SATELLITES UPON THE FORM OF SATURN’S RING. 
123 
As a solution we now take 
Ax — Aq +22A^ s 0 rs + , 
Xx = X^+22X^A, s + • • • , 
with the same restrictions as before. 
Substitute in (34) and equate to zero the terms involving no 0 but 0 JiO x and 0 5iO \ 
Then, for all values of X from 1 to n, 
—m 2 Aq — 2/a?nXo + 0* 0 -A-o + A2 Aq G /X x + +—Xq A |-0 3i , 
/X fX 
—m _ Xf + 2/ctmAo + 05 ,o Xo + /c 2 XAq G^ x+^ XXf, J^x = 0. 
• (35) 
These 2 n equations can be solved by the usual processes to give the values of the 
constants A* and X*. It is not necessary for us to work out the results in detail, it 
is sufficient to note that the determinant of the left-hand members will appear as the 
denominator in each case. The determinant is the following : 
m 2 + 0} 
2fi 
K VJ 2,1 ) 
K ' t 3, 1 , • 
2/U 
. . , K VjT ?l> ! 
— 2/ctm , 
AT,. i , 
AT 3i j, 
2 T 
• • , K ° n, 1 
2 Kim , 
2 rd / 
K Lx 2 5 ! . 
2 
* ^3,15 • 
2 pi / 
... /cu,^ , 
— m 2 +0j jO , 
AJ',,! , 
AJFi, 
2 T/ 
• • ) * 0 n, 1 
2GT 
K VJf] 2 5 
—m 2 + 0f iO , 
2n 
K VT 3)2 , . 
x- 2 G 
• • 5 ^ 71, 2 5 
— 2/am , 
aj 3i3 , 
2 T 
• • , K n, 2 
AG'j, 2 > 
2 Kim , 
2 rif 
l< Lr 3> 2, • 
2 pi/ 
• • ? % ^ 2 5 
+J / 1.2 ? 
-m 2 +0? o , 
/c 2 J' 3>2 , 
2 T' 
... , K J „ j2 
2 G 
K V - 7 1 , n i 
AG 2 , n , 
AG 3> „, 
.. , — m 2 + 0 "; o; 
2 1 
K 1 , n 5 
K Js, n J 
AJg, 
..., — 2 /am 
t 
*g 1iB , 
2 fir 
K ' jr 2 , n ) 
2 p 
K ' J 3, n • • 
.. , 2 /am ; 
2T/ 
* l,n 5 
2 7' 
* 5 2 . H , 
AJ'g, „, 
.. ,-7H*+ 0 ? o 
This determinant corresponds to the denominators in expressions (24). When it 
vanishes or becomes small, it is clear, as before, that the terms of the solution tend to 
become large, and instability follows. 
In estimating the values of F, G, IT and J, which appear in the above determinant, 
it is to be noted that mj M is exceedingly small for all values of /x. But the quantities 
in which it appears may be large by virtue of the small denominators which are 
involved. In the expression for F A , the term 3/sin (/u — X) i-fn may be neglected in 
comparison with the first term for large values of n. Also, A — . .. , -- — r will 
1 h „ M 8 sin t r/n 
lie between zero and ~ X —t-—- r — r- since all the signs are positive, if 777 is the 
M 8sm 3 (fx-\) ir/n s F 
greatest value of m IM appearing in the ring. Hence F A lies between zero- and 
0'0096 n 3 m/M in value. 
In the same way the value of G^x will arise almost wholly from the first term. 
The largest value it may have will be 771 A/StFM or O'OOIrdm/M . E A , IIx, J M ,x, Efi, Ffi and 
G^ x are seen to be one order lower in the reciprocal of sin (/* — X) 7 r/n and therefore 
may be neglected. Hfi has the limit — 0'0192n 3 m/M, and A the limit — 0'008rFm/M. 
