90 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
To find the jxirt of the r>otential involving /3/. 
The potential of the ring at P is given by 
,jr 
V = 
277 
0 0 
{e — T CO.S y) r dr d')(^ 
\/R~ — 2K?’COS (/) + 
= Stt 
0°:. p 
,3 
cpg 
3 
Pi (COS 0) 
Now 
tlierefore 
'-’""yri + ts - E= 
p = a{] + eos 2x + . . .), 
11 _p 2^, cos 2^ + • • • + ^ + ^ cos Ty + 
p3 — ^^3 11 _p 3^^ 2y + . . . + cos 4x + . . . 
&c. 
Therefore, tlie terms which will give rise to terms of highest order in at surface 
of ring are 
^ ^277 
-P 
ID 
27Tcdc/3J 
in + 4 ^5 cos 4x P 4 (cos 40) j f/0 
1 
I 
S ! 
= 2Trhdc^.fi ^ + I . fl cos 4a 
-=2,7=c*/3/.|4 + W'f 
Collecting these results, the potential of the ring at P is given by 
2!.% = r (1 + f) + ^(- j + s {- S+ + *-) - 
r 5(7® 
ID 1 1024 
+ &c. 
. fo-A + ift 1 + - McrP, + m + wm 
