296 Wellisch — Experiments on Active Deposit of Radium. 



with the conditions N = when r = b for all values of z (i) 



N = 7T)(^ 2 — ^ a ) when 3 = ±1 for all values 

 of r (ii) 



. r = ) £)N . , 2N 

 At ~ a "- = and __ = (in) 



The function N" = AJ (ar) (e*z 4- e~ a2; ), using the usual nota- 

 tion of the Bessel functions, is a solution of (3), which satis- 

 fies the condition (iii). We have the constants A and a at our 

 disposal. (i) will be satisfied if a be a root of J (Ab) = 0. 

 This equation has an infinite number of real positive roots, 



X n X 2 , . . . X n , . . . Now, if we can expand _?_(r a — 5 2 ) as a 



4D 



series 



00 



Jj2 A n (eU + e-U)J (Kr) 

 n = l 



it is easy to see that 



00 



N = ^ A a (e^z + e -2 Q z)J o (x n r) ( 4 ) 



n=l 



will be the solution of (3), which will satisfy all the conditions 

 of the problem. 



Carslaw* gives the following expression for the coefficient 



B n in the expansion f(r) — ^ B n J (X n r) where X n is as above: 



n = l 



B »=F " {J '(Kb)\ ' 

 So that in the case under consideration, we have 



2 for^(r'-^)J (Kr)dr (5) 



An ~ & ¥ («W + e-W)iJ.'(^)}" 

 We have next to determine the integral 



f\r>-b*r)J Q (Kr)dr (6) 



• / 



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



