ox OUR KNOWLEDGE OF THERMODYNAMICS. 73 



and this is obviously the same for all the co-ordinates /i,, thus 



Also by addition 



therefore each of the above mean values is one nth. of the mean value of T. 



If there exist other possible permanent distributions given by (10) for 

 which the function/ involves other integrals of the equations of motion 

 besides the energy, the argument will still hold good provided these 

 integrals onlij involve the co-m'dinates of the system, and not its velocities or 

 viomenta, because these co-ordinates are kept constant during integration. 

 Such integrals may, for example, depend on the equations of the paths of 

 the particles forming the system. 



But if y involves any function of the momenta or velocities other than 

 the energy, the integral in the numerator will assume different forms for 

 different A-'s, and the mean values of the difi'erent squares forming the 

 kinetic energy will, in general, be unequal. 



I would propose that the name Maxwell's Law of Partition of 

 Kinetic Energy be in future applied exclusively to the statement that 

 if the kinetic energy of a given system be expressed as a sum of squares, 

 the mean values of these several squares taken over a large number of 

 systems distributed in a given manner are equal. 



Hence Maxwell's Law of Partition of Kinetic Energy is only true 

 under the conditions stated above. 



Test Cases of the Law. — Motion of a Particle in a Plane. 



23. The test cases suggested by Lord Kelvin as apparently contra- 

 dicting Maxwell's Law of partition will be found, on examination, to 

 afford a valuable confirmation of all that has been said above regarding 

 the restrictions to which the law is subject. 



It is rather remarkable that the motion of a particle in a plane has 

 been employed by Boltzmann ' to furnish an illustration of the law, and 

 by Lord Kelvin ^ to furnish an apparent contradiction of it, which has, 

 however, since been met by Boltzmann.^ 



Lord Kelvin shows that it is impossible to give a general proof that for 

 a single particle moving in a plane the time-averages of x^ and ■ij'^ are equal. 

 This confirms what has been said in §§ 10-12 as to the impossibility of 

 applying Maxwell's law of partition to a single conservative dynamical 

 system. If, instead of a single particle. Lord Kelvin had covered the 

 plane with particles, and had projected them so that their co-ordinates 

 and velocities initially satisfied any law of distribution given by the 

 formula 



/ (E) dx dy dio dv, 



he would have found the same law to be satisfied at any subsequent time, 

 and the average values of it^ and v^ to be equal. 



' 'Eini^e allgemeine Satze iiber Warmegleichgewiclit,' Sitzb. dcr k. Wiener 

 Akad.' Ixiii. (ii.) (1871), p. 700. 



' Losung eines mechanischen Problems,' ibid., Iviii. (ii ) 



2 Nature, August 13, 1891. 



' Phil. Mag., March 1893. See footnote to § 20 above. 



