KM \IK. J. S. TOWNSEND ON THE DIFFUSION OF IONS INTO CASKS. 



2. If Equation (3) be written in the form 



the relation connecting any three consecutive coefficients becomes 



from which it is easy to see that the series we have found for M is convergent. 

 Let 2n be greater than (Fa*, and let /?_, + /3 n _ 2 = S. Then 



& < |, and A, +1 < |. 



Similarly, 



S 



Proceeding in this way, we see that 



&i+2, < 



2n (n + 2) (n + 4) . . . (n + 2m) 

 from which it follows that the series (3) is convergent. 



3. The Equation (2) has a second independent solution, N, which can be found by 

 using the solution <j> = M, the complete solution being aM + y8N. It will be seen 

 from what follows that, when r = 0, N becomes infinite ; so that it must be neglected 

 when the gas A, as in the present case, extends to the centre of the tube. 



Substituting for (j> in Equation (2) N = MM, we obtain 



HIT 2 i n .2 du , f du 



Mr 2 + 2r* + rM = 0, 

 dr dr dr dr 



or 



_i_ fa _2_ m i_ 



du/dr dr* ~ M dr ' r '' ' 

 which, on integration, gives rM 2 du/dr = c. Hence 



f dr 



u = c -TT 1 



J 



Expanding -rp- in partial fractions, and integrating, we see that u has a term 



c log r, so that N becomes infinite when r = 0. 



Thus p cannot contain N in its expansion, and we get 



p = Sc 9 M(,e" *v ' ......... (4). 



