CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
323 
Eliminating C and D, 
A (k cos nb—n sin nb)-\-B(i< sin nb-\-n cos nb) =0 
A(«r cos nb—n sin nb) — B(/< sin nb-\- n cos nb) = 0. 
From these we find 
( 1 ) 
( 2 ) 
B = 0, n sin nb — k cos nb— 0, C=Ae Kb cos nb=D 
A = 0, k sin nb-\-n cos nb—0, C=Bc k/ "cos nb —— D 
( 39 ) 
Putting all these together we obtain 
inside the plate P = £ (3) 
outside, z- f- ve , P=S (3> 
outside, z — ve , P=S <3) 
cos m<f>J m (Kp)(A cos nze A/ +B sin nze v/ ) 
-(-similar terms in sin m<j> 
cos m(f)J m (Kp)(A cos nbe~ Kt -\-B sin n'be~ y/ )e~ K(:: ~ b) -\- . . 
cos m(f)J m (Kp)(A cos nbe ~ k/ —B sin n'be~ Kl/ )e K{l!+i) -\- . . 
) ( 40 ) 
J 
where 
n sin nb — Kcosnb = 0, (k 2 4-A 2 ) 
47T 
n cos n'b-\-K sin n’b— 0, (k° n' : ) 
47r x 
The summations are to be extended, first over all the values of n and n corre¬ 
sponding to the roots of the above equations, then over all values of k from 0 to co , 
and finally over all integral values of m from 0 to oo ; the summation with respect 
to k will be of the nature of an integral, as will presently appear. 
§ 6. The investigation of the values of the coefficients is attended with some difficulty 
owing to the difficulty of interpreting the values of J m for infinite values of the 
argument. I have therefore sought to evade these difficulties by conceiving the plate 
as the limit of a spherical shell of finite thickness but of infinite radius, and keeping 
in view the general course which the solution for a spherical shell takes. It is 
possible to obtain the solution for any spherical shell, and it might seem therefore easy 
at once to adapt that solution to the present case: but unfortunately the adaptation is 
also beset with difficulties of a peculiar kind, and therefore we can do no more than 
take the steps of that investigation as guides in the present problem. The main light 
which the case of the spherical shell gives us is that we have to regard cos m</>J,„(fcp) 
as a degraded form of the spherical surface-harmonic cos ■jh^P^cos 6), obtained by 
putting 
sin 6 =-, 7i= kb, 
cl 
(A)) 
MDCCCLXXXI. 
2 u 
