SATELLITES UPON THE FORM OF SATURN’S RING. 
107 
^ m 
COS 
ju — X) 2Trjn {cos s 
p — 
X) 
27 r/n + t sin s 
p — X) 2nfn} — 1 
7 8a 3 
sin 3 
.P 
-X) 
n/n 
:« 3 V 
+ $ — 
7 8a 3 
— X) n/n j si 
cos s(p—X)27r/n + i sin s( 
p-X 
•27 r/n +1 
sin (/x — X) 77 / 
n 2 5 (/x—X) 7r/n cos 2 (p — X 
'n 
1 7r/n] 
m T 
=- 
a 3 
X) 7 r/n 4 sin 3 ( 
7 ~/n 
i 
cos (p — X) 
7 r/ll 
sin 2 1 
(p-X' 
) n/n 
——r—4—{cos s (p — X) nln + L sin s (p — X) irln— 1 
— X) it fn 
COS 1 
[p—x) 
1 n/n sin 5 ( 
,/x-X) 
• 7r /n 
sin 2 1 
[p— x) 
1 7T j 
In 
= £ — M • 
a 
8 a * 
V Wl COS (p — X) 7 rln r / _ \ / • / . \ / oi TO ii- 
? to’ sin’ („-x) ,/n {C0S 8 (,< “ X) 7r/,l + < Sln s ( "" x) 7r/ ’ l + S ' = ' «’ M - ; 
cos ( 
p—X)2n/n 3 sin 2 ( 
p- 
sin 3 
{p —x 
)n/ 
hi 4 sin 5 
(p 
{cos 6- (p — X)2-n-/n + i sin s (/x —X) 2 ~/n— 1} 
a 2 n. 
[ t sin 2 s \ 
(p —x) 
1 7r/n . COS 2 1 
[p— x) 
• 7 r/n 
l 2 
sin 3 1 
U—x) 
I./ 
n 
+ 4 
sin (p — A) t r/n 
N„ 
a 
The quantities K, L s , M s , N s can readily be found by direct summation when n, the 
number of particles, and s are known. 
In re-writing the differential equations ( 2 ), we may now omit the suffixes of panda-. 
Change the independent variable from t to cf> — (a/ —«) t + e— e— X . 2n/n. Also put 
m/M = v , m'/M = v, w'/co = k, (k — l ) _1 = k, and, to secure homogeneity, replace 
p by ap. Let us further assume that o 2 a A = M, and co' 2 a' 3 = M (the latter holds very 
approximately when m' describes a circle), so that we have a/a' = («'/«)'. 
The differential equations then become 
“jA — 2k -y- = KK A' — (T& 0 + ... + b { COS 1(f) + . . . ) — K KV — KK A' COS (p 
d(p “ <X0 pa 
+ 
+ 
3k" + /c“/c “v —— g ('^'^0 T ... U 6, COS i<f) T ...) — /caL s 
la" 
3 ~ 2 
2/%/ 3 
/c /c -v — (4 sin 0 + ... + ?’4 sin i<f> + ... — sin <p) + /c 2 wM s 
oa 
and 
+ 2 k~ — vkk u (b 1 sin (p + ... + ibi sin icp + ...) — vkk 1 ' 1 ' 3 sin <f> 
ci~<p a<p 
V, ! 
L 
r 
+ 
' 2 /‘/s 
VKK 
... +i~ sin i<p+ ...)— vkk 2,3 (... +ib { sin i<p+...) 
OCX. 
— mAM. 
— vk 2 k"^ ( ... + l 2 bi COS i(f) + ...) + vkk" COS (p + 1'K 2 N S 
O- . 
(3) 
