^g Art. 3.— K. Hirayaina : 



and therefore, the librations ultimately change to revolutions. 



We can prove also that the amjylittide of the libration increases 

 very slowly when it is very small, as in the previous case. 



Thus we obtain the same conckisions as for the general case 

 of the second order. 



Chapter IV. 



Peculiarities in the Distribution of the Mean Motions 

 of the Asteroids and their Possible Explanations. 



56. We shall first examine the nature of the gaps. The 

 ratios of the mean motions up to the seventh order and lying 

 within the denser portion of the asteroids are as follows: — 



The simplest way of determining the width of the gaps is to find 

 the difference of the two mean motions nearest to n^ in both 

 directions. But this will be very rough, especially when one or 

 both of the mean motions is not reliable, as in the case of (132) of 

 the class 3/1. The following method will answer for this defect. 

 Denoting by 



n_i5, n_u, ?i_.j, n-i, Wj, 7U, n^, v-i^ 



the mean motions arranged in ascending order and interposing no 

 between n.i and ni, the quantities 



Çi=-g-(ni + n.+ +W9— nis— Wu— »hs) — no 



