Wellisch —Experiments on Active Deposit of Radium. 297 

 By mating use of the integral/ x"J a _,(x)dx = x n J h (x) together 



with the known relations subsisting among the Bessel func- 

 tions of adjacent order* the following indefinite integrals 

 were obtained : (for the sake of brevity the analysis is omitted) 



I xJ (x)dx = xJ x (x) 



Cx z J a {x)dx = 2x 2 J (x) + (x> - 4x)J t (x) 

 Applying these to (6) and utilizing the fact that J (A n 5) = 

 we obtain - — J^n*) as the exact expression of (6) 



A n 



From (5) we obtain A n = - J J^ ^ + ^j^) 

 and from (2) and (4) in conjunction 



00 



U * 4D ( V) W ^ A n 3 eUTe-Ki J fab) 



Now the total number of active particles in the gas is given by 



G = 47r / / nrdrdz 



O \ t/o 



r h b 



Using (7) and also the integral / rJfar)dr = — J fab) 



we deduce, as the expression for the total number of active 

 particles in the gas contained within a cylinder of height 21 

 and radius b when the steady state is established 



e-Kz J far) 



00 



q _ vobH _ 87rq "X; 1 eKl 

 ^D ' ~D ~ x A? ~eKi 



(8) 



-Kl 



where \, X„ . . . are the roots of J (X5) = 0. 



So far no approximation at all has been made, so that the 

 series (8) is the exact solution of the problem ; the first term 

 was obtained in Section 3 as the approximate solution when 

 the diffusion to the top and bottom was neglected. 



* Cf . Gray and Mathews Bessel Functions, p. 13. 



