g4 -^rt. 3. — K. Hirayama : 



68. The first of these remarkable peculiarities may be 

 accounted for by a consideration of gravitation only. Taking the 

 cases of the first order, we have 



d = s{l-l')-{l-w) 



If we put l'=l 



in this equation, then 



l=w-d 



which shows that the conjunction of the mean longitudes occurs 

 at the point l=w—d. Now in the libration of Type 2 with a 

 negative value of C and in that of Type 3, the argument 6 oscillates 



about and the amplitude of the oscillation is less than —^ . 

 Hence in these cases the conjunction takes place near the point 

 l^w, that is, near the perihelion of the asteroid. Contrarily if 

 the type of the motion be the libration of Type 1 or the revolution 

 of any type, the conjunction maj" take place near the aphelion of 

 the asteroid. 



69. The linear distance of the asteroid from Jupiter when 

 the conjunction occurs at the aphelion of the asteroid is 



a' —a{\ + e) 

 This expression becomes zero when 



a \ n'l 



The values of the eccentricity satisfying this condition are 

 n 350" 400" 450" 500' 

 e 0-11 0-21 0-31 0-41 



This shows how the asteroid with a moderate eccentricity 

 approaches Jupiter when the conjunction occurs near the aphelion 

 of the asteroid. The eccentricity of the asteroid is generally 



