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



And we may thus assume an ascending series beginning with 

 m = 0, or a descending series beginning with m= —1, exponents 

 differing by 2 in each case. 



(1) The ascending series, 



00 



y = 2 Ar x™+-^ 

 



gives, by (E), 



(m+2r)2 Ar-(m+2r-l)2 A,_i = 



. (m + 2r-ir . ,. 



•■^^" (m+2r)2 ^'-' ^^' 



For m = 0, A, = \-,r^ K_, (3) ; 

 (2r)- 



(3) gives the A-series. 



Differentiating the m-factor of (2) we have 



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



\^ (m+2r)2 J \(m+2r) (m+2r-l)y 



And for m = 0, this is 



/(2r-l)^\ / 1 \ 



V (2ry^ J \r{2r-l)J 



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

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

 multiplying by 



r = n ^ 



,.fi?(273Tj W 



--' ^'^-i x''+ etc. 



By (3) the A-series is 



1- , 



y ■'- ' 92 ~'~ 92 42 ' 92 12 [i2 



And by (4) the B-series is 



il 2 7 P.31 , 37 . P.3151 

 2 ^ "^ 6 22.42. ^ "^ 30 2-A~.ir. ^ "^ 



(2) The descending series for the same equation. 



(m+l)2 x"^+i — m^ x'"-i = 0, gives, 



(m-2r+l)2 x--2^ Ar-(m-2r + 2) x-^-^^-^ Ar-i = 

 (m — 2r-|-2)- 

 For m = - 1 , Ar = -^Wy^ A,_i (2) 

 This gives the A-series. 



