40 



Ai't. 3.— K. ffiravama 



sufficiently small compared with «. Tlie amplitude of the 

 libration varies very slowly when the amplitude is small, and 

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

 with small eccentricities. 



46. General Case of the Second Order. Putting 6=- + d' in 

 the equations (27) of Chapter II, we have 



(11) 



x- = C + Q2hß'' cos, 6' 



1 dd 





n„ df 



= s'x — 'ipoGOSd' 



The equations for the variations of the constants become 



dC 

 dt 



= Sae'x + V2ßp,e' cos 6' -^' = -2(-^ + ;5J< 



The quantity JiJ always decreases as in the case of the first order. 

 G increases in the libration of Type 3, since x and cos^' are always 

 positive. We can pro\'e also that C increases if negative, that it 

 decreases in the revohdlon of Type 1 and that it increases in the 

 revolidion of Type 3 if E he less than a certain 'positive value. 



47. The limits of the mean motion and eccentricity, wlien 

 the path Ä is in contact with the limiting curve B^, may be 

 obtained as follows: — 



d' 



Ap, 



on 



^±vws i^+^P,±J§j,,E 



For the limiting case in whicli ro=^i=0 we have 



