the Active Deposit of Radium. 431 



Carslaw * gives the following expression for the coefficient 

 B w in the expansion 



/(r)=lB,J,(V) 

 where X„ is as above : 



2 fj b jf(r)J (\ n r)dr 



B "=¥ ' {J,/ pa)} 2 



So that in the case under consideration we have 



A _ w JO * U ... 



n b* (eW + e-MjiJo'Cknb)}'* 

 We have next to determine the integral 



(5) 



I 



t 



(r*-b*r)J (Kr)dr (6) 



) 



By making use of the integral 



^ n . 1 (x)da:=x n 3 n Qc\ 



r 



together with the known relations subsisting among the 

 Bessel functions of adjacent order f, the following indefinite 

 integrals were obtained (Tor the sake of brevity the analysis 

 is omitted) : 



]xJ (tv)dx = as Ji(x), 



$jjfa)da = 2.i' 2 JoGr) + (.y 3 -4 l i')J 1 (^). 



Applying these to (6) and utilizing the fact that J (\ n b)=- 0, 

 we obtain 4.7. 



as'the exact expression of (6). From (5) we obtain 



A -- 2 -2L - 1 



A> ~ b 2 D\ tt 3 (eX»'+ e-*J)Ji(kJ>y 



and from (2) and (4) in conjunction 



n =W^^ T ^~ bD^iXye^l+e-KljJx^y ' ' [n 



* § 126, * Fourier's Series and Integrals,' Macmillan, 1906. 

 f Cf. Gray and Mathews, ' Bessel Functions/ p. 13. 



