﻿Prof. P. Lowell on the Asteroids. 345 



both because the former is divided by e and because the 

 latter is originally at least one degree higher in e or its 



equivalent tan-/. For the asteroids e— sin $ and sin-i 

 Ji -| 2 



[which is approx. sin -y where 7 in Leverrier's notation 



signifies the mutual inclination of the orbits] = 77 are of the 

 same order of magnitude, the last being indeed on the average 

 about half as large only as the other. 



Owing to this divisor e the term in ~-=- , 



1 dR 



nd 2 e de ' 



is introduced with a large numerical coefficient, to the result 

 that instead of being negligible with regard to n — 2n'it 



sometimes, as in the case of f^Y actually exceeds the latter. 



10. Resuming now the equation 



dt = ViA[cos(0-y)+c], 

 if we put for cos (0— y) its value 



= 2cos 2 i(0-Y)-l, 

 we have for the criterion that libration cannot occur 



(6) f > VAcosi(*- 7 ), 



since in this case c must not exceed 1. 



A depends, it will be noticed, not upon e alone, as in 

 Tisserand's formula, but upon e' as well ; while 7 introduces 



OT — 1ST'. 



11. Applying this criterion to the several asteroids on both 

 sides of the gap n=2n' we have the following Tables I. 

 and II. 



d9 

 In this and the subsequent tables -y- and \/A cos f (0—y) 



have been calculated for all the asteroids where inspection 

 showed any possibility of libration. The apparent omissions 

 indicate that no libration occurs. L stands for libration in 

 the last column. 



