SATELLITES UPON THE FORM OF SATURN’S RING. 
105 
2 
H- 
8 2 F 
m! 0/7 - 7 \ 
— {b 1 sm <p+ ... + ibi sin i<p + ...) 
r ,2 r x 0 a 
- sin *>•••) 
rr K 
• 2m„ 
+ 2m„ 
3ry sin In-iy cos (O M -0 A )} 
I) 6 
J->A 
Pa 
sin (0„ —6> a ) _ 3?y si n ( fl M -0 A ) {r M - r A cos (0 M -flx)} 
T) 3 
J-'A/i 
Lb 5 
A/x 
+ ~ sin (0^—0 X ) 
Pm> 
2<t, 
0 2 F 
11 ?’a00 m 8^a 
m 
r r 
to i *27 * \ 
— ( ... JO, COS Up ...) + —f - 2 COS (ji 
(T\ 
+ 
r M cos (0 M —0 A ) _ Sr 2 r x sin 2 (0 m - 0 a) 
n 5 
J -'A/x 
- — COS (0,,-0a) 
(ay— 0 a). 
In the summations of the right-hand members, /x takes all integral values from 
1 to n, except ^ = A. 
The zero values of these derivatives are obtained by putting 
v' — a', 
r a = a, 
Whence 
where <p now is 
and 
Then 
(aF/ 3 n), = 
9' = o/t + e', 6 X = wt +e + \2iTr/fl. 
A a 2 = a' 2 + a 2 —2aa' cos <p 
(t o' — ft)) t-\-e —e — \2irjn ; 
D Am = 2a sin (,u — A) 7 r/n. 
\ _ M ml I-ydb 
db . 
m 
d -■ + ^b^ + - + sr 00S!>+ -J"^ 00 ^ 
— 
--—:- 7 -r- 7 — 4- 5 COS (yU —A)27r/n 
4« 2 sin (/x —A) 7r/n a 2 v 7 7 
8 2 F \ 
dr x drj 0 
"9 
(M + Wa) , to' 8 2 /1? i7 • \ 
-^ ^2(2^0 4 • • • +0 ; - cos 20+ ...) 
ar 
v J_ 1 3 V 
' x ,'>77, < - — - k 
n M [8a 3 sin 3 (/* — A) Trjn 8a 3 sin (yu —A) tt/wJ _ 
PA 
+ 
COS (jit — A) 2irj 
n 
+ 
_8a 3 sin 3 (/m — X) 7r/n 8a 3 sin (yu — A) 7r/ 
+ 
2 cos (/x —A) 27 r/n 
a° 
Pm> 
