April, 1916] Derived Solutions of Differential Equations 239 



Differentiating the m-f actor of (1) 



'm-2r+2\'\ / -"> 



^m-2r+l/ / V(m-2r+l) (m-2r+2) 

 For m= —1, this is 



,^^rj ) \~ 2r (2r - 1)^ 



And thus the n*'^ term of the B -series is derived from the 

 (n + 1)*^ term of the A-series, beginning with the 2d, by 

 multiplying by 



^ -2 



2 



^ (2r - 1) (2r) 

 The A-series is 



12 1- '^^ 



x-1 (1+ 2^x-2 + ^^x-^+etc.) 

 And the B -series is 



-2x 1 (^ -jjK - + 2^.[jj + -2a)'' +etc.) 



(b) Series starting with the same value of m, but not zero. 

 ,dV , dy 



X- 



^ -3x;^+ (4-x)y = 



dx2 dx 



y = x™ gives (m — 2)^ x"' — x^^+^^O. 

 .-. Two ascending series, starting with x^, 



Form = 2, A,= ,A,_Y-^^ (2); 



(2) gives the A-series. 

 Differentiating the m-f actor of (1), 



/ 1 \ / -2 \ 



\^(m+r-2)7 \m-\-r-2 J 

 For m = 2, this is 



(+) (- ') 



The n*'' term of the B-series is derivable from the (n-f 1)'^ 

 term of the A-series by multiplying by 



