350 ON THE STABILITY OP THE MOTION OP SATURN'S RINGS. 



This is a quadratic equation in x, the roots of which are real when 



(p-p'Y+W 



is positive. When this condition is not fulfilled, the roots are impossible, and 

 the general solution of the equations of motion will contain exponential factors, 

 indicating destructive oscillations in the rings. 



Since q and q' are small quantities, the solution is always real whenever 

 p and p' are considerably different. The absolute angular velocities of the two 

 pairs of reacting waves, are then nearly 



p p p p 



instead of p and p 1 , as they would have been if there had been no reaction 

 of the forced wave upon the free wave which produces it. 



When p and p' are equal or nearly equal, the character of the solution 

 will depend on the sign of qq'. We must therefore determine the signs of q 

 and q' in such cases. 



Putting f? = ,-, we may write the values of q and q' 



^ m 



n 



(lie): 



. /<u <<A <a w 



, X + 2u.m -, - 4i> - 



, n \n n) n n 



2 m" 4n"-2a>' 2 



Referring to the values of the disturbing forces, we find that 



X' fif v Ra 

 X p, v Ra'' 



q n 4n" 2o/" Ra 



Since qq' is of the same sign as , , we have only to determine whether 



2n -- , and 2n' -- , are of the same or of different signs. If these quantities 

 are of the same sign, qq' is positive, if of different signs, qq' is negative. 



