1048 MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING, 
and the equation must be regarded as an approximation to 
. fPv 
clx~ 
,2 + 
~ V = 0 
7 0 0 * 
az" c" 
or 
4_ 1 ^ . 1 1 f Y 
^2 + R ■ dll R2 dx^ ” c3 ^ 
r/I 
= 0 , 
/ipll\ 
of which cos nx . J» ( “^) ^ solution ; and starting from this, an approximate solution 
mav be obtained. 
t/ 
We shall only need the cases in which p is small. 
The equation may then be written 
S + 1 X + 3 b S+Fv) + + 2p’-^y) +. • ■ 
c dx 
dx 
(? V dx 
An approximate solution of this is 
R“ ct ^ 
cos j - cos nx + 7- cos {n - l)x 
+ 
4c ft* 
a- R*' + 3 UjP + 2?i + 1 
Sc^ ft'*+ 2 
‘2n + 
cos nx + 3 cos (n — 2)y j + &c, 
(^)- 
When n = 0 the solution is 
cosj4 ^ 1 ^ + &c. 
(5). 
§ 5. To find the Potential of a solid anchor ring at an internal point. 
At any external point whose jrolar coordinates are r, 6, <f, 
V 
= “|r_ 
TT 1 Jo v/ (r^ 
dcf) 
+ -^-^ 
8 cdc JI 
_ ^ 
192 UW Jo 
d<h 
U 0 \/ {d + c^ — 2cr sin 6 cos 
(d / d 
c/ Jo \/ {P + c^ — 2cr sin 6 cos 0) 
0 \/ + d — 2cr sin 0 cos (j)) 
— &c. 1.* 
Using the expansions of the integrals given in that paper, we find that at the 
surface of the ring 
Sti^^rd, p. 59. 
