NATURE 



THURSDAY, JUNE ii, ,1903. 



DIFFERENTIAL E(2UATI0NS. 

 A Treatise on Differential Eqiiaiions. By Prof. A. R. 

 Forsyth, Sc.D., LL.D., Math.D., F.R.S. Pp. xvi + 

 511. Third Edition. (London: Macmillan and Co., 

 Ltd., 1903.) Price \\s. 



THE value of this useful text-book has been increased 

 by the inclusion in the third edition of important 

 additional matter. 



The principal additions are an account of Runge's 

 method for the approximate numerical solution of or- 

 dinary differential equations, of Frobenius's method for 

 the integration of linear equations in series, and of 

 Jacobi's theory of multipliers. 



The chief modifications of the matter treated in the 

 earlier editions occur in the treatment of Lagrange's 

 linear partial differential equation of the first order, 

 in the discussion of the condition of integrability of a 

 total differential equation, and in the treatment of 

 Riccati's equation. 



Of the above-mentioned subjects the one of greatest 

 theoretic interest is probably the treatment of Lagrange's 

 equation, whilst the most useful is Frobenius's method of 

 integrating linear equations in series. 



The theoretic interest of the treatment of Lagrange s 

 equation arises from the fact that until Goursat published 

 his " Legons sur I'lntegration des Equations aux derivees 

 partielles du premier ordre" in 1891, the widely used rule 

 for the solution of Lagrange's eq uation had not received 

 adequate demonstration. ^ 



If u — a, V = b furnish values of z in terms of x, v 

 which satisfy the equation 



Vdzjdx + Qdzldy = R, 



where P, Q, R are any functions of x, y, z ; and if 

 ■^{x.yy s') = o be any other integral, then the condition 



r — — --1 = o must be satisfied, 7tot necessarily identi- 

 ially^ but in virtue of the relation beticeen x, y, z given 

 ^J^^i^^y^^) = o- It is only when the above condition 

 is satisfied identically that \|/- is a function of u,v. In 

 this case \//- is certainly included in the general integral. 

 But it is possible to take a case of the general integral, 

 and put it into a form in which the Jacobian does not 

 vanish identically ; e.^i^. if xdz/dx + ydz/dy — z, we may 

 take u = y'x, v = x/z, >//• =:y/x - x/z and the Jacobian 

 vanishes identically; but if we put \lr=yz - .v^, then 

 the Jacobian = - 2\j/ (xz-), which vanishes only when the 

 relation between the variables is such as to make yjr = o. 

 Finally, it is possible to have singular integrals, which 

 cannot be expressed in the form of the general integral 

 at all. In this case, let u = a, v = b be two integrals, 

 and let /{x,y, z) = o be any other integral, then by 

 elimination o( y, z express /(x,y,z) = o in the form 



hen if D denote partial differentiation when x, u, v 

 the independent variables, it can be shown that 



- e Chrystal, Transactions of the Royal Society of Edinburgh, 

 \xxvi. part ii., p. 551 (i8gi). 



ir by eliminating s, x in the form ^(y, ji, v) = o, or by eliminating 

 ill the form x{z, «, t») — o. 



NO. 1754, VOL. 6^'] 



PD0(.i-, u, v)!'Dx must vanish, not identically, but in 

 virtue of the relation between .r, 7, z given hy f{x,y, z) = o. 

 Prof. Forsyth proves that if P, Q, R are regular for values 

 o( x,y,z\n the vicinity of any point on the integral 

 f{x,y,z)^-o, then this integral is included in the 

 general mtegral. Taking as an example the equation 



(1+ s':-.v-^) dzldx + d^ldy=^2, 

 we may take 



u = 2y-z,v=y + 2^Jz-x-y; 

 and z=x+y is an integral not included in the general 

 integral. In this case 



<f>{x, M, &) = (I - x'l -! r-ii- .\ )-, 

 and PD0(.r, /^, 7/)/D.v = - ^/^-.i-i', which vanishes 

 when z = X + y. In this case it is at once seen that the 

 coefficient V ^ i + >Jz - x - y \s not regular in the 

 vicinity of points on the integral z=x->ry. 



A similar point, arising out of the conditional vanish- 

 ing of a Jacobian, comes up in connection with Art. 12. 

 It is there proved that an ordinary differential equation 

 of the first order and degree, with coefficients which are 

 one-valued functions of the variables, has only one in- 

 dependent primitive. 



As soon as the reader reaches the subject of 

 singular solutions, he is forced to ask himself why the 

 reasoning in Art. 12 is inapplicable. He wishes to have 

 an explanation of the fact that the many-valuedness of 

 the coefficients causes the reasoning to fail. 



Suppose the equation is idyldx -f .r -I- \/.r- + 4 y = o. 

 Two primitives of this are f2 -\-cx-y ^oand;*:^ -}- 4j = o. 

 Their Jacobian is ^{x -f- 2^), which does not vanish 

 identically, but conditionally, viz., at the point of con- 

 tact of the envelope .r^ -|- 4/ = o by the complete primi- 

 tive c^ 4- ex - y = o. 



The method of Frobenius for integrating linear dif- 

 ferential equations in series is explained on pp. 235- 

 249, and is applied to the solution of Bessel's equation. 

 It is of a more general character than the special method 

 applied to the same equation in chapter v. ; and it ex- 

 hibits the connection between the two solutions found by 

 it. The connection between the two solutions obtained 

 in chapter v. is difficult to perceive; and Frobenius's 

 method has the advantage both in directness and 

 simplicity. It is a valuable addition to the book. 



Runge's method for the numerical solution of dif- 

 ferential equations has suffered somewhat in the com- 

 pression which the author has found necessary. 

 Nevertheless, one cannot help regretting the omission to 

 state the geometrical meaning of the expressions 

 employed, and the connection of the method with 

 Simpson's rule for the approximate evaluation of an 

 integral. The student will probably be greatly perplexed 

 as to the origin of the various quantities introduced and 

 used in the investigation. 



There are several difficulties in the discussion of the 

 differential equation which is satisfied by the hyper- 

 geometric series in chapter vi. Although the subject 

 cannot be properly dealt with without assuming a know- 

 ledge of the theory of functions, which is not to be 

 expected of the majority of the readers of the book, yet 

 there are some very obvious difficulties which could be 

 removed by short explanations. 



