Researches ou the Distribution of the Mean Motions of the Asteroids. 31 



_ 2 - 



and x^—x^=6pi(ecosd—êu) é^—e^=-^(x—x) 



Differentiating these equations and eliminating -^ i?,ivl ^^ , we get 



Or substituting the expressions of [_-^J and |_-^J by (4) and 

 eHminating x and cos^ by the known relations, we get 



(10) -4p,ë^=s'Me{e' -e')+^s'C^a + s'ß)e{e'-eJ 



dt ^ 



where M=3ae^-2ßx 



as before. 



38. The second term of the right hand member of (10) is 

 positive for all values of e. The sign of the first term depends on 

 the sign of if and e'—e'. Now, if we confine ourselves to the 

 libration of Type 2 in which u is very nearly equal to— 1, M is 

 positive since x is negative, and the integral 





dt 



may be proved to be positive supposing the quantities C and E are 

 constants within a single revolution. To prove this I shall proceed 

 as follows : — 



We have, using the symbol of § 28, 



and therefore 



nj(e'-e')dt= - f^'~'^\ de 



Now, since d is very nearly equal to tt, 



