ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 315 



The tangential force may be calculated in the same way, it is 



T=- Jl{MAsm(ms + a) + NJBsin.(ms + P)} (10). 



The normal force is 



Z= --RJCcos(ms + -y) (11). 



6. We have found the expressions for the forces which act upon each 

 member of a system of equal satellites which originally formed a uniform ring, 

 but are now affected with displacements depending on circular functions. If 

 these displacements can be propagated round the ring in the form of waves 



/ fL 



with the velocity -, the quantities a, /3, and y will depend on t, and the 



complete expressions will be 



p = A cos (ms + nt + a) T 



o- = Bsin.(ms + nt + P) L (12). 



= C cos (ms + n't + y) J 



Let us find in what cases expressions such as these will be true, and 

 what will be the result when they are not true. 



Let the position of a satellite at any time be determined by the values 

 of r, <, and , where r is the radius vector reduced to the plane of reference, 

 < the angle of position measured on that plane, and the distance from it. 

 The equations of motion will be 



.(13). 



If we substitute the value of in the third equation and remember that r 

 is nearly =1, we find 



n*m8+-BJ. (14). 



As this expression is necessarily positive, the value of n is always real, 

 and the disturbances normal to the plane of the ring can always be propa- 



402 



