Researches on the Distribution of tho Mean Motions of the Asteroids. 43 



50. We can prove also, as in the case of the first order, that 



Y C §''-'''"' ' 



where S' is a quantity finite and positive. The amplitude of the 

 lihration therefore increases very slowly ivhen the amplitude is very 

 small. 



51. The conclusions for the general case of the second order 

 are as follows: — 



1. The revolution of Tj^pe 1 may change to the revolu- 

 tion of T3^pe 2 and finally to that of Type 3. The limits of 

 the mean motion and eccentricity increase discontinuously 

 Avlien the motion changes to the revolution of Type 2, whence 

 the variations of the constants become rapid and the asteroids 

 of this class will not stay long near the critical point. 



2. The libration of T^^pe 2, when (7 is not negative and 

 relatively great, changes either to the revolution of Type 1 

 or to the revolution of Type 2 which changes finally to that 

 of Type 3. 



3. The libration of Type 2, when C is negative and 

 relatively great, changes to the libration of Type 3 and finally 

 to the revolution of Type 3. The amplitude of the libration 

 increases slowly when the amplitude is small, and the 

 asteroids of this class will stay long near the critical point with 

 small eccentricities. 



52. Oeneral Case of the Third or Higher Orders. Restoring 

 e and putting 



s-s'=i and d={i-\)'J^d' 



in the equations (28) of Chapter II, we get 



(12) 



x'z=G + (Sp,é cos 6' e'=E + 4^x 



OS 



I dd' , 



^r-^=-8X — i-p,e 'COS 



7ifl at 



-2 



