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



gated as waves, and therefore can never be the cause of instability. We 

 therefore confine our attention to the motion in the plane of the ring ,-i> 

 deduced from the two former equations. 



Putting r = 1 + p and <j) = <at + s + <r, and omitting powers and products of 

 p, a- and their differential coefficients, 



(15). 



Substituting the values of p and cr as given above, these equations become 



f* * Tl T^ I * . f*. f*t * 



\ 



RM\ B cos (r>is + nt + /B) = (16), 



/* 



1 1 



RM) A sin (ms+ nt + a) + (n~-\ RN) Bsin (ms + nt + /3) = 0....(17). 



p. p. 



Putting for (ms + nt) any two different values, we find from the second 

 equation (17) 



=0 (18). 



nd 2 -RM)A >' RN B- 



/x, p. 



and from the first (16) (tf + ZS-- RL + n") A + (2a>n+ - RAf) B~Q (20), 



and a>*-S--RK=0 (21). 



V- 



Eliminating A and B from these equations, we get 

 n'-{3o>'-2S+~R(L-N)}n* 



1 111 



a hiquadratic equation to determine n. 



For every real value of n there are terms in the expressions for p and cr 



of the form 



A cos (ins + nt + a). 



