MI:. .!. s. TOW\SKNI> o\ mi-: DIFFUSION OF IONS i\ T o ,.\, 133 



positive and negative carriei-s are equal, the forces X, Y, and Z vanish, and the 

 equation for p becomes : 



wliich, expressed in oyHndrioal coordinates, becomes 



, dh) dp L'Vr* ,. dp 



r ^ + r -<&&-'*)% = ...... (1). 



We have to find a solution of this equation which will satisfy the conditions : 



p = Pe a constant when 2 = for all values of r, since A is distributed evenly 



throughout B on entering the tube. 

 i = when r = a for all values of z, since A gets absorbed by coming into 



contact with the tube. 



f K 



-<l"fc 



Letp = $e~ - v ', where $ is a function of r, and & a constant to be determined 

 afterwards. 



Substituting this value of p in Equation (1), we obtain 



* 2 -r*)< = (2). 



One solution, M, of this equation can be found in the form of a series. 

 Let 



be three consecutive terms in the expansion of M in powers of r. 



Substituting in (2) we find, by equating to zero the coefficient of the (m + 4)" 1 

 power of r, that 



(m + 4)' A, +4 + 0VA, +2 - ^A M = 0. 



If A M+4 r"' +4 is the first term in M, (m + 4) 2 must vanish. Hence, the first term 

 must be a constant, which we will take as unity. Thus 



&c ........ (3), 



\\ here B,, Bj, B 3 , &c., are found from the equations 



4B, + ffia* = 0, 



36B, + 0VB, - ^B, = 0, &c. 



