﻿350 Prof. P. Lowell on the Asteroids. 



17. The changes in the elements here are visibly systematic, 

 the slight variations from the systematic being in all cases 

 traceable to the different computers. They are, therefore, 

 approximately correct. Yet if we calculate from those of 

 1875 we shall find that then the action was not libratory, 

 while now we should conclude it to be so, and it will clearly 

 not be so again a few vears hence. Thus for 1875 we find 



d9 



t 



j t =U'o, \'A cos i (0-<y) =9-2, 



that is 



dO 



dt 



> \/Acos^(0~y). 



Thus the action was not libratory in 1875. 



18. Since the old criterion fails we must seek another. 

 At the limit of libra tion 



+ — = + \ZjX[cqs(0- 7 )- cos/3], 



whence 



0-J-/3 or = p + y. 



If /3-f 7 is over 90°, cos 6 has a — sign and the left side 



. d<& de 

 n ~ 2n + dt + Jt> 



cannot be for positive values of n — 2ri, since 

 ^ a 



— =r —CCOS&. 



at 



e being the whole coefficient. Permanent libra tion is thus 

 impossible for 



/3>90°~7. 



If ?i— 2n be negative, cos 6 must be negative; the libration 

 takes place round 1.80° and 0>9O°, 



/ 5>90°_ 7 , 



F° r dd /T-rf a \ HT 



— ^ ViA[-cos v <9-7)- cos/3]. 



