ON THE STABILITY OF THE MOTION OF SATUEN's RINGS. 329 



o_ 

 relatively to each particle in the time - , and travelling as a wave among 



v 

 the particles with an angular velocity , the angular velocity relative to fixed 



Ilif 



qj 



space being of course 01 - . The whole disturbing force may be split up into 



In* 



terms of this kind. 



17. Each of these elementary disturbances will produce its own wave in 

 the ring, independent of those which belong to the ring itself. This new wave, 

 due to external disturbance, and following different laws from the natural waves 

 of the ring, is called the forced wave. The angular velocity of the forced wave 

 is the same as that of the disturbing force, and its maxima and minima coin- 

 cide with those of the force, but the extent of the disturbance and its direction 

 depend on the comparative velocities of the forced wave and the four natural 

 waves. 



When the velocity of the forced wave lies between the velocities of the 

 two middle free waves, or is greater than that of the swiftest, or less than 

 that of the slowest, then the radial displacement due to a radial disturbing 

 force is in the same direction as the force, but the tangential displacement 

 due to a tangential disturbing force is in the opposite direction to the force. 



The radial force therefore in this case produces a positive forced wave, and 

 the tangential force a negative forced ivave. 



When the velocity of the forced wave is either between the velocities of 

 the first and second free waves, or between those of the third and fourth, then 

 the radial disturbance produces a forced wave in the contrary direction to that 

 in which it acts, or a negative wave, and the tangential force produces a positive 

 wave. 



The coefficient of the forced wave changes sign whenever its velocity passes 

 through the value of any of the velocities of the free waves, but it does so 

 by becoming infinite, and not by vanishing, so that when the angular velocity 

 very nearly coincides with that of a free wave, the forced wave becomes very 

 great, and if the velocity of the disturbing force were made exactly equal to 

 that of a free wave, the coefficient of the forced wave would become infinite. 

 In such a case we should have to readjust our approximations, and to find 

 whether such a coincidence might involve a physical impossibility. 



VOL. I. 42 



