348 ON THE STABILITY OF THE MOTION OF SATURN'S 



4th. By the tangential displacement of the second ring a tangential force 

 arises, depending on the relation between the length of the waves and the 

 distance between the rings. 



a a [ + xsinpx . 



If we make m - , =p, and TO I -f-j ax = U, 

 a, J - ( 1 + ar) 



TV 



the tangential force is ,, T\* H<r=v<r' .......... .............. (102). 



TTCt ^Ct ft j 



We may now write down the values of X, ^ and v by transposing accented 

 and unaccented letters. 



R (2a-a) R R ,.._. 



\=- Td-; p. = T-, - \; v- ,-, -- /\,n ......... (103). 



va (a ay ira(aa) va(aay 



Comparing these values with those of X', p', and v, it will be seen that 

 the following relations are approximately true when a is nearly equal to a': 



X" n~ v 



27. To form the equations of motion. 

 *The original equations were 



' 



Putting p = A cos (ms + nt), a- = Bain (ms + nt), 



p = A' cos (ms + nt), <r' = I? sin (ms + nt), 



then o>' 



. . 

 ' 



The corresponding equations for the second ring may be found by trans- 

 posing accented and unaccented letters. We should then have four equations 

 to determine the ratios of A, B t A', R, and a resultant equation of the eighth 

 degree to determine n. But we may make use of a more convenient method, 

 since X', //, and v are small. Eliminating B we find 



_ 

 - \'A' + n'm&) n' + (p'mA' - v'B 1 } 2omJ = 



* [The analysis in this article is somewhat unsatisfactory, the equations of motion employed being 

 those which were applicable in the case of a ring of radius unity. ED.] 



