﻿Prof. P. Lowell on the Asteroids. 351 



The criterion, therefore, for the gap n = 2n f is 

 n — 2n' = c cos #, 



where e= 551 , f 4j» + j 1 <»v 



2e v " y 



Taking the maximum possible value of cos = 1, we get 

 (11) „_2V<^'(4//-'>+^>). 



19. Applying this criterion we find that not a single 

 asteroid in the gap n = 2ri shows libration. The gap, there- 

 fore, is not due to that cause, but is permanent, and expresses 

 a secular action by Jupiter, which has cleared the gap and 

 keeps it so. 



20. For the gap n = 3n' 



^ = I n*a**m' 1 2e (21 b ^ + 10 b^ + b.p ) cos 



_V (40 &W + 20 b^ + 2 hp ) cos 0'~\ , 



whence 



1 nam' j 2 (21 &<»> + 10 lj® + bj® ) cos 



dm 



dt 



- 1 (4Q h (2) + 20 ^W + 2 & 2 < 2 ) ) cos & \ , 



the secular terms being as before.^ 



Introducing this condition for mlbA we see that the action 

 on it too fails of being libratory. 



21. This investigation shows, then, that the gaps con- 

 sidered are permanent and due to secular action by Jupiter, 

 libratory action playing no part in their formation. The 

 same can be seen from their position to be the case with the 

 gaps in the rings of Saturn, and, finally, in the positions of 

 the major planets, which I propose making the subject of a 

 separate paper- 

 In conclusion, I wish to express my obligation to Miss E. 

 S. Williams, of this observatory, for her valuable assistance. 



Boston, Mass., 

 August 10, 1911. 



