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



We must therefore take the larger value of to 1 , and also add the condition 

 that N be positive. 



We may therefore state roughly, that, to ensure stability, - - , the coefficient 



of tangential attraction, must lie between zero and i^V. If the quantity be 

 negative, the two small values of n will become pure impossible quantities. If 

 it exceed -fa?, all the values of n will take the form of mixed impossible 

 quantities. 



If we write x for - RN, and omit the other disturbing forces, the equation 

 becomes U=n t -(ot'-x)n l + 3ot t x = () (63), 



whence n* = (<u 9 - x) % -Jot 4 - 1 4a> a x + x* .................. (64). 



If a; be small, two of the values of n are nearly (o, and the others are 

 small quantities, real when x is positive and impossible when x is negative. 





If x be greater than (7 /48)o) a , or - nearly, the term under the radical 

 becomes negative, and the value of n becomes 



......... (65), 



where one of the terms is a real quantity, and the other impossible. Every 

 solution may be put under the form 



n=pJ^lq ................................. (66), 



where q = for the case of stability, p = for the pure impossible roots, and p 

 and q finite for the mixed roots. 



Let us now adopt this general solution of the equation for n, and determine 

 its mechanical significance by substituting for the impossible circular functions 

 their equivalent real exponential functions. 



Substituting the general value of n in equations (34), (35), 



... (67), 



(68). 



- A 



