Researches on the Distribution of the Mean Motions of the Asteroids. 35 



Hence -Tr^>0, and therefore the libration of Type 1 changes to the 

 revolution of Type 2. 



41. When the motion changes from the revokition or the 

 libration, of Type 1, to the revolution of Type 2, the limits of x 

 and e increase discontinuous^. Referring back to the table (6), it 

 may be seen that the lower limits change from the second row to 

 the third row, and the upper limits from the third row to the 

 fourth rOAV. Hence the lower limits of x and e increase by the 

 amounts 



respectively, and the upper limits by 



V12^ + ^ and J^^ 



respectively. 



The upper limits become minimum when os'^ei^=pu and the 

 minimum values of x and e are 



sf^^j and 3(-Äf) respectively. 



The corresponding lower limits are 



Numerical values of these limits for different cases of the first order 

 are as follows : — 



