ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 341 



the upper sign being understood to refer to the boundary at distance 4-c, and 

 the lower to the boundary at distance c from the origin. 



Having ascertained the potential of the undisturbed stratum, we find that 

 of the disturbance by calculating the effect of a stratum of density Jc and 

 thickness , spread over each surface according to the law of thickness already 

 found. By supposing the coefficient A small enough, (as we may do in calcu- 

 lating the displacements on which stability depends), we may diminish the 

 absolute thickness indefinitely, and reduce the case to that of a mere " super- 

 ficial density," such as is treated of in the theory of electricity. We have here, 

 too, to regard some parts as of negative density ; but we must recollect that we 

 are dealing with the difference between a disturbed and an undisturbed system, 

 which may be positive or negative, though no real mass can be negative. 



Let us for an instant conceive only one of these surfaces to exist, and let 

 us transfer the origin to it. Then the law of thickness is 



(83), 



and we know that the normal component of attraction at the surface is the 

 same as if the thickness had been uniform throughout, so that 



dV 

 ^ 



on the positive side of the surface. 

 Also, the solution of the equation 



d'V _ 



dx> + dz 1 ' ' 



consists of a series of terms of the form Ce'* sin ix. 



Of these the only one with which we have to do is that in which i= m. 

 Applying the condition as to the normal force at the surface, we get 



7= ZirJccfT^A sin mx ........................... (87), 



for the potential on the positive side of the surface, and 



F=2dkee"Msinna! .............................. (88), 



on the negative side. 



