20 Mr. W. H. L. Russell on Linear Differential Equations. [Nov. 21, 



IV. " On Linear Differential Equations."— No. VII. 

 By W. H. L. Russell, F.R.S. Received October 24, 1872. 



I am desirous to conclude this series of papers with some remarks on the 

 solutions of differential equations considered as transcendents. I shall 

 take the linear differential equation of the second order, 



which will be sufficient, as it wdl be seen at once that similar investiga- 

 tions apply generally. 

 Let 



y=s.« 4) +M l a?+M^* + . . . + u H x> l + 



We propose to investigate the general law" of the convergence of this 

 series. The general term of the series is given by the equation 



(n + 2)(n + l)au n+2 + (n -f l)n(3u n+1 + n(n — l)yu n 

 + (n + l)a'u n+1 + n(3'u n + (n — l))A* M -i 

 + o\-r/3"vi+y" w n-2=0. 

 As (ri) increases without hmit, this equation becomes 



(n + 2)(n + l)aW„ +2 + (n + l)nf$u n+1 + n(n — l)yu n — 0, 

 or more simply, 



This may be written 



a^+ 2 . !^±i + /3 . ^± 1 + y = 0. 



Now let !^±i converge, as (n) increases without limit, to a certain 



u n 



quantity p, then will also converge toward the same quantity p, and 



U n+1 



p will be given by the equation 



Let p v p 2 be the roots of this equation, ^the greatest root; then the series 

 w -f u x se+ u^+. . . . will be convergent if is in the limit less than 



U n 



unity, or if x is less than For large values of (a?) we must proceed 

 as follows : — 



^ = «0 + -+-2+ 1+ 



cc x- x n 



Then, substituting in the differential equation, 



(n — 2)(n — l)aw n _ 2 + n(n— l)/3w w _! + w(« + 

 - (n- l) a 'u n -i— w/3' Wb - ( n + l)y'w n+1 

 + a"t« n + /3% +1 + y"t< n+2 = 0. 



