﻿824 Dr. I. J. Schwatt on the 



"We now assume 



S-rr+V)+ff+"(«)+|?*'"(«)+-+^^(«), a) 



wherein the A's are functions of u and #, but independent 

 of <f)(u). 



To show that (1) is true, we must show that it holds also 

 for the n + 1st derivative. 



Differentiating (1) with respect to a?, we have 



d n +hj JA, , ,„, .FA 1 du , d Ao~l 



, ,m/, A rA! du , d A 3 -| . .rAj^A <2 A„_, -| 



+ * HaTS + ^ 3lJ + -* ^L(^2)7^ + S (^IT'.J 



L(^ — 1) ' dx dx n !J r v y ?? ! rt# 



= nfw+ff*"(«)+ £)*"»+..•+ ^TO+^-^d 



which is of the same form as (1). 



We shall now determine the coefficients represented by 

 the A's in (1). Since the A's are independent of y, they 

 will have the same value whatever <j>[u) might be. 



Letting therefore: 



y=u, then ^=A l5 

 d n u 2 



^ =w3j " ^r= 3u2A i + 3wA 2^ A 3; 



= *, n £^(l)^ 1A i + (2)^ 2A 2 + ---+(^l)^ A ^ + A ' 



Solving these equations for the A's, we obtain 



a - dn " 

 d;e a 



A _ '/'?/ 2 /2\ d n u 



