CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
325 
If we take into account that n is infinitely great compared to m, we see that 
(n—m)\=n\-r-n m and (n J rm)\ = n\ X n m ; also 
1 ,1.3.5... (2n—l) 
—=2 m m !---> 
C ” n ! 
whence 
( J m (Kp)fpdp= 
K*™a? m+ * 1 
(A 7 ) 
But n = K& and the successive values of n correspond to the successive integers 
. \ 8k=-; hence the above integral becomes 
jy„MI i P dp=^ x .(a 8 ) 
We are now in a position to find the values of the coefficients (A), (B) in the 
expressions (40) : for it may be easily shown that, if n 1} n 2 be two different values of 
n, n\, two different values of n from equations (39), then 
r+$ r+b r+b 
j cos npz cos n 2 zdz=0, J cos nz sin nzdz— 0, j sin n\z sin n\zdz=0 
1 
[ +b „ , 2nb+sm.2nb [ +b . 0 , 7 2nb—sm2nb 
cos z nzciz— ---, sir n 
J -h 2n J 
!> • (41) 
2v! 
From these results we may separate the different values of A, B: for since when 
<=0, p=-p 0 . 
We have in fact 
1 
A= — 
2 n 
7r 2 nb + sin 2 nb 
r2rr r 00 r + b 
k8k . cos \ pJ m (Kp)dp \ P 0 cos nzdz . . (42) 
Jo Jo* J —h 
B may be found in a similar manner, and likewise also the terms depending on sin m<£. 
Collecting all these results for the value of P (</>, p, z) at an external point on the 
positive side of the plate, we have (z positive) 
-j r2n r°° r 00 
P=— t m \ cos m{(f)—(f)')d(f)' Ke~ K(z ~ b) J m (Kp)dK p'J m (Kp)dpt 
TT J o Jo Jo* 
-xt Zncosnb [ +h ^, /7/1 _ x , t 2n'$i-nn’b [ +b ^, . /Jn ^ 
0 ~ , , • o t P o cos nz dz +e .x n . n P o sm nzdz) . . (43) 
2nb + sin 2nb) u 2nb — $m 2n b J u J v ' 
The last % is to be extended over all the values of n, n f which can be derived as 
roots of the equations in these quantities. The first 2 denotes that the summation is 
