56 Art. 3. — K. Hirayama : 



proximity of their orbits, never approach very near to each other. 



71. To explain the second character of the distribution of the 

 mean motions, I shall introduce the hypothesis of resisting 

 materials. Assuming the existence of the resisting materials 

 moving around the sun in circular orbits, it was proved in Chapter 

 III that, in the general cases of the second and higher orders, the 

 librations ultimately change to revolutions, while the revolutions 

 do not change to librations. In the general case of the first order 

 it was proved that the revolutions do not change to librations 

 except that the revolution of Type 1 ma}^ temporarily change to 

 the libration of Type 1, that the librations ultimately change to 

 the revolution or the libration, of Type 3, and also that, if the 

 constant a be sufficiently great compared witn /S, the libration of 

 Type 3 changes to the revolution of the same type. Now in the 

 class 2/1, the unique case of the first order with greater mean 

 motions, we may naturally suppose, as it was noticed in § 9, 

 that the density of the resisting materials rapidly decreases as the 

 distance from the sun increases, so that a becomes great in com- 

 parison with /9. Hence in general, in the cases of gi'eater mean 

 motions, the librations ultimately change to revolutions, while 

 the revolutions do not change to librations, except the revolution 

 of Type 1 in the general case of the first order, which may 

 temporarily change to the libration of Type 1. In the cases of 

 smaller mean motions, we have sufficient reason to believe that 

 the density of the resisting materials is very rare, as was noticed 

 in § 9, so that the effect on the librating motions is insignificant. 



The fact that the eccentricities of the asteroids near the gaps, 

 especially on the inner side, are generally smaller than those in 

 the other portions (§ 59), may be explained in the following 

 manner: — On the negative (outer) side of a:, the eccentricity 

 cannot be great so long as the type of the motion is confined to 

 the revolutions. Even when the eccentricity is moderate, the 

 motion will sometimes change to the libration on account of the 

 smaller inequalities and the asteroid will remove to the positive 

 side of X. Hence in order that the revolution of Type 1 may be 



