u^ . This formula results immediately from the defi- nitions of A and D (Art. 17) ; for as the operation denoted by A is equivalent to that denoted by -0 — 1, if these operations be performed n times upon the same function u^, we must have Ax = (i?-i)X. 20. If in the series just investigated, we assign a particular value to Mj5, we shall readily obtain an expression for its n"" difference. Thus let u^ = x"', .-. A- (a;") = (x+«)" - w (x+ w - 1)" + Uhl^ (a; + n - 2)" - &c., in which equation, if n > m, since the former member vanishes (Art. 13), the second member is zero for every value of x; and if m = n, so that A"x" = 1 . 2 . 3 ... m, we get 1.2.3...n=(a; + n)"-n(a; + n-l)»+^^4^^^(a; + M-2)"-&c.; 11 and making a; = 0, since the equation holds for all values of x, 1 . 2 . 3 ... «=n- - n (n - 1)" +^4^^ (n - 2)" - &c. J. • A If a; = 0, and A'O" denote the particular value of A"a;"' when X = 0, we have A"0'"=n"-n (w - 1)" + " ^f ^^ (« - 2)" - 1 . ^ &c. Of the numters comprised in the form A"0'", which are called the Differences of zero, we shall make considerable use in future investigations ; whenever n>m the value is zero, in other cases it may be computed by the above formula ; thus, A0'"=1, A''0'' = 2, AW = 6, A''0*=14, ... AV = 6, A'O* = 36, AW = 150, . . . 21. Reversing the order of the series in Art. 18, we find (- l)''AX = Mx - WWxvl + 12~ Mx+2 - • • • ± Mz4.n • Also putting Wj, = x", and then supposing m = n, a; = 1 , we find successively, (- l)-A-a;" = a;"- n {x + 1)" + "" 1" ~ ^^ (a? + 2)"- ... ± (x + n)", (-l)"1.2.3..«=a:"-«(a;+l)" + ^^^^(a; + 2)"-... + (a;+n)", (-l)"1.2.3..n=r-n.2'' +^-^P^ . 3"- ... ± (n + 1)". 22. Hence it may be proved that 1.2.3 ... (p- 1) +1 is divisible by j), if jp be a prime number, (Wilson's Theorem) Let M^ = a;" — 1 ; .-. A" (a;" - 1) = 1 . 2 . 3...n = (x + w)" - 1 - n {(a; + n - 1)" - 1} + !L^{(x + n-2r-l}-&c. 12 Let x = l, and n + l=p, then x + n=j>, and i.2.s...{p-i)+i=p''-iT-i){{p-ir'-i} + ^p-'^^^-\ p-2r-i]-&c. Now by Fermat's Theorem every term of the second member is divisible by p when p is a prime number ; consequently 1 . 2 . 3 ... (^ - 1) + 1 is divisible by p. 23. To express u^^ by u^ and its first n differences. We take h instead of imity for the increment of the principal variable, the investigation being precisely the same on either supposition. Wx+a = M^ + Am^ + A (m^ + AmJ =u^ + 2 Am^ + AV , Wx+tt = % + 2 Am^ + AX + A (m, + 2 Am^ + A'm J = M^ + 3 Am^ + 3AV + AV- Now suppose this law of coefficients, which as far as we have gone is the same as that of an expanded binomial whose index is the number of increments which the principal variable has received, to hold for n increments, so that «x+„» = M. +i?,AM,+^,AX+...+^^A''-'M,+AX, then m^^^„» = m^+^,Am,4-PjAV+."+PiA"-'m^+AX + A {u^ +p^£^u^ +... +i>^"^M + j9,A"-'m^ + AX) = M^ + (1 +p,) Am^ + {p^ +_pj aX + • ■ • + {p. +Pi) ^"'^. + [Pt + 1) AX + A"*'m„ which is the same alteration with regard to the coefficients as occurs in passing from (1 + a)" to (1 + a)""^'. Hence if the coefficients of the expansions of u^^ and (1+a)" are the same for any value of n supposed a positive integer, they 13 will always be the same ; but they are identical as we have seen when n = 1, 2, 3 ; therefore they are always the same ; ••• M.^„* = Mx + »Am=. + ^^ ~ ' A=M, + . . . + mA-V, + AX , or, supposing h = l, 1.2 Mx+n = «x + nAu^ + \ ' ^ A V + • • • + nA""X + AX- 24. This result, if we separate the symbols, may be written Wx+„= (1 + nA + "^"~^^ A' + ... + nA"-^ + A") u,, or M,^„ = (1 + A)"m,, each term of the development of (1 4- A)" being understood to be prefixed to u^. The same formula follows immediately from the definitions of the symbols D and A ; for as i) is equivalent to 1 + A, if the operations denoted by D and 1 + A be performed n times upon the same function m^, we must have Z>-M, = (1+ A)-M, ; but Z>"m, = u^„, ••• Mx^„=(l+A)X. 24*. Let M^ = or, then m^„ = (ar + w)" ; and making a; = 0, n" = (1 + A)"0"', each term of the development of (1 + A)" being prefixed, as said above, to af and 0", respectively. By the latter formula any power of a number is expressed by the numbers comprised in the form A"0'", i. e. by the diflferences of zero. 25. To deduce Taylor's Theorem from the formula Let nh = t, and let h be infinitely diminished whilst t remains finite; therefore n is infinitely increased; and since 14 h is indefinitely diminished, we have, regarding the differential coefficient as the limit of the ratio of the simultaneous increments of the function and the variable, h dx' A''u,_d_ /Au^ _ d'u, _ „ h' ~dx\ h )~ dx^' Hence, preparing the formula as foUows, , Am. nh (nk — h) A^m, „ u^^ = u^ + nh-f+ \., ' ^ + &o., and taking the limit of both sides by supposing A to be infinitely diminished and n infinitely increased, their product always re- maining equal to a finite magnitude t, we get The Differential Calculus is a particular case of that of Finite Differences; and the above investigation is introduced to show how, firom results in Finite Differences obtained with an indeterminate increment for the principal variable, we may pass to the corresponding results in the Differential Calculus. The theorems of Arts. 18 and, 23 have been proved by an in- ductive process ; they may also be established by the theory of Generating Functions, the principles of which we shall now pro- ceed to explain ; as it is a theory which, for its generality and power, especially merits our attention. Generating Functions. 26. Let (0 = . .. + M.ir' + M„+ M^< + . . . then Mj may evidently represent any function of x whatever, if we regard this equation as the definition of ^(<). The 15 function {t) consequently by its development generates the coefficients u^, u^, ...u^ annexed to their proper powers of t, and is therefore called the Generating Function of u^, and is denoted by Gu^, so that {t)=Gn,. Thus since log (!-<)-' = < + K+4«'+-+^«' + - log(l-r=G=^- Similarly, since t{i-ty=t + 2e+3f+...+xf+... t (1 - tr = Ga:. 27. To determine the generating functions of u^„ and A"m, from that of u^ . Let {t)= Chi^, then (<) = ... +uX + u^^e'-' + ... +u^J^+ (1) ; .-. r'^{t) = ...+u„ + u^,t+...+u,^^f+... .: r-ip (t) = Gu^^, or r'Ou^= Gu^„. Again, f {t) = Ghi^, 0TfGu,= Gu^,. Hence it follows that the generating function of Am,, or M^j - M, , is (^^ - 1 j ^ (<), for this function being developed will produce the difference of two series whose general terms are respectively u^^f and uj' ; ... ^(A«,) = g-l)^,. Similarly, G (A»M,) = (7 - l) G^ (^«.) = (7 - ^)Gu., and G!(AX) = (7-i)"g^«.. 16 Also G (AX J = g - lJGu^.„= (i - ijeGu^^ (1 - trCU,. 28. If instead of multiplying ^(e) by a power of t, we mul- tiply it by another function of t, -(^(i) similar to ^ (t), that is, capable of being developed in a series of integral powers of t positive or negative, so that ^{t) = ... +PJ-' +p^+pf+pf^... +^,f + (2), then in the product we find for the coefficient of f, which may be replaced (Art. 17) by • • • +P-J>u^ +Po^^ +2>J^' «x +jpj>'^u, + . . . ; and this, separating the symbols, may be written or i^ljAu^, (each term of the development of •^ ( -= J being mi- derstood to be prefixed .to mJ ; which shews that •^ (<) x <^ {t) is the generating function oi yjr (-j^] u^ ; in other words, And if in (2) we replace t by <"' it may be shewn in exactly the same way, that "^fr [-] x {t) is the generating function of f]r{D)u^, that is Suppose for example, ■\}r{t) to assume successively the forms T", (t^ — 1)" ; theii, as before, we ^et (r' -irxGu,= G{{D- !)»«,} = G (AX). 17 29. To investigate the expression for A'm, in terms of Mx+n> ^x+n-u &c., by Generating Functions. = irOu,- «r^' Gu, + ^Lzi) r"« G^M, - &c. /-Y /-» n in — 1) y-v „ A« n{n — \) p .-. AX = M„„ - WM:^„_, + \ ^ Mx+«-2 - &c. for, the generating functions of both being the same, the co- efficients of f in the developments of those functions must be identical, however those developments have been effected. 30. To investigate the expression for m^^^ in terms of u^ and its first n differences, by Generating Functions. Gu^, = rGu, = G^. + nG'(A«J +^^^^ G (AX) +&C. = Gf K + «A«. + ^^-^^ AX + &c.) ; . m (w — 1) .- , » .-. «^^„= M, + n Am, + -y72- ^ V + &c. 31. It is obvious that by transforming the expressions (1 - l") G«M^, and rGu„ H. D. E. -^ 18 in different ways, we may obtain various other expressions for A"Mj and u^^„ besides the above. Thus to express AX in terms of AX_n. ^"'^V-n_i. &c., we have ^{l-tyGu. + nil-ty^'Gu, + 'ii!5±l)(i_()"«G=«, + &c. = G (A»«_) + nG (A»^'«,.„.J + "^-^^ G (A--«_,) + &c. = G {AX.n + «A"^'m^,-. + ^^J^ A-««.,_, + &C.1 ; .-. A"«, = AX.„ + «A"X-^, + ^^^ A""V^, + &c. 32. Again, to express u^^„ in terms of u^, Am^.^, A'u^„, &c., we must transform <~" into a series of powers of f ( - — 1 1 , that is, we must develop r" in powers of x from the equation which may be done by Lagrange's Theorem ; and we find (see Herschel's Examples) n(n + 2r-l) ., «(n + 3r-l)(m + 3r-2) , „ + -5^ j-yj-3 A Vsr + &c. This method is obviously not confined to the function u — M^ ; it is equally applicable to any other combination of the successive values w„ %+i, m^+s, &c., of the first degree. If we 19 take Am^ to mexa au^, + bu^^^ + eu^, then the generating func- tions of AMj, and A"m^ will evidently he and the expression for A'u^ in terms of u^ and its successive values, might be obtained as in the preceding case. Separation of the Symbols of Operation from those of Quantity. 33. We have seen (Arts. 18 and 23) in the formulse JTu^ = (1 + A)"«., AX = (^ - 1)X, instances of the method which consists in separating the symbols of operation from those of quantity ; the use of which is not con- fined to simple cases like those just noticed, but may be extended with remarkable effect to a great variety of investigations con- nected with this subject. By symbols of operation are meant certain characteristic letters placed before any functions, to denote that certain opera- tions have been performed on them: thus I) placed before a function of any variable x, denotes the operation of changing in it X into x + h; and A placed before the same function, implies that two different values of the variable have been substituted in it and the results subtracted from one another. By symbols of quantity are meant the subjects of the operations just mentioned ; that is, letters taken to represent numbers, or algebraical expres- sions ; and it must be remarked that these latter may be also regarded as symbols of operation. For if a be a number, then a denotes that the unit employed in the investigation, whatever it be, is to be added to itself n times ; and a" or a . a denotes that the same operation has been performed on a that was performed on unity; whenever therefore the two kinds of symbols occur together in the same formula, as in {Da — 1 )", a must be regarded as a symbol of operation. 34.* The expressions (1 + A)", {B - 1)" must be taken as abbreviated forms for their developments ; and when prefixed to the function m^., each term of these developments is understood to 20 be applied separately to that function. And, in general, if F (A) be a function of A capable of being developed in a series of powers of A such as ^A« + B6F + &c. ; then for Ah^, + -FA^m^ + &c. the expression i^(A) u^ is used as an abbreviation ; and the same notation is applicable to other characteristic symbols, such as that in their combinations are subject to the same laws as alge- braical quantities. Also if we replace A*^^, A%j., &c. by their equivalents, we get F (A) 'u, = A{D- \Yu, + B{D- l^u, + &c. = {A{D-iy + £{I)-l)P + &c.}u, = F{D-l)u^, which shews that in any formula the symbols A and D — 1 may be interchanged. Hence also the successive performance of two or more series of operations represented by F{A), F' (A), upon the same function u^, is equivalent to the performance of that series of operations denoted by their product. The method of separation of symbols is in every case capable of a strict inductive proof, and does not rest merely upon accidental analogies; and it deserves great notice on account of the facility with which it enables us to conduct many intricate processes. But as the gene- ralizations which it offers, may present some difficulties to the student, we shall continue, as we have done hitherto, to obtain several of the principal results by an elementary process ; before investigating them by the method of separation of symbols, or pointing out how they arise from that method. We shall now give instances of the application of the method of separation of symbols, to obtain several important results. 34. To find the »* difference of the product of two functions. We have seen that A {uj)J) = ^u^v^^ + w^Ar, = AuJ)v, + u^v, = (a2)' + A') MA, 21 separating the symbols, and supposing in the second member A to affect Mj only, and A' and D' to affiect v^ only, then A'(m,«,) = (AD' + A') A K«J = (AZ>' + A')V,7;,; and, generally, a" (m^vJ = (aZ>' + aTm^w^ = (A-i?"'+wA"-^A'i)'"-' + !L(!Li11 a"-*A'^Z>'-* + &c.) M,v, = AX . v^ + «A"-M^ . A»„,^. + -JT^ A-^M.A'v^^ + «S:c., which may be also proved inductively by shewing, as in Art. 1 3, that the coefficients of the developments oi l^iyi^^ and (l+z)", which are identical when« = l, undergo the same changes in passing from w to w + 1. If the series be reversed or, which is the same thing, if we develop the formula A" {u^v^ = (A' + aD'^u^v^, we get A" {u^v,) = M^AX + nAM^- V. + ^^^~^' AXA"-^v^, + &c. Since A {u^vJ)=u^^^v^^—uJ}^=DuJ)v^ — u^v^ = {DB' — 1) u^v^ where D affects m^ only, and D' affects v^ only, we obtain another development of A"(mjvJ, viz. = (D-i>'" _ „i)"->z>"'-' + '^^^-^) jjr-^j)"'-^ - &c.) ujo, w (w— 1) s In the formula A" {u^v^ = {DB' - \Yu^v^ suppose v^ = a', then B'cf = a'^S so that the operation upon c^ denoted by B' is multiplying it by a, .-. A" (mX) = (J5a - 1)"mX = a" (^« - 1)"Mx. Also M^„A"«j. = (A'i?)X««= (^'-0 + ^ - ^)'*«A = {(A'2) + A)"- w {A'X> + A)"''A + &c.j v^u^ = A" (va) - nA"-' (v^%) + i« (« - 1) A"-'(y^X) - • • • + I'xAX, a formula by which m«+„AX is expressed by a series of diifer- ences with constant coefficients. 22 In the above instance we use an accent not to imply that the operations denoted by A, D, are altered at all, but merely that A', D', affect v^ only, whilst A, D affect u^ only. The above result is sometimes written, A" («,«,)= [DU- l)Xfx= {(1 + A) (1 + A')- l}"M.f., replacing D, D' by their equivalents; or, if there are more func- tions Wj,, z^, &c. and we use A", A'", &c. to imply that in the second member these symbols only affect w„ z„ &c. respectively; and similarly for D, D', D", &c. ; we have A" {u^v^w^^...) = {DD'D"... - l)XVx«'.- = {(l + A)(l + A')(l + A")(l + A"')...-irM.r,w^.... 35. To shew that A"m^= {e^—\Yu„ in which the symbols of operation are separated from those of quantity. By Taylor's Theorem, we have and separating the symbols of operation from those of quantity, we get r d n^ fdV w' /dY p ] "1- bimilarly, m^^,^, = e '^u^, &c. m^^,= e'^u^; An "S '"-"^ m(w— 1) (1-2) # or, again separating the symbols of operation from those of quantity, An f "S '»-"^ nln-1) (n-2)/- - ] /^ \» a celebrated theorem first given by Lagrange. 36. To find a general expression for the n"" difference of a function in terms of its differential coefficients. 23 The development of the second member of the equation will consist of a series of terms of the form and A^ is evidently the coefficient of f in the expansion of (e'-l)". Now (e' - 1)" = e^ - «e<"-'" + " ^f '^^ e'"^" - &c., 1 • i2 and the coefficients of f in the developments of e"*, e'""'", &c. are respectively I OT ' \m , &c.; from Art. 20. Now so long as m < n, this vanishes ; and when n = m, A"0'"= [w ; • • "^ "' ~ die" + | n+l da;"« ^ | w + 2 rfa;"« ^ °^^- Ex. Let u^ = x'', then A" (x") = «i (m - 1) ... (m - n + 1) aT'"" Anrvn+l + P-^.m(m-l) ... (w-«) a"*-"-' +&c. + A"0". In + l 37. If in Lagrange's Theorem for A"m^, w = 1, we have Am,= (e*^— 1) Mj;, or A = 6*^— 1, the meaning of which is, that the operation denoted hy A is equivalent to the series of opera- 24 tions denoted by e"^ — 1. And generally, the series of operations denoted by /(A) is equivalent to that denoted by/(6'^— 1). For lety(A) be developed in a series of the form /(A) = ^A« + £AP + OAv + &c., then /(A) u^ = A^.'^u^ + BiaPu, + CA^m^ + &c. = A{e^- 1)X+-S (e^ - 1)^m^ + C (e^ - 1)i'M:,+ &c. or, separating the symbols of operation form those of quantity, /(A)m.= [A (e^- 1Y+B{^- ly + G {^- l)v + &c.} m. =/(e^-l)w.. Thus, suppose /(A) = (1 + A)", then /(e^ -!) = (!+/- 1)" = e^ ; therefore, annexing a function u^ for the symbols to operate upon, as already proved. Similarly, making w = 1 in the formula m^„ = e '^m^ (Art, 35), we find Mj^j = e'^M^, or Du^ = e^u^ ; therefore D = e^, the mean- ing of which is that the operation denoted by D is equivalent to the series of operations denoted by e*" ; and, generally, the series of operations denoted by f{D) is equivalent to that denoted by /(C^), which is expressed by the formula 38. To express the n"" differential coeflScient of any func- tion by its differences. Suppose f{D) = (log DY, then /(e«) = {\ogh'= (^)"; .-. ^^ = aog D)X = {log (1 + A)}X. 25 39. To find the general term of the expansion of y(e^ in a series ascending by powers of t. Writing /(e') in the form /{I + (e* — 1)}, and expanding by Taylor's theorem, we find /(«') =/(l) +/' (1) (e*- 1) + j-^/" (1) (e'- 1)' + ... + j^/'»'(l)(e'-ir+... Then taking, as in Art. 36, the coefficient of C in each term of the second member, and observing that in /(I) it may be represented by \ — , this quantity being /(I) when m = 0, and zero in all other cases ; and that in 1 1 A"ft"' we have for the coefficient of <" in the expansion of /(e'), the value = ^/(l+A)0». a remarkable theorem first given by Herschel ; for the applica- tions of which, see his Collection of Examples. Hence in the development of e"' the coefficient of f is J-e"^0'" = ,— fo-* + AO-'+rri^ A'0"+ ... +^P) ; |ni \m\ 1.2 \m / and in the series for -r—z the coefficient of f is e +1 11 I/O" AO" A'O" . A'-O^N |m 2 + A I™ V 2 2" 2 i / the advantage of using the differences of zero being that any series of them necessarily terminates at A^O". H. D. E. * SECTION II. INVERSE METHOD OF DIFFERENCES. Integration of Explicit Functions. 40. The Inverse Method of Differences has for its object to determine the primitive function from its given difference; or from given relations between it and its differences. We shall begin with the simplest case, Am^=/(x), in which it is required to determine a function whose difference is given explicitly in terms of the principal variable. 41. Since Am^, is the difference of u^ + C, as well as of u^, it will be necessary, in passing from the given difference Am,, to the primitive function, to annex an arbitrary constant G, in order to give the result all the generality of which it is capable. Also C may be a function of a; as well as an arbitrary constant, provided its value remains unaltered whilst x changes to a; + 1. For if Cx denote such a function of x that C^, = C^, or A C, = 0, we shall have A(m^+Q = Am^. It is evident that G^ = <}> {2\itx) has the property in question, ^ denoting any trigonometrical function, sine, cosine, &c., and X any integer. We shall see further on the importance of this remark. 42. The symbol S is used to denote the operation by which we pass from the difference Au^ to the primitive function ; so that S (Am J = u^ + constant ; hence S and A denote operations the reverse of each other. 27 Also, as the same function admits of successive differences, so a function may be integrated any number of times ; the second integral of m^, or 2 (Swj,), is written 2'mj,, and the n"" integral If in the formula X (AuJ) =u^, we suppose the symbols of operation to be separated from those of quantity, we find zAm^ = M^ ; but on the same supposition A'^Am,, = A'u^ = Wi ; so that X produces exactly the same effect as A"\ Similarly S" may be shewn to be identical with A"" ; so that in any formula a negative power of A may be always replaced by the same positive power of %, and vice versa. We now proceed to deduce the integrals of various expres- sions ; chiefly, by reversing the processes given in Section i. for finding the differences of functions. 43. It is evident that S (m, + v^ + w^ = %u^ + Sv^, + Sm^ ; for if we take the difference of both sides, we get the same result, viz. u^ + v^ + w^. And in the same manner it appears that S {au^ = a%u^ , and 20 = C. 44. To find the integral of any rational integral function. Since the difference of a rational integral function is a func- tion of the same kind one dimension lower, it follows that the integral of a function of that description is a similar function one dimension higher ; hence, to find the integral of we may assume it equal to ax"*'' + Ja;" + ... + fee + Z; then upon taking the difference of each side, and equating the coefficients of like powers of x, there will arise n + 1 simple equations to determine the n + 1 quantities a, b, c,...k; the last term I will remain indeterminate, being in fact the arbitrary constant which must be added to make the integral complete. Ex. To find S {x* + 1). Assume X {x* + 1) = ax" + hx* + (^ + dai' + e-x ; .-. X* + 1 = a (5a;* + lOx' + 10a;' + 5x + l) + b (4a;' + 6x' + 4a; + 1) + c{Sa? + Sx + \)+d{2x + l) + ei 28 .-. 1 = 5a, = 10a + 4J, = 10a + 6& + 3c, = 5a + ib +Sc+2d, l = a + b + c+d + e. Ill 1 J /^ 29 •'•«=5' ^ = -2' " = 3' '^=^' '=30' ^ ^ ' 5 2 3 30 45. To find the integral of the product of consecutive terms of an arithmetic progression, we must annex one more factor at the beginning, and divide by the number of factors so increased and by the common difference. For let Ux = a + bx, then we have seen (Art. 11) that Am^m^^j . . . m^„ = M^,M^+s, ...u^.{n+l)b, therefore, taking the integrals of both sides, and wiiting a; — 1 for X, we get iU,U^, ... U^n.,- (^ ^ j^ J + t', which proves the rule stated above. Ex. :S(2. + l)(2. + f)(2. + |) Each factor of an expression capable of being integrated by this rule, must be derivable from the preceding factor by changing x into 35 + 1. If one or more factors be deficient in a factorial of this kind, it may be resolved into others which are complete, as in the fol- lowing instance ; (2XH-1) (2x + 5) (2a; + 7) = (2a;+3-2) (2a! + 5) (2a; + 7) = (2aj+3) (2a; + 5) (2a;+ 7) -2 (2a; + 5) (2a; + 7). 29 46. A rational integral ftinction may often be resolved into factorials of the above form, and in this way its integral more conveniently found, than by the method of Art. 44. Ex.1. x'+a^ = a?{x+l) = {x-l + l)x{x + l) = {x-l)x{x + l)+x{x + l); And in general, any quantity of the form may be resolved into factorials, by the method of indeterminate coefficients ; thus, if we assiune ax' + bx + c = A{x + l){x + 2)+B{x + l) + 0, making a; = — 1, we get a — b + c=C; .-. a{x^-l) + h (x + l) = A {x + l){x + 2) + B(x + 1), or a{x-l) + b = A{x + 2)+£; make x = — 2, .'. —3a + b = B; .: a{x — l) + 3a = A{x + 2), .*. A=a. In practice, however, it is generally easier to resolve a func- tion by inspection, as in Ex. 1, than by this method, which is theoretically certain. 47. To find the integral of a fraction whose denominator is the product of consecutive terms of an arithmetic progression, and numerator constant, we must efiace the last factor, divide by the number of factors remaining and by the common difference, and prefix a negative sign. For let u^ = a + bx, then we have seen (Art. 12) that 30 therefore taking the integral of both sides, W:tMx+, • • • %+» 'nbu^u^^^ . . . u^^^ ' which proves the rule just stated. 48. If the proposed fraction, instead of having its nume- rator constant, be Ax"-^ + 5a;"-' +...+KX + L (the degree of the numerator being at least lower by two imits than that of the denominator,) we must reduce the numerator to a series of terms each of which is the product of consecutive factors reckoning from the beginning of the denominator ; that is, assume Ax''-^ + Bx"-^ + ... + 1^ + L = A' + B\ + C'u.u^, + ... then, developing the second member, and equating coefficients of like powers of x, we obtain « — 1 equations for determining A', B', C, ...K'; and the fraction resolves itself into the fol- lowing, each of which is integrable, A' B' „ K' + + &C. +- If the degree of the numerator were the same as that of the denominator, or only lower by one unit than that of the deno- TJ minator, we should arrive at a term , of which we are able to find the integral, only approximately. Hence also, if any of the factors of the denominator of the fraction in Art. 47 be wanting, they may be supplied by intro- ducing them into the numerator and denominator at the same time ; and then the resulting fraction may be treated as in the present Article. 31 Fx 1 1 {x-l)x{x+l) ' ' a;'-4~(a;-2) ...(« + 2) (a; + l)(a; + 2) a;(a;+l)(x + 2) (a;-l)a;(a;+l)(x + 2) + 6 {x-2){x-l)x{x + l) (x + 2)' which is got by assuming {x-l)x{x + l)=a{x-2){x-l)x + b{x-2){x-l)+c{x-2) + d, and making a; = 2, 1, 0, successively; taking care to reject the factor common to both sides, after each substitution. Ex.2. 2- ' ' 12x'-12:r-l 4^-d 6 (2x-3)(4ic'-l)" 49. The fraction in the preceding Art. may be also inte- grated when the denominator is the product of any number of factors X, x + mh, x + nh, x + rh, &c., m, n, r, &c., denoting whole numbers, and h the increment of x. For by taking the difference of both members, we perceive the truth of the result (which, although expressed in a series the number of whose terms is variable, is often useful) V ^ - ^ [1 I 1 I ... I I l...(i). x{x + mh) mh\x x + h x + {m—l)h) B t JgH-^ - ^ I -^ (2) X {x + mh) {x + nh) x{x + mh) x[x + nh) ' where A + B=l, {An + Bm) h = a; so that this fraction is integrable by formula (1). Next multiply X -^h both sides of (2) by j-; then the second member of the result can be resolved by (2) into fractions having a constant niimerator, and the product of two simple factors for denomi- nator, and is therefore integrable by (1); and so on to any number of factors, the dimension of the numerator being always less by at least two xmits than that of the denominator. 32 50. To find the integrals of o", and log v^. We have seen (Art. 14) that Aa" = (a — 1) a*; .•.ta^ = -^ + a Also S"a' = - " a-1 ' """ (a-1)"' suppressing the part Introduced by the constants, which would be a rational integral function of the (w — l)*"" degree, and might be represented by ^"0, since So= C, S'0=Cx+ C", &c. The formula seems to fail when a = l, as it gives infinity for the value of "Za' in that case, instead of x ; but if we give the con- stant C the form C , then a — 1 a— I now let a = 1 + A, where h is very small, then Za = — ^^— T — = x + ^x{x~l)h + &c. ; therefore, when a = 1, Xa" = x+ C. Next to find the integral of log v^. If M:, = log {v^v^v, ... v^^), then Am^ = log (v.v, ...vj)- log {v,v^ . . . v^,) = log IV ; .-. S log v^ = u^ + log C= log (<7. VjV, ... v^J = log CPv^^, using Puj, to denote the product of all the successive values of the function v^, fi'om some fixed term v, (or more generally v„, n being independent of a;) to Wj, inclusive. 61. To find the integrals of cos x0, sin x0. Since A cos a;0 = — 2 sin ^fl sin (a; + ^) 6, .'. A cos (a; — ^) = — 2 sin ^^ sin x0 ; _j . „ cos (a; — 1) , ^ 2 sin ^6 33 Also since A" sin {x6 + a) = (2 sin ^6)' sm. [x6 + a. -^ \n {-rr + B)], (Art. 16), integrating both sides n times, and replacing a by a — jM(ir + ^), we get S» sm (.5 4- a) ^—L-^^^ ^ ; the same result as if in the value of A" sin {x6 + a) we had changed the sign of n ; as we should expect, A"" being equiva- lent to 2". Again, A sin a;5 = 2 sin ^0 cos {x + \) 6, .•. A sin (x — ^) 5 = 2sin^^ cosx5; _i y, sin (x—i)6 „ .: tco3x9= \ ^' +C; , <., , a \ COS \xd + a — in (-rr + 6)} and S"cos(x5 + a)=^ H, changing the sign of n in the value of A" cos {xd + a) in Art. 16. 52. The preceding expressions may also be integrated by substituting for them their exponential values ; as in the follow- ing instance. 2a' cos x0=it (a'e^^'^ + a'e" *^') ae: —1 ae ^ '—1 .J _ ^ o cos (a; — 1) g — cos xd ^ ~ a' — 2a cos ^ + 1 Hence, putting a* - 2ffl cos + 1 = c, and denoting a* cos xd by u^ , we have c^u., = a*u^_^ — u^ , c*2 V = aV-s — 2a'M^.j + w, ; and generally c"SX= a*""^.- wa'"-'w^n*,+ """^ a'""""...*. - &c. + «, H. D. E. 5 34 This result may be obtained immediately by separation of symbols ; for we have tu^ = - (aV. - «x) = - («'-»" - 1) «x; c c consequently S"mi = -^ (a'-O"' — 1)"m^ ; and this when developed produces the preceding result. Exactly the same formulae hold for u^ = a" sin x6. In the same manner the integrals of 0° (sinx^)"*, a" (cos xOy may be obtained, when the powers of the sine and cosine of x9 have been replaced by the sines and cosines of multiples of x6. 53. To find the integral of .^^^g J(^+ ^^ g - Since A tan xB „ ^™ , -r-s , (Art. 16*.) cos xa cos [x+lj a , _, 1 tan xd , ™ we have 2 ^ -. — -—t-t. = . „ + V. cos xa cos [x+\)d sm 54. To find the integral of tan"* „ . ° p + qx + rx Since A tan"' (a + Z>a;) = tan"' ; -. t-t-? v jt- . (Art. 1 6*.) ' 1 + (a+&a;) (a + 6x + 6) • we may assume 2 tan"' » = tan"* [a + 6x) , '' p-^-qx-k- rar ' and take the difference of both sides ; then if the proposed func- tion is capable of being integrated, the indeterminate coefficients a and h will become known. Also by differencing n" tan"' {0n~^) we may find an expression of which it is the integral, when w = 2, 3, &c. 55. The integral of a'uji^^^...u^^^, where u^=pa'^+q, may be determined by assuming it equal to the same expression (only with another factor at the beginning instead of a") multi- plied by an indeterminate coefficient, and taking the difference of both sides ; for Am^.jM^ . . . M^„_j = M^M^^j . , , M^^^^ (m^„ - M^j) 35 a! Similarly, to find 2 , for the assumption we must efface the last factor in the denominator, and write instead of a" an indeterminate coefficient ; for A ^ u^n-tix _ pa" (a" - 1) . 56. In like manner, if m^ = a + hx, expressions of the forms [p + qx) f {p+ qx + ra?) C can sometimes be integrated, by assuming their integrals equal to expressions of the same form, except that the last factor in the denominator is effaced, and the polynomial in the numerator is replaced by another one dimension lower with indeterminate coefficients. It is of course only when a certain equation of con- dition between the quantities a, b, p, q, t is satisfied, that this method succeeds. Ex.1. Let 2 ^+' --^-(*l-^ (2a: - 1) (2a; + 1) 3"^ 2a; -1' ' ■ (2a;-l) (2a; + 1) 3" ^^' \2x +1 2a; - l) ,,. -4(a;+l) . =^ar .-. ^ = -1, and 2M,= C-f Ex. 2. Let 2 (2x-l) (2x+l)' 1 (2a; -1)3' a?" + 6a; + 12 A+Bx a;(a; + l) (a;+2)2'' a!(a; + l)2'"' £C + 3 then A = -&, 5= - 2, and 2„^ = ^~ xirx+l)r 36 57. Since A (m^vJ =u^^v^ + v^^Au^, we have 2 (m^O = u,v, - 2 (!!:,« Am,), the formula for integration by parts, corresponding to the for- mula , 7 dv I. , du J axu -j-=uv —J dxv -J- . Change v^ into "Zv^, and consequently Ad, into v„ then 2 {u^v,) = u^J-v^ - 2 (Am,2v,^j) ; hence, by successive substitutions, we get 2 (AM,2t,,,J = Am,2V. - 2 (A'u,2'0. 2 (AX2=0 = A V 2'v, - 2 (A^S'O. 2 (A"M,2Xn) = AX2""t;,,„- 2 (A''"w,2"«t,,,^.) ; .-. 2 (m,v,) = M,2r, - dLU^^v^, + AX2'»Vj - &c. + A"M,2"««,^„ + 2 (A"««,2"+^v,^^^). 58. The above formula, replacing in the second member S by its equivalent A~', may be written 2 (M,r,) = uA-\ - Am, . A^ (Z?«,) + AX • A"' {I)\) - &c. = (A' + AZ>r'«A, if we restrict A to affect m, only and A' and D' to affect », only ; hence 2" (m,«,) = (A' + AD')'^u^v^ ; and generally • 2"(M,v,)=(A' + AZ)'r«.*'x (1) = (A'-"-«A'-'-'Ai)'+'^^^A'-"-'^A»i)"-&c.)M,v„ or 2" {m,v,) =M,2"v,-nAM,2"+'«,^j n (« + 1) 1.2 -AX2"«r^,-&c...(2), which may be also proved inductively, by shewing that the coefficients of the developments of 2" (m,w,) and (1 + «)"", which 37 are identical when «=1, undergo the same changes in passing from n to n — 1. This appears from diflferencing both sides of equation (2). Both the formula (1) for 2" {u^v^), and its de- velopment (2), result immediately, as we should expect, from the expressions for A" {u^v^ in Art. 34, by changing the sign of w. If in the formula 2" [u^v^) = (A' + £^D')~'u^v^ where A afiFects «, only, and A', D' affect v^ only, we suppose v^ = a' ; then A'v^ = (o — 1) a", D'v^ —a.a''; and the formula becomes 2' (mX) = («-!+ Ao)-"M,a' = a" [Da - 1)""m^, which agrees with the result in Art. 34 when n has its sign changed. If in this last formula we change a into - , we get and {D — a)~^u^ = oT^ 2 {u^'^). 59. The above formula for 2" {u^v^) always enables us to find the integrals of functions made up of two factors, one of which leads to zero, as the value of one of its successive dif- ferences, and the other admits of successive integrations. Sup- pose, for example, that u^ is a rational integral function of the w*" degree, and that v^ = a' ; then -. -yi , &c. ; A'u^ = const, A'^'m, = ; [a — 1) Am^'^' , AXa^ ^ A-M^'^ 2«. = « , 2 a-1' ,-. 2 {u^') = a-1 {a-iy^ {a-iy '"-(a-l)" Again, suppose u^ to be a rational integral function of the n* degree, and v^ = cos x0 ; then taking the value of 2*'v^ from Art. 51, 2 K cos xe) ^^^ coB{{x+l)0-{->r+e)} . ^,,. cos((a;+2)g-|(ff+g)) - ^"' (2 8ini0r (2^mp? *''•' 38 the series terminating with A"m,. Similarly, if t;^ = a' cos a;^ ; or if v^ = a' cos"'x6 .sm''xO, since the product cos" a;^. sin" x^ may be replaced by simple dimensions of sines and cosines of multiples of xd; and it will be noticed that the fraction of Art. 48 may be brought under this case. 60. Since the performance of the operation 2 upon any series of terms AA."'u^ + BA'u^ + ... , reduces it to ^A'"-'m, + 5A"-X+ — ; it appears that prefixing 2 to (J.A" +-BA" + ....) m^ has the same effect as prefixing A"' ; in other words, % is equivalent to A"'. And in like manner, since integrating A^m^ n times, reduces it to A'^^'m^, S" must be equivalent to A"". -j-j ; whenever therefore, in separating the symbols of operation from those of quantity, as in the expression F{A)u^, fl-j-ju^, terms containing negative powers of A and -j- occur, they must be understood to be replaced by the corresponding positive powers of 2 and Jdx. This being premised, we proceed to investigate a general series for %u^ ; preparatory to which the following pro- positions must be proved. 61. To determine the generating functions of from that of u^ . By virtue of the relations r=(ioge^r=(^ and changing the sign of n, and replacing f ^-j by {Jdx)', 63. To investigate a general series for 1u^, involving only Jdxu^, u,., and the differential coefficients of m„. G (2w,) = (i - 1)" Gu, = (e"^ - 1)" Gu, + (-ir^(iogiyv...}G^. assuming, as will be proved in the next Art., that — — - can be expanded in a series of the same form as that within brackets, and denoting by 1.2' 1.2.3.4' ' ^ ' [2« • the coefficients of d', u*, ... v*" in that expansion. Hence 41 .■.S„..;&..-K+j5^§-fSf-- m+1 Ex. 1. Sa;" = -^-— -iar+ IB.nuT-' m + 1 ' ■^ ' 6 2 ^ 12 12 ^ Ex.2. 2 \ =Tlog(a + fa)- „, ^ , ■ ^■* + /gf &C.+ C. 2(a + &a;)' 4(a + 6x)* Similarly 2"mj, may be expressed in terms of the integrals and differential coefficients of u^ by making n negative in the d formula of Art. 35, and expanding e**, which gives _/{t)-i>{-t)=-t; which shews that the only term in the development of if) (t) which involves an odd power of « is — ^* ; for if any higher odd power entered, it would occur in (— t) with its coeffi- cient doubled ; we may assume therefore for (t) the form of development given above, viz. Now ^y.^iM^^i-w-^)^w-iy-- and if A.^^ be the coefficient of f in this development of the second member, ■I Sn nZn n J»« "*" 2w + 1 • I 2w all the terms after A^'O*^ vanishing, since A"*0'*' is zero when m > 2n. Hence ^i^i = (- 1)"^' {- iAO*" + J A'C" - &c. ... +^-l-j A'-O'"}. By this formula B^, B^, B^, &c. may be readily computed, supposing the numbers comprised in the form A"0**, or the dif- ferences of zero to be known ; we find "^•=6' -^» = 30' -^'=42' -^' = 30' ^• = i'*^- values that may be easily verified by applying Maclaurin's Theorem to obtain the development (1). 43 65. Also, since A"'0'- = m'»-OT(m-ir + ^^^^^^(m-2r-&c. (Art. 20) we may, if we please, eliminate the differences of zero from the expression for B^^^ ; and we find [2re 2n+l' (- l)"^'^^i = - i+i (2'" - 2) - i (3*- 3 . 2=-+ 3) + &c. + Besides serving to express the general value of Sm^, the numbers of Bemouilli have various other uses, of which we shall now give one or two of the most remarkable. Any function that can be put under the form < -=- (e' — 1) may be expanded in terms involving those numbers. 66. To find the general terms of the expansions of cot 6 and tan 6 in powers of 6. cot I Now the general term of the expansion of Jg^ is (- 1)- ^ (2^ vCTl)- or - ^^ ^ ; 2^B .". the general term of the expansion of cot 6\a '^' ^"~', andcot5 = i--=-5,5 " g°-&c. e 1.2' 1.2.3.4 Also since tan ^ = cot 5 — 2 cot 2d, the general term of the series for tan 6 is 2'"P^.g^-' , , r'B^, (2g)'"-' _ 2» (2" -\.)B„ + 2 -1. [2n [2n [ 2» Hence, by differentiating the above expressions for cot 6 and tan 6, we may deduce the general terms of the expansions of 44 cof 6 and tan' 6 ; and by integrating them, the general terms of the expansions of log sin 6, log cos 0. 67. To find S the sum of the infinite series i-4.i- - Sn "T* o2n "" Q2n *^ • • • Sincesin^ = ^(l-5)(l-^)(l-35.)... changing ^ into ^0, we get »'«-*(■ -?)(>-S('-S- therefore, differentiating the logarithm of each member, ^ . 1 20/^ efy' 20 [^ 0X' 20 f^ ff\-' X ^cot,r5 = g-^(^l-pj -2i(l-2«j --p(l-3ij -&°- But the coefficient of ^"' in the expansion of ir cot tt^ is ■• 1.2.3...2n' XT 1,11, Tr'lll TT* Hence p + 25 + 3-,+ ...= g. ^ + 2-^ + 31 + - = 90- Calling ;S^ the sum of this infinite series, we haye /8U ^ (2« + l)(2n + 2).g^, ^ 45 B n* Now suppose n veiy great, then ■^'^ = -5 , which proves an-l the diyergency of the series formed by the numhers of Ber- noTiilli; these numhers increase very rapidly, beginning with B^^. 68. To find 8 the sum of the infinite series — J-4- — l2n "T" Qi«* ' cSi . J I 1- „.. ■» 2n T^ Qi«* ' c3n ' changing into ^, we get cos ^ = (l " la) (^ " 32) (l — 5?) • • ■ therefore, differentiating the logarithm of each member, 2*^T = t(i-pJ +3^l^-3^J +T'(^-fO +*°-' and equating the coefficients of ^"'""* in each member, which are respectively, weget^=i5^,^!^;i^. TT 1,1,1.^ I 1 Sit* TT^ Hence p + 3. + 55 + &c.=- .-. j-^ = -. .,11 11 69. Also TaS ~ oan "^ 'Ssn ~ Ti™ ■'■ • • • _i .JL+Jl. _J_ri_4.JL4.i.+ ^ ~ l"" ■'' 3"* 5»" 2'"V1'^ 2*^ 3^ 7 1 2"- - 1 ^ P 1 2'"-'7r^ „ ~2^^ "^^ 2*' )2« »^' _ (2'"-'-l)7r^^^. |2n 46 Since log (l + e-) = «- - ^ 6"*'+ J e •*- &c., integrating this 2n — 1 times between the limits 2 = 0, a = co , we find for result the series jnst summed ; .♦. Udz: )-- log (1 + e-") = ^ ~ ^^ '^^«-> |2w 70. To find an approximate value of r (x + 1) = 1 . 2 . 3 ... X, when x is very large. Making Mj, = log x in the formula we find, (Art. 50) adding log x to both sides, log {1 . 2 . 3 . . . (x — 1) a;} = C+ a; log sr — a; — ^ log a; 5, 1 i5, 1 , +T:i5-3:i^+-+i°g^ = C + (a; + i)loga;-a; + log(l + A), putting log (i + ;i)=^-^ + 3_.-&c., so that A is a quantity continually approaching to zero as x increases. Now to determine C, suppose x very large so that A may be neglected, and change x into 2a;, then log (1.2.3...2a;)= (7+(2a; + i) log 2a;- 2x = C+(2x + i) (log a; + log 2) -2a;, and log (2.4.6...2a;) = log (2M .2.3 ... a;) = a;log2 + <7+ (a; + i) log a;-a;; .-. log {1.3.5... (2x-l)}=a;loga;4-(x + i)log2-a;; 47 ■•■'°« i.l:5:!pf-i) -g-^*"'g'-*'°^^ = (7+i log 2a; -log 2; ••• 2a-2log2=2log ^_^3-^^-^"ff^p log2. _, 2. 2. 4. 4 .6. 6... (2a; -2) 2a; ~ °^ 1.3.3.5.5.7... (2a;-l)(2a;-l) = log — , by Wallis's Theorem, since x is indefinitely large ; .-. C'=ilog2,r; .-.log (1.2.3 ...a;)=ilog 2Tr+(a;+ J) loga;-a; + log (1 + A) = log V2^ + log (jj +log {1+h); .-. 1.2.3...a; = V27ra;.[-j .{l+h), where h is to be calculated from the series and in general it will suffice to take the fu'st term only, which gives 1.2.3... a;=: Vifl^afe"^^; or, more accurately, 1 .2.3 ... a; = V27ra;a;''e '*", where /i de- notes a quantity lying between and 1. Obs. The preceding series, even for large values of x, becomes divergent after a certain number of terms; this will happen after n terms if (2n + 1) (2n + 2) ' oT*^ (2n - 1) 2« aj"'* ' J^ (2r»+l)(2« + 2) ^'^^^^^ (2n-l)2n '^ ' 48 but the first member of this inequality never exceeds (2n + l)(2n+2) -^ , (Art. 67.) .'. (2w— 1) 2m>47r*ar', or n>7ra;. It can, however, be proved that an approximate value of the series will be obtained by taking the aggregate of the con- vergent terms only, as will be seen in the next Article. 71. An approximate value of "Zu^ will be obtained by taking the aggregate of the converging terms only, in the series for %u^, invoWing Jdxu^, u^, and the differential coefficients of u^; and the error will be less than the last of the convergent, or the first of the divergent terms. We have by Art. 63, (omitting the index of u,. in the second member,) ^ n . rj 1 , B, du B, 8^u or since in general, by Art. 67, |2« |(27r)»" ^ (47r)" ^ (Ctt)'* ^ '^''•j ' 7?-i-J: , 1 ,1, l '^"" '^''~ l(27r)'"« "^ {^.ttT^ ^ (eTT)""^ "^ ■•• I since, a; being an integer, the term multiplied by sin 2'mvx disappears, and cos 2r>nrx = 1 ; and the arbitrary constant is unnecessary, being already introduced in equation (1) ; • J? - (dx i ^^"" ^"^ sin 4a^ sin Gxtt . da^'^ therefore, numerically, 2-B„ is less than -/ ^ , 1 1 1 ] d'-'^'u B^.cP'u (27r)''" ^ (47r)''" ^ (Btt)^ ^ ' ' ' j ^Zar""-^' [2^ rfcc^" ' 7? /T***"*"^/ which last quantity lies between t|^* ^^ ^.nd '^o ^ a-tn if these be the last of the convergent and first of the divergent terms respectively of the series for Sm^. Consequently the sum of all the divergent terms in the series for Sm^. is less than the last of the convergent or the first of the divergent terms. H. D. E. SECTION III. EQUATIONS OF DIFFERENCES. 72. We now come to the case in which the relation between the principal variable and any function of it is to be determined by means of an equation between x, u^, and one or more of the successive values, or differences, of u^; that is, from equations of the form or, /(as, Mx, Awx. ••• AX)=0, since, by the theorems of Arts. 18 and 23, these forms are con- vertible one into the other. An Equation of Differences is said to be of the n* order when the successive value, or the differ- ence, of the highest order which it involves is the n"". An Equation of Differences of any order is said moreover to be of the first, second, &c. degree, when the successive value, or the Difference, which marks its order, is raised at most to the first, second, &c. power ; or, when it involves a product of successive values or differences at most of two, three, &c. dimensions, it is said to be of the second, third, &c. degree. 73. The complete integral of an equation of differences of the n**" order will contain n arbitrary constants. Let ... Mj,, u^^, Mj4.2s, ... be a series of terms corresponding to the successive values x, x + h, x + 2h, ...; and let F{x,u„ a)=0 be the equation by which the general term is determined as a function of x and a, or the equation of the series, a being an arbitrary constant. Since this equation must hold for all the succeeding terms, we shall have F{x + h, te,^, a)=0. 51 Eliminating a between these two equations, we get an equa- tion between x, u^, and m^^; or, substituting u^+ Au„ for u^^, an equation between x, u^, and Am^, which is the equation of differences of the first order whose primitive equation is F{x, u^, a)=0. In like manner if the equation of the general term con- tained two arbitrary constants a and I, as F{x, u^, a, h) = 0, we might eliminate a and h by means of the two succeeding equations, F{x + Ji, u^, a, l) = 0, F{x + 111, v^^^j,, a, h) = 0, and thus get an equation between a;, u^, u^^, u^^; or, substi- tuting u^+ Am^ for «,^, and u^ + 2Aw^ + AX foi" m^+im an equation between x, u^, Am^, A^, without the constants a and b, which is an equation of differences of the second order, having for its complete primitive the equation F{x, n^, a, b) — 0. Hence it appears that every equation of differences of the first order, or between two successive terms of a series, will introduce one arbitrary constant into the equation of the series; every equation of differences of the second order, or between three successive terms, will introduce two arbitrary constants into the equation of the series ; and, generally, every equation of differences of the n^ order will introduce n arbitrary constants into the equation of the series. Linear Equation of Differences of the First Order. 74. The general equation of the first order and degree is 52 A^ and JS, being functions of x. To integrate it, assume Mx = •"xW^, ••• ■"x+x (Wx+ Aw^) - A^v^w^= B^ ; and in order that this equation may resolve itself into two others each of which admits of being integrated, assume (as we are at liberty to do, having made only one supposition respecting v^ and w^ or, dividing by v^w^ -Sii= A^. But A log v^= log -^±3- , .-. A log t)^= log A^, .: log v,= 2 log A,= log FA,_^ , (Art. 50.) or v^ = PA^^, the constant being unnecessary. The other part of the equation gives u^,Aw^ = 5^, the complete integral, involving one arbitrary constant. Taking the diflference of the result PA^^ - ^ [paJ + ^' "^^ S^* ^PA: PA^' which shews that „ . is a factor which makes each side of the proposed equation integrable; and it is generally the most con- venient way of integrating the equation to multiply it by this factor. Ex. 1. M^, — au^ = of. Here A^= a, PA^ = a^ ; 53 .•.^ = sg) = 2(xV), putting i = a, „I+2 ^ri - («_iy^ + (-^t:t)-3+ ^; (Art. 59.) a;" 2a; 4- 1 2 ^ ^ 1 — a (1 — a) (1 — a) Ex.2. «^,=2.?^ ^' a; + 2 W=r. 1.3.5... (2a;- 1) ' •■ 1.2.3... (a; + 1) ■ Ex. 3. w_ , , — aw. = cos x9. »+i cos (a; — 1) 6 — a cos x0 „ , a' — 2a cos » + 1 Ex. 4. Two vessels which hold a and t gallons respec- tively are filled, the one with proof spirit, the other with water ; c gallons are taken from each and poured into the other ; and this is repeated such a number of times as to make their con- tents of the same strength ; find the number of times logifl-"' logf,-£-" a b Indirect Integrals of Equations of DiflFerences. 75. Since the equation of dififerences of the first order, fix,u^, AmJ=0, is formed by eliminating the constant a between the equations u^ = F{x,a), u^ = F{x+h,a), it follows that we shall arrive at the same equation of differ- ences, whether a be constant, or be a function of x such as a^, provided it satisfies the condition F(x + h,a,^)-F{x+h,a,)=0 (1). 54 Now this equation is satisfied ty a^^=a^, which gives Aa^=0, and a^=a, a constant, and leads to the ordinaiy or direct equation to the series u^ = F{x, a). Also the first member of (1) will be divisible by a^^— a^, because a^ is a value of a,^, which satisfies equation (1) ; and if dimensions of a^^^ and % superior to the first are involved in it, the result of this division will be an equation involving a^^ and a^; i.e. an equation of differences of the first order with respect to a^, the solution of which will give one or more values of a^ in terms of x and arbitrary constants ; and these being substituted for a^ in the equation u^ = F{x, a^, will furnish equations of series, which are primitive equations of f{x,u^, AmJ=0, and each involves an arbitraiy constant. If equation (1) does not involve higher dimensions of a^ and a^^ than the first, they will disappear from the result when it is divided by a,^ — a^. In this case a^ will have only one value, viz. a^= a, and there will be only one equation of a series cor- responding to the proposed equation of differences. The mode in which the indirect solutions just treated of are obtained, is analogous to that in which the singular solutions of differential equations are obtained; but whereas the latter can contain no arbitrary constant, indirect solutions of equations of differences may contain as many arbitraiy constants as the complete inte- gral itself from which they are deduced. Ex. u^ = ccAm^ + F (Am^) . Taking the Difference, we find Am^ = Am^ + (a; + 1) AV + ^F (Am J , or 0=(a; + l) A'M,+ Ai?'(AMj, which is evidently satisfied by Am, = a, a constant ; .•. M, = aa; + a' ; 55 and substituting in the proposed equation ax + a =ax + F{a), .•. a' = F (a) ; .'. u^=:xa + F{a), the complete integral, containing one arbitrary constant. For the indirect solutions we shall have u^ = xa^ + F{a^), rtj, being determined from the equation {X + 1) a^. + F {a, J _ (x + 1) a, - i^ (a,) = 0. Suppose, for instance, that F{a^) = a\, ■■• (a; + 1) K« - aj + aV, - a'^ = 0, or, rejecting the factor a^j — a^, «x+, + Ox = - (a; + 1); or«, = -?^+C(-ir; .-. u, = xa^ + «^. = m^' -iCi-ir+ C\ Linear Equations of Differences of all Orders. The linear equation of Differences of the /i"" order is all the coefficients being functions of x ; the first step towards its integration is to establish the following theorem. 76. If there be n particular values v^, w^, ... z^, which, when substituted for u^, satisfy the equation 56 that has no term independent oiu^, its complete integral is M^ = a^v^ + a^w^ + ... + a„z^, a^, a^, ... a„ being arbitrary constants. For let this value be substituted in the expression and it becomes, (collecting the terms multiplied by the factors «1 (^'x+n +PiVx+,u.i + ■ • • +PnVx) + «J (w^+n + ^i^x+n-i + • • • +P^'>^x) +■'• Now since «;,:, lo^, ... z^, satisfy the proposed equation, each of the quantities included within brackets is equal to zero, there- fore the whole is identically zero; consequently the assumed value of Mj, satisfies the proposed equation, and it contains n arbitrary constants, therefore it is the complete integral of that equation. 77. To integrate the equation of differences, all the coefficients and q being constants. Assume u^ = v^ + k; then by substitution we get Vx+,+i',Vx+n-i + ••■+I>nV, + k{l+p^+... +p„) -q = 0. Let k = , then the equation becomes ^'x+n +PiV^n-i +i'2'^x+»-2 + • • • +PnVx = (l). Let v^ = a"^, then oT (a" +^^0,"'' +p^a'''^ + ■-. +Pn) is the value of the first member ; now this will vanish if a be any root of the equation (called the auxiliary equation), /(a) = a" +J9.a"-' +p,a"-^ + ... + p^,a+p„ = 0. 57 Hence the n roots of this equation a,, a,, a,, ... a„ will give n particular values of v^, a', a/, a^ ...a^ which saitisfy equa- tion (1) ; therefore its complete integral is and the complete integral of the proposed equation is "x = Ci V + c,a/ + . . . + c^a^ + -^. . This shews that we may treat the proposed equation as if it had no term q independent of u^ , provided we divide that term ^y y(l) t'le sum of the coefficients, and add the quotient to the value of Mj obtained on the supposition that q = 0. If we replace c, by c„ — g- -r-/(l), and a, by 1 + A where h is small, so that very nearly o,'" = 1 + hx, and /(I) = - A/'(l) since 1 +h is a root of f{a) = 0, then reducing and making A = 0, we get the form which the solution takes when /(I) =0, viz. M«: = Co + :^\ + &c. "'o /(I) 78. If the auxiliary equation have equal roots, the above ceases to be the form of the complete solution ; because, in that case, it does not involve the due number of arbitrary constants ; and it must be modified as follows. Suppose two roots a,, a^, to be very nearly equal to one another, so that a^ = a^ + h, h being a very small known quantity ; then cfi,' + c^a," = (c, + c,) a: + c, {xa^'h. + ^yi^^ ""^^^ + ^^^ he replacing the constants c, + c, by (7, , and — ^ by O, ; now this equation continues true however small Ti be taken, and therefore when A= ; in which case the second member becomes .'. u, = {C,+ C,x) a,' + c,a; + &c. H. D. E. 8 58 Similarly, if the auxiliary equation have r roots equal to aj, the complete solution will be u^ = {Co+ c^x + c^ + ... + c^.i**"') a^" + c^^ia\^^ + &c ; of the correctness of which, we may be assured by the following reverse process. Assume % = d'w^, then «.+„ = a^ . w^ = a^ a" (1 + A)"«,, = a''{a + aATw,, and the first side of the proposed equation becomes, when divided by a% = {{a + aA)" +p^ (o + aA)""' + &c. + p„} w^ =f{a + aA) w, = {/(a) +/'(«) . aA +/"(a) ^-^ + &c.} w^ =f{a) w^+\ .f\a) . A«. + ^ /"(a) • AX + &c. + a'AX- Now suppose a=aj, and/(a)=0 to haver roots equal to a^ ; this makes the terms as far asy"~""(a) vanish ; and if «'x = c„ + CjO; + . . . + c^_^d^^, then A'w^ = 0, A'^'w^ = 0, &c., and all the remaining terms vanish ; and consequently the equa- tion is satisfied by "x = (c„ + CjCC + . . . + Cr.jX*"^) a^. Hence we see that every root a^ that occurs r times in the auxiliary equation, gives rise, in the complete integral, to a term of the form f^_Jix)a^, where ./r-i(^) denotes a rational in- tegral fanction of x of the (r — l)"" degree involving r arbitrary constants. The root a, may be either real or imaginary. 79. Also if the auxiliary equation have a pair of imaginary roots, m±n V— 1 = p (cos 6 ± V— 1 sin 6), putting p = Vw" + 71", tan ^ = — , m the corresponding terms in the value of u^ will be Cp' (cos e + V^ sin ey + Cy (cos e - V^ sln^)"^ = p" (Cj cos a;^ + Cj sin a;0) , changing the arbitrary constants. 59 And if the same pair of imaginary roots occur r times in the auxiliary equation, the corresponding terms in the value of u^ will be (a„ + a^o; + .. . + a,_,a;'~') f (cos Q + V^ sin 6)' + (60 + ^a; + . . . + J,_,a!-') p'^ (cos Q - V=l sin &f ; or changing the arbitrary constants, (c„+c.a;+ ... +c,_^a;'-')p^cosa;5+ (c'„ 4 o> + ... + cV_j »'->'■ sin a;^. Ex. 1. M=r«-2M,„-13M^,+ 14M^, + 24M,= 0. The auxiliary equation is (a + 1) (a - 2) (a + 3) (a - 4) = ; ••• «- = c. (- 1)' + c,2^ + C3 (- 3)^ + c,4^ Ex. 2. M:^3-5m^j+8m^j-4m, = 0. The auxiliary equation is (a - 1) (a - 2)' = ; .-. M»=c,+ (c+c'a;)2''; and if the second member be q, Mx = c„ + ja; + (c + c'ic) 2"^. Ex. 3. u^-u^ = q. The auxiliary equation is a'- 1=0, whose roots are 1, and 27r , , — - . 2Tr cos -g- ± '^- 1 s™ "3- ' 27rx , . 27rx , .-. M^=C(,+ c cos — + c sm -— + i qx. 80.* The Differential Calculus being a particular case of that of Finite Differences, a strict analogy exists between the methods and results in the two Subjects, as the reader cannot fail to have observed; indeed whenever in the latter a result is obtained with an indeterminate increment for the principal variable, it is possible to pass to the corresponding result in the former, by a method similar to that pursued in Art. 25 ; and which we shall further illustrate by the following instance. 60 In the equation u^^ - 2mu,^ + {m" + w") u^ = 0, putting Mj, = a', we find a={m±n V— 1)* ; = (mH«')^{c,cos gtan-£) + c,sin (ftan-^)} . Now the proposed equation is or, if we replace the known quantities m—1 and n by other known quantities m^h and nji, the equation and its solution take the forms c„ sin T tan ' X \c^ cos ( T tan"^ ; — - — 5- ) + v„ ».^ , , i,,*!* y , , . Now take the limits of these expressions when A = 0, and we i_ find the well-known results, since (1 + 2m fi)'^ becomes e^i, Ma, = e"*!"" (c, cos w,a; + c, sin WjO;) . 80. Before proceeding to the general case of linear equa- tions, we shall consider the case of a linear equation with all its coefficients constant, but having a function of x, X, for the term independent of m^. This equation can be shortly treated by the method of sepa- ration of symbols. For if we substitute B'^u^ for u^^,^, it be- comes {IT +p,JT-' + pjr-^ + ... +p„) u, = X; or, if fflj, ttj, ... ffl„ be the roots of the auxiliary equation, 61 ■'• "^=(i)-a,)(i>-a,)...(i)-a.)^ ^^^- Now resolve this function of D into partial fractions (see Integral Calculus, Art. 34), so that the values of^,, A^, &c., are all known in terms of a^, a^, &c. respectively ; then ^^.= (7^ + 75^ + ...+ n^)x But we have seen Art. 58, that (D - a)"'X= cT^t [XaT) + Ca"^'. .-. u, = Afir"S. {Xa^) + Avar's, [Xa^) + ...-\-A^ar'S, (Xa„-^) aij arbitrary constant being introduced by each of the integra- tions, and the product of two or more constants being replaced by a single constant. The part of the above expression involv- ing the arbitrary constants is called the complementary function ; and it is the value of u^ which satisfies the proposed equation when X=0. 81. If the auxiliary equation have r roots equal to a,, then from (1) 1 y and this function of Z> may be resolved into partial fractions, (see Integral Calculus, Art. 36) so that -ii^. I) -a, -0-aJ *r+i But {D - a)-" Z = d^T {a'X) + a^ {C,+ C,x+ ... + C^.x"-') , C , 0,, &c. being the constants introduced after each inte- gration ; .-. u=A,ar-^{arX)+A,art' (a.-Z) + ...+^A"'^^ (V^ + (c„ + c,a; + . . . + c,.,**"') «/ + Cr+i«%+i + . • • + c„a„'', 62 a single constant, as before, being substituted for the sum or pro- duct of several constants in forming the complementary function. 82. Again, suppose «! = p (cos 6 + V— 1 sin 6) to be an . A . imaginary root of the auxiliary equation; then since — ^ is a ^1 function of a^ , it will be of the form R (cos a + */— 1 sin a) ; and the term involving a^ in the value of % will consequently be i?/)" {cos (x0+a) + V^ sin (x0+a) } 2p"^Z (cos a;0 - V^sinx^) ... (1 ) . Now the term in u^ involving the conjugate root to a,, will result from this, by changing the sign of V^ ; and therefore the sum of the two terms introduced into the value of u^ by the pair of imaginary roots p (cos 6 ± V— 1 sin 6) will equal twice the real part of expression (1) ; that is, 2iip^ cos [xB + a) (Sp'^'X cos xB + 0) + ^Rp' sin {x6+a) (Sp'^Xsin xQ + C) ; where iRp" C cos [xd + a), ^Rp" C sin {xd + a) are the terms introduced into the complementary function ; and, if we alter the constants, may be replaced by p'' (c cos xQ + c' sin xd). 83. Exactly in the same way, if the imaginary root a, = p (cos 6 + V— 1 sin ff), occur r times in the auxiliary equation it will produce in the value of M^, r terms of the form Ajx^"Sr(a^X) ; or, since — ^ is a function of a, and may be assumed «i = i?„ (cos a„ + V- 1 sin aj , of the form i?^^ {cos {x6 + a„.) + V^sin {x6 + a J } 2 V (cos xO - vTT sin xd) X . . (2) . But the root conjugate to a, will produce a term exactly the 63 same as this, except with — V'^l instead of + V— 1 ; conse- quently the sum of these terms will equal twice the real part of (2) ; that is, 25„p"' {cos (cc^+O £■" {p-''cosxeX)+sbi{xe+(x^)-sr{p^Bmx0X)}, and to get all the terms introduced by the pair of imaginary roots p (cos 6 ± V^^ sin 0) that occur r times in the auxiliary equation, m in the above formula must receive all values from 1 to r. Also we see that the part of the complementary function introduced by these roots, by substituting a single constant for the sum or product of other constants, will take the form (c„ + c,a;+ ... +Cr_^x'~'^) p''coaxd+{c\+c\x+ ... + c'^^^x''^)p'Bmxd. We shall now give an instance of each of the cases that have been examined. It may be observed, that for equations capable of being reduced to either of the forms fin) «, = a^ or f{D) u, =^„x"' +j>^x-' + &c. +p^, the process may be greatly simplified. For in the former case, since Btf = a.) u^ = (f is u^= jr^+ com- plementary function. And in the latter case, if jTjy: = ttz — ly. can be expanded in the form A^ + A^A + A^^ + &c., then + complementary function. Ex. 1. «^3-9M^„ + 26M^+,-24M^=5^ or {ir-diy + 26D - 24) m, =/(J9) u, = 5'; 5^ _/l 1 1 1 1 \^^ •■• "--(i)_2)(i)-3)(i)-4) [2l)-2 l)-3'^2D-ir .l..".g)-3".(5)Vl.."S0)- 64 a result we could have foreseen because/ (5) = 3.2.1 = 6. Ex. 2. a'Mjt^.j — 2aUx+i + u^= a'x, or (aZ> — lyu^ = a'x ; .-. M, = fi) - -^x = ^ S' (xa^) = 4s («^V- 2S'a^') V aj a ^ a ' ^ 1 { xa" la^^ „ ^,] 'a'-^\{a-\Y (a-1) a^x 2a' _ a^x 2a' , .1_ OtheiTvise, ..= (l-l+Af..= |(l-lf-2(l-l)"A + &c.}. Ex. 3. %^.2 — 2a cos Qu^^ + a^w^ = X, or (i)'-2acoseD + a'')M^ = X; X 1 / JT X \ •'■ "- - (D _ as) (i) _ az-') ~a{z- s"') Vi> - az -D - ag-V ' putting 2 cos ^ = a + s~\ and consequently 2 V^ sin 5 = s — z~^ ; Now the value of the former of these integrals is — T=z fcos (x—l)0 2^-1 sin 61^ + V'^ sin {x-l)0}'ZXar^{cosx6-'^~lsmxd); 65 hence, taking twice the possible part of this expression, we have M« = -^^ (sin (a;- 1) 6t (AVcosa;^) -cos {x-\) 6% (Za-^sin x&)\, the complementary ftmction being o£' (c cos xQ + c sin xB). If we suppose X = a*, then as we could have foreseen Wi = irr-, Tin + <'^ (c cos xd + c' sin x&). 2 (I — cos w) ^ ' The case of ^ = wiU be presently noticed. Ex. 4. Mx+«+i',Mx+3+i'8Wx+a+i>3M^, + P,M:, = X, where the auxiliary equation is supposed to have two pairs of imaginary roots of the form a (cos 6 + V^^ sin 6), and conse- quently the proposed equation is of the form [P - "ia cos QB + a^u, = {D- azf {D - as"') V = X. Now by the preceding example we have 2a V^ sin 6 _ _J ] {D-az){D-az-^)~ D-az D-az'"' •*• {D - azY {D - az-'f ~ {D - azy V^ / 1 1 \ 1 ■*■ ^^m5 Vi> - as D-az-')'^ {D-az-'f' .: - 4a' sin' du, = [az)'^ S' {az)-^X + ■ \, (az)'^'!, (a2)-^Y+tenns in az'' = a"^{cos {x-2) e + V^sin {x-2) 0} X 2'a"^X (cos xd — V^ sin xff) + " ■ ~^ {cos (a;-l)5+\/^sin(a;-l)5} X ta'^'X (cos a;^ - V-^sIncc^) + &c., H. D.E. 9 66 and taking in these terms only the possible part, and doubling it, we have — 2 sin'^a-'^M^ = sin e cos (a; - 2) ^S' {a"^Xcos xO) + sin ^ sin (a; - 2) 6%^ {a' X sin x6) + cos (a; — 1 ) 6% {a" X sin x6) —sm{x — \) 6% {a'" X cos x6) , the complementary function being {b + h'x) a" sin x9 + {c + c'x) a" cos xd. If X = a", the result will be, as we are aware, a " 4(1 -cos 6')' + complementary function. 84. To integrate the linear equation of differences of the w"* order, the coefficients being functions of x, on the supposition that it can be solved when X= 0. If s^, '^z^, ... '~^Zx be n particular values of v^ in the equa- tion with which the proposed coincides when its second member is zero, we have (Art. 77) ^z = c^^x + c^\ +- c.,% +... + c^i'''\. If we now divide both sides by z^ and take the diflference, we shall eliminate c^ ; next dividing both sides by the coefficient of c, in the new result, which suppose y^, and taking the dif- ference, we shall eliminate c^ ; again dividing by w^ the co- efficient of Cj and taking the difference, we shall eliminate Cj ; and proceeding in this manner till all the constants are eliminated, our final result will be of the form (each A affecting the whole of the expression that follows it) AiAl...AlAfl^), 67 in which expression the coefficient of v^„ is evidently 'x+i War+2 * • • 3'ii:+n_i ^i+n therefore, dividing by this coefficient, we get the expression .,,_.,.„AiAl...AlA(j), ilx which must be equivalent to the first member of equation (1). Therefore the same expression, only with u^ instead of v^, must be equivalent to the first member of the proposed equation, and consequently equal to X; hence, equating these equals, and integrating, we get 'n-I W^, . . . yx+n-i ^^.n (each S afiecting the whole of the expression which follows it) which is a general formula for ,the integration of any linear equation of difi'ereuces whose solution can be effected when X=0. In the case of constant coefficients if the auxiliary equation contain equal or imaginary roots, this method is still applicable ; it is only necessary to assume for v^ the value belonging to the case of equal or imaginary roots, as will be seen in some of the foUovring instances. Ex. 1. Suppose the coefficients of the equation to be con- stant, and a^, flj, ... a„ to be the « roots of its auxiliary equation. Then V, = Cj a\ + c,a\ + c,a\ +...+c„a\, changing, both in this, and in the similar succeeding steps, the arbitrary constants ; -©■-^.-©"--■©" 68 Mi)^©^?.-©----e in which expression the coefficient of v^^„ is evidently therefore, dividing by this coefficient, and replacing v^ by m,, we get (-.,>X.A(^)"^(^)-...A@-A^.Z. or, if we choose to introduce the arbitrary constant after each integration, u^ = cy^ + c^a"^ + ... c^a'^. -<-)-F.«-^©'^©'-^fcT^© This result may be readily obtained by separation of symbols, and so shewn to agree with that obtained in Art. 80. For the proposed equation is i>X +p^ir-\i, + ... +p,^,Du, +p„ii, = X, or {D - «„) {D - a^,) ... (Z» - «.) u, = X; .: {D-a^,) {D-a^,) ... (D-aJ «, = (i)-a„)-'X=ar'2 {a^X) = a\X„ suppose, where X„ = — S (a„~'-30 ; = a^r t (;^T^» = «V,^^, , where X,^. = -^ 2 (^)\, 69 and 80 on till we arrive at «. = {D- aya\X, = ar t [^Jx,, when X, = i S g) X,. Hence restoring the values of X^ , Xg . . . X„ , we get since «,«,... a„ = (- 1)>„, .=,->,- i<.g)-xg)-....(^)-x©, where each 2 affects the whole of the expression that follows it. Ex. 2. aX+2 - 2aM^, + m^ = a'X. Here v^ = (c + c'a;) — ; .-. A" (aX) = 0, in which expression, the coefficient of v^^^ is a"^; therefore, dividing by this quantity, and replacing v^ by %, we get ^ A» (aX) = ^; .-. «. = ^ S' (Xa-^). Ex. 3. u^^ — 2mu^^^ + (ni' + n^ u^= X. Here v^ = c^p" sin x0 + c^p" cos x0, ft where p' = m!' + m°, tan ^ = — : V. ■ — — ^— 3 = c, tan a;^ + c, ; p cos X0 ^ " • A f — ?^5^^ - CiS in [^p" cos a;^/ cos x0 cos (a; + 1) ^ ' .-. Acosa,ecos(a, + l)eA(^-,^)=0, COS f 3? "4" 1 1 V in which expression, the coefficient of v^^ is \^ - ; there- fore, dividing by this coefficient, and replacing v^ by u^, we get p- cos (a; -— T-5 A cos a;5 cos (a; + 1 ) ^A ( ^— ^-^ = X ; + 1) ^ \p cos X0J ' 70 .-. u^ = p cos xdZt sec x6 sec (a; + 1) PZ \ \^ — — ■ ; to which we may add the terms c^p'' sin x6 + c^p" coaxB, if we suppose a constant to be added after each integration. 84*. Having given a particular integral of the equation ^x+2 + ^xU^l + Sji^ = 0, to find its complete solution. Let v^ be a particular integral, then Hence, eliminating A^, we find, putting — = z^, 'Xf^l^^'^''' appose; or Alog(Az^) = logw^, .■.Az,= CFw^„ (Art. 50) 85. If we know a particular integral of a linear equation of any order that has no term independent of u^ , we may reduce it to another equation of the same kind of the order immediately inferior. Let Mj. = w^ be a particular integral of a linear equation of differences of the w"" order reduced to the form AX + ^.A--^ + ?, A-'-^M, + ...+q^u, =/(A) M, = (1) . Assume u^ = v^Xw^ ; then AX = (A + i)A')XtM7„ where A and D affect v^ only, and A' affects Sw» only (Art. 34), and the proposed equation becomes 71 Now /"(A + i)A') is a rational integral function of A and A', ■which we wish to arrange according to powers of A' , and this may be done at once hy Taylor's theorem, which gives /(A) +/ (A) D^' + ^A (^) ^^'^+ ... + ^"A'". Hence, observing that /(A) v^ = 0, since v^ substituted for w_,. satisfies the proposed equation, we get the depressed equation wj, (A) Vi + ~ Aw J, (A) V,,, + ...+ Ar-\o, . v^,„ = ; or, reversing the order of the terms, {since f^ (A) means the same function of A, that '{ r ^, +P/i'^u^^^ + ... +p„aru^ = 0. Assume u^ = v^a^^'^^-*'\ then any termj?XX+n-r becomes — I'r'^x+n-r"' • " > therefore the equation becomes divisible by a^^"*^*^', and is reduced to the linear equation with constant coefficients, 3. u^, + {a + b{-iy}u^, + cu,= 0. Let u^ = «^ Va + i (- 1)% then M^, = v^, 'Ja — b{- If, 4. Mj+iMj; + aw^i + Jmj, + c = 0. 73 Assume «. + a = -^^ , then w^^, + a = -^^ , or u^2 - (a - 5) «^^j + (c - a5) v^ = 0. The two arbitrary constants which will appear in the value of t)j, must be reduced to a single constant by the condition of the proposed equation being satisfied. Ex. ujii^^ - 2m^ + 1 = 0. M, = 1 + (a; + c)*\ 5. M^„ . M^^^i ...«/= X If we take the logarithm of both sides, and assume log u^=v^, we get Vx+» +i'«^n-i + 2'«^x+a-2+ . . . + rv^ = log X ; from which we can determine v^-, and then we have m^ = e*'. Thus, if u^u^^^ • Wx = 1> then u^, - 2j;^, + v^ = ; .*. v^ — c^ + c^x, andM^= C". C^ Again if u^^ . m^, . m"" = o, then v^^ + 2mr^j + wX= log a ; which is the value of log u^ . 6. m'^, + m. = 2. Assume u^ = 2 cos r^ , then 4 cos'' v^„ + 2 cos v^ = 2, .-.2 cos" u^i = 1 - cos «;^ = 2 sin' J t), , or cos v^, = cos i {■"■ — v^ . ••• Vx+, + i v^ = if, which gives v^ = | . i^ + c (- J^)'', .-. M, = 2cos{j7r + c(-in. H. D. E. 10 74 Similarly, m^^, - 2m', + 1=0, by assuming u^= cos r,, gives cos v^^^ = COS 2v^, .'. ?;,^,, — 2v^ =0, v^ = C2°', and M, = cos r, = i (e"-^^ + e- •"^) = i (c-''' + c^) putting c = c*^^'. 7. M^,M:, - X (m^i - mJ + 1 = 0. Let M^=tan?;,, .-. X(tan v^, -tanv,) =1 +tanv^, tanw,, or tan {v^^^ ~ ''^^ ~ T ' •*• ^'"'' ~ ^^~^ T' and u^ = tan f (7+S tan"' -^j . If X=l + a; + a;', «x = i3^- Similarly m^+jM^,Mx = aw^ + « ("i+i + "x«)> by assuming u^ = Va . tan v^ , becomes tan v^ (1 — tan v^, tan Vj^.^) + (tan v^^ + tan r,^,) = 0, or tan v^ ± tan [v^^ + tv^.,) = 0, .*. v^^, + v^^ ±v^= ■nr, where r is any integer ; , 2wa; , . 27ra; .-. v^ = Jm- + c cos — - + c . sm-— - , o o or v^ =r7r + c ^2 sin ^j + cW - 2 sin -j^j , according as the upper or lower sign is taken ; and then u^= Va tan V,. 87. We shall next give some examples of equations which by particular substitutions are reduced to linear equations of the first order. 1. {ax + h) A^u^ + mx^u^ + mnu^ = 0. Assume «, = A"~'»x ; .-. {ax + b) A''*^ + »»a; AX + mn A''\ = 0. 75 But putting u^ = x — n in the formula for transforming M^^„A"t;, (Art. 34), we have a;AX = A" {(a; — n) vj — nA""V^, .-. oA-"-' {(a; - w - 1) v^| - a (w + 1) AX 4- iA"""?;^ + wA" {{x-n) v^] = 0, or A {a {x - n - 1) +b} v^- {a {n + I) -m{x-n)}v^ = '%*(), .'. {a {x — n) + b] v^j + {{x — n){m — a)—na — b] v^ = S"0, a linear equation of the first order, which will famish the value of Vj:, and thence the complete integral of the proposed ; the supernumerary constants contained in S"0 = c„ + Cjo; + . . . + c„_ja;"-' being determined hy substituting the value of u^ in the proposed. For a particular solution, supposing S"0 = 0, and putting . _ {a — m) (x—n) +na + b '^ a{x-n) + b ' we get v^= C.PA^^, and m^= CA""' (P^^J. 2. {ax^ + h) AX + (»• + mx) Am, + (»n - w) (re + 1) u^ = 0. Let M, = A""'dj, then (03^" + J) A""'X+ (y + wa;) AX+ (w-w) (« + 1) A""X = 0-"(l)- But if in the formula for transforming m^^AX (-^^ 34) we make % = a; — w, (a; — w)', successively, we get values of xA'v^ , ar'A"'^'Wj; and substituting these values in (1), we find aA"-"' [{x-n-iy V,} - a (« + 1) A" {(2a; - 2m - 1) v,} + a (m + 1) wA""'«;, + iA""*''«, + rAX + 'w^" {(a; - n) v^] - «jwA"''«, + (»i - w) (m + 1) A''"X= 0, or A"-''{a(a;-w-l)'' + 6}v, - A" \a {n + 1) {2x - 2n -l)-m{x-n) - r} v^ + {(« -l){n+l)n + vi} A"-'i., = 0, 76 which is integrable in several cases by the disappearance of one of its terms ; thus suppose o = 1, m = 0, then A{{x-n-iy+b]v^-{{n + l){2x-2n-l)-r}v^ = f0, or {{x- ny +b}v,^^~{x^-n{n + l) + h- r] v^ = S"0, a linear equation of the first order, from which v^ and % = A''"Vj, may be determined; and the arbitrary constants contained in %"0 must be reduced to the proper number by substituting the value of Wj. in the proposed equation. But if we suppose the complementary function ]S"0 to vanish, we may readily obtain two particular values of v,, ; for let oc'—n{n + l) + b — r_ . [x-nY+b ~ "' then «,„ - A^v^ = 0, v^= G . PA^^^ . Now replace n by —(« + !) which amounts to assuming u^ = A~""^v^ , then v^= C . PA'^_^ , where a? — n{n + l) +b — r _ ., {x + n + iy+b "' then M, = CA- (P^^,) + C'S"« {PA'^J, the complete integral of the proposed equation. Several other examples of the like kind may be seen in the Cambridge and Dublin Mathematical Journal. 87*. The solution of /(Z>) u^ = b" in the form (Art. 83) u^ = jTy: + Ca^ + &c. fails, if b be equal to a root of the auxiliary equation f{a} = 0. Suppose that J = a, a root that occurs twice in /(a) = ; then the value of u^ by changing the constants may be put under the form u^ = jrgr {b" - a/ - xha^'^^) + (c„ + c^x) a^ + &c. 77 Now suppose i = a, + A, where h is very small ; then since /{«,) = 0, /'(a,) = 0, we have ^a (a: - 1) AV^ 4- &c. , x « p ^-= U'/"K)V&c. +(^o + c.^)< + &c.; therefore, making A = or S = a^, we get a; (x — 1 ) M« = -^77^ a,^ + (c„ + c,a;) a,' + &c. And if the root to which h becomes equal occur r times, the first term in the value of % will be x{x-l)...{x-r + l)ar -/"■' («.)• Hence in u^^^ - 2a^u^^ + a^u^ = a^, since f{a) = {a- a,y f{a^) = 2, w^ = ^x{x- 1) a,"^" + (c„ + c^x) a^ ; and in examples 3 and 4 of Art. 83 if ^ = 0, the first term in the value of Mj. becomes ^x{x-l)a''^, and —x{x-l){x-2){x-3)ar*, respectively. 88. In simultaneous equations, instead of one equation be- tween the independent variable x, and the successive values or differences of the dependent function u^, we have given two or more equations between x and two or more of its functions u^ , v^, &c. and their successive values or differences; the object, as before, being to determine each of the imknown functions in terms of x. When the proposed equations are linear with con- stant coefficients, we may separate the symbols of operation fi-om those of quantity ; and then obtain by the ordinary processes of elimination an equation containing only one of the unknown functions ; for as the symbols of operation here employed com- bine according to the same laws as ordinary algebraical quanti- ties, they may be treated precisely as if they were symbols of quantity. 78 1 . Mj^j + cv^^., — c'm, = a' , = a . These may be written {Ir-^)^l^+cDv^ = a'', which give by the elimination of v^ {If - &jy + c*) u,= (P- c) a' - cDaT ; ••• *^- = a- - cV + 0- - 1 - cV + cV + ^^"'"Pl^'^^^t^^y f^°^faon. Now if c be greater than 2, i/- c»i)^+ c*= (i?- a') {ly-^), and consequently the complementary function = c, a^ + c, (- ar + 03/3^ + c, (- /Sr . If c= 2, the values of a and /3 become equal to one another; or imaginary if c < 2 ; and the corresponding values of the com- plementary function may be determined in the usual manner. The value of u^ being known, we have 2. M^^, = 2z^ + 12m, , y^^ = Uz„ Upon introducing the symbol D, and eliminating m, and y^, we find {D - 6) (-D + 2) (D - 14) s, = 0, which gives z^=a.G'' + b. (-2)^ + c(14)'; then y, = 14^^, = ^ 6' - 76 (- 2)' + c (14)% «x=-?6^-|(-2)^ + «(14)'. 79 89. It may be observed that the knowledge of the genera- ting ftinction of the second member of a linear equation will sometimes lead to the solution of the equation. For since (Art. 28) Now suppose \lf{I>)u^ = X to be an equation of Differences, and let <}){t) be the generating function of X, and therefore of its equal yjr {D) u.,, then and if this value of Chi^ can be developed in a series of powers of t, and have w^ for the coefficient of f, then the solution of •^ {D) Mj. = X is Mj, = to^ + complementary function. Thus, for M^j — 2 cos 0u^^ + Mi = sin x0, *W = n:iS^^=- + *^---^+- (Trig- Art. 155) ^fA . I (^\ <' sin it) -ty = (i_2 - 1)" w,,„ , Z>X,„ = (1 + A)" «,_„ ; which become by development AX , = i>X,„ - wi?"-X,v + "^"~^^ i>"X. - &c. ^«xj, = Mx,» + «am, „ + ^^"2 ^Xt/ + '^°- ; where A, D express total operations that refer to both of the independent variables. 81 If there be given Am^„ =f{^: y), a rational and integral function of x and y, we may find the value of m^„ by the method of Art. 44 ; thus if Am^„ = x+y + \, then where G^_y denotes any function oix — y, the increments of both variables being taken equal to unity. 91. Again, if in u^y we change x into x + h without alter- ing y, the result u^^^ is called the first partial successive value relative to x, and is denoted by B^u^^, so that JDji^y = %^ „ ; consequently and as M,j^j, = e '^m^,„, -Dj, is equivalent to e '^. Exactly in the same way we have, relative to the other variable y, ■^V^x.tj — ^x,v+t > -^y ^x,v = '^x.y+nk ) d_ Dy being equivalent to e ""; hence D = D^.Dy. Also the dif- ferences M;c4ji,y — Mx.yj ''x.K+i — W3,_y, are called the partial dif- ferences of u^y relative to x and y respectively, and are denoted by A^, A„; so that ^xWx,» = "x+».» - Ux.v = (^x - 1) Wx,!/ ; \Ux_y = 2(^.„+i - u^,y = (D„ - 1) u^_y ; and generally A.X„ = (2>x - 1)" %..> K = (A - 1)" %., ; and the symbols S^,, S^, which are equivalent to A^"\ A^"', are used to denote the operation of integrating w^„ with respect to the variable x only, and the variable y only, respectively. In the investigations which. follow, the values of h and k will be taken equal to unity, unless the contrary be stated. H. D. E. 11 82 92. By Equations of Partial Differences are meant relations between a function of two independent variables m^_„, some of its partial successive values or differences, and the variables x and y. As the symbols of operation here employed combine in subjection to the same fundamental laws as algebraical quantities, we may, by separating the symbols, resolve these equations when linear or otherwise, by the same processes as ordinary equations of differences of the like class: but we shall first employ an elementary method. Ex. 1. M^^„,„ +i'iW^+n-i.»+. + ••• +i'nMx,v+» = (l). t^C COCffi- cients being constant, and the sum of the subscribed indices the same in every term. If the second member be q instead of zero, it is easily seen that the value of %,„ determined from (1) would have to be increased by -2^ . Assume u^^ = d'. G^y, where Cj.^„ is any function of the binomial a; +y, then the first member of (1) becomes a'' (a" +p,a"-'+ ... +p„) C^^^,. which is reduced to zero if a be taken equal to any one of the roots real or imaginary, a^, o^, ... a„ of the auxiliary equation /(a)=a"+^,a"-'+...+^„ = 0. Hence, if Oj, a^, &c., be all unequal, the complete value oiu^^^ involving n arbitrary functions is • M,,, = a^ C'^y + 0/ C"^y + . . . + a/ C (1) which shews that arbitrary functions of the binomial a; + y here stand in the places of the arbitrary constants that enter into the solutions of ordinary equations of Differences. But if a^ = a , first suppose a^ = a^+h where h is very small, then 83 = < • c'^y + c\^, . xa,' {l + \{x-\)- + &c.}, replacing the arbitrary functions C'x+« + C"^ and - C"^„ by other arbitrary functions c\^ and c"^„; now make h = 0, then the equal roots a,, a^ produce in the value of m^„ the terms '''i (<'W» +^c"^J. If a^, a^ be a pair of imaginary roots and = p (cos 6 ± V^ sin 6), then it is easily seen that a' G'^^ + a/ C"^^ takes the form p" {c^y cos a;0 + c'^y sin a;^). It may be observed that the assumption u^,y — a" . C(^„ is symmetrical with respect to x and i/, for it may be replaced by ay n..y=[^.a^.G^y = V.C'^y. b Ex. 1. u^+i,y — O'Ux.y+i = b. Here u, ■x.v — « • ^x+y + J _ 3 • Ex. 2. M^j,„ - 2aM^^.,.„^, + aV^,„^j = h ; W:r,„ = «" (Cx+„ + a; • C';^„) + -r-, . ^1 — «j Ex, 3. u,^,,„ - 2a cos 0m^„,,+, + aX, «+2 = i- u..y = a"^ (c^« cos a;0 + c'^„ sin xd) + ^_g^^^gg^^, . 93. The preceding results can of course be readily obtained by separating the symbols; but that method may be applied with still greater advantage to the case where the second member is Fa function of x and y. For the proposed equation may be written {{Dj)ir+p, {Bj)-r-'+ ... +Pn] D;u..y= v, instead of Wa^+n,!, +i'i«*+,i-i,i/+i + ..• +i'»%.i/+n= ^- 84 Let a be a root that occurs singly, and h a root that occurs m times, in the auxiliary equation z" +^,3""' + . . . + jo„ = : then by resolution into partial fractions we shall have . B„ . -^m-i , I -"i I (fee z ''+p,z"-' + ...+p„~ z-a^ {z-hr^ {z-br-'^-^z-b where ^, jB„, &c. are known in terms of ^,, p^, &c. ; hence sub- stituting D^D~^ for z, we get •■-" li).-ai)/ {D^-bD,r {D^-hD,r . B,D.. i>rv; but A (Z>, - aD.YD,-^^' V= A {aD^y't, {aD,)-^D,-^*' V; &c. ; since the operations denoted by B^, By, are independent of one another : .-. «,,„ = A (aB„r 2. {aB^r A"^" V + BJJ>B,r^ tj" {bB,r B^*"" V+ &c. Instead of reserving the complementary functions under the sign of integration, we may obtain them explicitly by supposing V= ; then since 1,0 = c„, S/0 = c\ + xc\ + a?c\ + . . . + a^^-V""", and B^''\ = c^y_^, X)„""c„ = c^^„.„, the part of the complementary function due to the root a and the m roots h will be, changing the arbitrary functions, Aa^c^, + BJf {cV« + xc',^, + . . . + aj'^'-^S^"}, and similar terms will be introduced by the other roots of the auxiliary equation. If any of the roots a, h, &c. be imaginary, these results may be transformed into real expressions, as shewn above. 85 Ex. 1. M^,„ - M^ „^, = xy, or {D^-D,)u,^,=.xy, .: u^.y = {D^-D,rxy = {D,r't,D;^ ^ • but D^xy = a; (y — a;) = X (^ + 1) — a; (a; + 1) ; . _j,^A {x-\)x{y^-\) {x-\)x{x^\) \ = J (a; - 1) a? (a; + 33^ - 2) + C^,. Ex. 2. M^2, „ - 2m^,_^^j + M^,„^j = a; + y, or {D^-D^Yu^_y = x + y; .: u,,,= {D,-D,r{x + y)=B,'^V^r (s^ + v) = i?/-^S/y = A"^ {i (a; - 1) a;y + a:C, + C;| = ^ (a;-l)a;(y + a;-2)+ax;^„ + c',^„. Ex. 3. M^^.3,, - 12m^^.,.^^, + 1 6m^ „^3 = c''"', or (D, - 2Z)J' (i). + 4Z)„) M,.„ = c^ ; and since D^, Z>„, applied to c°^ are each equivalent to multi- plying it by c, Mx„ = + &c. =10^-^ + 2^ (c^, + a^'^,) + (- 4)'c'; Ex. 4, M^^.„ - 2a cos ^M^r+i.^+i + oX.n.2 = J^- «... = «^ ^^fe^^rS.(a-^ cos x0i?,— F) - q^ cos(a;-l)g sin«02?„--' F) Binp 1/ i\ tf + a* (cos x^ . c^^y + sin a;5 . c'^^y). Ex. 5. [D, - a) [D, - 1) «,., = V, or M^i.y+, - OM^.,^, - iu^+i,y + ahux.y = ''^5 .-. {Dy-b) «,,„ = (Z),- a)- F= a-'S^-^F+ «-(?„, If V= (f^, since D^, D^ are in that case equivalent respec- tively to c and c~' ; en {c-a){c-'-b) Wx.» = /. _ „s ,„-! _ j.^ + ^'c'x + a'c"r Equations of Mixed Differences. 94. Equations of Mixed Diflferences axe those in which the differential coefficients of the dependent variable u^, as well as its differences or successive values, are involved along with the prin- cipal variable x. When the equation is linear with respect to the successive values and their differential coefficients, and the coefficients of the several terms are constant, solutions may be obtained by proceeding in the same way as for linear equations of differences of the like kind. It is plain that any differential equation /(^' y> S ' ^^ ' &C-) = 0, dx ' da? which admits of a solution y = X, will lead by substituting y = Ux^n +PiU^n-i + ■ • • +i'»Mx , where ^,,^j, &c. are constants, to a linear equation of differences of the kind that has been solved. Multiplying both sides by e"" and integrating, we get an ordinary linear equation of differences. Thus suppose the proposed equation to be then u,: = Cm" + C (- a)", where m = e"'. 87 then u^, - 2m, = ^^-^ + C/^ ; 3. (1 + ae"^') «;,+! - 6 (1 + oe"^) ^ M^ + hu^ = c. Assume u^ (1 + ae") = v^ ; .-. aeX+(l + «O^«.-5 = 0, and multiplying this by h and adding it to the proposed, we get Now, letv^=e"" + A; let k + hk = c, then e" — (m — 1) 6 = ; and if ^ be a value of m which satisfies this equation, then a particular value of u^ is nb(^^+iT-^)' 95. The following are examples of Equations of Mixed Partial Differences. or I>A» -^ «:».» = <'» .■.».,.=(i>.-i)-.=(ir^(ir-i''.}' putting -7- (7„ for an arbitrary function oiy; 88 •■• *'-="+(!;) ^'"" and if c = 0, we have for the equation 2. M,^j.„-a^M,.,= F; Suppose V=x^, then the effect of any power, positive or negative, of -5- upon V, is to multiply it by 1 ; .•...(«|)-V.«..-«=^.--<|ji^+.4c., ...... ..(-.-«^/-(«|P'("-|)>- Let F= xe", then proceeding as in the last example. 4. A^M,.„-a^M^.„ = c. As this may be written A. (Mx,» -"')-'* ^ (''-.v -«")=<>> 89 we may integrate it supposing c = 0, and then add ex to the result. Assume ! °^' "^m-i ("x + «) = c ; V Assume ■u^+a= -^±i , then v^^^ — av^^^ — cy^ = ; .'. v^= c^ol" + c^^, a. and /3 being roots oih'' — a'k — c = 0; = -^^^rjrzfw~' smcea + /9 = «; c 1 + C3 /3 H. D.E. 12 90 If a and /9 be imaginary, so that l^-ak-c = ¥ -2kp co%e + p\ sin x6 then Ux = — p —. — 7 — , t\ a • '^ sm {x+lj 2. To find the value of the continued fraction c c c c "' "" ^ a+... ^ 6 ' to a; fractional terms, the last term being -j . Proceeding as in the last example, we find a^' + c,^-' and M, = T = c If c = — 1, a = 2 cos ^, b = cos 0, then it will be found that _ cos(a!— 1)^ * ~" cos xd 3. To find the value of the continued fraction to x fractional terms, 1 1 1 1 '''~a+ b+ a+ b+ ...' 1 u Here M^, = —, or- — ^—=b + u^. a' + c,^ ' a + 0,/3' •• ^3^ b ~ b- — a j(a-^_^>) + c {a'^- -^-^) Z' + Mx Assume 1 - au^ = -^=*^ , .'. 1 — aM,^, = ^^2±i % Ws 91 05^ ^x+3 - (2 + ah) V, + r^, = 0, of which the auxiliary equation is A*- (2 + fli) t=+ 1 = (i'- V^A;- 1) (F+ V^ife-1) =0. Let a, , — a, - , be the roots of this equation, then v^ = a" K + c, (- 1)''} + ^ {c. + c, (- 1)^}, ' 1+ab a' + »,a-" .•.^, = l=C-0', .-. (7 = 0, C7' = -l, and^, = (-l)', .-. au^ = l "^ '- a-*> +(_!)-«--'' 4, Suppose h to be essentially negative and = — c, so that the continued fraction is a+ — c+ a+ —c+... and let VoF = V^ Vac = 2 V^ sin 6, then ;t»_2\Airi sin ^ A- 1 = 0, anda = co3^ + V^ sin^; 92 therefore, according as x is odd, or even, sinfx — 1)^ , cos (a;— 1)0 aw, = 1 — : — 7 -4-7i , or 1 ) — — rra • sin(x + l)0' cos(a;+l)0 Functional Equations. 97. If a function <^ (x) is of such a nature that when it is twice performed on a quantity, the result is the quantity itself, that is {(^ [x)] or ^^ {x) = x, then it is called a periodic function of the second order ; and if ^ {x) be such that ^" {x) = £c, then ^ (a;) is termed a periodic function of the m"" order. Any sym- metrical function of x and y, put equal to zero, will ty resolution give for the value of y a function of x, which is a periodic func- tion of the second order. Consequently, a X 1 — x' ' x-V 1 + x' Vl-x", are all periodic functions of the second order. Examples of periodic functions of the third order are , ^/l — x"': and of 1 — x 12. the fourth order ^ , . Functional equations involving these expressions may be frequently solved by a simple elimi- nation ; as will be seen in the following instances. Ex.1. {fia^)Y.f(lz£j^c'x; 1 — X . here, since is a periodic function of the second order, 1 — x replacing x by , we get {f{\^)]'fi^i'^-\?.- hence dividing the square of the proposed equation by this, we have {f{^)Y = \^^c^x\ or-/(x) = (l±|c'-^* Ex.2. /(.)+../(_!_) =1; 93 here since is a periodic function of tlie third order, re- placing X twice in succession by , we get Now multiply the second of these equations by — a, and the third by a', and we find by adding them together, 98. Other functional equations may be sometimes solved by the aid of the Calculus of Finite Differences, or by particular artifices. Ex.1. f{x)+f{a-x)=c\ Assume x = u,, a — x = m^, ; .-. V, + Mr = a, or M. = C'(-l)' + ia. But /(m^i) +/(m.) = c', which may be written v^,. + n = c' ; ... ^, = C (- 1)' + ic' =-^ («. - ia) + \e =f{u,) ; •'• /W =m{x-^a) + ic^ m being an arbitrary constant. In the same way we may find a particular solution of the equation f{a + x).f{a-x)=a'-Qc'; or we may use for that purpose the following artifice, f(a + x) f(a — x) , ± / \ J / \ 1 '' ^ — '- /-^ = 1, or A (a;) .d>{—x) = l, a + x a — x 1- \ / r \ 94 which is satisfied by (f) {x) = C ; .: f{a + x) = {a + x) C^ ={a + x) C'"*'^-' ; .■.f{x)=xC''-\ Ex. 2. f{bx) = nf{x}. Let bx = !/^j , x — u,; .-. M^, — bu, = 0, OT u, = -b' = x; .-. v. = n'.<^(cos27ra)=n'°"x (cos 27r«) = 1 , Ex. 4. /(x) ./(^) =/(x + y) +/(a; - 3/) . Expanding the second member by Taylor's theorem and dividing both sides hjf{x), we get /W -^l^ + ty^^)- j^2 +1.2. 3. 4 -/(a;)- t^x* + '^•^j • 95 But as f{y) does not involve x, the coefficients oiy\ y\ &c. in the second member must te constants ; let b be the value of the coefficient of y^, then ••• /W = 2 (1 + ^by' + 172^ ^y + '^^•''' or replacing i by another constant - c^ f{y) = 2 (1 - icy + Y^gig^ cy - &c.) = 2 cos c^,. 99. We shall conclude this Section by two or three Problems, the solution of which is effected by equations of Differences. 1. To find the value of m, = v 2 — v'2 — ViTT to x terms. Since u^, = V2-M, , u\, + u, = 2; of which the integral is m^ = 2 cos {c (- ^)'' + ^tt} ; (Art. 86) ••.^ = 2cos(-| + |) = V2 = 2cosf, .-.0 = 1; .-. «, = 2 cos 1^ (- i)^ + j| = 2 sin I {1 - (- i)^. 2. A swan breeds three cygnets in its second year, and four every succeeding year ; and the young ones all breed according to the same law ; required the number at the end of the cc"" year. Let u^ = number at end of a;*'' year ; then in (a; + 1)"* year, there will be bred three each by those in their second year, and four each by all the others ; .'. ?<^^j = M, + 3 K_j - Mi.,) + 4m^_, = m, + 3m^, + M,., ; ■■• M«3 - «i+2 - 3«;r+, - Mx = ; 96 /. m'-nt'-Bm-l=m {rri' - 1) - {m + 1)' = ; .-. m = - 1, and 1 + V2; .'. M,= a(-l)'" + 5(H-V2)-+c(l-V'2)^ u^= = a+b + c, M, = 1 = -a + & (1 + V2) + c (1 - Vi), Mj = 4 = a + 6 (3 + 2^2) +c(3-2V2); If the law of increase be one in the second and every succeeding year, V5u,= i(i+V5r-i(i-V5r. 3. If a person possessed of a given capital of £n, spends in the first year the whole of the interest receivable at the end of that year ; and in the second year twice the interest receivable at the end of the second year; and in the third year thrice the interest receivable at the end of that year, and so on ; to find how long it will be before his capital is exhausted. Let u^ = reduced capital in pounds at the end of the a;"" year, r = interest of £l for a year ; then ru^ = interest receivable at end of (aj+l)*"" year, and (x + l) rM^ = expenditure in (x+l)"" year, therefore reduced capital at end of {x + 1)"" year, is M^ + ru^ — (x+l) ru^ = (1 — rx) u^ ; •■• Mx+i - (1 - '•^) u^ = 0; .'. M, = n (1 -r) (1 -2r) ... {I - {x - 1) r], since when a; = ] , m. = n. Now this vanishes when a; = 1 + - , which expresses the number of years the capital will last. 97 100. The follomng are instances of Geometrical Problems, which admit of laeing solved by Finite DiiFerences. 1. To find a curve in which the chord QP joining the extremities of any two ordinates PN, QM at a given distance from one another, when produced shall meet the axis of the abscissae at a constant distance from the foot of either ordinate. then (y,^ - y,) -r /< = «/, -=- ~^ ; ov y= C»i* is the equation to the curve. 2. To find a curve such that the portion of a sfraight line drawn through a fixed point in its plane and terminated by the curve, shall be of a constant length. Let p =fifi) be its equa- tion, then we must have for all values of 9, As IT is here the increment of ^, let^ = 7r«, and let/(7rz) be denoted by «,, then m,+, + m.= c ; ••• w.=ic+c,(-ir, where C. denotes any function of z that does not alter in value by the change of z into z+ 1. Now {—!)'= cos irz, and F (cos 2irz) ftilfila the condition to which C, is subject ; .-. p =/(5) = i c + ^ (cos 26) . cos 0, the equation to the required curve. If i^(cos iff) = b, this be- comes p = ^c + & cos ^ ; which represents a circle if 6 = 0. 3. To find a curve such that any n radii drawn through a fixed point in its plane so as to make equal angles with one H. D. E. 1-^ 98 another, and tenninating in the ciirve, shall have theii sum invariable. ^^P=f{^) be the equation to the curve, and A denote the angle between two consecutive radii, we have f {0) +f {0 + h) + ...+f {6 + {n-l)h}=c. ..{!), or making = hz, and f{hs) = «,, M^^i + M»+„-2+... + M. = C (1), the solution of which is 1 „, iiTZ „„ 4t7rz «, = - c + O, cos h C, cos 71 71 n + cr cos ^ + &c. + Cr^ cos ^llt3ll , n n for it is seen by summing the resulting series that any one of the values cos , cos , &c. for u,, satisfies (1), when c = 0. „ , 27r ,, 27r» mhz md j /-,, ■, xsovT suppose ft = — , then = = — , and (J. may be replaced by f^ (cos 27r«) =f^ (cos mff), &c. ; consequently the required equation to the curve is 1 , ^, as ''^0 , xf a\ 2to0 p=- c+f^ (cos mvj cos h/j (cos mff) cos 1- . . . , - , ^\ (n — l)m0 +j^, (cos?«t/) cos-^^ . If in = 71, f^ (cos mff) = J, and if the rest of the arbitrary functions vanish, we get 1 - p = -c + b cos „ n 0, 4. To find a curve such that the product of the two seg- ments of any straight line drawn through a fixed point in its plane to meet the curve, shall be constant. li p=f{0) be the equation to the curve, then 99 f{e) .f{6 + tt) = c* ; let = trz, and /(ttz) = u, ; .•. M^j., . w, = c', which gives (Art, 86) u, = cCy^y = c . {F{coB 27rs)}"»'", since G, is any function of z that does not alter in value when z is changed into z+1, and (— 1)' = cos ttz ; .-. p = c[F {coa20)}">*\ Now i^ (cos 20) denotes any function of cos 20 whose value does not alter by the change of into 9r+ 0; and may therefore be replaced by < ( F{co3 26) + cos ) ""sfl ti^(cos20)-co8 0J ' which evidently has the property of not being altered by the substitution of tt + for ; p _ r .F'(cos2g) + co3 |" ■' c ~ti?'{cos20)-cos0j ' which gives algebraic curves by assigning an algebraic form to the arbitrary function F. Thus if « = ^, and F(coa 20)= J , then <- = , b ' c 'Jd'-b^ the equation to a circle. Several other interesting Problems of this description may be found in Herschel's Examples. The following example supplies an omission at the end of Art. 84*. It is evident that v^ = a* satisfies the equation M^j- a (1 + B^) M^i+ a'Bji^= 0, .-. w,= CB, and u,= a' [C'+Ct (P-B^J). „ „ x+1 ., „„ 2. 3. ..03 1 Suppose B,=—^ , then PF^^=^^^_^-^ = (^^^^^^^^^ , C f C \ .: S (P-BLi) = r ; and m, = a" ( ' + — — - ) is the solution of {x + 3) u^-2a {x + 2) u^^ + a' (x + 1) u, = 0. SECTION IV. SUMMATION OP SERIES. Integration of the Greneral Term. 101. One of the most direct and important applications of the Calculus of Finite Diflferences, is the general method which it furnishes of assigning the sum of any number of Terms of a Series, of the general term of which we are able to take the inte- gral. There will be two cases to consider, according as the general term is given explicitly in terms of the index, or is only given by means of an equation of diflferences. We shall begin with the former case, by shewing that the sum of any number of terms ending with the general term u^, is equal to the integral of the following term, together with a constant. Let iS, denote the sum of the first x terms of a series whose general term is u^, then S^ = u,+Uj+ ... + u^, Making x = 0, 8„ = = tu^^+ C; or, as it is usually written, = Smj + C, In general the arbitrary constant will be determined by the term with which we make the series commence. Thus, if we 101 make it begin with m, instead of u^, this amonnts to supposing the sum of the terms preceding m,, that is, Xu^ to be zero ; and therefore to determine the constant, we have 102. As the summation of series, where the general term is given explicitly as a function of the index, thus resolves itself into the cases of integration treated of in Section II., it will only be necessary to give a few numerical examples; every expression integrated in that Section gives the sum of a series of which it is the general term. 1. To find the sum of the first x terms of any progression of fignrate numbers. In the r* order the general term is x{x+i){x + 2) ...{x + r-2) '*'" 1.2.3...(r-l) ' •^t..., = "^"Y.^,••^"!;-^^ C7, (Art. 45) tu^ = 0+ C=0; „ _ x{x + l)...{x + r-l) 1.2.3...r Similarly, for the series of inverse figurate numbers, except when r = 2, that is, for the series l + i + J + 1+... + -. 1.2.3...(r-l) For we have »x = ^(^ + i) ...(^ + ,_2) ' 102 "^^ ^'^ ^^V-2 (a;+l)(a; + 2)...(a; + r-2)J' and ;SL = r. r — 2 2. To find the sums of the squares and cubes of the natural numbers. Here u^= x' = {x — 1) x + x; . _. _ {x-l)x{x+l) . x{x+l) , ^ J < n ■ /-« A . c a: (a; + 1) (2x + 1) and Sm, = 0+ C/ = 0; /. 0^ = — ^^ -^ -. 6 Again, «,. = a' = a; (a?" — 1) + a; ; _, (a; - 1) a; (a; + 1) (x + 2) , £B (a; + 1) , „ ••• ^M:r4.i = ^ + 2 "^ ' 2m. = 0+ C=0; ... s = '"(^+^) j(x-l)(a;+2) _!_ 1 ^ (a;(x + l) l'_ Similarly, l"+ 3'+ 5=+ ... + (2a!- l)' = ?i^^^ . l' + 3'+5'+...+ (2a;-l)' = 2x*-x'. 3. To sum the series l'-2' + 3'-4''+... ± ar'. HereM„={-l)^V; S„,.(i:iCli-P£±^(:il):+iyr+c, (Art. 59) 103 .-. 5f,= (-ir'^^^ii-^ Similarly, 2 . 1*+ 2' . 2'+ 2' . 3'+ ... + 2V = 2'*' (a; + 1) (x- 3) + (2""-!) 6. 4, To find the sum of x terms of the series 1 1 1 » 2'^_3* 4'- 3' 6''- 3' "- = 1^^^ = {2x - 3H2^ + 3) ' ""^^^ ^^"^ "^^'^ ^'^- ^^' „1 x+1 1 6a; +5 18 (2a;+l)(2a;+3) 6' (2a;- 1) (2a;+ 1) (2a; + 3) ' and 'S;»= — . Similarly, + -z—x — 7 + ;; — -, — r + &c. to X terms = 2 — - 1.2.32.3.43.4.5 (a; + l)(a; + 2)' 5. 2. 4. 7 + 4. 7. 13 + 8. 13. 25 + 16. 25. 49+&C. to aterms. Here u^ = 2^ (3 . 2*-' + 1) (3 . 2' + 1). Assume Sm^^ = ^ (3.2^'+l) (3.2=' + l) (3 .2-^'+ 1); (Art. 55) .'• «.« = ^ (3 . 2' + 1) (3 . 2"*' + 1) {3 . 2*" + 1 - (3 . 2^-' + 1)} = ?i^.2'^'(3.2^+l)(3.2^'+l); which gives ^ = ^ , 2«„. = ^ (3 • 2'-'+ 1) (3.2' + 1) (3 . 2^^' + 1) + C; 2«,=^.f.4.7 + C=0; 104 ... Sr, =^ (3 . a'-' + 1) (3 . 2' + 1) (3 . 2"=*' + I) -^. ^" 3T3~3.9"'"9. 15 15.33 Here m^ = {(-2r-ii{(-2r'-i}' •'■ '^^ "" 9 "^ 3 ■ ^2)""'^ ' ^^'^^' ^^^' ^^^ '^-' " 9 ■ 16 21 2 , 26 /2V „ ^ , Here « - L^Jlli^^i) (_!)- Assume St..,. = (^^ 3/(^^3) (- 1)% (Art. 56) then^=-3,and -•• ^. = i - (, ^,)%^ 1) (- r- Similarly for the sum of x terms of the series 19 1 28 1 39 J_ 52 1_ 1.2.3"4 2.3.4'8"^3.4.5'16'^4.5.6"32"^"" "^ a;(.T + l)(x + 2)^^^ ' *"• (x + l)(x + 2)^*^ • 8 1 - I 1 L &C COS cos 20 cos 20 cos 30 cos 30 cos 40 to a; terms. Here m^ = ^ r — , ,, „ ; therefore (Art. 53) cos x0 cos (a; + 1) ^ ' _, tan (a; + 1) , ^, .^ tan , ^ ^ +' sm0 ' sm0 105 „ _ tan {x+l)6 — tan 6 9. 1 cos + 2 cos 26 + 5 cos 30 + &c., to x tenns. Here m^, = (o; + 1) cos {x + 1) 0, S.^. = (. + l) ^^^(-+^y -^Ssin (.4-1)0 (Art. 59) ^ (a! + 1) ain (a: + ^) g cob (a; + 1) „ 2sm^0 "^ (281040)" ^"• = *+(2sini0)'+^ = ^' •■• ^=~(2lmW •••^-- "tsint0^' +2iI^0^^^"(-+*)^^^'^^^--^°'^('-+^)^^ a; sin {x + ^) 6 . /sin ^ a;i / sin^aig x" * Uin i y ■ ~ 2 sin 4 Hence also, if /S, = 1 cos' + 2 cos' 2 + 3 cos' 3 + &c. then 28^ = 1 (1 + cos 20) + 2 (1 + cos 40) +&c. „ _ x{x + l) X sin (2 X + 1) _ . / sin xO V •••*.- I + 4gi^^ *Vsin0/'' 10. aco80 + 2a'cos 20 + 3a'cos 30 + &C., toa:terms. Here % = a^" cos a;0 = xv^ , suppose ; .-. Xu^ = xXv^ — S'Wi+i = ? (a'i>-' - 1 ) 1,, - i {a'D-' - 1)' ij,„ ; (Art. 52) c c 1 -I -t c c H. D. E. 14 = ± (o» - o cos 0) - ^ (a* - 2a' cos + a' cos 2 0) + C ; 106 ■where v^ = a" cos x9. Similarly, if the general term be oe'a'- cos x0. to a; terms ; M, = tan : — » = tan ■ l+x + x'' l+x{x+l)' .: tu^ = tan"' x+C, (Art. 54) .-. S, = tu^, - %u, = tan-' (x + 1) - Itr. Here m^, = 2' tan'' ^33x^3 ^.g. J .-. t u^, = 2nan-' p+ C, (Art. 54) •. S^ = tu^^-tu^ = 2''txDr''%-t&n-^e, and ^„ = ^-tan-'^. And we may similarly shew for the series whose (x+1)* term is ^ ^ ^27.3*^ + 180\3"-^;' that S^ = S'' tan"' |i - tan"' ^, and 8^= d - tan"' 0. 13. log cot + ^ log cot 26 + 1 cot 4^ + &c. to x terms. Here m,^ = ^ log cot 2^^ = 2A ^ log (2 sin 2'0) ; .'. S,= .^k log (2 sin 2^6) - 2 log (2 sin 6). 107 Recurring Series. 103. We next come to the case where the general term is not given explicitly in terms of its index, but only certain rela- tions between the consecutive terms, or these and their indices are expressed. The equation of differences which determines the form of the general term, cannot always be solved ; when however the equation is linear with constant coefficients, its solution, as we know, can be effected ; and this happens for Recurring Series — the most remarkable class of this sort of series. This equation when integrated will involve the same number of arbitrary constants as is expressed by its order: implying that that number of consecutive terms may be regarded as inde- terminate, and the remaining terms formed according to the law which the equation to the series expresses. A recurring series is a series in which an equation of the first degree with constant coefficients, holds good between a cer- tain definite number of consecutive terms, in whatever part of the series they be taken. For example, in the series 3+ 5 + 9 + 17 + 33 + «&;c. we have 9 = 3.5 — 2.3, 17 = 3.9 — 2.5, &c. ; and in general M««=3w^, -2m^. (1) The integral of this is m^. = c . 2"^ 4- c' ; and assuming a and h for the first two terms of the series of which m, is the general term, a = 2c + c', J = 4c + c'; .•. c = \{b — a), c' = 2a — b, so that u^={b- a) 2'^' + 2a - J ; and the series that has (1) for its equation is, in its most general form, o + S + (3& - 2a) 4- (7J - 6a) + &c. But for the series — ^ + -^ + —-5 -I- &c. we have 71+ 1 n +2 n + 6 (x + n + 2) u^-2a {x + n+l) «„,+ a' (x + n) m,= 0, . • a' {c + c'x) an equation with vanable coefficients, givmg m,= — ^^ — • 108 104. The general equation of every recurring series is The series of coefficients which connects any term with the preceding ones is called the Scale of Relation. Thus is the scale of relation of the recurring series whose equation is Mx+»+i'.«^+n-l+"-+i'»''=» = (!)• 105. A recurring series may generally be resolved into two or more geometric progressions. For if a,, a^, ttj, ... a„ he the roots of the equation a'-+p,ar-'+p^ar-^+... +p„ = (2), the complete integral of the equation of the series is u, = c^a,' + c,a,'^+ ... + ca", (Art. 77) .-. M, = Cja, + c,aj + C3a,+ ...+c„<7„, u^ = c^a' + c^a^ + Cfi^ + . . . + c„rt„', t/„ = cfi^ + Cji^ + C3< + . . . c^a„', and the proposed series consequently is transformed into c, (ai + a/+...+0+ <^Mi + ^i+ ■•• +««") +&C. 106. In the particular cases in which equation (2) has equal or impossible roots, the recurring series can no longer be resolved into geometric progressions ; for the complete integral of the equation of the series becomes in those two cases, respectively, Mx = K + CiX + c^aj" + . . . + c,_^x'-') a^ + c,^,<^, + . . . + c^a^, u^ = (f „ + c,a; + . . . + Cr.iX*"') f cos xO + (fl'o + c> + . . . + c',.,*'""') p" sin xe + Cj,^,a'j,+, + . . . + c„a/. 109 107. Hence, to find the general term of a recurring series, we must integrate the equation expressing the relation between its successive terms, and determine the arbitrary constants by making the general term u^ coincide with a sufficient number of given terms. When a series is known to be recumng, its equation may be determined by assuming it to be of the form (1) ; and then forming a sufficient number of equations for find- ing the coefficients p^, j>^, &c. p„, by substituting the given terms in order. 108. To find the sum of x terms of a recurring series. First, let its general term be of the form u^ = c^a^ + c^a^ +... + cji' ; then 8^ = Sm>+i - Sm, a, — a, . a„ — a. "'■ a,-l + ^ a,-l +••• + "'•• a„-l Secondly, let the general term be Mx = (Co + c,a; + c,a^ + . . . + c^^a;""') o/ + «■,+,«%+, + &c. ; then since tx^a' = - 7 TT^ + 7 TTF - &c. (Art. 59), a — 1 (a — 1) (« — 1) 2«.« = o„ ^ + c, |--^i-^ - ^^-y,} + &c. ; .'. ;S, = tu^^ - Smj , is determined. Thirdly, let «, = p" (c„ cos xd + c„' sin xd) + p'x (c, cos xd + c,' sin xd) + p'x* (CjCOS x9 + Cj' sin xff) + &c., then each term to be integrated will be of the form x'p'caa {x6+a.), no the integral of which may be found by Art. 58, because Xycos(a;^ + a) is always assignable. Ex. 1. To find the sum of x terms of the series 1 + 5 + 17 + 53 + &c. Let the equation be u^^ +i'Mi+i + 2'Wi = '■> then 17 + 5p + gr = 0, 53 + 17p + Sgf = 0, which give f — —\, q=3, so that m^^., — 4m^, + Su^ = 0. Let u^ = d'; then a''— 4a +3=0; a = 3 orl; .: u^ = c,S'' + c.^, l = 3Cj + c„ 5 = 9c.+ c„ which give Cj = f, 0^ = — 1, so that m^ = 2 . 3"^'— 1; .-. S,= 2.^-x+C, 0=l + (7; .-. S, = 3'-a;-l. Ex. 2. 2-a-a'+2a^-a*-a^+2a°-a' -&c. u^ = 20^^ cos .x-i„_(2«-2)7r. „ _ 2a'^' cos ^ (2x - 2) -TT — 20" cos § Tra; + a + 2 "~ a^' + a + l Ex. 3. 1 + 2 + 3 + 8 + 13 + 30 + 55 + 116 + &c. «x4, - 3«^> - 2m, = 0, M, = ^ 2^ + i (3a; - 4) (- 1)^ *^~9^ 4+^ ^^ • 36 • Ex. 4. 1 4- 4 + 18 + 80 + 356 + «S:c., to x terms. Ill Here m^^j — 4^^^, — 2u^ = ; .-. 2V6m^={2 + V6)"^-(2-V6)==; and o ^/qS - (2 + '^6)'^' -_(2 + ^^) (2 - Ve)""' - (2 - V6) i + Ve 1-V6 Application of the Integral Calculus to the Summation of Series. 109. As integrals are often expressed by Series, so, con- versely, tte latter may be represented by integrals; and it is often desirable to find the integral of which a proposed series is one of the developments, in order to subject it to the methods which we possess for calculating, at least approximately, the value of any integral taken between assigned limits. We proceed therefore to notice one or two processes given by Euler for effecting this; they consist chiefly in performing certain operations on the series, by which it is transformed into another series which we are able to sum, or which is similar to the pro- posed one. 110. Series which proceed according to the powers of some quantity t, affected with coefficients consisting of factors in arithmetic progression either in the numerator or denominator, may be summed by the aid of the Integral Calculus, the deno- minators being taken away by differentiation and the numerators by integration. t 2t° 3t° Ex.1. Let5 = - + -g- + -j +«S:c. (toco), log(l-<) 1 . ri .-. s = -2-i; <- + - — -, smce C = -l. 112 _, „ a + b a + 2b „ a + nb - _ . . Ex. 2. s = 5<+ ^a ^+--- -, r+... to »i terms; /. ^^(s^)= ... + {a+rd,)rr'+ ... dt t-1 ... + e *+... = i-:if <* ' 111. Series like the above in which the coefficient of every term is the same function of the index of that term, may be readily summed by separation of symbols. Thus if we put t = e*, and denote j^ by d, so that/(<^ e^ = f (m) e'^', the series in Ex. 2, taken to infinity, may be written ■e ,jde- ^ J 1-e' ^/3"l-e» a a^-ba -? (j f b t which may be easily shewn to coincide with the preceding result when m is infinite. And in general the series /('O «" +/(« + r) t"^ +f{n + 2r) <"«' + &c. to m terms, d_ dd' becomes by putting < = e*, -ja^d, 1 «mrfl f{d) (e«» + eC-^)" + c<«+2rw + &c.) =/(^ e»« . ^j^ . 113 Thus the infinite series 5(1° 6<* ^^ ,, X. : + TT-^ — 7 + 7. — ; — r + <*c. becomes 1.2.3 2.3.4 3.4.5 ^3» IS _ /I _ 3 2 \ e^ ~U (Z-l''" tb j^jdtst -„_j+„^.^+(„ + ^)^ + 2)8) o (o + _ , '^(a+/3)(a + 2/8)'(a + 3^)''' +&«•; H. D. E. 114 hence, performing the diflferentiation, we find (/3<-J0§+(«-«*)« = a, a linear equation of the first order for finding s. For the sum of m terms, we shall evidently have from equation (1) I dt f"' fdtsr^=-±-^r J j _ a{a + h)...{a+{m-2)b} ,] ^ T {a + /3)(a + 2y3)...{a + (m-l)/3l' j' M j/i+b jO+26 ^^•^- a + ^FT^ + a(a + &)(a + 25) +'^^-(^°'")- s = e^' (c+jdte~^'\f-'y If we suppose a = J = 3, we get s = e — 1. Ex.3. . = i + -.-«+-^-^-^-^«^ + r:273 7(7+l)(7 + 2) '^ + **'•^*°*^• («'-<)5+{(«+/8+l)«-7}|+a/8.=0, which is satisfied by s= (1 — <)"*, when /3 = 7. 113. When consecutive denominators have only a single factor in common, or none at all, the following is a convenient mode of summing the series. Ex 1 I 1 ■ a (a + b){a + 2b) {a+2b) {a + 3b) {a + 4.b) + &C. (to 00). Since 115 (a + od>) {o + (a; + 1) h] {a + (x + 2) Jj a + xb 1 + c + (x + 1) 5 a + (x + 2) J ' resolving all the terms by means of this formula, by putting x=0, 2, 4, &c., we find 2J»^=i_^ _2 __2 _2 _^ a a+h a + lh a + ib a + ib J„ l + a;" a Similarly, -^—^^^—^^^^—^ "^ (a + 3J) (a + 45) (a + 5i) {a + 6b) "^ '^'^' ^*° * ^ _ 1 /I p x'*'-' •] ~6b'\a~^J, l + x' + xV- ll^. We may obtain an expression for the sum of the series a; #(^) x' cf {{z) x' d^mzW , . dy 1 where -^4>{y) must be obtained in terms of x and z, fi:om the dy^ equation y = z + x^ {y). Suppose, for example, '^(3^)=i(3''-l), then^ = y, 116 a.ni t/ = z + ^{y'-l), .-. 1 - scy = Vl - 2zaj + »'; X I rf(g'-l) x* 1 (2D = C {a^- y) {a^-y) ... {a„-y), 1-^^^y) 11 1 Now dcYeloping the two members of this eqiiation in powers of y, the general term of the second member is and by Taylor's theorem the general term of the first member, putting 1 - ^ «^ (y) =f(y), and regarding f{y) -i-iz-y) as a function of z and ^ {y) as the increment of s, is f(y)'-{y) ^ ( 1 — f— V -Rufxi fiy) = A, + A^y -^ ... + A^ + ..., 1 1 V 3^" , since =- + -^ + ... + 4h+ •••, z—yzz z therefore the general term of "^-^ is "^-^ y", whatever be the fiz) form of /(z), if -s?r ^ restricted to such terms only as in- volve negative powers of z ; consequently the general term ^^fJMM is tm^ .y, and therefore of z — y z " 1 dr ( /(y)0-(y) l.,. 1 d' im^M,r. i; dz' t 3-y r \r'd^ 1^=^' \y ' hence equating coefficients of y* on both sides of equation (1), *-"-• - i^ + e?4 ;r-« j +1.2 ^1 3"« [ + *°- 118 Now putting for f{z) its value, the general term may be resolved into or 1 (T' { '{z)4,'{z) \ \rdz^\ 2"« J' of which the first and last will be destroyed by the correspond- ing parts in the preceding and succeeding terms; therefore, changing n into n — 1, 1 n (j)iz) n d <'' (z)] n d'^^ ((!>'■ {z z \ z 1.2 dz tg"" j ••■ \r_dz'-'- !«"*' &c. But the second member of this equation is what Lagrange's Theorem gives for the development of ^~", in the equation ^ —y + ^ iy) = ; therefore s_„ is equal to the sum of those terms of the development of y~" which involve negative powers of z. 117. By the help of the preceding Proposition it may be shewn that Lagrange's Theorem, applied to the solution of the equation gives an approximation to the least root. By what precedes, we have 1 n {z) n d {^^ (z)| «_„ 2" 1 z"*' 1.2 dz z"-^' J &c. s_^^ ~ 1 n-irr ^{z) n + r d (ify'{z) z"^ 1 a"^' 1.2 ^^z {s'^\ • — &c. where each series is restricted to those terms which involve negative powers of z ; and the number of those terms increases with n, and if n be supposed very great, then each series may 119 be taken ad infinitnni. But whatever be the value of n, if both series go on ad infinitum, the value of the second member is ^ = / = .' + ^-.-<#, (.) +^^ I {.-^» (.)] + &C., as given by Lagrange's Theorem, and which may be verified by actual multiplication ; therefore when n is infinite, limit of -^=/, (y denoting that function of z which is given for the develop- ment of y*" by Lagrange's Theorem). But if Oj be the least of the roots a^, a,, a„ &c., we have also (Theory of Algebraical Equations, Art. 156) s limit of -^^ = a,'', when n is infinite ; .•. y' = o,', and y = a, the least root. 118, To find the sums of the series sin a 2 sin 2a 3 sin 3a I' + F^ 2'' + A' ^ B^ + fe' - + «&c. (tooo ), COS a cos 2a , cos3a , „ u ^\ , n sm na i r,m cos na or the values of */S°° ^ , „ , '^°° , , ,« • First, we have {Triton. Art, 155) 0- ~P^ ^ , g-fa + 2 (»cosa;+ycos2a;+/cos3a; + &c.) c"**, 1 — 2p cosa; +^ and integrating both sides from a; = to a;= a, and putting j? = 1, we get •7r=^(l-e"*') + -wsinna „, -taiam cosna . . , „« k 2e-'^»!^I^«_2yfee-'5" ,^^+2'5'- ^^^-^«e « FT»?"^ " ^ + «' 120 Again, integrating from a; = to a; = 2ir, and putting jt = 1, we find which gives 2 '8- j^-^.=-'r . \±^ - ^ , and substituting this value in the foregoing result, and calling the two sums we are in search of, 8^ and 8^, we have 27re ■IL_, = l-28, + ,k8,; and changing the sign of k, -2^^"*° }:^28-2k8- i-e^ ~ k ' " therefore, adding and subtracting, wsinna ,r e^"^"' -€"*<'"" 8, = '8° 8 = '8' f£' + n' 2 e*'-e-*' ' , coswa _ TT e*''-' + e~*""" 1 ^ + w" ~ 2k e" - e'" 2A^ ' These formulae were first given by Poisson, and may be con- sidered as embracing the chief results which have hitherto been obtained relating to the summation of series of the sines and cosines of multiple angles. 119. The definite integrals required in the preceding in- vestigation may be found as foUows. Suppose _ (l-y')e^ 1 — 2p CQS X +p* to be the equation to a curve ; and let it be proposed to find the limiting value of its area, from a; = to a; = a, on the supposition that p approaches continually to unity. First, we observe that 121 all the ordinates will be ultimately evanescent except those cor- responding to a; = 0, 27r, iir, &c., which will be very large, because for those values - — ^ becomes a factor of the expression for y. We will begin by supposing that a < 27r ; and therefore the only ordinates that we are concerned with are those imme- diately succeeding that through the origin; so that, making ^ = 1 — p, and then supposing x very small, we get successively p (2 - p) e-*" 2p , , . , , y= p^ + 4(l-p)sin'ix ='7fi^' ultimately; .*. I dxy = 2 tan ' - , and I dxy^\ = tt, making p = 0. ■' 9 •'o Next, suppose that the area is to be found from a; = to a!=27r; then, besides the area just found, there will be another portion immediately preceding the point for which x = 2'jr; to find this latter portion put x = 2ir—x', and call the ordinate y ; then y' = } J' , , , = ^^y, ^ 1 — 2pcosa; +p^ ^ therefore the portion of the area immediately preceding the second limit = Tre-***; /, 2S-p»cosn (c -x) - .f{x) • l — 2p cos {c — x) — y-p .-. f "^<%p=i = j^dxf{x) + 2'8°' r^dx cos n (c -x) - .f{x). J _a J —a. J —a ff Now, suppose c to be less than a, then for all values of x between x= — a and x = + a, y is evanescent, except when x = c; if therefore we write x = c-\-z, supposing z exceedingly small, H. D. E. 16 122 and then integrate with respect to z, from g = to 2 = 7 any- small finite value, and double the result, and make p = \, we shall obtain the value of dxy^^i. But making ^= 1 — p, we have ^^ p(2-p)/(c + .) ^J^ ^^^.^^^^^^ . irz p» + 4(l-p)sin^^ p»+-^ .•.rcZ.3,=^tan->^J; f^.y.^. = ,/(c) ; Jo T pa Jo .-. 2a/(c)= rja;/'(a;)+2'fi'- T'c^ajcosn (c-x)-./(x). (1) J —a J —a ^ 121. Hence ■^^"^ " ^ / /"^-^^"^^ "^ a /"^.^ ^*^°^ ^"^ " *^ a ■■^'•^^ + cos (c — a;) — -/(a;) + cos (c — x) — -fi^) + •••}; therefore, making a infinite, flc)=— limit of — I dx {cos (c — a;) — + cos (c — a;) 1- &c.| fix), 1 r°° r r*" i or, /(c) = - / — a is divided. If the intervals be very small, f{x) may be considered constant from x = a + n — 'l to x = a + n, or A'y(x) may be neglected ; and the value of the definite in- tegral is 2/(a + w — ^), where n is to receive in succession the values 1, 2, 3,...b-a. For greater accuracy, suppose A^y(x) to be constant from x = a to x = b, or A'/(a;) to be neglected. Then f{a + n-^ + z)=f{a+n-i)+zAfia+n-is) + ^^Ay{x), and the integral for the interval in question is Jdgf{a + n—^ + s), from a = — ^ to z = i, =f{a + n-^)+l(^l + l)Ay{x)=f{a + n-i,)+^Ayix). The integral, consequently, from a to J, is n receiving the same successive values as in the former case. In calculating the value of a definite integral by this method, the intervals are to be taken smaller as the variation of the func- tion is more rapid. 124 Convergency and Divergency of Series. 123. A series u^ + u^ + u^+ ...+ M„+...(to oo ) is called con- vergent, if the sum «„ of any numter n of its terms approaches continually to a finite quantity s as its limit, when n is indefi- nitely increased ; and divergent in the contrary case. When a series is convergent, the sum of any number of consecutive terms after the n"" continually tends to zero, as n increases. For therefore, as n increases, the value of the first member con- tinually approaches to s—s, or zero, as its limit. This being true when m = 1, we see also that u^^, or the general term m„ continually tends to zero as n increases ; that is, each term is greater than the following one ; but this, although a necessary condition, is not sufficient to insure the convergency of a series. Thus in the series M„ = - continually approaches to zero as n increases ; but 11 1 + — --r+...+- '^'" " w + 1 n + 2 n + m is evidently greater than > A, if m = n ; and therefore ■^ ° n + m the sum of n consecutive terms after the w"", does not diminish indefinitely as n increases ; consequently the series in question is divergent. 124. In the geometrical progression a + ax + ax'+ ... + ax"~^+ ... (1), l-x" _ l-a;""^ 1 -x'" *iinn ~ *n — '"•'' • 1 ^ I — X 125 Hence if a; < 1, when n is infinite x' = 0, .-, s = - ; also M„ = 0, Vm-«n = 0; 1 — X both which results shew that the series is convergent. But if a; > 1, then m„ = ax"'^ increases indefinitely with n, which alone shews that the series is divergent. Hence the geometrical progression (1) is convergent or diver- gent, according as x is less or greater than unity ; and it may be used as the test of the convergency or divergency of other series. For if a proposed series can be shewn to have no term greater than the corresponding term of (1) when x < 1, then that series is convergent; or if a proposed series can be shewn to have no term less than the corresponding term of (1) when X > 1, then that series is divergent. Thus in the series for e the base of the natural logarithms 1 + 1+1+1^+...+.. 1 . 11. 21. 2. 3 | w-l ■■■ the terms which follow the w*, viz. are evidently less than the corresponding terms of the geome- trical progression 1,1 1,1 1 , ^ In \n n \n n the sum of which is 11 11 \-l\_\ VV-L 1 ' n-l' and continually tends to zero as n increases. Therefore the proposed series is convergent; and the error in taking the aggregjtte of its first n terms for its sum, is less than the quo- tient of the n"" term divided by n - 1. 126 125. From tke measure of convergency or divergency which a geometrical progression furnishes, we shall now proceed to deduce one or two other tests as given hy Cauchy {Cours d'' Analyse Algiirique). The series m^ + m^ + ... +m„ + m^i ... is convergent if the limit of the ratio of u^^ to m„ when n is infinite, be less than unity ; and divergent in the contrary case. Suppose that as n increases indefinitely, the ratio -^^ continually tends to become equal to a finite quantity k as its limit ; and let i denote a num- ber less than the difference between k and 1, so that the quanti- ties Jc — i and Jc + i are both less than 1 when k is less than 1, and both greater than 1 in the contrary case. Then by giving n a sufficiently large value the ratio — ^^ may be made to lie between k — i and k+ t for all superior values of n ; and in the series w„ + u^{k- i) +u„{k- iy + ... u„ + M„ [k + i) + M„ [k + »■)"+... every term of the second will be intermediate in value to the corresponding terms of the first and third. But the first and third series evidently decrease indefinitely if m„ does so, as n is increased without limit, provided k be less than 1 ; therefore the second series which is intermediate in value to these is conver- gent under the same circumstances ; and therefore the proposed series is convergent provided the limit of the ratio of u^^ to m„ when n is infinite, be less than 1. Also the three series above written will all increase indefinitely if k be greater than 1, and the proposed series is in that case divergent. The test of con- vergency or divergency is here presented in the form most convenient for application ; but it may be changed into the fol- lowing : if A; = 1, the test gives no result in either form. 126. The series m, + Mj+ ... + m„+ ... is convergent, or will become so, if the superior limit of (m„)" be less than 1, when n is infinite ; and divergent in the contrary case. 127 Let h denote the superior limit of (m„)" when n is_ infinite ; and first suppose h<\; also, let a be any magnitude between Tc and 1, so that A\; and take, as before, a between 1 and k, so that k>a>\. Then, when n is indefinitely increased, 1 (m„)° cannot approach indefinitely near to k without finally becoming constantly > a ; we shall therefore be able to satisfy the condition (m„)° > a, or m„ > a", by taking n sufiiciently large ; and consequently we shall always find in the series M, + M^ + ... + M„ + M„+, + ... , ail indefinite number of terms greater than the corresponding terms of the geometrical progression which is divergent, a being > 1 ; and therefore the proposed series will end by being divergent. 127. It may be shewn that if as n increases indefinitely m. remains positive, and the ratio -^^^ continually tends to become 1 equal to a finite quantity k as its limit, the expression (m„)" 128 continually approaches to the same limit. For as in Art. 125, by giving to m a value large enough, we may make each of the ratios -^^^ , -^^^a^ , ... -^^^!±5- differ from ;fc hy a quantity as small as we please; and consequently the geometric mean between these ratios r?f5±5 j" may, by sufficiently increasing n, be made to differ from yfe by a quantity i as small as we please. "We shall therefore have, taking n sufficiently large, Now suppose n infinite, so that ^ becomes equal to zero, and we get limit of (m„)L= = ^ = limit of C^") . 128. Suppose that the series u^ + u^ + u^+ ... consists of both positive and negative terms; then if v^, v^, -y,, &c. be the numerical values of these terms, so that m, = + Vj, u^= -t *'s) &c., it is evident that the sum of the proposed series can never surpass that of the series v^-\-v^ + v^ + ...; if therefore the latter series be convergent, that is, if (-^^' J <1, the proposed series will be convergent; or if the latter series finish by having terms greater than any assignable magnitude, that is, iff-^^j > 1, the same thing will happen to the proposed series, which will consequently be divergent. Hence the above test is applicable to series consisting of both positive and negative terms, provided we use the numerical values of the terms without regard to signs ; and it fails, as in the preceding case, when ^ = 1. 129. Let a^+a,x+a.ff+...+a„x'-+ ... (1) be a series arranged according to positive and ascending powers of the variable x ; the coefficients being positive or negative ; then, by what has been proved, this series will be convergent or divergent, according as he, 129 where h = (0.)"= -^^^ when n is infinite, is numerically less or greater than 1. Hence for all values of X tetween — 5; and + -r the series will he convergent, and for all values of x beyond those limits it will he divergent. Here -^** = = — 1, when n is infinite ; o„ « + l therefore the series is convergent or divergent, according as x lies between + 1 and — 1, or without those limits. 3 8 T-, - a a a „ M. = — , .". -!t!i= = a when n is infinite ; n M„ n + 1 therefore the series is convergent or divergent, according as a < or > 1. T^ „ 111 1 Ex.3. T + TH + i?+-- + r' 1 ^ |2 ^ 1^ « 0, when n is infinite, M, n+ 1 therefore the series converges. Similarly, it will appear that the series for sin x and cos x are always convergent ; and that for tan"* x convergent for all values of x situated between + 1 and — 1. 130. The series Mj + «,+ •••+«« + ••• is convergent, or will become so, if the inferior limit of ^ " be > 1, when n log- ° n is infinite ; and divergent in the contrary case. H. D. E. 17 130 Let k denote the inferior limit of " when n is infinite, log — ° n and first suppose ^ > 1 ; also let a be any number between k and 1, so that k>a>l. Then when n is indefinitely increased, , log — — aJ^ or its equal , cannot approach indefinitely near to 1 1 ^ log n log- k without finally becoming constantly greater than a. There- fore it will be possible to take for n so large a value, that for that and all superior values we may constantly have log — > a log n > log n", or — > n", or u„<—:i. ° u„ ° °- M„ n Consequently the proposed series will finish by always having its terms less than the corresponding terms of the converging series (Ex. Art. 131) 111 1 1 „ 1 2 3 n {n + 1) and therefore will itself end by being, h fortiori, convergent. Similarly, if i < 1, it may be shewn that the proposed series will finish by being divergent. 131. The series M„ + Mi + Mj+M3 + M^4-M5 + &c (1), each term of which is less than the immediately preceding term, and the series u^ + 2Mj + Am, + 8m, + 16m„ + &c (2) , are convergent or divergent at the same time. Suppose series (1) to be convergent and that its sum = s, then M„ = M„, 2Mj = 2u^, iu^ < 2u^ + 2m„ 8m, < 2m^ + 2M5 + 2m, + 2m„ &c. ; .'. M(, + 2Mj + 4Mj + 8m, + 16m,5 + &c. 2mj>Mj + Mj, 4m, > Mj + m^ + Mj + Mj, (Sbc.; .-. Mj + 2m, + 4m3 + 8m, + &c.>m, + ?*j + Mj + m, + &c., and is therefore divergent. Ex. Let series (1) he ^ + ^+-^+ L + &c (3), then series (i) ia — + .^^ + —^ + .^^ + &c., a geometric progression convergent when m > 1, and divergent in the contrary case ; consequently series (3) will be convergent ii m>l, and divergent if jw = or < 1. 132. The alternating series u^ — u^ + u^ — u^ + &c., is con- vergent, if the numerical value of the terms decreases without limit. For, by writing it in the forms M, - (Mj - M,) - (m, - «,) - &C., Mi-M,+ K-M,)+&C.; we see that it is > m, — Mj and < m, , and therefore is convergent. Thus the sum of the series 7 — 5 + 5 — 7 + &c., 1 A o 4 lies between 1 and - ; also the series ^ — rs + oS - Tsr + &c., is convergent for all positive values of j». 133. It is also shewn in the Work from which these tests of convergency are taken, that two converging series all whose terms are positive, will by their addition or multiplication pro- duce new converging series, whose sums result from the addition or multiplication of the sums of the former. If the proposed 132 aeries be imaginary, having its general term of the form v„ + w„ V— 1, it may be resolved into the two series v, + Vj +...+«„+..., V^(w,4-W2+ ... + «?„ + ...) ... (1), to which the foregoing tests may be applied separately. If we assume v„ + w„ V~l = p, (cos 0„ 4- V^ sin 0„) , then v°„ + »'„ = p\, so that v„ and w„ are both less than p, ; if therefore the series of moduli p, + Pj + ••• +p» + ... be con- vergent, then each of the series (1) will be convergent. Interpolation of Series. 134. When a series of values of a quantity has been obtained either by observation or calculation, it is of great importance to be able to insert other values between them, such as would have resulted from a similar observation or calculation, without the labour of performing these. This is the object of Interpolation ; and in this the Calculus of Finite Differences finds one of its chief uses. More strictly, Interpolation of Series is the inserting among the terms of a given series, new terms subject to the same law as the first. In doing this, the terms of the series are considered as particular values of the function which expresses its general term, corresponding to a given regular succession of indices; and it is the business of Interpolation to discover that general term ; or at least to assign such a function of the index as shall represent the given series of values, and, approxi- mately, all intermediate values. The problem thus requiring us to assign the analytical expression of a function from a limited number of its numerical values, is plainly indeterminate ; it is the same as to form the equation to a curve which shall pass through a limited number of points, whose abscissae represent the values of the independent variable, and the ordinates those of the function, vrithout giving the species of the curve ; which, as is evident, may be done in an infinite variety of ways. But if the given terms are numerous and near to each other, the expression 133 for the general term, within the limits of the given quantities, may be found to a great degree of accnracj. 135. There are two principal cases to be considered ; first, when the given values of/(x), namely, /K). /K + A), f{x, + 2h),... f{x,+ {n-l)h}, as we shall write them, correspond to values of the independent variable, a;,, x^ + h, x^ + 2h, &c., x^+ {n—l)h, in arithmetical progression. And secondly, when the" given values /"(ar,), f(x^, ...f{x^, or M,, M„...M„ correspond to values x^, a?,, ...a;,, of the independent variable, not obeying any assigned law. 136. Having given m,, «,, «,, &c. m„, n values of a function f{x), corresponding to the n values of the independent variable a;,, x^ + h, ...x^+{n—\) h, to find an expression for any inter- mediate value /{a;i+ k). Since f{x + nh) =f{x) + nAf{x) + "^"~^^ Ay{x) + &c. (Art. 23), changing x into a;,, and then replacing nh by k, and n oy I ^^ get k{k-h)...{k-in-2)h} . + |n-l.A-' ^ "" in which, since Am, =Mj-m„ A\ =m,-2m, + m,, A.-1 / ,\ (n — 1) (n— 2) p A"X = ««-(«-l)".-.+-^^ f7^ ^W,^-&C....±M„ (2) if we make k = 0, h, 2h, &c., (n — 1) A, the second member assumes the n values u,, u,,...u„; and not only this, but if we assume for k any value whatever between and (n — 1) A, we 134 shall obtain the value of the fiinction corresponding to that value of the independent variable. 137. In applying the above formula, the simplest mode is not to calculate Am^, A'm, , &c., by equations (2), but by continued subtraction of the given terms ; that is, we must write down the series of given values m,, m^, ... m„, and subtract each from the succeeding one ; next subtract each of these dif- ferences from the succeeding difference ; then perform the same operation upon the new differences ; and so on, till the process terminates ; the first terms of these series of diflferences are the values of Am,, A'm,, ... A""'u,. Unless the terms of the given series by continued subtraction lead to a constant difierence, the expression for /(a;, + k) will have as many terms as the given series of values has. Ex. I. Having given the values of sin 30', sin 31", sin 32°, sin 33°, to find sin (30° + k'),k being between and 180'. Here u,= .5 A M, = .5150381 150381 A' u, = .5299193 148812 -1569 A° u^ = .5446390 147197 -1615 -46; .-. Am, = 0.0150381, A\ = - 0.0001569, A\ = - 0.0000046, .-. sin(30» + A') = .5+|A«, + ^^=^A\ ^(^-60)(^-120) , + 27376^ ^"•• If i = 20, it will be found that sin 30°. 20' = .5050299, which is too large only by a unit in the seventh place of decimals. Ex. 2. Having given log 3.14 = 0.496929, log 3.15 = 0.498310, log 3.16 = 0.499687, log 3.17 = 0.501059 ; shew that log 3.14159 = 0.497149. 135 Ex. 3. Having given u,, u^, u„ u^, four right ascensions (declinations, longitudes, &c.) of the Moon at intervals of 12 hours, to find its value t hours after the time corresponding to the second value. Here A = 12, 4 = 12 + <, and if the required right ascension = m, + S, then «. f- t\A (< + 12)«A. (<+12)«(«-12) ., 12 Va 6^2 \UJ ^6 \12j ' when developed in powers of — , A' being written for Am^, A' for A*«j, &c. 138. Between every two consecutive terms of a given series, to interpolate any number of equidistant terms. Let Mj, M,, Mg, &c., M„ be the given series, and let m — 1 be the number of equidistant terms to be inserted between every two consecutive terms ; then the new series will be m mm "• If therefore r,^., denote the r + 1** term of this series, we r . r (r — m) .f ,» have Hence, taking r from 1 to m - 1, we get the terms inserted between u^ and m, ; next taking r from »n + 1 to 2»* — 1, we get the terms between m, and «, ; and ao on. The differences Aw„ A'«„ &c., are to be computed by continued subtraction as in Art. (137) ; and the series for «,^., will have as many terms as the proposed series has, unless those terms by continued sub- traction lead to a constant difference. Ex. To insert three equidistant terms between every two 136 consecutive ones of the series 1, 7, 15, 28, 49, &c. Here »» = 4, and A, e, 8, 13, 21, A» 2, 5, 8, A' 3, 3, _ 6r r (r-4) r (r- 4) (r- 8) _ 128 + 192r-4r'+r'' _ ■'■ "••+•" ■*"T '''■^^6 "^ 128 ~ 128 ' ,,, . . , 317 504 695 „ 1113 „ andthesenesisl, — , — , — , 7, — , &c. 139. The formula of Art. 136 may be presented under a dif- ferent form by changing k into x — x^, which gives ., . x — x (x-xj (x — x—h) , „ f{x) = M, + — ^ Am. + ' i,2.h* • "^ *'^- jx-x,) {x-x,- h) ... {x-x^-{n- 2) h] ^ where f{x) is a function of a; which, as x assumes the n values a;,, x^ + h,&c., x^+{n — 1) h, successively assumes the corre- sponding values Mj , Mj , . . . M„ ; and for any other value of x within, or not far beyond, the limits x^ and x^+ {n — 1) h, it gives the value of the corresponding interpolated term. If we put A = 1, the formula is adapted to the case where the increment of the principal variable is unity. 140. In any series of consecutive equidistant values of a function, where one is deficient to insert that one. Let M,, M,, Wg, &c., u„ be the values of the function cor- responding to the values x,, x, + A, ...a;, + (« — 1) 7* for x. Then assuming that A""'Mj = 0, or that the (»— 2)"' differences are constant, which will almost always be the case in tabulated results, we have A-X = „„-(„- 1) „^, + {n-l){n-2) ^^^ _ ^^ ^^^^^^ 137 an equation of the first degree from which any one of the values as u, may be found, if the rest be known. Having thus com- pleted the system of values, we may interpolate any intermediate term /(a;, + k) by the method of Art. 136. If two values out of n are deficient, then we must suppose A""*Mj = 0, A"'^Mj = 0, or / „^ (n-2)(n-3) „ , M^,- {n - 2) w^+ ^ ^ -' ««_,- «fec. ± w, = 0, / c»\ (w - 2) (n - 3) p which equations will suffice to determine any two of the values in terms of the rest. In the same way any number of deficient terms may be inserted. Ex. 1. Given the cube roots of 121, 122, 124, 125, to find that of 123. «,= 4.946088, M. = 4.959675, m, = 4.986631. m, = 5, A*Mj = Mj — 4m^ + 6m, — 4m, + Mj = ; •'• "s = fi {* K + "<) - «i - "si = 4.973190. Ex. 2. Given M„ «,, m,, m,, to find m,, u^, A\ = Mj — 4m^ + 6M3 — 4m, + m, = 0. A*M, = Mg — 4m, + 6m^ — 4m, + m, = ; ••• "a = iQ (- 3m, + 10m, + 5m, - 2m,), M, = ij (- 2m. + 5m, + 10m, - 3mJ. Ex. 3. Given the logarithms of 121, 122, 125, 126 equal respectively to 2.0827854, 2.0863598, 2.0969100, 2.1003705; shew that the logarithms of 123, 124, are equal respectively to 2.0899051, 2.0934217. 141. We next come to the case where the given values /(a-,), /(a;,), «Scc., /(x,), or m„ m„...,m„, H. D. E. 18 138 correspond to values x^, x^, ...x„, not obeying any assigned law; and it is required to determine a rational integral function of n — 1 dimensions, /(x), which shall assume the n given values M„ Mj, ... M„, when for x the values x,, aj^, a?,, &c., x„ are succes- sively substituted. Since f{x) is of (w — 1) dimensions, we may assume {x — x^){x — x^)...{x — x„) x — x^ x — x^ '" x—x^' ••• /(*) = C; (a; - ajj (aj - a;,) ... (x-x„) + C^{x-x,) (x-x^ ... (a3-a;„)+&c.+ (7„(a:-a;J {x-x^ ...{x-x^^). Now make a; = a;,, ajj, &c., a;„, successively ; and observing that the corresponding values of the first member are m„ u^,...u„, we get ^1 = <^i (*i - ^s) (».-»;,)... (a;^ - x„), w, = a, (a;, - xj (ajj - x,) . . . (a;, - x„) Mn= C>, (x,-a;,) (x„-Xj) ... (x„-x,^J; • • -^ ^ '' ' {x^-x^ (a;,-a;J ... (x.-a;„) + „ {x-x;){x-x,) ...{x-x„) ^ ^^ ' (»i! - ».) K - a:,) . • . (a;, - Xn) ^^ (x - xj (x - xj ... (x - x,^,) " (a;» - a;,) («„ - Xj) . . . (x„ - x^i) ' which is Lagrange's Theorem for interpolation. Ex. To find a function of x which, when x= 1, 3, 6, 12, shall assume the values 1, 7, 10, — 8. (x-3)(x-6)(x-12) (x-1) (x-6)(x-l2) •'^*^~ 2.5.11 "^'" 27379 (x-1) (x- 3) (x- 12) (x-1) (x-3) (x-6) 3.3 ■ 11.9.3 142. To determine the maximum or minimum value of a function, from three of its values near its maximum or mini- mum, and the three corresponding values of the independent variable. 139 If Mj, u,, M3 be the given values of «, and a;,, a;,, x^ those of X, we have ' K - 25,) (», - a!.) » K- a,) (a;,- a;,) » (x, - a;J (a;, - x,) " Hence, putting -5- = 0, we find «i (^s - a^a) (2a; - a;, - a;,) + m, (a;, - a;,) (2a; - a;. - a;J + u, [x^-x^ (2a;- Xj - x,) = 0; . ^ _ M, (a;,'- X,') - M, (x,' - x,°) + M. (x.' - x/) 2Mi (x, - X,) - 2m, (Xj - xJ + 2m, (x, - x,) ' the value of x at which u is a maximum or minimum. This formula is usefiil in various Astronomical problems, as for instance, to determine the meridian altitude of a heavenly body, when an observation exactly on the meridian cannot be ob- tained. Cambridge; Printed at the UnivertUy Praa. MATHEMATICAL PUBLICATIONS REV. J. HYMERS, D.D. LATI TBLLOW AKD TUTOK OP ST JOHN'S COLLBGB, OAMBRIDGK. THE ELEMENTS OF THE THEORY OF ASTRONOMY. 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Fourth Edition, re- vised. 8vo. 8<. 6d. jLiC5i 01 WUJIKJS m GENEEAL nTERATUEE rUBLISHBD BY Messrs. LONGilAN", BEOWjS", GEEEN, LONGMANS, and EOBEETS, 39, PATEENOSTEE BOW, XONDON. CLASSIFIED N D EX. AcTieultxire and Rural Affairs. Paget. BajIdononTaluing RenU, Ac. - 3 Cecil's StQd Farm - - - « HoBkyns'i Talpa - - - - 10 LoudoD'i Apicoltare - - - 12 Xiow'c Eiementi of Agricoltnn - l;i ArtB, Manufactures J and Architecture. BourDC on the Screw Propeller - 4 Brande's Dictionary orScicnce, ftc. 4 " Organic Chemistry- - 4 CbcTreol on Colour - - - 6 Cmj't Civil Engineerine - - 6 Fairbaim's Informa. for Engioeers 7 Gwilt'aEncycIo. of Architecture - 8 Harford's Plates from M. Aogelo - 8 Humphreys's Parables lllummated 10 JamesOD'u Sacred & Legeudary Art 11 *' Commonplace- Book - 11 EoQig's Pictonal Lif>- or Lather - 6 London's Rural Architecture - 13 MacDougaU'B Theory of War • 13 Ualan'a Aphoriams on Drawing - 14 Uoaeley't Engineering - • - 16 Pieise'a Art of Perfumery - - 17 Richardson's Artof Horsemanship 18 Scrivenor on the Iron Trade - - 19 Stark's Printing - - - - 22 Steam-Engine,DytheArtisan Club 4 Ure'»Dicti(iluryof ArtSf&c. - 23 BiosTaphy. Arago's Autobiography " Lives ofScientific Men - Bodexutedt and Wagner's Schomyl Buckingham's (J. S!) Memoirs Bunsen's Hifipolrtui - - . Cockayne's Marshal Turen&e Crosse's (Andren) Memorials Forater's De Foe and Churchill - Green's Princesses of England Harford's Life of Michael Angelo - Harward's i faesterfitld and Selwyn Holcrod's Memoirs . - . Iiardner's Cabinet tyclopxdia Mannder's Biographical "Treasury - Memoir of the Duke of Wellington Memoirs of Junes Montgomery llerirale's Memoirs of Cicero Uonntain's (Col.) Memoirs - Parry's (Admirall Memoirs - Rogers's Life ana Genius of Fuller Rnasell's Memoirs of Moore - Bontbey's Life of Wesley " Life and Correspondence " Select Coirespondence - Stephen's Ecclesiastical Bio^aphy Strickland's Queens of England - Sydney Smith's Memoirs Syznond's (Admiral) Memoirs Taylor's Loyola - - - " Wesley - - - - Waterton's Autobiography & Essays Books of General Utility. ActoB*s Bread-Book " ' o " Cookery - " " " ? Black'sTreatise on Brewing- - • Cabinet Gaietteer - - " ' * " Lawyer - - - - ° Cost's Invalid's Own Book ' ' I Cilbart'fl Logic for the MilUon o Hints on Etiquette - - » How to Nurse Sick Children - JO Hndson'sExecntor'B Guide - - '0 " on Making Wills - - JO Kesteren's Domestic Medicine 11 Lardner's Cabinet Oyclopadia 12 London's Lady's Country Compa- nion - - - - t,";. Uaunder's Treasury of Knowledgs 14 «« Biographical Treasury 14 " Geographical Treasury 14 Maunder's Scientific Treasury - 14 " Treasury of History - 11 " Natural History - - 11 Piesse's Art of PeTfumery - - 17 Pocket and the Stud ... 8 Pycrofl's English Reading - - 17 Beece'B Medical Guide - - - 18 Rich's Comp. to Latin Dictionary IS Richardson's Artof Horsemanship 16 Riddle's Latin Dictionaries - - 18 Roget'a English Thesauius - - 18 Rowton'» Debater - - - - 18 Short Whist ----- 19 Thomson's Interest Tables - 21 Webster's Domestic Economy - 24 West on Children's Diseases - - 24 Willich's Popular Tables - - 21 Wilmot's Blackstone - - - 24 Botany and Gardening:. Hassall'B British FreshTvater Algv 9 Hooker's British Flora - - - 9 *' Guide to Kew Gardens - 9 '* " " Kew Moseum - 9 Liodley's Introduction to Botany 11 " Theory of Horticulture - 11 Loudon's Hortus Britaonicus - 13 *' Amateur Gardener - 13 ** Trees and Shrubs - - 12 ' * Gardening - - - 12 " PUnU - - - - 13 " Self-lnBtmction for Gar- dener*, &c. - - - - 13 Pereira'B Materia Medica • - 17 Rivera's Rose-Amat)^ur's Guide - IS Wilson's British Mosses - - 24 Chronology. Blair's Chronological Tables - 4 Brewer's Historical Atlas - - 4 Bunsen's Ancient Eeypt - - fi Calendars of English StJite Papers 5 Haydn's Beatson's Index - - 9 Jaqucmet's Chronolopy - - 1 1 Nicolas's Chronology of History - 12 Commerce and Mercantile Affairs- Gilbart'a Treatise on Banking - fl Lorimer's Young Master Mariner 12 Macleod's Banking - - - 13 M'CuUocli'sCommerce & Navigation 14 Scrivenor on Iron Trade - - 19 Tliomson's Interest Tables - 21 Tooke's History of Piices - 21 Criticism, History, and Memoirs. Blair's Chron. and Hiitor. Tablei - 4 Brewer's Historical Atlas . - - 4 Bunsen's Ancient Egypt - - 6 " Hippolytus - - - 6 Burton's History of Scotland - 5 Calendars of Engluh SUte Papers 5 Chapman's GustavuB Adolphus - 6 Conybeare and Howson's St. PuoJ 6 ConnoHj's Sappers and Miners - fi Gleig'B Leipsic Campaign - - 22 GuToey's Historical Sketches - 6 Herschels Essavs and Addrcsats - S Jeffrey's (Lord) Contributions - II Kemble's Anglo-Saions - 11 Lardner's Cabinet Cyclopedia - 12 Macanlay's Crit. uid Hist. Essays 13 " History of England - 13 " Speeches - - - 13 Mackintosh's Miscellaneous Works 13 " History of England - 13 M'Cnlloch'BGeosrapnicalDictionary 14 Maunder's Treasury of History - 14 Memoir of Uie Duke of Wellington 22 Merivale's History of Rome - - 15 " Roman Republic- - 15 Wilner's Church HisUwy - - IS Moore's (Thomas) Memoirs^&c. 15 Mare's Greek Literature - - 16 Perry's Franks - - - -17 Raikes's Journal - - - - 17 Ranke's Ferdinand ft Maiimillao 22 Riddle's Latin Dictionaries - IB Rogers's EssAvs from Edinb. ReriewlS Roget's Enelish Thesaurus - - 18 Schmiti'B History of Greece 16 Southey's Doctor - - - - 20 Stephen's Ecclesiastical Bioeraphy " Lectures on French BistoiT Sydney Smith's Works - - - " Select Works " Lectures *' &Iemoirs Taylor's LoyoU - - - " Wesley - _ - Thirlwall's Flistoryof Greece Thomas's Historical Notes 20 ; - 22 ThombuiT'sStiakfipeare's England 21 ToH-nsend's State Trials - - 21 Turkey and Christendom - - 22 Turner's Anglo-Saxons - - 23 Middle Ages - - - 23 " Sacred Hint, of the World 23 Vehse's Austrian Court- - - 23 Wade's England's Greatness - 21 Whitelocke's Swedish Embassy - 24 Young's Christ of History - - 24 Geocrntphy and Atlases. Brewer's Historical Atlas - . 4 Butler's Geography and Atlases - 6 Cabinet Gazetteer - - - - C Cornwall: Its Mines, ftc. - - 22 Durrleu's Morocco - - - 22 Hoghes's Australian Colonies - 22 Johnston's General Gazetteer - 11 M'Culloch'sGeographicalDictionary 14 " Russia and Turkey - 22 Maunder's Treasury of Geography 14 Mayne's Arctic Discoveries - - 22 Murray's Encyclo. of Geography - Sharp's Britisn Gazetteer 1 Juvenile Books. i Amy Herbert - 19 CieveHall - - - _ ]9 Earl's Daughter (The) - - 10 1 Experience of Life - - - 19 ! Gertrude - - - . 19 1 Hewitt's Boy's Country Book - 10 i " (Mary) Children's Year - 10 [ Ivors - - - - - -I9| Katharine Ashttin - - - 19 ' Lane ton Parsonage - - - 19 I Margaret Percival - - - 19 , Medicine and Surgery. I Brodie's Psychological Inquiries - 4 . BnU's HinU to Mothers - 4 , '* Management of Children 4 j Copland's Dictionary of Meilicme - 6 I Cust's luTalid's Own Book - - 7 | Holland's Menul Physiology - 9 Medical Notes and Reflect. 16 I 19 ! How to Nurse Sick Children - Kesti>ven's Domestic Medicine Pereir^i's Materia Medica Reece'R Medical Guide - Richardson's Cold-Water Cure West on Diseases of Infancy - Miscellaneous and General literature. Bacon's (Lord) Works - - - 3 Carlisle's Lectures and Addressei 22 Defence of £c(ipieo/Ffli(A - - 7 Echpseof Failh - - ■ - 7 Greg's Political and Social Essays 8 Greyson's Select Correspondfcce - 8 Gumey's Evening Recreations - ti Iia6salVsAdDU''ralions Detected,&c. 9 Haydn's Book of Dignities - - 9 Holland's Mental Physiology 9 Hooker's Kew Guides - - 9 Howitt's Rural Life of England - lo " TisitstoRemarkablePlaccBlo Hutton's 100 Years Ago - - In Jameson's Common place -Book - ] | Jeffrey's (Lord) Contributions ij CLAflSIFIED INDEX. Johns's Lands of Silence and of Barknefis ----- 11 Last ofthe Old Squires - - 16 UacaulBj'a Crit. and Hist. Euaji 1 3 " Speeches - - - 13 Mackintosh's Miscellaneoas 'Works 13 Memoirs of a Maitre-d'Armcs - 22 Maitiand'B Church in the Catacombs 14 Mailinean'B MiFcellaniea - - M Moore's Church Cas^s - 16 PrinUng: Its Urigin^&c. - - 22 P^'croft's Englieti Reading Kich's Comp. to Latin Dictionary Kiddle's L.itin Dictionaries - Rowton's Debater - Seaward'e Narrative of tui6hipwxeckl9 Sir Roger De CoT^rley - - - ' " Hiiutb'«(BeT. Sydney) Wotlia Sonthey's Common-place Books - " The I>oclor*c. BoTiveEtre's Attic Philosopher " Confessiooaof a Working Man Stephen's Essays - . - - StoWfl Training System Thomson's Lavs of Thought TowDRend'a State Trials "Willich's Populir Tables yo&ge'G English-Greek Lexicon - 17 • 18 Latin Grados Zampt's Latin Grammu 20 21 24 MatTiral HistoTyln ereneral. Catlow'B Popular Concbology - fi Ephemeraand Yonng on the Salmon 7 Garratt'8 Marvels of Instinct - 8 OoBse'e Natural History of Jamaica 8 Kenip'B Nutural History of Creation 23 Eirbyand Spence's Entomology - 11 Lee's Elemeuu of Natural History 11 Maunder's Natural History - - 14 Turton'p Shrlls oftheBritishlslanda 23 Van der Hoever's Zoology - - 23 VonTschudi's Sketches in the Alps 22 Waterton'sEssaysMtlilBturalHUt. 24 "Vouatl'l Th» Dog - - - - 24 " Tlieliors* - - 24 }-Volnme Encyclopsedias and Dictionaries. Blaine's Rural Sports - - - 4 Brande'E Science, Literatttre,and Art 4 Copland'G Dictionary ofMedicine - G Cresy's Civil Engineering - S Gwilt's Architecture - - - 8 Johnston's Geographical Dictionary 11 Loudon's Aenculture - - - 12 " Rural Architecture - 13 '* Gardening - - - 12 " Plants - - - - 13 " Treeaao'lShfBlM - - 12 H'Culloch'sGeoj^raphicalDtctionary 14 " DictionaryofCommerce 14 Murray's Eucvclo.or Geography - 16 Sharp'!) British Gaietteer - - 19 Vre's Dictionary of Arts, Ac. - -23 Webster's Domeatic Economy - 24 Religriotis & Moral IVorks. Amy Herbert - - - - 19 Bloomfif ld'« Greek Testament - 4 Calvert's Wife's Manual - • G CleveH*U - - - - - 19 Conybeare's Essays - _ _ g Conybeare and Hovaan's St. Paol £ Cotton's lustnicUons in Christianity € Dale'B Domestic Litonry - - 7 Defence of Eclipse of Faith - - T Discipline - - - - - 7 Earl's Daughter (Tbel . . - 19 Eclipse of (-aith - - _ 7 Englishman's Greek Concorda>ee 7 " Heb.&Chald. Concord. 7 Experience (Tbe)orLifc - - 39 Gertrude - *- - - 19 Harrison's Light of the Forge - B Hook 'a Lecture? on Poaaion Week 9 Home's Introduction to Scripture* 9 " Ahnilgmentof ditto - 10 TIuc'B Christianity in China - - 10 'Hu-nphreys's ParabUt Illuminated 10 Ivors 19 Jameson's Sacred Legends - - 11 " Monastic Legends - - 11 " Leeendxiif the Madonna 11 " LcLtures on Female Em- ployment - - - - . 11 JerMny Taylor's Works- - - 11 Katharine Ashton - - - 19 Konig's Pictorial Life of Luther - B Laneton Pjrsonaee 19 Letters to mv Unknown Friends - 11 " on K»ppiness - - - 11 Lyra Germjnict - - - - 5 M-tcnsHsht on Inspiration - - 14 Maguire's Rome - - - - 14 Mnitland'srhurchinCatacomlw - 14 Margaret Percival - - - - 19 Martineaa's Christian Lire - - 14 " Hyrnns - - - 14 Merivale'6 Christian Records - IS Milner'B Church of Christ - - Ifi Moore on the Use of the Body - 15 " " Soul and Body 1& " 's Han and his Motives 16 Mormonisin ----- 22 MomiDg Clouds - - - - 16 Neale'a Closing Scene - - - 16 Ranke's t'erdinand & Maximilian 22 Headings for Lent - - - 19 " Confirmation - 19 Riddle's Household Prayers - - 18 Robinsou'a Lesicoa to the Creek TesUment - - - - - IB Saints our Example - - - IS Sermon in the Mount - - 19 Sinclair's Journey uf Life - - 19 Smith's (Sydney) Moral Philosophy 20 " (G.V.}ABByrianProphecie6 20 " (G.) Wealeyan MethodiKm 19 " (J. )St.Paul's Shipwreck - 20 Southey'B Life of Wesley - - 20 Stephen's Ecclesisstical Biography 20 Taylor's Loyola - - - - 21 •' Wesley - - - - 21 Tbeologia Gennanica -1.-5 Thumb Bible (The) - - 21 Tomline's Introduction to theS»&Ie 21 Turner's Sacred History - - - 23 Young's Clinst of History - - 24 " M)Elery - - - - 24 Poetry and tbe Drama. Aikin'B {Dr.1 BriUsh Poets 3 Arnold's Poems - . - - 3 Baillie's (Joanna) Poetical "Works 3 Calvert's Wife's Manual - - 6 De Verc's May Carols - - - 7 Estcourt's Music of Creation - 7 Fairy Family (The) - - - 7 Goldsmith's Poems, illustrated B L. E. L.'e Poetical Works - 1 1 Lmwood's Anthologia Ozoniensis- 12 Lyra Gerraianica - - - - 5 Macaulav'6 Lays of Ancient Rome 13 Mac Donald's Within and Without 13 " Poems - - - 13 Montgomery's Poetical Works - 15 Moore's Poetical Works - - 15 " Selections (illustrated) 15 " Lalla Rdokh - - - 16 " Irish Melodies - - - 16 " Songs and Ballads - - 15 Heade'B Poetical Works - - 17 Shakspearc, by Bowdler - - 19 Southey's Poetical Works - - 20 " British Poets - - - 20 Thomson's Seasons, illustrated - 21 Political Econonty and Statistics. Dodd's Food of London - - 7 Greg's Political and SociftlEvHys S Lama's Notes of a Traveller- - 22 M'Culloch'flGeog. Statist. ac.Dict. 14 '* Dictionary of Commerce 14 '* London - - - 22 Willich's Popular Tables - 24 The Sciences in general and Iflatbematics. Arago's Meteorological Essays S " Popular Astronomy - 3 Bourne on the Screw Propeller - 4 " '8 Catfictiiam of Steam-Engine 4 Boyd's Naval Cailet's Manual - 4 firande'n Dictionary of Science, Ac. 4 " Lectures on Organic Chemistry 4 Cresy's Civil Engineering - - 6 DelaBeche'sGeologyofCamwall^c. 7 T)e la Rive'B Electricity - - 7 Grove's CorreU. of Physical Forces 8 Herschd's Outlines of Astronomy 9 Holland's Mental Physiology - 9 Humboldt's Aspects of Nature - 10 " Cosmos - - - 10 Hunt on Light - - - - 10 Lardner's Cabinet Cyclopedia - 12 Marcet'R (Mrs.) Conversations - 15 Morrll's Elenients of Psychology - 16 Moseley'sEngineering&Architerture 16 Our <;oal Fields and our Coal-PiU 22 Owen's Lectureson Comp. Anatomy 17 Percira on Polarised Light - - 17 Peschel's Elements of Physics 17 Phillips's Fossils of Cornwall, &c. 17 " Mineralogy - - - 17 Guide to Geology Portlock's Geology of Londonderry 1 Powell's Unity of Worlda 17 Smee'8 Electro- Metallurgy - - 19 Steam-Engine {The) - - - 4 Wilson's Electric Telegraph- - 22 Rnral Sports. Baker's Rifle and Hound In Ceylon Berkeley's Forests of France Blaine's Dictionary of Sport* Cecil's Stable Practice - - - " Stud Farm - - - - The Cricket Field - - - - Davy'sFishing Excursions, 2 Series Ephemera on Aneling . - - " "a Book of the Salmon - 7 Hawker's Young Sportsman - - 9 The Hunting-Field . _ - 6 ldle'3 Hints on Shooting - - 10 Pocket and the Stud - - .- 6 Practical Horsemanship , . 8 Richardson 8 Horst-manship - - 18 Konalds' Fly-Fislier'a Entonu^ogy 16 Stable Talk and Table Talk - - 8 Stonehenge va the Greyhound 20 Thacker's Courser's Guide - - 21 The Stud, for Practical PurpoHS - 6 Veterinary Medicine, frc. Cecil's Stable Practice 6 '< Stud Farm - 6 Hunting-Field (The) • 8 Miles's Horse-Shoeing - - - 15 " on the Horse's 1: oot - - 15 Pocket and the Stjid - - _ 8 Practical Horsemanship - • 8 RichariJson'B Horsemanabip - 18 SUble Talk and Table Talk - - 9 Stud (The) _ ^ - - 8 I'ouatt'sTheDt^ - ' - - - 24 " The Horse - - - 24 Voyages and Trarela. Auldjo's Ascent of Mont Blanc - 22 Raines's Vaudois of Piedmont - 22 Bilker's Wanderinvs in Ceylon - 3 Burrow's Continental Tour - - 22 Rarth's African Travels - > 3 Berkeley's Forest* of France - 4 Burton's East Africa - . > 5 " Medina and Mecca - - 6 Carlisle's Turkey and Greece 6 De Cuatine'B Rusbia - - 22 Eothen ------ 22 Ferguson's Svriss TraTel* - - 32 Flemi-h Interiors - - - - 7 Forester's Rambles in Norway - 22 " Sardinia and Corsica - S Gironi^re's Philippines - - - 22 Gre?oroviuB'e Corsica - - - 22 Halloran'E Jnpan - - - - 8 Hill'3 Travels in Siberia - - 9 Hinchlitfs Travels in the Alps - 9 Hope's Brittany and the Bible - 22 " Chase in Brittany - - 22 Hewitt's Art-Student in Munich - 10 (W.) Victoria - - - 10 Hue's Chinese Empire - - - 10 Hue and Gabet's'TartarT & Thibet 22 Hudson and Kenneoy's Mont Blanc - - - - - 10 Hughes's Australian Colonies 22 Humboldt's Aspects of Nature - 10 Hurlhut's Pictures from Cnba - 22 Hutchinson's African Exploration 22 Jameson's Canada - - - - 22 Jerrmann's St. Petersburg - - 22 Laing's Norway - - - - 22 " Notes of a Traveller 22 M'Clnre'a North- West Passage - 17 MacDougall'sVoyageoftheJKesoIu'e 13 Mason's Zulus of Natal - - 22 Miles's Rambles in Iceland - - 22 Osborn's Quedah - - - - 16 Pfeitfer's Voyage round the World 23 " Second ditto - - - 17 Scherzer's Central America - 19 Seaward'fi Narrative - - 19 Snow's Tierra del Fuego - - 20 Spottiswoode's Eastern Russia - 20 Ton Tempskv's Mexico and Gua- temala . . - . a* Weld's Vacations in Ireland - - 24 " United States and Canada- 24 Weme'8 African Wanderings - 22 Wilberforce'B Braail ASlttve-Trade 22 Works of Fiction. Cruikshank's Falstaff - - - 6 Howitt's Tdllangetta - - - 10 Macdonald'a Villa Verocchlo - 13 Melville'c Confidence-Man - - 15 !Moore s Epicurean - - - 15 Sir Roger be Coverley - - - 19 Sketches (The), Three Tales - 19 Southey's The n«ctrtr Ac - - 20 Trollope's Barchester Toweia - 23 " Wapden - - - 23 ALPHABETICAL CATALOGUE of NEW WORKS and NEW EDITIONS PUBLISHED BT Messrs. LONGMAN, BEOWN, GEEEN, LONGMANS, and EOBEETS, PATERNOSTER ROW, LONDON. Miss Acton's Modem Cookery for Private Famibes, reduced to a System of Easy Prac- tice in a Series of carefully-tested Seceipts, in which the Principles of Baron Liebig and other eminent Writers have been as much as possible applied and explained. Newly-re- Tised and enlarged Edition j with 8 Plates, comprising 27 Figures, and 150 Woodcuts. Fcp. 8to. 7s. 6d. 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