356 ON THE STABILITY OP THE MOTION OF SATURN'S RINGS. 



Now Professor Stokes finds */- = 0'0564 for water, 



' P 



and =0'116 for air, 



taking the unit of space one English inch, and the unit of time one second. 

 We may take a = 88,209 miles, and 6 = 77,636 for the ring A; and a = 75,845, 

 and 6 = 58,660 for the ring B. We may also take one year as the unit of 

 time. The quantity representing the ratio of the loss of energy in a year to 

 the whole energy is 



l_d_E = 1 . 



E dt "60,880,000,000,000 



1 

 811(1 39,540,000,000,000 fo1 



showing that the effect of internal friction in a ring of water moving with 

 steady motion is inappreciably small. It cannot be from this cause therefore 

 that any decay can take place in the motion of the ring, provided that no 

 waves arise to disturb the motion. 



Recapitulation of the Theory of the Motion of a Rigid Ring. 



The position of the ring relative to Saturn at any given instant is defined 

 by three variable quantities. 



1st. The distance between the centre of gravity of Saturn and the centre 

 of gravity of the ring. This distance we denote by r. 



2nd. The angle which the line r makes with a fixed line in the plane of 

 the motion of the ring. This angle is called 6. 



3rd. The angle between the line r and a line fixed with respect to the 

 ring so that it coincides with r when the ring is in its mean position. This is 

 the angle <j>. 



The values of these three quantities determine the position of the ring so 

 far as its motion in its own plane is concerned. They may be referred to as 

 the radius vector, longitude, and angle of libration of the ring. 



The forces which act between the ring and the planet depend entirely upon 

 their relative positions. The method adopted above consists in determining the 



