SATELLITES UPON THE FORM OF SATURN’S RING. 
103 
§ 2. Formation of the Equations. 
Let M be the mass of the primary and m' the mass of the principal satellite which 
is assumed to describe an unperturbed circle round the primary. Take the origin at 
M. Let there be n particles forming a ring round the primary, subject to attraction 
from M, m', and one another, and let the mass and co-ordinates at time t of particle X 
be m A , r A , 0 A - If the co-ordinates of m' at the same time are r\ 0', then the motion 
of particle X will be produced by forces which are the derivatives of the function 
where 
and 
F 
M + m\ ^ m' m'r K 
n A a r 
if cos(0'-0 A )+ Z 
Da, 
-2^cos( 0,,-0 a ); 
r, 
A A 2 = r /2 + r A 2 —2rV A cos {6'-Of, 
D Am 2 = rf + r K 2 -2r lx r x cos (0 Jli -0 A ). 
The equations of motion of m K are then 
I §l 
r K dt 
(d0 a\ 2 
\dt) 
{r/0f 
3F ] 
0r A ’ I 
t 
fdF | 
n dof J 
(i) 
As we are assuming that m! describes an unperturbed circle, 
r' = a' and 6' = wt + e', 
where w' 2 a' 3 = M + rn' — M, with sufficient approximation. 
Let us assume now that the remaining particles are moving in the vicinity of the 
vertices of a regular polygon of radius a. Then we may put 
r a — a + p\, 
6x — dot + e + X . 27 rjn + cr A , 
for all values of X from 1 to n, where p and a are assumed small, so that squares, 
products, and higher powers of them and their first derivatives with regard to the 
time may be neglected. 
The equations (l) now reduce to 
Q 2 
