Kosoarches on the Distvibutiou of the Mean Motions of the Asteroids. 39 



Consequently the variation of â is slow when â is small. The same 

 proposition may be proved for the libration of Type 1 when 6 is 

 very nearly equal to tt. Hence in general the amplitude of the 

 libration varies very slowly lohen the amplitude is very small. 



44. Now when the amplitude of the liljration is very small x 

 and e are connected approximately by the relation s'ex=p^ in the 

 libration of Type 3. Hence the point (x, e) moves in a rectangular 

 hyperbola, and therefore when e is great the point moves down 

 nearly parallel to the axis of e until e becomes small and consequent- 

 ly the variation becomes very slow. 



In the case of the libration of Type 1 the point {x, e) will 

 move in another rectangular hyperbola represented by the equation 

 s'ex=—j)i. It will move nearly parallel to the axis of x with a small 

 eccentricity until reaches at the point 



and the motion abruptly changes to the revolution of Type 2. 



45. The conclusions for the general case of the first order 

 may be stated as follows: — 



1. The revolution of Type 1 will change to the revolu- 

 tion of Type 2 either directly or indirectly and finally to that 

 of Type 3. The limits of the mean motion and eccentricity 

 increase discontinuously when the motion changes to the 

 revolution of Type 2, whence the variations of the constants 

 become rapid, and the asteroids of this class will not stay 

 long near the critical point. 



2. The libration of Type 2, when C is not negative and 

 relatively great, changes to the revolution of Type 2 either 

 directly or indirectly and finally to that of Type 3. 



3. The lil)ration of Type 2, when C is negative and 

 relative^ great, changes to the libration of Type 3 which may 

 change to the revolution of Type 3 if the constant ß be 



