ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 337 



relative magnitude of these two opposing forces determines the destruction or 

 preservation of the ring. 



To calculate these effects we must begin with the statical problem: To 

 determine the forces arising from the given displacements of the ring. 



The longitudinal force arising from longitudinal displacements is that which 

 has most effect in determining the stability of the ring. In order to estimate its 

 limiting value we shall solve a problem of a simpler form. 



21. An infinite mass, originally of uniform density k, has its particles 

 displaced by a quantity parallel to the axis of x, so that g=Acosmx, to 

 determine the attraction on each particle due to this displacement. 



The density at any point will differ from the original density by a quantity 

 k', so that 



(73), 



k'= k~,~ = Akm sin mx .......................... (74). 



The potential at any point will be V+ V, where V is the original potential, 

 and V depends on the displacement only, so that 



ffiV d'V d*V 



Now V is a function of x only, and therefore, 



sinmaj .......................... (76), 



m v ' 



and the longitudinal force is found by differentiating V with respect to x. 



dV 

 X= j = ^TrkA cosmx = iirk ..................... (77). 



Now let us suppose this mass not of infinite extent, but of finite section 

 parallel to the plane of yz. This change amounts to cutting off all portions 

 of the mass beyond a certain boundary. Now the effect of the portion so cut 

 off upon the longitudinal force depends on the value of m. When m is large, 

 so that the wave-length is small, the effect of the external portion is insensible, 

 so that the longitudinal force due to short waves is not diminished by cutting 

 off a great portion of the mass. 



VOL. I. 43 



