ON THE FURTHER TABULATION OF BESSEL, ETC.. FUNCTIONS. 45 
As another physical application, consider the potential V of a circular 
dise treated as a plane circle ; then, with Maxwell’s notation in § 701, 
‘Electricity and Magnetism,’ V may be considered a homogeneous 
function of a, A, b, of the first degree ; 
dV dV dV 
(49) Mee otha tO 
Here - is the potential of the rim circumference, and so is given by 
2a 
__ dV _ f{ adé 
(50) a dies tis | PQ’ 
i) 
(51) PQ?=a? + 2Aa cos 6 + A? + 6? =7,? cos*h6 + 7, sin?40, 
7, and r, denoting the greatest and least distance of P from the circle ; 
and putting 6 = 2, 
(52) PQ = V(r,’ cos *w + 7,7 sin’w) = r,/ (1 — c? sin?w) = 7, Aa, 
To? 7 
e=1—-%, Ya ee 
: a T; 
_fa(de _40F yay 
(3) R=™ (= cE = F(bm, 0), 
0 
Next — - is the component of attraction norma to the disc, or the 
magnetic potential for uniform normal magnetisation ; and so is given 
by Q, the conical angle subtended at P, which depends on the complete 
Elliptic Integral of the Third Kind. 
And Maxwell’s M, the coefficient of mutual induction between two 
parallel circles of radius a and A, a distance b apart, is the Stokes 
function of Q; 
aie dV _ { — 2rAa cos 6d0 
(54) M= a , 
dV _[(acos0d0_ _ 
where Q is the component attraction of the disc parallel to a diameter, 
or the magnetic potential when magnetised uniformly in its plane. 
Then 
2r dr ‘ 
5 R = (4(1 — cos 6)d6 __ (8a sin?w dw 
ae [ PQ 7, Aw 
_8afl—A%w~,  8aF—E 
== tae =, go? B= Eno). 
(67) == (225% 2). 
Tr) c 
Thus 
(58) V=aR — AQ — 30, 
a OEE 
