58 MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
Therefore 
cos i// = a sin" (f) + cos </> \/( 1 — a- siir ^). 
Therefore 
1 ~ '2a cos xfj a^ = 1 -\- a~ — 2a' sill" (j^) — 2a cos (p — or siu" (p) 
= [v/(l “ '^"siir p) — a cos pj^. 
I’herefore 
v/(l — 2a cos p 4- ^') = {\/{^ ~ sill" “ cos p\. 
Thus 
d'p 
d(f) 
and 
giving 
\/(l — 'lucos-p + «") \/’(l — 
cos -p dp u .sill p + cos (b \/{\ — ad siir p ) 
^(1 — 2« cos p + «") 
y/(1 — a- sill- p) 
dp, 
C’^ 
dp 
Tlierefore 
I 0 \/'( 1 — 2« cos p 4 a-) 
siid p dp 
= 2 [ 
.‘o a/(1 —2a. cos 'll?’ 4 a") Jo \/(l — «' sirr p) 
cos p dp ^ f-d sin' p dp 2 
I 77^ o ^ t \ I ” 
U-) Jn 
dp 
= 2F 
Jo a/(1 - a'shi'P) 
(F - E). 
Jov/{i- — .ia cos p 4“ «"} o 
4 
n = P-‘F-:('“ + h(i''- 
3 + 
Tlierefore the potential 
Now 
and 
1-’= S{(i-5)1^+ ('+?) e}- 
E = j{l- 
1.1 
a~ 
1 o .-> 
2'\^ 
a* — &c. 
Substituting, we find 
V = 
M 
i; 
T ^ 
1 - 8 “ M “ - 
— a® 
1024 
— &c. — 2 
lb :3b.. (271 - 3)2. (2a - 1) 
a~ 
I 1 rd 1 od 5 (d 
22 . 42 .. . (2a - 2)2. (2a)2.(2a 4- 2) 
12 . 32 .. . (2 a-3)2. (2a-1) a2'* 
81F''“64IF 10241h ^ 22 . 42 ...(2a-2)2.(2'a)2.(2a + 2) Pd-'+i 
This series is less than the series whose general term is 
1 ndp 
2id ’ 
and is therefore convergent if a = E or a < E. 
