326 ON THE STABILITY OF THE MOTION OF SATURN'S EINOS. 



15. We come next to consider the effect of an external disturbing force, 

 due either to the irregularities of the planet, the attraction of satellites, or 

 the motion of waves in other rings. 



All disturbing forces of this kind may be expressed in series of which the 

 general term is 



A cos^vt+ms + a), 



where v is an angular velocity and m a whole number. 



Let Pcos(ms + vt+p) be the central part of the force, acting inwards, and 

 Q sin (ms + vt + q) the tangential part, acting forwards. Let p = A cos (ms + rt + a) 

 and o- = -Bsin (ms + vt + ft), be the terms of p and <r which depend on the 

 external disturbing force. These will simply be added to the terms depending 

 on the original disturbance which we have already investigated, so that the 

 complete expressions for p and a- will be as general as before. In consequence 

 of the additional forces and displacements, we must add to equations (16) and 

 (17), respectively, the following terms: 



{3<w' - - R (2K+ L)+tf}A cos (ms +vt + a) 



+ (2<av + -RM)BcoB(ms + vt + @)-Pcos (ms + vt+p) = ...... (49). 



(2a>v + - RM) A sin (ms + vt + a) 



+ (v' + - RN) E sin (ms + vt + /3) + Q Bm(ms + vt + q) = ......... (50). 



Making ms + vt = in the first equation and - in the second, 



8 



co8a + (2<av + -RM)BcoBp-PcoBp = Q ...... (51). 



fii 



-RM)Acoaa+(v > + -RN)BcoBp + QcoBq = ...... (52). 



Then if we put 

 U' = tf-{<S+- 



-4 -RMv 



(53), 



