xMR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING, 
1101 
mj-Cj 8^3 7\ , (^2 G 1 (^h 
«0 
logP-i +T^?log ?-i 
Cj 2 
C;^t “3 cos (f) d(j) 
\/{('^2 "I” 2c^C 2 COS + Cj"} 
8 Cn 
= 771o2Co log —° - i- 
(U7). 
where a,j and Cq are the values of a and c, when the ring and sphere are a long way 
apart. 
Let the radius of the sphere be Z’, 
Changing to polar coordinates, let 
Co = r sin 6. 
z, = r cos 6. 
c, = — sin 6. 
r 
Co = - cos i). 
■" r 
Then 
(sj — 4" c.^ — 2 C 1 C 3 cos ^ 4 - = ?’” 4* "7 — 2Z;" cos*^ ^ — 2/;^ sin^ 6 cos^ </> 
z= Ir — + 4F sin 0 sin^ ^ 
Therefore 
r sin 6 f loof — ')'k sin^ 6 | 
cos (f) dfp 
\/{{t — k~lry + 4:k^ sin^ 6 sin^ (} 
• A n sin ^^0 7 
= Tq sin 6 »o log - I 
Also 
ah' sin 0 = a^^VQ sin 0q. 
Substituting a from the second equation, we have the path of the ring from infinity 
up to the sphere, i.e., the path of any point in the ring. 
Let 
Vq sin Oq — Cy and cja^ = n. 
Then the above equation becomes 
’ sin 6 log 8n — 
- f log ' 1 — hr sin^ 0 j 
cos ^ d^ 
- khrf + 4:¥ sin3 d sin^ (kcj))} 
= Co (log 871 - I).(118). 
When the centre of the ring coincides with the centre of the sphere, let 
?• sin 0 = CT, sin 0=1; and an easy transformation of the integral gives 
