548 Mr. A. Stephenson on 



There is nothing in this method peculiar to the trigono- 

 metric functions, and we now apply it to the Bessel's expan- 

 sions to obtain a generalization which is essential for the 

 complete solution of a certain type of physical problem. 



If B n (fir) is a particular solution of 



d 2 u , 1 du . I 9 n 2 \ A 



& + tw + \? -?r= 0; 



then, for fx k ^^i, 



which is 



Pk~-M 



5—, ^ A W)»« M - f^,(wWW } > 



if B»W0=B„W) = O; 



and therefore 



if the m's are roots of 



a M^) = ^ +g L. ..... ( ii.) 



Also from (i.) by approaching the limit Pi=t u k, we nnd 



f VCWO**-* { Wfaft) -a»B,V*») - (a 2 - ^) B* (*a) } 



Now consider the problem of expanding a function of r, 

 for the range between a and 5, in a series of Bessel's functions 

 of the first order 



/(r)=2AB Or), (iii.) 



where the //,'s are determined by (ii.) and the particular 

 solutions B^r\ of Bessel's equation are so chosen that 

 B (/ub) =0. Both sides of (ii.) being odd functions of fi, the 

 negative roots are numerically equal to the corresponding 

 positive roots ; and therefore it is sufficient to consider the 

 positive roots alone in the summation. To determine A* 

 multiply (iii.) by rB (fi k r) and integrate between a and b '< 

 then by the preceding results 



I rf(r)B (/i k r)dr=pB (fi k a)%AB (fja) 



+ A k i \FB %J>) -a*Bo\fi k a) - (a 2 + 2p) B 2 (^a) \ ; 

 and therefore, since there is no reason to question the 



