1066 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING-. 
§ 17. To determine we must first find Vg. 
Let rr', z, (f)' be the coordinates of any external point, ct, 2 , of any internal point. 
Then 
cr dcj dz d(f) 
T- — 2cjci' cos ((j)' — 0) -f (z' — z)~} 
Let ta7 = c — X. 
Then 
V.= 
(c — x) dx dz d<p 
^/{(c — xy + cj'" — 2 (c — x) cj' cos ((/)' — (p) + iz' — z)-] 
— g - Wdc) - s(d,dz') 
c d (j) 
{c^ + 2cuj' cos {(p' — <^) + 
=nf{' -(*i+Li 
c d(p 
-y/— 2cm' COS {(p' — (p) + z'^} 
where the double integral is taken over a circle of radius p. 
Therefore 
,'277 
fdP . d- 
Vg=: Up-^+8VbH + 
\dc- ‘ dz'- 
Now 
j g /bP fP 1 _ c dp _ 
+ To 2 -r + . • • I ^ 1^,3 + _ 2c57' cos (p' -P) + z'^-} 
pp = cW 1 1 + (ot/, sin n p -j- cos n p) 
^ + (a« “7^+1 + Ai A/+i) COS + 
(P d^ 
Substituting this, and writing — + — = 
Vn = 
c dp 
+ mi- jj, + -j ^*j ^{s'a + c« _ 2 c 5 ,'cos (<f.' - ^) + s'-} 
\/+ c” — 2 cot^ cos [p' — p) -{- z''^} 
c S («,» sin n p + I3n cos n p) dp 
3a~ 
5rd 
c dp 
. o <^n + J , , V72 I 'AL V74 
Jo 2 1 ^ 4 ^ + 64 ^ i + c 2 _ 2cny'cos (0' - <^) + z'~} 
