] (89); 



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



Calculating the potentials of a pair of such surfaces at distances +c and c 

 from the plane of xy, and calling V the sum of their potentials, we have for 

 the space between these planes 



F,' = ZirkcA sin max-"" ("- + e""*) 

 beyond them V t '-2irkcA sin mxe* au (t me + 



the upper or lower sign of the index being taken according as z is positive or 

 negative. 



These potentials must be added to those formerly obtained, to get the 

 potential at any point after displacement. 



We have next to calculate the pressure of the fluid at any point, on the 

 supposition that the imaginary planes protect each slice of the fluid from the 

 pressure of the adjacent slices, so that it is in equilibrium under the action of 

 the forces of attraction, and the pressure of these planes on each side. Now 

 in a fluid of density k, in equilibrium under forces whose potential is V, we 



have always 



dp , 

 = 



so that if we know that the value of p is p t where that of V is F , then at 

 any other point 



Now, at the free surface of the fluid, jp = 0, and the distance from the 

 free surface of the disturbed fluid to the plane of the original surface is , a 

 small quantity. The attraction which acts on this stratum of fluid is, in the 

 first place, that of the undisturbed stratum, and this is equal to 4-irkc, towards 

 that stratum. The pressure due to this cause at the level of the original 

 surface will be &irk?ct>, and the pressure arising from the attractive forces due 

 to the displacements upon this thin layer of fluid, will be small quantities of 

 the second order, which we neglect. We thus find the pressure when z=c to be, 



p, = lirJc'c'mA sin mx. 

 The potential of the undisturbed mass when z = c is 



and the potential of the disturbance itself for the same value of z, is 



F/ = 2nkcA sin mx (I + e-""). 



