ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 371 



magnitude without limit, and the other continually diminishing, so that one 

 of the waves in each ring will increase in violence till it has thrown the ring 

 into a state of confusion. 



There are four cases in which this may happen. The first wave of the 

 outer ring may conspire with the second or the fourth of the inner ring, the 

 second of the outer with the third of the inner, or the third of the outer with 

 the fourth of the inner. That two rings may revolve permanently, their distances 

 must be arranged so that none of these conspiracies may arise between odd 

 and even waves, whatever be the value of m. The number of conditions to 

 be fulfilled is therefore very great, especially when the rings are near together 

 and have nearly the same angular velocity, because then there are a greater 

 number of dangerous values of m to be provided for. 



In the case of a large number of concentric rings, the stability of each pair 

 must be investigated separately, and if in the case of any two, whether con- 

 secutive rings or not, there are a pair of conspiring waves, those two rings will 

 be agitated more and more, till waves of that kind are rendered impossible by 

 the breaking up of those rings into some different arrangement. The presence 

 of the other rings cannot prevent the mutual destruction of any pair which 

 bear such relations to each other. 



It appears, therefore, that in a system of many concentric rings there will 

 be continually new cases of mutual interference between different pairs of rings. 

 The forces which excite these disturbances being very small, they will be slow 

 of growth, and it is possible that by the irregularities of each of the rings the 

 waves may be so broken and confused (see 19), as to be incapable of mounting 

 up to the height at which they would begin to destroy the arrangement of the 

 ring. In this way it may be conceived to be possible that the gradual dis- 

 arrangement of the system may be retarded or indefinitely postponed. 



But supposing that these waves mount up so as to produce collisions among 

 the particles, then we may deduce the result upon the system from general 

 dynamical principles. There will be a tendency among the exterior rings to 

 remove further from the planet, and among the interior rings to approach the 

 planet, and this either by the extreme interior and exterior rings diverging 

 from each other, or by intermediate parts of the system moving away from the 

 mean ring. If the interior rings are observed to approach the planet, while it 



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