ON THE STABILITY OF THE MOTION OF SATUKN's RINGS. 355 



If the angular velocity had been uniform, there would have been no sliding, 

 and the velocity would have been 



<ar + <a8r. 

 The sliding is therefore 



d< ~ 

 r -7- Sr, 

 dr 



and the friction on unit of area perpendicular to r is pr -y- . 



The loss of Energy, per unit of area, is the product of the sliding by the 

 friction, 



or, 



dr 



Sr in unit of time. 



The loss of Energy in a part of the Ring whose radius is r, breadth 

 Sr, and thickness c, is 



dear R 



27rrcu -j- or. 

 dr\ 



In the case before us it is Tr/i^Scr" 1 Sr. 



If the thickness of the ring is uniform between r = a and r = b, the whole 

 loss of Energy is 



in unit of time. 



Now half the vis viva of an elementary ring is 



irpcrBr 1*0)* = npcSBr, 



and this between the limits r a and r = b gives 



irpcS (a 6). 



The potential due to the attraction of S is twice this quantity with the 

 sign changed, so that 



E= irpcS(a b), 



dE /I 1 



and 



1 dE_ _ p !_ 

 E dt = " * p ab ' 



452 



