76 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
The equations to determine A^, Ag, &c., are, therefore, 
Ai 1 + cr- + —o-M + A 30 " I + o- 
Ai 0 -- + 2 A 3 + SAjo-' = 0 
6 T 
+ Agcr^ -g- = — a-c~(02 
A, (- A + o^) + AAi - T-is + A 3 (6 + \i <r=) + A, ^ 3 '^ = « 
~ 6 ^ -^1 + ^3 + 2 -i^r = d 
~ ~ xts ^3 + 1-6 -^3 + f + 120 Aj — 0 
These equations give 
= - 1 - 
12\ - 1 , , 144X2 _ 168 X + 21 T\ , , 
+ -“TRYI-O' « ; 
32 
1024 
. o 6 X + / o 5 .1 
^'2 — ^ -12^ o-.crc'wa; 
A. 
f ^ , 42X + 7^\ o\ 0 o 
64 
512 
o-'' I crc'Wo ; 
A,= 
1 o, 9 
3 8 4 (X C W-i’, 
A. = - 
^ 9 9 
The kinetic energy is given bj" the equation 
n 2n 
0 
ccoo sin ^{c — a cos x)(Af+ A3a^f+.. . j cos® (f) a dx 
rZn ^ 
= 77«c®w 2 (sin X - 2 sin x) (Ai • 
12X-1 „ . 144X2 _ I 68 X + 21 ^ 
= 7 r®tt®c^Wo®-I ( 1 
32 
o-® + 
1024 
o-M 1 
12 X 4-15 
32 
" I 
o-' + 
6 X + 11 
256 
cr 
36x + 7 o , 1 74 
J + 64-8 0 -^ 
,, c2a),;2 f, 12X + 7 . . 18X2 _ 3X + 5 ^ 
= M-^ 1 -■ — 0 -+-- 0 -* 
