242 The Ohio Journal of Science [Vol. XVI, No. 6, 



And the B -series is 



"^ ^ 4 1.4 "^ ^ 20 1.4 2.5 



To get the terms of the B -series preceding the A-series, 



_ (m-r) (m-r-3) .^. 



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



Differentiating 



(m-r) (m-r-3) . / 3(m-r)''+2(m-r)-3 \ 



(m-r)2+l) ^^+1 (^ (m-r) (m-r-3) ((m-r)2+l)y 



And when m= —1, this is 

 ^^ ^ (1+r) (4+r) .^^ / 3r2+4r-2 



^r— „2 I o„ I o -"-r+l 



r2+2r+2 '+^\^(H-r) (4-fr) (r2+2r+2)y 



.,_, f (l + r)(4 + r) 3r^ + 4r-2 ) 



•• ^^-^^+1 1 r2+2r+2 {r''+2v^2Y ^^^+1^+ ^^^j 



.-. A_i = Ao (0+3 (m+1) 



A_2 = A_i (1+ki (m+l)+ etc. 

 = Ao (0+3 (m+l)+ etc.) 



A_3= A_2 (| +k2 (m+l)+ etc.) 



= Ao(0+^(m+l)+etc.) 



A_4 = A_3 (O+kg (m+l)+etc.) 

 = Ao(0+0(m+l); 



and all subsequent terms vanish as to m + 1. 

 .-. terms to be added are 



Bo(3+3x+^x2) 

 o 



Vanishing of the first term in A_i prevents using the pro- 

 cedure used heretofore, since in this case a summation no 

 longer holds. 



(2) 4x(l-x)|^-4^-^-y = 



dx^ dx 



y = X™ gives 



4m (m- 2) x--i- (2m- 1)^ x'" = 



/. series start with x" and x-; exponents of powers differ 

 by unity. 



