08 
THE HON. J. W. STRUTT ON THE THEORY OE RESONANCE. 
We proceed to investigate the value of §§<pda, which is the potential on itself of a cir- 
cular disk of unit density. 
Potential on itself of a uniform circular disk, 
r denoting the distance between any two points on the disk, the quantity to be eva- 
luated is expressed by 
Eiar. 5. 
~i n> /» /"» 
* 1 i ch ' 
. , j r 
The first step is to find the potential at any point 
P, or | | — . Taking this point as an origin of polar 
coordinates, we have 
potential =jj\ — = ~^=^rd0= j(PQ-f-PQ')d^. 
Now from the figure 
i(QQ') 2 =R 2 -c 2 sin 2 0, 
where c is the distance of the point P from the centre of the circle whose radius is R. 
Thus potential at P 
:2R 
1 — ^2 sin 2 ^ d0=i.R 
1 — j^sin 2 0d0. 
(30) 
Hence potential of disk on itself 
= 4ttR 3 | dx (* \/\—x sin 2 0 dt 5, 
Jo Jo 
if for the sake of brevity we put ^=x. 
In performing first the integration with respect to b we come upon elliptic functions, 
but they may be avoided by changing the order of integration. 
f dx \/ 1 — x sin 2 0 = j — g g N> g (1 — x sin 2 0) 
tJ 0 v, 
2 l 
— (1 — cos 3 $)=ih— ; *4-1- cos 0 ; 
3sirr0 v ' 3 l + cos0 1 3 
.-. potential on itself 
=frfy| o ^jiw 0 r«+ cosfl }=4 5rE M 1 +ii =¥*»■ (31) 
This, therefore, is the value of j)<pc?<7 when the density is supposed equal to unity. 
The corresponding value of 
dtp 0 
