MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
Substituting in the above expansions 
r d(j} 
jo\/{?’" + — 2cr .sin^ cos</)} 
9 
r 
COS (j) d(f) 
COS 3% 
32 ^ 24 
, J501 + 15 . 4/ + 1 ^ \ , 
128 
\/{r^~ + c- — 2c?’ sill 6 cos 
= I {^ - P~ " + cos V - * cos 3x 
, ^/322/ + 35 , 20/- 3 ^ , 
+ oA.o + ^^cos 2x - 6\cosx)+ • . . 
\ 2048 
Now 
2 1 
R, ~ ^ 
= ?L + 
.? COS X + 
H cos X 
+ ^ 116 + IG Au cos X + cos 3x 
64 
64 
/ q 90 35 
+ (1024 + bl4 2x + 77^:11 cos 4x) + ... 
Therefore 
f 
J 0 
1024 
dcj) 
y/{r~ + c~ — 2c?’ sin 0 cos 0} 
1 
I + 1 
= -|/ 4- 2 + 6’ —- cos X + S' 
^9/ + 3 . 1.5/ + 7 
2/ + 1 . 3/ + 2 
''"\6^ + "i6'"^®2x 
+ s' (-^ cos X + cos 3x 
+ s' 
10.5/ + 27 , 60/ + 13 ^ , 105/ + -34 , , , 
H-4FTr- cos 2x 4- gQ - ;^ ' ^ cos 4x) + &c. 
2048 
768 
And 
cos (f)d(f) 
I {?’" + c~ — 2c?’ sin 6 cos <^} 
/- 1 
= : 1 ^ COS X + 
6 / + 1 _ 3/ - 4 
16 
+ 
16 
- cos 2 
X 
4- 
+ 
3.3/ + 1 , 15/ - 23 
4- 
64 ■ 192 ^^"2X 
/540/ + 27 , 240/ - 23 
V 2048 
4 - 
MDGCCXCIII.—A. 
768 
H 
^ , 105/- 176 , \ , 
^X + —^ -cos 4x j 4- &c. 
