ON OUR KNOWLEDGE OF THERMODYNAMICS. 99 



The equations of motion rfefcvred to thd principal axes are of tlie form 

 A'^'^'-(B-C) 0*2^3=0 .... (35) 



and the kinetic energy is 



T=l(A<o,2 + B.;.,' + Cw3') .... (66) 



Let 0|, fJj, fij be the initial angular velocities about the principal 

 axes, and let 



8 (<-.,, Wo. '"s) /R7V 



Then 



d^ 9 ((i)|, (>i„ »•>■,) 9(w,, 0)2, Wa) I 9 (••'i, W>> ti>3) 



(fi!~"o(0,, 12,, fis) 9'(ni, fio) i^a) 9(n,, Oo, lis)" 

 But by (i) 



^9j'^bJ'^<"3) _/g_Q\ / <^ C^ ('"3. '■^2) ^^3) I ^^ 9 (<■>;, <.)2, "'3) 1 



9 (ii„ n„ O3) ^ ^ \ - 9(fii, fij, fig) '9 (n,, n„ 123) J 

 =0, 



since each of the two determinants has two rows or columns equal. 

 Therefore 



dt 

 and 



A=const. = l, (its initial value) . . . (68) 



Since T^constant for any body, it follows that if a very large 

 numbey (N) of such rigid bodies have their angular velocities initially 

 so distributed that the number with angular velocities between fl, and 

 Oi + dili, Do and I22 + f'^^2> ^3 ai^d ^3 + dil3 is 



i:if(T)dnidn,dn3 



the distribution of any subsequent time will be given by 



'Nf(T)d>^id,02d,o^ (69) 



and therefore distribution will be permanent. 



With this distribution it is easy to see that at any instant the average 

 values of 



AAo),'-, {,Bwo^ iCw3^ 



over the different bodies are equal to one another, so that Maxwell's law 

 of partition of kinetic energy is satisfied. 



But the equations of motim have a second integral expressing the 

 constancy of resultant angular momentum, namely, 



A2<u,HB'^a),,HC2u)32=G2=const. . . . (70) 



Hence any distribution given by 



N/{G)d<oydu,.,du>3 (71) 



will (in the absence of collisions bctAveen the bodies) be permanent. 



H 2 



