MR. F, W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
1071 
displaced to the point whose cylindrical coordinates are c — </>, 
where 
and 
^ = c 2 (a,, sin nff) + A* cos ncf)), 
i = ct {y„ sin n4> + cos ntj)), 
^ and ^ are taken to be small quantities compared with a. 
Let c — i ^ I bti coorciiDcLt0S of poitit of tldo j^ncl co ^ ^ ^ f of 
any external point. 
The j)otential of the ring at ct', z . (/>' is given by 
(c — X — dx dz d^ 
J J J \/ {jjj'' — 2{c — X — cos (0' — 0) + {e — X — + {z — z — ^)-} 
= \ dx dz 
c d(f) 
\/+ c“ — 2ct^' cos — <^) + 
= j 1 _ a; £ - 2 
+ . . .’^dx dz 
+ • • • 
c d(f) 
y/+ c" — 2ct5i^ cos — 0) + 
- 0 
c dcf) 
y/Ipt'- + C- — 2 cpt'C 0S {(f)' — (/))+ .2'^} 
^ ^ 8 ( dc^ dz'-^) ‘^ ■ ■ ■ J Jo r ^ fL-' ■*“ 2 dd^ 
+ . . 
c d (f) 
y {pj'" + — 2cpj' cos {(j)' — (p) + 2 ^"} 
• (37)- 
Iletaining only those terms of the second order in the small cjuantities a, /3, y, 8, 
which do not involve </>', 
Vr, = na^ 
c d(j} 
\/{^''^ + — 2cpj' cos (p + z'^} 
- TTcd ^ 1 + £ £|j +£,-.)+■■• j I ^ + A. COS ncP') £ 
do 
■j- 2 {y;, sin ncp' + cos n(p') j 
e cos 7icp d(p 
o? / dd 
d^ 
\/+ d — 2cpj^ cos <P + z'"} 
+ 'x,n„ + /3„S,j d? 
d-8— 
4 dd 
dc dz 
+ 8/ d? i 
c d^ 
4 dz'y Jo + c3 _ 2cnT' cos (P + z'y 
(38). 
