SATELLITES UPON THE FORM OF SATURN’S RING. 
Ill 
We now substitute the assumed values for A, X, c, 0 M) in equations (7) and equate 
to zero the coefficients of each term in G r s , 0 ns , Q p q , &c. It will be found that the 
relations (8) are satisfied identically. Two conditions further must be imposed in 
order that all the unknown coefficients may be determined. These are : 
(i) The term cos (n<p — r) must not appear in the series for A ; 
(ii) The solutions for A and X must be purely periodic with period 2tt. 
The condition (i) amounts to a definition of r, and condition (ii) secures that no 
part of the exponent shall appear in the periodic series. Further, these conditions 
determine uniquely the undetermined coefficients in the series for 0 1>o and c. The 
work from this point is purely mechanical though long. The following sample 
sufficiently indicates its character. 
On equating to zero the terms involving 0 1-r we find 
2c Ur nA 0 cos (n<p — r) + A! f ^ r — 2/cc Jir X 0 cos (n<p— t) 
— 2/cX'j, r + a h „Aj. r + r A u sin (hep — r) + A 0 cos r<p sin (rup—r) 
— 2c h r nX„ sin ( n<p — t) + X \ r + 2 kc u r A„ sin (n<p —r) + 2k A' h + ©g^X^ r 
In the case when r is not 2 n or n, it is clear that 
o y (12) 
- 0 
"1, r 
and 
«i , r = 0. 
Equation (12) then reduces to 
A" hr — 2«-X / lir + a li0 A lir + ^A 0 '[sin {(w+r)f-T}+ sin {(n—r) <p — r}] = 0 
X /r i,r+ 2/vA / 1 , r + 0 5tO X lr 
Solving in the usual way we find 
= 0. 
■ (13) 
■A-l.r — 
X 1>r = 
A 0 n 2 {(n + r) 2 — 0 5- o} sin | ( n + r) A,/n 2 {(m —?■)-’ — Or.,,} sin {(n-r) 0-t} 
2r (2 n + r) («i,<A,o— n 2 (■n + r ) 2 ) 2r (2 n—r) {eq „0 5 , 0 —n 2 (n-r) 2 } 
A 0 ?i 2 k (n + r) cos {(n + r) <p — r } A„nk (n — r) cos {(n—r) <p — r }' 
r (2n + r) {a h0 Q 5t0 —n 2 (n+r) 2 } r(2 n-r) {a liO 0 5fO — n 2 (n — r) 2 } 
In the special case where r = n, we have 
c Un = 0, n Ur , = 0, 
and 
A"!.,, —2/r,X / liB + a li0 Aj ill .+ 2 -A 0 [sin (2 rnp—r) — suit} = 0, 
X ,/ ],fl + 2/fA , li „. + 0 5>o X ljW = 0. 
From which 
a = A„ (4n 2 -©,,,„) sin (2 n<p-r) A 0 
n l a -A Cl \ ' ’ 
Y — 2 
Ai,« — 3 
6 (4n 4 —a liO 0 5 , o ) 
2 «- nA,) cos (2 U({> — t) 
4n 4 — a 1)0 © 5 ,i 
L 0 
2«j, o 
. 0 
n 2 
(14) 
and 
