102 
THE HON. J. W. STEUTT ON THE THEOEY OE EESONANCE. 
some finite value ; so that when we consider the motion as a whole it will be a finite 
value of p that gives the least vis viva. 
The vis viva of the motion outside the ends is to 
be found by the same method as before, the first 
step being to determine the potential at any point 
of a circular disk whose density = [hf - ; 
where 
potential at P= /xOP' 2 , 
OP' 2 =c 2 +£ 2 -j-2c§ cos Q; 
potential at P= j jc 2 £-f | cf cos $j ; 
or if previously to integration with respect to 6 we add together the elements from Q to Q f , 
= i 4^(PQ+PQ') [c 2 + ~ Q2 + FQ 3 PQ - F ^ +ccos 0(PQ— PQ')j. 
Now 
PQ + PQ' =2 v^R 2 — c 2 sin 2 0, 
PQ— PQ'= — 2PN= — 2c cos B y 
PQ . PQ'=R 2 — c 2 . 
4R 3 C*2 — 
Thus potential at P= dd \/ 1 
— c 2 sin 2 9(1 -f-2c 2 sin 2 9), c being written for c-riE„ 
To this must be added the potential for a uniform disk found previously, and the result 
must be multiplied by the compound density and integrated again over the area, the 
order of integration being changed as before so as to take first the integration with 
respect to c. In this way elliptic functions are avoided ; but the process is too long to 
be given here, particularly as it presents no difficulty. The result is that the potential 
on itself of a disk whose density 
r 2 
= 1 +^E 2 
is expressed by 
16ttR 3 
(1+Tf^+irrp 2 )*- 
( 45 ) 
* [Hr. Clerk Maxwell has pointed out a process by which this result may be obtained much more simply. 
Begin by finding the potential at the edge of the disk whose density is 1 + gr 2 . Taking polar coordinates (p, 9), 
the pole being at the edge, we have 
r 2 =p 2 +a 2 —2ap cos 9 
and 
V=|j{l + ^(p 2 + a 2 — 2ap cos 9) jdddp , 
the limits of p being 0 and 2a cos 0, and those of 9 being — ~ and + 
2 2 
We get at once 
Y=4 a + SA H a 3 . 
