SATELLITES UPON THE FORM OF SATURN’S RING. 
121 
chosen. It will be seen that k/u increases slowly from unity as increases from 
zero, until jL, reaches the value 0‘039. At this point the curve turns back and rises 
rapidly to an asymptote at i/L s = 0. 
In the case when expression (30) is exactly zero, it is seen from elementary principles 
that the independent variable 0 would appear explicitly. With passage of time, 
therefore, p and <t would increase linearly in magnitude and there would be complete 
departure of the particles from the vicinity of r = a. 
§ 5. Case where the Particles forming the Ring are of Unequal Masses. 
The previous equations (2) were reduced to the form (3) on the supposition that all 
the masses m K were of the same value m, = j/M. We now proceed to the modifica¬ 
tions introduced when these masses are all distinct in value. 
Equations (2) with the same reductions as before, but maintaining the separate 
values m x , become : 
// o ' 2 eu / J d i i 
a — 2 /ot a — k k 3 " J — 
P A 
v —j (|f> 0 + ... + b, cos icj>+ ...) — cos (/>, — k 2 E 
a a “ J 
+ 
rl 2 
3 A + A'V {if + ...+f cos i<p + ...) - ^ 
p K + 2/c 2 G ja> xp,j_ 
+ [/tY*V -j-[b 1 sin 0 + ... + ib l sin i<p+ ...) — kk'\' sin <p — /c 2 H x ] <r K 
clot 
+ 2/c 2 J, 
/X, A^”/X 5 
(t ,, a +2 Kp\ — A-V' /s (6 X sin 0 + ... +ib i sm «0+ ...) — vff' 3 sin 0 + wc 2 Eh 
' 2 
V K K 
... +i C ~ i sin i<p+ ...) + iYV(... + ib { sin ?0 ...) 
-PF\ 
+ [ — v kk ^ (... + i 2 b, cos i(p T ...) + vkk 2 cos 0—/AHA] v x 
+ 2/e 2 J' 
*^"A 1"" 2 / c ' G / x , AP/x 
/x, A^/x 
(31) 
In these equations 
E 2 WR /. 
M [4 sin (/x—X) 7i-/w 
+ cos (/x — X) 2~/u f ^ 
U 1 _ y m v- 
J- A - 
i M [8 sin 3 {/a— X) 7r/n sin (/x—X) 7r/n 
G ^ ( cos (/x —X) 2 7r jn 
+ 
M [8 Sill 3 (/x — X) 7r//4 Sill (/x — X) irjn 
+ 2 cos (« — X) 27rf 5 
