CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
337 
Putting all the results together we find for the value of P at an outside point 
i oo oo /h\»+i rb p' r' 2 
P= — m S n X('j ^ cos m(<i>-<f>')d<f>'. P m *(0)P m *(0’) sin O'dd’ H 
TT 0 0 \ ^ / J 0 • 0 a m 
A V x 
xn;'.(71) 
From this formula the action of the shell on an external point due to any inducing 
system may be found. 
We proceed to inquire what will be the result when the shell is supposed to become 
infinitely thin. 
Infinitely thin shell. 
§ 13. Putting b — a=c, <r = Eq the exponential-function e~ K * c '+^. will be finite only for 
such values of A as make A 3 c finite : if A 3 c become indefinitely great the corresponding 
terms will rapidly disappear. More accurately, the terms for which \~c is finite will 
die away infinitely less rapidly than those for which it is infinitely great. 
W in the equation 
Ac = a'— ft' -\-in, 
we take as a first approximation \=—, we may readily expand Ac in a series of the 
• • IT 
form fTr+Ej.c + EyP-b . . . ; but in this case A 3 c= — approximately when c is very 
great. 
The corresponding system of currents will rapidly decay. It is therefore by putting 
i=0 that we obtain the currents which have the greatest persistence. The equation 
in A is 
A' 2 c = 
(n + 2)(n+l) (n — l)n 
2 a. 
2b 
-f terms in A A" 
and, for a first approximation, b = a and 
A 3 c=-(2r+1) 
(72) 
Hence, by equations (58) and 
for points outside the shell, P^=S(F(-j CJ„ 
H + l 
_ K 
e 47ra' 
(2n + \)t 
within the shell, P = £(i£(-) U„c ^ 
(2,i + l )t 
(73) 
To interpret these results : 
