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



27 



and 



s'eiXi+Pi=0 



Evidently, the limiting values of x and e will be attained when 6^0 

 or 7t. Denoting these values by x and ê, we have 



x^=G±6pie 



-é=E+p 



in which the upper sign corresdonds to ^=0 and the lower to <?=;r. 

 Eliminating C, JE and e, we have 



{x'-x,J-l2pM^'-^')-^Pi\^-^ù=^ 



which becomes by the use of the relation s'eiXi+pi=0 



(x-x,y[(x+x,y-np,e,} =0 

 Hence, 





that is, 



Pi_ 



s'e. 



or — a;i(±)V'12p,ei 



or ^(±)VÎ2^ 



se. 



the double sign enclosed in the parentheses meaning that it is 

 independent of the value of d. Similar^, we obtain 



e=q=ei 



or 



±e 



ri±)J 



^Pi 

 3s% 



The limiting values of x and e will thus be tabulated as 

 follows : — 



0, ;r 



(6) 





h-JM;) 



-P. 



se, 



+ 4i-+Vl2i?iei +ei+7 



^Pi 



3s'\ 



