MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
V/ i 
§ 8. For the purposes of calculation, it is convenient to transform the integrals of 
Si 6 by Landen’s theorem. 
# __ o p— __ 
(' 
J (\ 
J 0 v/“ 2cr sin 6 cos <p + 
2 i 2 
d(f) 
where 
and 
F = 2- 
H- 
4F 
h + Ih ’ 
d(f) 
J 0 \/l — sin - ^ ' 
Ih - R 
lij + h 
> ’ 
R and R^ being the least and greatest distances of the point r, 0, from the circular 
axis of the ring. 
o 
Now, 
and 
But 
Therefore 
Similarly 
Therefore 
4F 
fZF dfi 
E 
2r d,x 
r dc 
djji c dc 
dd" 
c dc 
dV. 
rfE dfx 
E 
- F 
dfjL 
c dc 
dfj, cdc 
d 
c dr 
[Ca 
R2 z: 
0 1 0 
_ y- _j_ _ 
2cr sin 
0. 
(Hi 
r — r sin 0 
4(^2 + 
R2 
— 
T> 2 
J'l 
dc 
i; 
4(4 
i 
(?Ri 
c + r sin 6 
4(;2 + 
Ih 
0 
- ];2 
[Cayley, ‘ Ell. Func.,’ p. 48.] 
(fr i; 4cli. 
, /(HL 
dr 
dc 
(R, + RE 
/dll . (HP 
( dc + ^dcj , 
(H?, (/R 
(Ri + R)- 
_ Ih - R Vd + Rd - 4(;2 
~ Ih + R 2cEIh 
d- I 
= - cos xb ; 
c 
where if/ is the angle between R and R^. 
