MR. F. W. DYSOI^ ON THE POTENTIAL OF AN ANCHOR RING 
1059 
§ 13. The exhaustion of potential energy is given by 
U = -I I Vi dm 
=im- L + i (1 - + <r cos t! - ^) H'’ “ ® '=®®x) R # 
2 a 8fl^ 
lP» + i 1 p,«-i\ 
n a 
+ TTCd 
1 
+ -TT—rrcos0i4- 1 )y 
n (n + 1) a" ' ^ ^ ^ 
c — R cos x) R o^R r^x ^^4* 
2L + 1 ^ ^ ^ L-1 
1) s v-o-S A^Ai + i- 
+ O- 2 A® (f cos X + 5 f) I (c - B cos x) B c^B («x # 
= 2irV 
f 
+ i) 
i\p: 
8 a^ 
— cos X 
+ i) 
nf- _ -P- 
3 10«~ 
+ -cosxiV£ 
o-/3« 
+ 2TThdc S A. cos nx t cos nx - cos {71 - 1) x 
+ 
cr/3n 
4:71 (n + 1) 
2L + 1 
cos (?i+l)x|(l“ 0 - cos x) dx 
+ 2^%‘c{(L-l)2A-^i^o-2AA.i 
In the first integral, substitute 
pp = a?’ 1 1 +_p 2 A. cos nx + + cos x ^ AA +1 +•'•)}' 
On integration this gives 
27rVc |l + i + (L-1)2§-“- o- 2 I3„l3 „,} ■ 
The second integral may be written in the form 
2^%*c £'2 A cos «x 2 (f - “x ( 1 - ^os x) a^, 
6 T 2 
