CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
(■ b .) In like manner, the equation in n, which is 
n'b cos n'b-\-Kb sin n'b = 0, 
, 2i+l 
is satisfied, approximately, by n'b = -- . 7 r=a, say: and more accurately byn'b = a~\-x, 
where 
(afi-x) sin x—kL cos ,r. 
This is precisely the same equation as before, and gives 
n'b = a + -. K b - 1 ,rfh 2 - * 3 & 3 
a a'* or 
X/ / 2,0 7 I — 1 .77 0 18 + 2 a 2 37 3 
k=T^ a- + 2/c6+ — K-lr— — k b iJ r . 
Zirb\ a or 
j 
. . (47) 
When we make b indefinitely small, retaining It as a finite quantity, all the values 
of X and X' become indefinitely great except those which arise from the approximation 
Ea: 
in n, n z b 2 — i<b-\- . . . ; X is then = —. Rejecting therefore all other values of n, n', 
X and and confining our attention to the parts of P outside the plate, we find 
[writing A cos nb under the single term ^H], 
z positive, P=2 cos m<£.J w (/qo).^le *( 2+ ^) 
y 
z negative, P = £ cos m(f).J m (Kp). < ^ie +K ( z ~^) 
(48) 
the term in b in the exponential e being neglected. 
If, at the surface of the plate, P —fix, y, t), then at any point for which 2 is 
positive, 
and at any point, z negative, 
P =/(«, y, f - + 
P =f(x, y, t — 
2 -\~) 
r ; 1 
h 
(49) 
This is Maxwell’s result. 
I have dwelt somewhat fully upon this case because it brings out very clearly the 
mode in which the terms depending upon all but a certain number of the values of 
X and X' disappear when b is made indefinitely small. When we consider the case of a 
spherical shell, we shall find that a similar reduction takes place. 
