MR. R. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
1047 
Writing sines instead of cosines, we obtain another set of solutions of Laplace’s 
equation applicable to space inside a ring. 
Writing — n for n solutions of Laplace’s equation applicable to space just outside 
an anchor ring are obtained. They can, however, only be obtained as far as the nth 
term. At this point log E, arises in the series. In obtaining each term from the pre¬ 
ceding, as an integration is performed, a constant should be added ; and for the series 
to be convergent these constants must have definite values. To find the series furtlier, 
the method of my former paper is necessary. 
The following are the solutions of Laplace’s equation inside a ring for n = 0, 
1, 2, 3, or 4. 
Constant. 
R , a 
cosy -f 
a 
1 5* + 
zc 
3 
R3 
f • ,7 cos X + 
/a\3rp 
c 
(I COS 2x + -h) 
+ ( 7 ) 7 Ws cos 3x + M cos x) + . . ., 
3 cos 2x + 1cos X + {.y:) ( 1 % cos 2x + |) 
a 
«■* 
+ (|) 7(Mcos3x + fcosx) + 
IF 
cos 3x + 
A L 
2 
IF 
cos 2x + 
a \3 R 
2c a 
y (ft cos 3x + I cos x) + 
J>5 
^cos 4x+^.i.^;.cos3x+... 
1 
c. . (3). 
J 
§ 4. In discussing the stability of a fluid ring, approximate solutions of the 
equation 
1 dV 
+ 
cPY 
d~Y (PY _ 
clx^ dx" c — X dx ' (c — xf d(f) 
- = o 
are required. 
Writing V cos for V, it becomes 
d~Y d^Y 
dx "- + .7,2 
1 dY 
dz 
- V = 0. 
c — X dx (c — x) 
If p is of the order cja the last term becomes of the same order as the terms 
I d^Y 
dx^ dz^ ' 
itT + equation may be written 
dx? d?} 
— See. = 0 
