ON THE STABILITY OF THE MOTION OP SATURN'S KINGS. 351 



Now there are four values of n, which give four corresponding values of 



a.' 



In 



n 



nj= a + &c., 2, is negative, 



"i 



w, = a small quantity, 2n 3 -- is positive, 





, = + a small quantity, 2n, -- is negative, 





n 4 = ft> &c., 2n 4 -- is positive. 



n t 



The quantity with which we have to do is therefore positive for the even 

 orders of waves and negative for the odd ones, and the corresponding quantity 

 in the other ring obeys the same law. Hence when the waves which act upon 

 each other are either both of even or both of odd names, qq' will be positive, 

 but when one belongs to an even series, and the other to an odd series, qq' 

 is negative. 



29. The values of p and p' are, roughly, 





(118)- 



Pi = ft) H &C., p. = ft) + &C., p t = ft) &C., p t = (O 1- &C. 



/ / 



7> 1 ' = ft)'H &c., ,' = ft)' + &c., ,' = ft/ &c., / = &/ H&c. 



m m 



ft)' is greater than <a, so that j9 t ' is the greatest, and p t the least of these 

 values, and of those of the same order, the accented is greater than the unac- 

 cented. The following cases of equality are therefore possible under suitable 

 circumstances : 



P*=Pt' (when m=l), p t =p,', 



In the cases in the first column qq' will be positive, in those in the second 

 column qq' will be negative. 



