254 Mr Bryan, A simple test case of [Nov, 26, 



8. When the different systems move about freely and collide 

 with each other, as the molecules of a gas, it has been shown by 

 Watson*, by Burburyf* and by Boltzmann^ that if the velocities 

 and angular velocities about the principal axes be arranged 

 according to the Boltzmann-Maxwell distribution 



JVe~ ''^ du dv dw da^ doo^ dw^ , 

 that is 



iV^exp [- i/i {M(io' + v' + w') + A<o^' + Bay^ + Gay^W du. ..dco^, 



this distribution will be unaffected by collisions. From the present 

 paper we see that it will also be unaffected by the free motion of 

 the bodies between collisions, and therefore it satisfies all the 

 necessary conditions of permanence. The mean values of Mu^, 

 Mv^, Mw\ Aw^, Ba^, Cw^\ are equal. The other distribution of 

 § 6 above, in which the mean kinetic energies due to the three 

 principal rotations are unequal is, in general, no longer permanent 

 when collisions can take place. In fact Boltzmann starts by 

 assuming a distribution of the form 



^^/h^jcjcjc^ exp [- h {u' + v' + to'} - k,<o^ - h^^^ - h^fo.^] du. ..da,^, 



and deduces that 



il K. /C„ Hj„ 



M^A^B^G 



9. An exceptional case is that in which each molecule (regarded 

 as a perfectly smooth hard body) is symmetrical about an axis 

 through its centre of inertia. The angular velocity about this 

 axis will be unaffected both by collisions and by the free motion 

 of the molecules, this angular velocity might, therefore, follow 

 any law of distribution whatever. 



Any distribution of the form 



/(wg) exp [- \h [M{u^ + w' + lu^) + A(o^^ + A(o;']] 



will therefore be permanent and /(wg) will not necessarily be 

 proportional to exp [— \hGco^\ 



The chief interest of this case lies in the fact, pointed out by 

 Boltzmann, that since partition of energy only takes place among 



* Nature, Vol. xlv. March 31, 1892, p. 512. 

 t Ibid. Vol. XLV. April 7, 1892, p. 533. 



X 'Ueber das Gleichgewicht der lebendigen Kraft zwischen progressiver und 

 Eotations-Bewegung bei Gasmolekiilen,' Sitz. d. k. Akad. zu Berlin, lii. Dec. 1888, 



