Resoarehes on tlig Distribution of tlio Mean Motions of the Asteroids. X3 



where C'is an arbitrary constant. 



16. By the last equation of (8) 



dt " la e ^e 



Neglecting m'x, me and higher powers we may write 



or introducing Uo by (10) 



(12) 4^=,^^^(,,_^J+(,_,yv^o ^Bc 



dt "' "' "■ ' e ?e 



The equation (9) may be written also 



(13) e'=E + ^-^^(x-x,) 



where E is an arbitrary constant. 



17. The constant .ro is a quantity depending on -^ and F, 

 multiplied by w'. Now JS is supposed to be small, of the same 

 order as e^ Hence, neglecting m JE in Xq, the latter becomes a 

 constant depending only on -J-. 



Since x is always diminished by Xo, if we change the origin 

 by that amount, we may write x for x—Xo and Xo disappears in the 

 equations (11), (12) and (13). Thus 



x- = C—6a(fRc 



•> ET , 2 s-s' 

 ^" = ^ + T^^^ 



Wo at e ce 



(1^) 



18. The expression of Xo in terms of the powers of-^may be 

 obtained easily by the usual process. The result is 



