Eesearches on the Distribution of the Mean Motions of the Asteroids. 25 



'K ^-^0 -^ôo ./o -^ -%/ s'ex-p, COS d 



where T is the period of the libration. Writing x and e for x and 

 e in the part in which -jr is negative, we have 



nJ' e'xdt=2f '(-, ^^ 5— , , ,^"^' n]dd = 2f Xdd 



X is evidently positive when both % and x are positive. Also, 



Y_. s'{e^—e'^)ee'xx'—px oo^d{e*x—e''^x') 

 (s'ex—piCosd)(s'e^x'—piCOsd) 



The denominator is negative, e^— e'^ is positive, cos and e^a:— e'*:c' 

 are also positive and therefore JT is positive when x' is negative. 



-nT^i is 



dit 



positive. Thus C increases algebraically when negative. 



dC 

 31. The sign of the first term of -jr is exclusively negative 



in the revolution of Type 1 and positive in that of Type 3. For 

 the second term we have 



fiol e-coQ ddt=±/ — ^dd 



/g ./y s'ex—pi cos a 



in which T is the period of the revolution and the double sign cor- 

 responds to the positive and negative values of -^, i. e., to Type 

 3 and Type 1 respectively. We may write 



J g sex—p^ cos ^0 sex—pi cos o sex +Pi cos o/ 



n _ 



= 2f^Ycosddd 



