SATELLITES UPON THE FORM OF SATURN’S RING. 
113 
Terms involving argument 0, r , where r is not n nor 2 n: 
«1 ,r = 0, C Ur = 0, 
_ _ A o n 2 {(w + r) 2 -0,,„} sin \(n + r) c /j- T } A„w 2 {(n-r) a -0 5iO } sin {(n-r) >/,--} 
2r(2n + r ) («i. o 0ft.o-»*(» + *’)*) 2r(2n—r) \a U0 O 5 , 0 -n 2 (n-r) 2 } 
A 0 71 2 k {n + r) cos {(n + r) </> — t} A u n\ (n — r) cos {(n —r) </> —t } 
X 1>r + - 
r(2n+r) {a u0 O 5!0 -n 3 (n + r) 2 } r (2n-r) {« j!O 0 5iO -n 2 (^— 
Terms involving argument 0 ljfl : 
«l.n = 0, C, „ = 0, 
A] , n 
X,„, 
A„ (4n 2 —0 5 , o ) sin (2 n<p — r) A () sin r 
6 (4n 4 -«, iO 05,o) 2a 1>0 
2/mA„ cos ( 271(f) — t) 
Terms involving argument 0i, 2 «-' 
\ cos 2t, 
], 2 /? 
C l,2» 
_ n (0 fl , o — n 2 ) sin 2 t 
4 (®i, o0,5,o 'A ) 
A _ ( 05 .Q~ 9^ 2 ) Ap sin (3n0 — r ) 
l ’ 2 " 16 ( 0 , 000 , 0 - 9 ;^) 
Y 3n/cA 0 cos (3n^ —t) (n 3 —a L0 ) (0 5 , o — n 2 ) sin 2 tA 0 sin (n0 —r ) 
^ ~ ~~ 8 (« liO 0 5 .o-9n 4 ) _ ’ 8 kU («,, o 05 ,o-9w 4 ) 
Terms involving argument 0 2 , rJ where r is not n nor 2n: 
= 0, Co r = 0, 
A, r = 
W 2 •! 0 fl .o— 1 
(n + r) 2 } X 0 sin {( 
n + r) 
<t>~ T| 
r + n 3 {0 5 ,q 
4/0 
-1 
^~ r / 
OX„ s :,i{( 
( p-r} 
4r (n + r) 
00.5,0 tt? 
(n+r 
) 2 } 
!*- 
- c) {«, 005 , 0 “ n 3 
(n-r 
Y} 
Y _ X 0 n 2 /c(n + r) cos {(n + r) L X„ n\- (n- r ) cos {( n-r) «/> — r j- 
2,r — 2r(n + r){a lt0 Q-„' Q —n a (n + i') i } ' 2r (n- r) {« 1 , o 0 5iO — n 3 (n—r) 3 } 
Terms mvolving argument 0 2 ,„; 
«2,„ = 9,. Co, „ = 0, 
Ft 
i, o 
_ x„ (0 5 ,o 4;i 2 ) sin (2 iup-r) X, sin 2r 
2 ’" 6Ko0 s ,o-4n 4 ) 
Y _ 2X n rn cos (2n<ft — r) 
2 ’ 5i_ “ 3(a 1 ,A, 0 -4w 4 ) 
