Researches on the DistriVtutidU of the Mean 'MotionK of the Asteroids. 33 



and 

 whence 



.^(6,3_M^i^_3..)(^-3..) 



The quantity —^ — 3ê£^ is positive within the limits and 3ë^— -^ 

 is also positive by (7). Hence the integral 







is positive. Now 



I eiê-e^dt^-lj {e--e')dt^ [e-è){é'-ë')dt 



Hence 



/ e{é'-e')dt>0 



err 



and therefore /* au ^^^^^q 



J . dt 



This result shows that the quantity w, Avhen it is very nearl}^ equal 

 to—], approaches to— 1. Hence we may conclude that the libra- 

 tion of Type 2 in which the maximum value of is very nearly equal 

 to TT changes to the revolution of Type 1 or Type 2. 



Now it has been proved that the revolution of Type 1, when 

 é"i is greater than a certain positive value, approaches the posi- 

 tion of contact, and that the revolution of Type 2 recedes from 

 that position. Hence it may be seen that the revolution of Type 1 

 changes to the libration of Type 2, when Ci is greater than a certain 

 positive value, and immediately later to the revolution of Type 2. 



