A WEEKLY ILLUSTRATED JOURNAL OF SCIENCE. 



" To the solid ground 

 Of Nature trusts the mind which 'builds for aye." — WORDSWORTH. 



THURSDAY, NOVEMBER 6, 1902. 



LINEAR DIFFERENTIAL EQUATIONS. 

 Theory of Differential Equations. By A. R. Forsyth, 

 Sc.D., LL.D., F.R.S. Part iii. Ordinary Linear 

 Equations. Vol. iv. Pp. xvi + 534- (Cambridge: 

 The University Press, 1902.) 



IN this volume Prof. Forsyth deals with a part of his 

 subject which, for many reasons, is full of interest. 

 Ordinary linear differential equations concern the physi- 

 cist, on the one hand, by their occurrence in the analysis 

 required for many of his most important problems ; on 

 the other, they offer the pure mathematician an attrac- 

 tive field of research which appears to be almost 

 inexhaustible. 



Thanks to the contributions of a host of analysts, the 

 theory of linear equations has now reached a high stage 

 of development, and, as in other like cases, it is extremely 

 interesting to see how different parts of it, which at first 

 seemed isolated, are being gradually brought into 

 organic connection. One of the first great steps in this 

 direction was made by Gauss in his memoir on the 

 hypergeometric series ; this is another example of the 

 extraordinary and almost uncanny way in which Gauss 

 transformed and generalised every subject that he 

 touched. It is as if his predecessors had been hewing 

 stones for him to fit together into the lower courses of a 

 stately building which he left for others to complete. 

 And worthy successors have not been wanting, of whom, 

 perhaps, Riemann is as yet the chief. For his brief 

 memoir on the P-function marks an epoch by introducing 

 several new notions of the very highest importance— 

 the indices associated with the critical points, the 

 analytical continuation of a branch of the function 

 which satisfies the equation and the group of linear 

 substitutions generated by describing cycles including 

 critical points. 



The real significance of Riemann's paper became 



fully evident only after the appearance of the celebrated 



memoir of Fuchs. It is, of course, impossible to say 



how Fuchs arrived at his discoveries ; very likely he 



NO. 1723, VOL. 67] 



could not have explained his induction completely him- 

 self. In the introduction he refers to Briot and Bouquet 

 as well as to Riemann, and acknowledges his obligations 

 to Weierstrass. Fuchs deals with an equation of quite 

 general order, the coefficients being functions of ,r. with 

 a limited number of singularities. He shows that in the 

 neighbourhood of each critical point a there is a solution 

 of the form (x-a) k <j>, where is a one-valued analytical 

 function and /• is a constant determined by an equation 

 which can be constructed from the differential equation 

 itself. He also shows how the simplest independent 

 solutions group themselves according to the multipli- 

 cities of the roots of the indicial equation. 



The importance of these expansions near the critical 

 points is that, besides giving us information about the 

 analytical properties of the function defined by the 

 differential equation, they enable us to investigate the 

 group of substitutions associated with it. Suppose, for 

 instance, we have an equation of the second order, and 

 that in the neighbourhood of a there are two solutions of 

 the form (> -«)*<£ and (x-af^r; if the independent 

 variable starts near a and describes a small circuit round 

 it, the solutions, by continuous variation, are multiplied 

 by e- nhi and e'-" ki respectively ; thus with these solutions 

 we have a substitution of the form y\ = sy l , y' 2 = ty 2 , 

 where s, t are constants. When the indicial equation for 

 a has multiple roots, the associated substitution is less 

 simple, but can always be determined. If we start from 

 any ordinary point with a set of independent solutions, 

 then by Weierstrass's principle of continuation we can 

 (in theory at least) follow up their values as x approaches 

 a critical point a, then find the substitution which takes 

 place as x goes round a, and finally bring back x to its 

 original position. The effect of any closed circuit can 

 thus be determined ; and we have, on the whole, a group 

 of linear substitutions, with generators corresponding to 

 the critical points. 



The singularities of an integral are determined by the 

 coefficients of the differential equation ; they may be 

 poles or they may be essential singularities. One of the 

 most remarkable things in Fuchs's paper is the deter- 

 mination of the form which a differential equation must 

 have if all its integrals are regular in the neighbourhood 



B 



