ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 359 



The object of the investigation is to find the conditions under which this 

 compensation is possible. 



It is evident that when SRO becomes straight, there is still a difference 

 of angular velocities between the rotation of the ring and the revolution of 

 the centre of gravity, so that there will be an oscillation on the other side, 

 and the motion will proceed by alternate oscillations without limit. 



If we begin with r at its mean value, and <f> negative, then the rotation 

 of the ring will be retarded, r will be increased, the revolution of r will be 

 more retarded, and thus <f> will be reduced to zero. The next part of the 

 motion will reduce r to its mean value, and bring <f> to its greatest positive 

 value. Then r will diminish to its least value, and <f> will vanish. Lastly r 

 will return to the mean value, and <f> to the greatest negative value. 



It appears from the calculations, that there are, in general, two different 

 ways in which this kind of motion may take place, and that these may have 

 different periods, phases, and amplitudes. The mental exertion required in follow- 

 ing out the results of a combined motion of this kind, with all the variations of 

 force and velocity during a complete cycle, would be very great in proportion to 

 the additional knowledge we should derive from the exercise. 



The result of this theory of a rigid ring shows not only that a perfectly 

 uniform ring cannot revolve permanently about the planet, but that the irregu- 

 larity of a permanently revolving ring must be a very observable quantity, the 

 distance between the centre of the ring and the centre of gravity being between 

 8158 and '8279 of the radius. As there is no appearance about the rings 

 justifying a belief in so great an irregularity, the theory of the solidity of the 

 rings becomes very improbable. 



When we come to consider the additional difficulty of the tendency of the 

 fluid or loose parts of the ring to accumulate at the thicker parts, and thus 

 to destroy that nice adjustment of the load on which stability depends, we 

 have another powerful argument against solidity. 



And when we consider the immense size of the rings, and their comparative 

 thinness, the absurdity of treating them as rigid bodies becomes self-evident. 

 An iron ring of such a size would be not only plastic but semifluid under the 

 forces which it would experience, and we have no reason to believe these rings 

 to be artificially strengthened with any material unknown on this earth. 



