ON THE STABILITY OF THE MOTION OF SATUEN's RINGS. 343 



So that we find the general value of p at any other point to be 



p = 2-rrJf (a* - z 2 ) + ZvfccA sin mx {2cm - 1 - C" + 1 (f + c'" 12 )} . . . (90). 



This expression gives the pressure of the fluid at any point, as depending 

 on the state of constraint produced by the displacement of the imaginary planes. 

 The accelerating effect of these pressures on any particle, if it were allowed to 

 move parallel to x, instead of being confined by the planes, would be 



_ldp 

 ~k dx' 



The accelerating effect of the attractions in the same direction is 



dV 



dx' / 



so that the whole acceleration parallel to a? is 



X= -ZirkmcA cosmz (2mc-e" smc - 1) .................. (91). 



It is to be observed, that this quantity is independent of z, so that every 

 particle in the slice, by the combined effect of pressure and attraction, is urged 

 with the same force, and, if the imaginary planes were removed, each slice 

 would move parallel to itself without distortion, as long as the absolute dis- 

 placements remained small. We have now to consider the direction of the 

 resultant force X, and its changes of magnitude. 



We must remember that the original displacement is A cos mx, if therefore 

 (2mc c" 2 1) be positive, X will be opposed to the displacement, and the 

 equilibrium will be stable, whereas if that quantity be negative, X will actr 

 along with the displacement and increase it, and so constitute an unstable 

 condition. 



It may be seen that large values of me give positive results and small 

 ones negative. The sign changes when 



2,mc = 1-147 .................................... (92), 



which corresponds to a wave-length 



o_ 



X = 2c = 20(5-471) ........................... (93). 



The length of the complete wave in the critical case is 5 '471 times the 

 thickness of the stratum. Waves shorter than this are stable, longer waves 

 are unstable. 



