Mr. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 1079 
It is necessary that 
a /K 
^TTcrah <{K + -- •+~(M~F)cosn-l- 
2+12 + 1 
2 71 \ 
92/^0 
COS 2v 
« o o 5 «“ , 
— — COS — 7 - 6 % “Y COS 4y 
J4c c“ 
1 , a cos V . ct^ cos^ V 
+ 3 7r/3rN-+ - +... 
+ — {c^ — 2ac cos V cos^ v] 
(53) 
should be constant at the surface of the ring. 
Therefore 
and 
Thus 
and also 
fd5 4K-7 , 4 3 „ 
TTCr-7-+ ^wp~ „ — = 0 
C *x O'* 
cro)^ 
^ 7 f 1 , 80K + 271fd} , , 
2^aah j- i + 192 -^+ = Oj 
4K - 7 b w 
TT 4 a c' 
= 4 I - 
TI O' + 3 yP 
0 
80K + 271 a2] I 
96 c2 J ^ — 3 ^3 F J 
(54). 
As a rough approximation to Saturn’s rings, take 
- = 1 and - = Then K = log^ 64 = 4’] 5 
c c 
and 
and also 
therefore 
(0~ 
TT 
= -15-17 +ip 
= 2-44^0- - ip 
.k 
p/a- = 4‘58 h/a. 
(55), 
For a ring, about the thickness of Saturn’s, vve shall have roughly p/cr — 0 ; or 
the ring would need to be 100 times as dense as the planet, if it were a continuous 
fluid mass. 
