252 Mr Bryan, A simple test case of [Nov. 26, 



4. Now let there be a very large number N of such rigid 

 bodies all perfectly independent of one another. Let them be 

 initially set in motion, in such a way that the number of bodies 

 whose angular velocities about their principal axes initially lie 

 between 12^ and O^ + cZOj, O^ and n^ + d^^, Og and Og + c^flg is 



NfiT)dn^.da,.dn^, 



where the "frequency factor "/(T) is any function whatever of 

 the kinetic energy. 



Then at any subsequent time t, it follows from the determi- 

 nantal relation proved above and the constancy of the kinetic 

 energy for each body that the distribution is given by 



Nf{T)dco^.d(o,.d(o„ 



and is therefore the same as before. 



Hence the distribution is a permanent one, being independent 

 of the time. 



5. Now T = ^{Aco^' + B(o^' + Gco^'}, 



and we shall now show that with the above distribution the 

 average kinetic energy is distributed equally among the moraen- 

 toids corresponding to w^ w^, (o^, or that the average values of 



iAco,\ ^Bco,\ iCo,/ 

 are equal. 



For the average value of ^Aw^^ for all the bodies at any 

 instant 



J —00 J — OoJ — 00 



faAco^' + ^B<o^' + ^Ca>;)dco^dco,d(o^ 



J — CO J — OO J — 00 



Writing (o^^/iA = ^, w^s/^B = 7], (o^sJ^G= ^, 



.,• . JHf{^" + v" + ^')Vd^drid^ 



tnis becomes - ^^^^^p ^ ^, ^ ^,^ d^dr^d^ ' 



and the average values of J^co./, ^Gco^ are evidently equal to the 

 same expression. This expression may be put in the form 



-1 /• GO /» 00 



i f{T)THT^\ f{T)T^dT. 



O J Jo 



Hence Maxwell's law of partition of energy holds good in this 

 case. 



