1894.] Discussion of Linear Differential Equations. 205 



Therefore 



00 

 Km 



and (X, - \y =-, {{h^ + grjf - 4a (5f + cv' +f^v)]. 



(Jj 



Consequently 



2m-(\ + \) _ ^(/i-2ap) + 7;(gr-2ag) 



= const. 



Thus we have ^f{m) = 0. 



We see, therefore, that for the sohition of the equation Aw — 

 we may write 



a a 



w = e ^^ ^1 .F{^,'n), 

 where F {^ 7^) = A + B^+Cr^ -^ ^^ {D^' + 2E^r] + Frj') 



+ ^; (G^r + ^Wv + ^KH + W) + &c., 

 the coefficients A, B, G, &c. being matrices. 



We will now expand the operative symbol, and make ^ and rj 

 zero after the operations have been performed. The only term 



which gives any value when operated upon by ^^ ^ is, leaving 

 out the matrical coefficient, 



1 (^ + «)Ur s 1 fcr s 



{r + s)\ ' r\ si rlsl 



This is reduced to unity by the said operation. Also, the coeffi- 



cient of ^ ^ in the expansion of the operating symbol contains 



the sum of all the products that can be formed involving u and v 

 respectively r and s times. Thus we shall finally obtain as our 

 solution 



w = A+uB + vC + Y] b''^^ + (^^ + vu)E + v^F] 

 + ^ {w^(r + {uh + uvu + vu^) H 

 + (uv' + vuv + vhc) K + v^L] + &c. 



