52 
MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
From this d^ljdc^ might be obtained, and then d?\^ld(?, &c., but in the question of 
an attracting ring there is a special advantage in using functions of the form 
1 
c clc) 
Call this I«. Thus 
Also denote 
]„ = 
/ 1 
dcf) 
' ''^^7 J Q d — 2cr sin 0 cos (f >} 
by J, 
1 d \"-l G 
cos <f) d(j) 
'' dc) Jq \/{d + c- — 2cr sin 6 cos 0} 
The values of the functions ai’e collected here for tlie sake of reference. 
(A) 
T 1 Jz L O L ^ 1 , /2/ + 1 3/ + 2 , 
Ii==-V + 2+ 2 cosx6'+ + - ^-cos 2x)s' 
L = 
+ 
+ 
cosy + 
2 
91 + 3 
64 ^ 192 
108Z + 27 60/ + 13 
16 
15/ + 7 
cos 3x 
2048 ’ 768 
2/ + 3 cos 2x 
105/ + 34 i \ j. I 
+ ' 3072 ~^^^ ^ 
+ 
■ ,12^ + 11 . S 0^0 
S + ( —37;— cos X + #2 cos 3x 
, /12/ + 5 , 12/ + 9 , . , , \ s , 
+ (—H 5T- cos 2x + tIs cos 4x + 
128 
64 
h = 1 COS 2x + (I COS X + i cos 3x) 6- 
/36/ +51 0,3 , \ L 
+ ( — + i cos 2x + 3 % cos 4x) S' + 
O ! 
1.1. = ^ {cos 3x + (t cos 2x + i cos 4x) S + . . .] 
^5 — ;j[7i [cos 4x + . . .j. 
(B) 
rH_i _ 
dl ~~ 
- jsin X + i sin 2x s + o'^'" siiiy + 
’3/ + 3 
+ (-^3- 2x + rls sill d X) Y 
, /6/ + 3 . , 
+ X + 
120/ + 101 . ^ \ . 
—- sin 3x + offs sin ox ) + 
