April, 1916] Derived Solutions of D iff ere?itial Equations 243 



_ (2(m+r)-3)^ " 



^^~4(m+r) (m+r-2)^'-^ '^^'' 



For m = 0, the series would fail for r = 2. 

 For m = 2 



_ (2r+l)^ , 



^^"2r.2(r+l)^'-^ ^^^ 



This gives the A-series. 

 Differentiating (1) 



_ (2(m+r)-3)^ 2(m-r) -6 



4(m+r) (m+r-2) '"Hm+r) (m+r-2) (2(m+r)-3) ^ 



And when m = 2, this is 



_ (2r-l)-^ . 2(r-l) 



^ 2r.2(r+2)^-' r(r+2) (2r+l) 



The B -series derivable from the A-series is gotten thus: 

 The n*^ term of the B-series from the (n + 1)*'' term of the 

 A-series by multiplying by 



S 2(r-l) 



^ r(r+2) (2r+l) 



And the terms of the B-series preceding the A-series 

 come from 



_ /2r^2(r+2)__8(r-R)_ \ 



'-' ^^'\ (2r+l) (2r+l)3 ^ ^^ ^ 



.-. A_: = Ao (0+8(m-2)) 

 A_2 = A_i(-4-16(m-2) ) 

 = Ao(0-32(m-2) ) 

 A_3 = A_2(0+k(m-2) 

 = Ao(0 + 0(m-2) ) 



.-. the terms are 



Bo(-32 + 8x) 



