ON THE STABILITY OF THE MOTION OP SATUBN's RINGS. 309 



7* 



Assuming - = A sin vt, and ^ Q cos (vt ), 



Of 



{-v* + j,<a s (5 - g)} A cos vt + hot 3 A sin vt +/co 3 (3 - g) Qcoa (vt -0) + ^fhv'vQ sin (vt - ft). 

 Equating vt to 0, and to - , we get the equations 



{S-ivo? (5 -g)} A =fo?Q {(3 -g) o> cos ft - %hv sin ft}, 

 - ho? A =fo>*Q {(3 -g)ownft + $hv cos ft}, 

 from which to determine Q and ft. 



In all cases in which the mass is disposed symmetrically about the diameter 

 through the centre of gravity, h = and the equations may be greatly simplified. 



Let & 1 = P cos (vt a), then the second equation becomes 

 (i/ 3 + frS (3 + g)} A sin vt = 2Pfo>v sin (vt - a), 



whence a = 0, P = * + *?(*+9) A ........................ (38) . 



2jotv 



The first equation becomes 



4.4 o)v cos i>< - 2Pfv* cos i/ + Q/" (3 -g) o? cos (vf - ft) = 0, 



whence /f=0, ^ = 11 ^ ..................... (39)- 



In the numerical example in which a heavy particle was fixed to the cir- 

 cumference of the ring, we have, when f='82, 



* f'5916 Pf3'21 Qf- 1-229 

 (a~\'3076' A~ \_572' A\- 797' 



so that if we put o> = = the mean anomaly, 



-/8) ..................... (40), 



& 



6, = 3-21 A cos (-5916 0,-a) + 5725 cos (-3076^-^).... ........ (41), 



^= -1-229 ^4 cos (-5916^ -a) -57975 cos ('30760,- ft) ... (42). 



These three equations serve to determine r,, 1 and fa when the original 

 motion Js given. They contain four arbitrary constants A, B, a, ft. Now since 



