Researches on the Distribution of the Mean Motions of the Asteroids. 



23 



(3) 



ra- 



The differential coefficients of x, e and 6 are composed of two 

 parts, viz; the differential coefficients due to the perturbation and 



Ldx "1 

 I ax \ 

 Hence, if we denote the former by [-jrj^ 6^^- 



dx _/ dx \, r dx ~\ 

 dt~\dtrLdtJ ®^^' 



Or, substituting the expressions of -jj- , etc. we get 

 dx _( dx \ 3a 3 



dd_ldd\ 

 dt \ dt J 



(4) 



dt 



de 

 dt 



Since x is supposed to be very small, we can write 

 As it will be seen in Chapter I, ß is essentially ^oÄ'^e and a may 



OL 



be positive or negative according as ki is less or greater than — - 



which is positive. We shall consider the case in which kx is 

 negative and consequently a is positive. 



29. Oeneral Case of the First Order. Differentiating the equa- 

 tions (15) and (16) of Chapter II completely with respect to t, we 

 get 



2 _^_.^;E , _2_ dx 

 dt dt 'as' dt 



