68 REPORT— 1894. 



To discover, if possible, a general class of- dynamical systems satisfying 

 the assumption would form an interesting subject for future investigation. 

 It is, howevei', doubtful how far Maxwell's law would be applicable to 

 the time-averages of the energies in any such system. We shall see, in 

 what follows, that the law of permanent distribution of a very large 

 number of systems is in many cases not unique. Where there is more 

 than one possible distribution it would be difficult to draw any inferences 

 with regard to the average distribution (taken with respect to the time) 

 for one system. 



Thus the proof of Maxwell's Law of Partition of Energy furnishes no 

 general conclusions with regard to the average distribution of energy in a 

 xingle conservative dynamical system with a finite number of degrees of 

 freedom, independently of initial circumstances, except by making as- 

 sumptions which are nearly tantamount to assuming the law. It may 

 reasonably be inferred that no such conclusions exist. We shall therefore 

 assume in future, unless otherwise stated, that we are dealing with the 

 distribution at any instant in a large number of systems. 



12. It is probable that the molecules of a gas in the ' special ' or equi- 

 librium state, in consequence of their frequent collisions, satisfy the 

 assumption, which Boltzmann ' has employed to give a simple proof of the- 

 e~''^ law of distribution. But he was careful to point out that no proof 

 had been given that the assumption either was satisfied or could be 

 satisfied by gas-molecules, and he therefore referred for an alternative 

 verification of his results to an independent but longer proof ^ based on a 

 consideration of the collisions between molecules. Similar questions are 

 raised by Boltzmann in his Appendix to this Report. 



The Evahiation of the Fimctional Determinant. 



13. Watson^ has raised an objection to Maxwell's evaluation of the 

 functional determinant on the ground that the result takes the form 



d{pu . . . q,) A' 



where A and A' are separately zero ; and, therefore, the investigation 

 leaves the value of the determinant undeterminate. This, he shows, 

 follows from the fact that the condition E constant supplies no independent 

 relation between the initial and final states. 



Both Watson'* and Lord Rayleigh* have therefore given an inde- 

 pendent proof based on the substitution of Hamilton's ' principal function ' 

 S, for the ' action ' A, where 



S 

 and 



= ['{T-Y)dt 



^''■-~d^' ^'~~d^^ ■ ' ' ' ^'' 



' * Einige allgemeine Siltze iiber Wiirmegleichgewicht,' SitzungslerleJitfl der 

 It. Wiener Altademie der Wissenschafte/i, Ixiii. (part ii.), pp. 707, 711. 



-'Ueber das Warmegleichgewicht zwiscben mehratomigen Gasmolekiilen,' Sitzher. 

 der Ti. Wiener Akad., Ixiii. (ii.), p. 397. 



' Nature, May 12, 1892 ; Kinetic Theory of Gases, new ed., p. 22, footnote. 



* Kinetic Tlieorij of Gases, § 8. 



' Phil. Mag., April 1892. 



