85-89] Maxwell's Treatment of Equipartition 79 



AN ALTERNATIVE TREATMENT OF THE PARTITION OF ENERGY. 



88. The doctrine of the equipartition of energy in a system of molecules 

 of varying masses was discovered and enunciated by Waterston* in 1821, 

 in the paper which has already been referred to. He states the doctrine in 

 the following form : " In mixed media, the mean square molecular velocity is 

 inversely proportional to the specific weight of the molecule. This is the 

 law of the equilibrium of vis- viva." Lord Rayleigh, in a footnote, says 

 " This is the first statement of a very important theorem. The demonstra- 

 tion, however,... can hardly be defended." Exactly the same theorem was 

 brought forward independently by Maxwell in 1859, in the British Association 

 paper already referred tof. He states the proposition: "Two systems of 

 molecules move in the same vessel ; to prove that the mean vis- viva of each 

 particle will become the same in the two systems." The question was again 

 brought into prominence by the publication of a paper by Boltzmann in 

 1861 J. In 1879 Maxwell also published a paper on equipartition in which 

 he regarded the whole question from a somewhat different standpoint^. In 

 what follows we shall treat the question from Maxwell's point of view, the 

 only difference being that the mathematical analysis can be put much more 

 concisely by the help of the conception of a generalised space. 



89. Let us, as before, represent a general dynamical system having 

 n degrees of freedom in a space of 2w dimensions. We shall now find it 

 convenient to take as variables 



where q 1} q z ... q n are generalised coordinates of the system, and 771, rj 2 ... rj n 

 are the momentoids of 77. Since 



d(p 1 ,p 2 ...pn) 



it is clear that any assemblage of systems will be represented in the present 

 space by fluid of the same density as that by which it was represented in 

 the former space. If, therefore, the fluid in the present space is taken to 

 be initially homogeneous, it will remain homogeneous throughout all time||. 



* Phil. Trans. CLXXXIII. p. 1. 



t "Illustrations of the Dynamical Theory of Gases," Phil. Mag. Jan. and July, 1860. 

 Collected Works, i. p. 878. 



J ' ' Studien iiber das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen 

 Punkten," Sitzungsber. der K. Akad. Wien, LVIII. 



" On Boltzmann's Theorem on the average distribution of energy in a system of material 

 points," Camb. Phil. Trans, xn. Collected Works, n. p. 713. 



II This treatment seems to obviate, in a simple manner, a criticism which has often been 

 urged against Maxwell's original proof. Maxwell takes coordinates in which the kinetic energy 

 is already expressed as the sum of squares, and assumes these to form true Lagrangian co- 



