Researches on. the Distribution of the Mean Motions of the Asteroids. 53 



of the librating asteroids near the point 3/1 is doubtful. 



65. Only two asteroids will be taken near the point 5/3 with 

 the following results : — 



5/3 no=498'55 p.}'^xW =0-^222 



No. Epoch -uT I I' n a;Xl02öXlO 8 C'XlO' JE X 102 FoX 102 FiX 102 Type 



522 1913 IV 6-0 13 228°3 280°6 513'62 +3-02 078 4-5 + 9-93 -0-74 1-4 R3 



721 1911 X 18-5 29-0 19-2 2362 526-85 +5-68 1-18 14-6 +34-05 -1-13 2-6 R3 



Both make the revolution of Type 3. 



66. In the cases of the third or higher orders, the effect of 

 the neglected terms being relatively great, it becomes very hard to 

 determine the types of the motion accurately. As a rough 

 approximation however the same method was applied for the 

 classes 5/2 and 7/4, arriving at the following results : — 



All the asteroids near the point 5/2 (?io=747"'82) make the 

 revolution of Type 1 or Type 3, except (464) which seems to make 

 the libration of Type 2. But this asteroid has not been observed 

 since its discovery in 1901 and therefore full weight cannot be 

 assigned to its result. For the class 7/4 (?io==523"'48) two asteroids 

 (522) and (721) only will be taken. The former seems to make 

 the revolution of Type 1 and the latter, the libration of Type 2 

 with a positive value of C. 



67. It is a well known fact that all of the four asteroids of 

 the class 1/1 make the libration about the triangular equilibrium 

 points. Hence, including this case, the results will be summarized 

 as follows: — 



1. All the astreoids with smaller mean motions than 600" make 

 the libration and form a series of the groups near the commensurable 

 points Ijl, 3l2 and 4j3. 



2. The asteroids with greater mean motions than 580" do not 

 make the libration (except a few cases which are mostly doubtful) 

 and form a series of the gaps at the commensurable points 2jlj Sjl, 

 512 etc. 



