1)4 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Substituting the values of A^, A^, &c,, these equations are 
•2 ..2 
A 
- .y^ 1 +? + 
iira^ ' I / 
4X + 3 
8 
12X - 1 4X-1 
H-HT3 O' + 1C Pi — 
128 
16 
3X + II- o , - if .1 , X n \ 1 1 1 -2 1 Ih 
+'* 128 ^'" ' 8+^-1 " '"■+ 16 ‘"+™=4 
+ A j(^ ~ IV) J. ~ 
n 0 
a wC" O’ 
4 Tra^ 4 
0, >- 
A + 
64 
a-'+ U + 1 % 
68(X + t'*) .1 I 3X + 1 , 
- 0 -" +- 
1024 
16 
cr 
+ 
A-I- i 
4 ^ 
A 
9 ^ , 
COV/CT'- 
cr/ 
A-iA.+ il + ;f4ir^^-ift)=f' 
7ra" \ 8 
(7 
4 A 
These ecjuations give 
(D 
IT 
Aa 
A 
A. 
= (\ + |) cr—i (\ + fl) 
f (A + tV) 4-iA(A-ii)^^ 
1 — (A + D cr- 
(X — - 2 ^-) 0 -=^ 
75A2 80X 4- 21 , 
- 
2o6 
■5 
12 8 
where cr = ajc and X = log,, ( 8 c/«) — 2, 
The result u^jv = (X + f) cr is given by Mme. Kowalewski in a paper on Saturn’s 
rings in the ‘ Astronomische Nachrichten ’ for 1885. She finds A 2 = i (^ + f) 
M. Poincare also gives oy^ln = (X + |) cr®, and finds A:j = | (X + f) cr^. Both papers 
are given in Tisserand's ‘ Mecanique Cdleste,’ vol. 2. The value found above has 
been kindly verified for me by Mr. Herman, Fellow of Trinity College, Cambridge. 
The numerical values of w®/ 7 r, Aj given below when cr = 1 %. 
11 
b 
X = 2-3820 
A-- -0189 
As — 
-0001 
CO 
0 
0 
0 
11 
= -0313 
irp 
cr = ‘2 
X = 1-G889 
A = -0570 
11 
-0004 
Ai = -0023 
— = -0970 
TTp 
cr = *3 
X = 1-2834 
A 3 = -1208 
A- 
-0010 
Ai, = -0078 
— = -1836. 
TTp 
A figure is given for the case of cr = '3. 
The cross-section is roughly an ellipse whose major axis is perpendicular to the axis 
of the ring. The eccentricity of this ellipse increases with the angular velocity. 
