acted on by small Forces. 215 
Expanding this in a series of the form 4+ B cos. 6 + C cos. 20+ &e. 
and integrating every term through a circumference, we get 
ae 1 13 84a Re 12-3°.5. 7 | loa. he 
3n JSR +r 22.4 (Ro + rt 2°.4°.6.8 (R® +4 ryt 
1°.3°.577.-9. 11 me o4agke 4 1?.3°,5°.7°.9.11.13.15 | 256. a 
DAO .o LO a2 “(Re4r’)? 2° 47.6%.8°.10.12.14.16 (R? ery 
if we stop ata’. Let e, f, g, h, be the coefiicients of a’, a*, a°, a®; then 
putting r°-c’ for a’, dividing by 27, and neglecting the term in- 
dependent of c, we find that this contributes to «(c) the following 
ark 
terms = {—(e+2f+3g¢+4h) c+ (f+3g4+6h) c—(g+4h).c+h.c'}. 
(25.) Our expression will not be very erroneous, if for R we 
put the mean radius of the ring. Suppose then Boo. The last 
expression then becomes 
2) 09233 .c° + 04849. °,—.01768 .°. + .00295 ie 
= —.09 oo 9. a 7 “pat: 9 a 
and adding the term for centrifugal force, 
_ 09233 _ Do. 04849 | i td ponanee! F- 
00295 
1 n Tp nr? nr’ nro * 
6 
Substituting these values in the expression of (22), 
7 4476 .1487 c* 0568 c* .0107 a) 
A OFA =<) {5 - ‘ET? n n Tr nr. n -PS 
(26.) To exterminate & from the first term of this expression, 
let ¢ be the periodic time of Saturn’s 7th satellite, s its distance; 
the motion of this satellite is nearly the same as if all the matter of 
Saturn and his ring were yg into aries center: hence 
rk (it: er hencez7s = oie s2f 5 1+- ~); 
ra 
3 2 
SEP a3 PCH) = A= 8 oe: (42) = (142) x 419. 
