508 JAMES CLERK MAXWELL. 



4352 times that of the ring in order to secure stability for 

 all displacements. " If this condition be not fulfilled, . i . 

 then, although the motion depending upon long undulations 

 may remain stable, the short undulations will increase in 

 amplitude till some of the neighbouring satellites are brought 

 into collision." The satellites of such a ring, when the con- 

 dition of stability is fulfilled, admit of four vibrations in 

 different periods in ellipses about their mean positions, and 

 these vibrations will be transmitted as waves with different 

 velocities round the ring, so that the form of the ring at 

 any instant resembles " that of a string of beads forming a 

 re-entering curve, nearly circular, but with a small variation 

 of distance from the centre," forming a number of " regular 

 curves of transverse displacement at regular intervals round 

 the circle. Besides these, there are waves of condensation 

 and rarefaction, the effect of longitudinal displacement." 

 Any external disturbance, such as a satellite or wave in 

 another ring, will produce a forced wave in the ring, and if 

 the angular velocity of the external disturbing cause around 

 the ring coincide, or nearly coincide, with the velocity of one 

 of the free waves of the ring, the amplitude of the forced 

 wave will increase indefinitely until the ring is destroyed ; 

 but if this condition is not fulfilled, the forced wave will 

 accompany the disturbing cause around the ring as the tide 

 follows the moon. 



The effect of one ring upon another is to produce in it 

 a series of forced waves travelling with the same angular 

 velocity as the free waves in the disturbing ring. Hence in 

 a system of two rings " there will be eight waves in each 

 ring, and the corresponding waves in the two rings will act 

 and react on each other, so that, strictly speaking, every one 

 of the waves will be in some measure a forced wave, although 

 the system of eight waves will be the free motion of the two 

 rings taken together." 



The dynamical stability of a ring of satellites is ex- 

 plained by the consideration that when one of the satellites, 

 in consequence of its oscillations, moves with more than its 

 mean velocity, and thus gets in front of its proper position, 



