SATELLITES UPON THE FORM OF SATURN’S RING. 
1 17 
We shall form the solution by taking only F'"*’ in the term cos r</>. Changing the 
sign of r will then give the other part. Assuming that A, r and X lir vary as e^, we 
have 
A lr ( —m 2 —2mr—r 2 + 0 ]O ) —2/a (m + r) X 1<r = —|-A 0 , 
A ltr (m + r) 2/a + X 1>r (— m 2 — 2tnr — r 2 + 0 5>o ) = 0. 
From these 
and 
-|A 0 {0 5 ,o- (m + r) 2 } -r-[{0 JiO - (m + r) 2 } {0 5>o - (m + r) 2 }-4/c 2 (m + r) 2 ], 
+iK ■ 2/a (m + r) -i-[{0 1)O — (m + r) 2 } {0 5)O - (m + r) 2 } - 4/c 2 (m + r) 2 ]. 
On determining the corresponding values for the term and combining the two, 
we have 
A,, r = -|A o e t, ' </> { 05 . o - (m + r) 2 } + [{0 liO — (m + r) 2 } {0 5 , o - (m + r) 2 } -4/c 2 (m + r) 2 ] 
—iA 0 e~{0 5 ,o—(m—r) 2 } ^[{0i, o -(w->\) 3 }{ 0 o,o- (m-r) 2 }-4/c 3 (m-r) 2 ], 
X lir = A 0 e‘ r Vi (m + r) 4- [{0 liO — (m + r) 2 } {0 5 , o — (m + r) 2 } — 4/c 2 (m + r) 2 ] j 
+ A 0 e- 1, A« (m-r) 4- [{0 1>o - (m-r) 2 } {0 5 , o - (m-r) 2 } -4/c 2 (m-r) 2 ]. J 
(23) 
Expression (23) shows that A 1>r and X lir are factored by A 0 , which is a multiple of 
0 8 _ m . Now the terms in the expansions of A and X that we are seeking are A ] r 0i r 
and X l r 0i r . Since both of these involve the product 0 3> m 0 lr , if is clear that they 
may be neglected in comparison with the values of A 0 and X 0 . 
We have further to determine the parts of A and X arising from a term — 0 6 , 
2 1 
in the right-hand member of the second equations (4). These can be written down 
from the results already given, and are 
X 0 = ^ 0 6 , m (0 5il) -m 2 )-+ {(0,., -m 2 ) (0 5 , o -m 2 )-4/c 2 m 2 }, 
A 0 = /cm0 6iM + {(0i, o —m 2 ) (0 5 , o -m 2 )-4/c 2 m 2 }. 
Hence to the degree of accuracy we are using, wa may summarise the results as : 
P = X [0 3jm (0 5jO — m 2 ) cos m<j> + 2/cm0 6i m cos m^] ■+[(©!, 0 —m 2 ) (0 5iO —m 2 ) — 4/c 2 m 2 ], 
m 
o- = 2 [2/cm0 3i m sin m0 + 0 6 , m (0 5 , o -m 2 ) sin m<j>] +-[(0 liO -m 2 ) (0 5 ,o-m a )-4K ? m 2 ]. 
Ill 
Except when the denominators are small, it is seen that, owing to the very small 
factors 0 3 m and 0 6 m , the values of p and a derived from the above equations are very 
small. 
VOL. CCXXII.-A. S 
