April, 1916] Derived Solutions of Differential Equations 241 



(Compare Johnson, p. 185-9). 



xMl+x)^+xg+(l-2x)y = 



(Compare Johnson, p. 185). 



y = x"', gives 

 (m + 1) (m-2) x-+i + (m2 + l) x- = (1) 



And we may select two descending series, beginning with 

 x-i and x^, with powers differing by unity. 



CD 



By (1), v= 2 ArX™-"^ gives 

 1 



(m-r) (m-r-3) A,+i+( (m-r)^-+l) Ar = ._ 



. (m-r)^+l , ...^ 



••A'-+>- (m_i.) (m-r-3) ^^ ^-'^ 



For m = 2, this gives, 



(2-r)'^+l 



(2-r) (r+1) 

 And this will fail when r = 2 

 For m= —1, (2) gives 



^^+^~ (r+l)(r+4)^^ ^"^^ 



This gives the A-series, from which, by the method already 

 used, we can also get the B -series. 

 Differentiating the m-factor of (2) 



/ (m-r)^+l \ / 3(m-r)^+2(m-r)-3 \ 



\^ (m-r) (m-r-3)J 1^ (m-r) (m-r-3) ( (m-r)2+l)J 



When m= —1, this becomes 



/ (l+r)^+ l \ / 3r^+4r-2 \ 



1^ (r+1) (r+4)y V (r+1) (r+4) (f^+2r+2)y 



Thus the terms of the B -series (except as to those preceding 

 the A-series) are gotten, the n*^ term of the B-series from 

 the (n + 1)*'' term of the A-series by multiplying by 



5 3r2+4r-2 



-^r+i— (ck \ / I i\ -^r 



Q (r+1) (r+4) (r2+2r+2) 

 The A-series is 



x-Kl-^x-+-^-^-x-+etc. 



