MOTIONS OF A SET OF NON-HOMOGENEOUS ELASTIC SPHERES. 507 



fraction comparable with 10"°, the rate at which the special state would be 

 attained is still extremely rapid, 



Note. — With regard to the form 



/ 



7T3 



for the number of particles per unit volume with angular velocities between 

 given limits, the fact that gj x , w 2 , o) 3 are periodic functions while the particle is 

 moving freely, suggests at first sight a difficulty. The period for each particle 

 is a function of its energy of rotation and of its angular momentum. Suppose the 

 particles whose energy of rotation lies between given close limits divided into 

 sets, the angular momentum in each set also lying between given close limits, 

 Then (compare Kirchhoff's Vorlesungen, p. 64) 



where p, q, r, A and the modulus of the elliptic functions are constants for any 

 one set. 



The individuals of each set are distinguished by the values of /x. If then the 

 circumstances are such that /x may be regarded as a uniformly varying quantity 

 between limits separated by a period for each set, the number of particles cor- 

 responding to the product dojx doi % da) S will be independent of the time. 



