ON OUR KNOWLEDGE OF THERMODYNAMICS. 65 



It may be safely asserted that a large portion of our progress in the 

 present subject has been made, first, by showing that Maxwell's demon- 

 strations are faulty and unsatisfactory, and subsequently by discover- 

 ing fresh methods of proof, which, while leading to the same general 

 conclusions, show more clearly the limitations a,nd conditions under which 

 these conclusions hold good. In this process of destruction and recon- 

 struction a large amount of literature has accumulated, and I shall 

 endeavour in the present Pteport to unearth from the general mass the 

 main results to which these papers tend. 



3. Maxwell claims that his theoi-em is applicable to any dynamical 

 system whatever. 'The material points may act on each other at all 

 distances and according to any law which is consistent with the equation 

 of energy, and they may also be acted on by any forces external to the 

 system, provided these also are consistent with that law. The only 

 assumption which is necessary for the direct proof is that the system, if 

 left to itself in its actual state of motion, will sooner or later pass through 

 every phase which is consistent with the equation of energy.' * 



4. Instead, however, of a single system. Maxwell considers a large 

 number of independent dynamical systems, similar in every respect, 

 each defined by its n co-ordinates (g,, . . . q„) and the corresponding 

 71 momenta (^^i, . . . ^J„). Each system is capable of passing through 

 every phase which is consistent with the equation of energy, and it is 

 thus assumed that all the systems have the same energy. In the case of a 

 free system unacted on by external forces, the six components of linear 

 and angular momentum remain constant, and Maxwell assumes that these 

 are the same for all systems. 



5. Taking the ' action ' of the system during any period of the motion, 

 he employs this function to establish the determinantal relation between 

 the multiple differentials of the co-ordinates and momenta at the beginning 

 and end of any interval, and thus establishes the relation 



S(y/, . . . f.l.qx',. . • An ) _.^ ,j. 



8(PU • • • Vn^ <1\, • • • In) ' ' ' ^ ^ 



from which he deduces that, if the energy E be kept constant, so that j;, 

 can be expressed as a function of the n — 1 other ^'s, then 



dilh, • • • l^n, <1\, • • • (In) ?! • • K y 



6. Hence it follows that, if the systems are so distributed that the- 

 number which initially have their co-ordinates and momenta within the 

 limits of the multiple differential d!/;2 • • • dVndl\. • • • dq„he 



-^dp.y . . . dp„ dq^ . . . dq„ . . . (3) 



their total energies being all equal and C a constant, then the same 

 expression gives the law of distribution at any subsequent time. Maxwell 

 says : ' We have found one solution of the problem of finding a steady 

 distribution ; whether there may be other solutions remains to be inves 

 tigated.' 



' Zoe. cit., p. 518 

 1894. F 



