MR. J. S. TOWNSEND ON THE DIFFUSION OF IONS INTO GASES. 135 



The boundary condition, p = when r = a, requires that such values of 6" be 

 chosen as will make M, = when r = a. 



Substituting (</) for (;) in the function M, we obtain a function of Pa* with 

 numerical coefficients. Let x lt x^, x t , &c., be the values of (Fa 4 which satisfy the 

 equation M r=0 = 0, and let t , 0~, 3 , <fec., be the corresponding values of 0; and 

 equation (4) becomes 



,oK ,WK 



+, &c (5). 



4. Before proceeding to determine the coefficients c,, c 2 , &c., it is necessary to prove 

 some general properties of the solution of the equation V<f> + 0; f(x, y, z) <f> = ; 

 f(x, y, z) being any function of x, y, z. Let <f> n and ^ n - be solutions corresponding 

 to values n and #. of the parameter 6. 



By GREEN'S theorem, we have 



fff O^< - *.***] ** *y dz = ff (*,* - f 2) rf s. 



Let B and #. be such values of as will make < and <. vanish at the surface S ot 

 the region throughout which the above volume integral is taken. The surface 

 integral vanishes under these conditions, and we get, on substituting for V 2 ^, and 

 n , their values, 



/(*, y, z) dx dy dz = ..... (6a), 



- ) fff 



which shows that the triple integral vanishes when n and #. do not coincide. 



Let us suppose $ n > to be got from < by changing 01 into ffi n + d(P, and GREEN'S 

 theorem gives 



- \\\ 



so that 



We also have 

 from which we derive 



=-'dS ,,.,. . . (6c). 



5. Let/(a;, y, z) be (a 2 r 2 ) and < a function of the cylindrical coordinate r. The 

 equation V 3 ^ + 0>f(x, y, z) <j> = then reduces to -^ * f (r ^ + ^ (a 2 - >- 2 ) <^ = 0, so 



that we can substitute M for < in the three equations (Ga), (66), and (6c). If the 

 surface integrals be taken over the surface of the cylinder of radius a, we obtain 



