Eeeearches on the Distribution of the Mean Motions of the Asteroids. 29 



Substituting these expressions and the expressions of -^ and -^ of 

 (5) in the equation of -^, 



%rex-Vi cos ^)-^=(3s'ë2 + 5)(ffe=^a; + e/9pie' cos (?)-(5H3j?iê cos ^)(a + 2s'/9)e' 



(XL 



Or, reducing by the known relations we obtain finally, 



(8) 2^=Jlfêe + 3(a + 2s'/9)êe(e^-ë2) + ^[ilfe(e2_ê2)+|_(2a+//9>(e^-ê7] 

 at Au 



where 



s'ex—p-, cos d 



35. Putting 0=7: and changing x, e and ^ to x^,, e^ and ^i 

 respectively, apply the above result to the case of the revolutions 

 of Type 1 and Type 2 very near to the libration of Type 2. In 

 this case, (a) se^x^+p^ is very small, negative in Type 1 and 

 positive in Type 2, (b) e^—e^, is always negative in Type 1 and 

 positive in Type 2, (c) 31 is positive by (a), (d) the quantity 



^s'e,' + x,= ^^"^]~^' 



s'e, 



is positive by (7) and therefore (e) N is negative in Type 1 and 

 positive in Type 2. Since N is very great, we may write 



In the revolution of Type 2, 3f, iV and e^—e^, are all positive and 

 therefore —jr- is always positive. In the revolution of Type 1 , N 

 and e^—e^, are negative. Writing 



