MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
45 
Section I .—Preliminary Analysis. 
§ 1. Take a fine ring in the plane of x, y, centre at the origin, and consisting of 
attracting matter of density h cos n where </>' is the azimuth of any point. 
2 " 
The potential of this ring at a point P. whose coordinates are r, 6, ([>, is 
Jn 
Jc cos np cclp 
Iq \/[A + — 2cr sill 6 cos — p)} 
Put (j)' = (f) then the potential 
ck (cos n(p cos n-v/r — sin n(p sin nylr) d-^ 
=r 
Jo 
= cos ncj) j 
v/{r- + c“ — 2cr sin 0 cos-yfr} 
ck cos d-\Jr 
0 \/{A + — 2cr sin 0 cos -v/r} 
as the second integral vanishes between the limits. 
Therefore 
cos n(f) d(j) 
COS n 
J n 
or 
cos 
n<fy^ 
Jo 
0 •/{A + c" — 2cr sin ^ cos 0) 
cos n(f> d<j) 
\/ ( 2 ^ + — 2cot cos cp + cj^) 
is a solution of Laplace’s equation. 
