340 



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



According to this law of displacement, certain alterations will take place in 

 the distances between consecutive planes ; but since the fluid is incompressible, 

 and of indefinite extent in the direction of y, the change of dimension must 

 occur in the direction of z. The original thickness of the stratum was 2c. Let 

 its thickness at any point after displacement be 2c + 2, then we must have 



= 2c (82), 



or 



= c -~ = 



sinmx ........................ (83). 



Let us assume that the increase of thickness 2 is due to an increase of 

 at each surface ; this is necessary for the equilibrium of the fluid between the 

 imaginary planes. 



We have now produced artificially, by means of these planes, a system of 



n 



waves of longitudinal displacement whose length is and amplitude A ; and 





we have found that this has produced a system of waves of normal displace- 

 ment on each surface, having the same length, with a height =cmA. 



In order to determine the forces arising from these displacements, we must, 

 in the first place, determine the potential function at any point of space, and 

 this depends partly on the state of the fluid before displacement, and partly 

 on the displacement itself We have, in all cases 



Within the fluid, p = k; beyond it, p = 0. 

 Before displacement, the equation is reduced to 



Instead of assuming F=0 at infinity, we shall assume F=0 at the origin, 

 and since in this case all is symmetrical, we have 



within the fluid 



at the bounding planes 



beyond them 



dV 



^ = 2irh? ; - ' = fakz 



dV 

 V Ztrkc' ; -y- = T 



dV 



F, = 2irkc ( T 2z c) ; -7- = T 



(86); 



