Expansions of BessellFunctions. 233 



Solutions of the Associated Equation. Values of R, p. 

 If y denote y± or y 2 in the substitutions for Bessel functions, 



G^)"+(i-^=i)^i=o. 



Thus 1 = 1- ^i, (24) 



and the associated equation becomes 



z 3 u'" + (lz*-±n 2 z + z)u' + (±n 2 -l)u = Q. . (25) 

 Writing u=% r a r z r , 



the relation between successive coefficients becomes 

 r + l.r + 2-2/4.?' + 2 + 2n.a r+2 =-4ra r , 

 with an indicial equation 



s- 1.6-1 4- 2n. s- l-2n = 0. 

 The following series solutions therefore exist, 



i , , 4n 2 -l 2 1.3 4>i 2 -l 2 .4n 2 -3 2 

 u >= 1 + i'-&-+J7i-~ ~J^ + (20) 



z z 1.3 z 5 

 w 2=- + i^2^y2 + 274/i 8 -l 2 .7i 2 -2 24 " ( 27 ) 



. >ft+1 / 2/i-fl ; 2 2». + 1.2» + 3 S \ 



Ws-* ^4 M + n,2n + l + »+l.n+2 2 .4.2n+l .2n+2 "P 



. . . f2S) 



where if n be an integer, u 2 must be multiplied by the 

 evanescent factor of the denominators, thereby ceasing to be 

 distinct from u s . For positive real values of n 9 u 2 and u 3 

 are convergent for all finite values of z, but U\ is ultimately 

 divergent except when 2n is an odd integer, in which case it 

 terminates. 



Proceeding to an examination of m 1? it is seen that when 

 z is infinite in comparison with ?i, u } = 1. 



But comparing (8, 9) and (12, 23), in this case, 2/i 2 +#2 2 = l, 

 or R=l from the first asymptotic substitution. Thus ui = R 

 when 2 is very great, and being always a linear combination 

 of R, S, and T, which are of similar magnitudes, it must 

 always be R either identically or in an asymptotic sense, not 

 necessarily that of Poincare. 





