506 PROFESSOR BURNSIDE ON THE PARTITION OE ENERGY IN THE 



therefore the problem of determining at what rate the system tends to reach 

 the special state would be intractable even if it were legitimate to suppose the 

 second assumption to hold throughout. 



If specially constituted spheres, however, are taken in which 



/3 2 +y 2 = y 2 +« 2 ^ a 2 +/3 2 . 2_ 



ABC A+B + C 



an attempt may be taken to determine the rate in question, for the forms of the 

 average changes in energy at a collision then shoAv that if the equations 



A = B_C 



«! k, 2 k 3 



hold at any one instant, they will always hold. 



Suppose, then, that in this case x, y are the whole energies of translation 

 and rotation per unit volume ; so that 



Sn 3?iA 



By § 14 of Prof. Tait's paper in conjunction with the forms found above 



y V h ns A+K + (kk-Ti) 



ySirnx 8c' 2 s 2 y — 2x 

 "T^A+B+C "~ 3~ * 



If 3E be the whole energy per unit volume, and if 



/8tto. 



8c 2 s 2 



3 A + B + C~T > /E 

 *"T \^-S( E - a )i 



the complete solution of which is 



2\/E i 



—7= 7=r + 1 = constant eT . 



six— \'E 



on the supposition that the energy of translation is originally greater than one- 

 third of the total energy. 



The ratio of the quantity T found here with that in § 23 of Prof. Tait's 

 paper, supposing there the numbers and masses of the two sets equal, is 



6c 2 



A+B + C 

 Hence it would seem that if c is of the order sxl0\ and therefore this 



