Researches on the Distribution of the Mean Motions of the Asteroids. 55 



smaller than 0"20, but this will not remain always small if the 

 orbit of the asteroid be sufficiently close to that of Jupiter. So the 

 motion of the asteroids with smaller mean motions, something 

 like 450" or less, if the conjunctions occur near the aphelia, will 

 be disturbed a great deal by the attraction of Jupiter. 



In the cases of the asteroids which make the libration about 



TT 



zero with the amplitudes less than -0-, since the conjunctions always 

 take place near the perihelia, they will not suffer large disturbances 

 and the motions will be stable. Contrarily the asteroids librating 

 about TT, or making the revolution will sufïer large disturbances, 

 since the conjunctions may take place near the aphelia. So their 

 motions will be unstable. 



Theoretically speaking the libration of Type 3 is possible for 

 any positive value of x. But if the value of x be great the range 

 of JE for the libration becomes very small. Consequently the 

 libration may be changeable to the revolution by a slight variation 

 of ^ due to the smaller inequalities. The libration of Type 3 is 

 thus practically impossible when x is not small. The fact that the 

 asteroids do not exist at the intermediate positions is thus 

 explicable. 



70. The asteroids of the class 1/1, in spite uf the proximity 

 of their orbits to that of Jupiter, never approach very near to the 

 latter and consequent!}^ their motions are stable. This fact is in 

 perfect agreement with the above explanation. 



An instance analogous to the librating asteroids of the 3/2 and 

 4/3 classes may be found in the Saturnian system. Hyperion, the 

 seventh satellite of Saturn, has a period of revolution very nearly 

 commensurable to that of Titan, the sixth and the largest satellite 

 of Saturn. It has been shown in theory and by observations that 

 the argument 4/— 3/'— 0?'-' of Hyperion makes a libration about tt, 

 so that the conjunctions always take place near the aposaturnium 

 of Hyperion. Consequently these satellites, in spite of tlie 



1) Here I, ur and l' denote the mean longitude, the mean longitude of perisaturnium, of 

 Hyperion, and the mean longitude of Titan, respectively. 



