ON OUR KNOWLEDGE OF THERMODYNAMICS. 97 



as 1871/ and by Burbury in 1876.^^ Burbury now treats the converse 

 problem of determining the law of distribution from the Second Law. 

 He finds that if the law of distribution of the co-ordinates x^yX^, . . . x-„ be 

 given by the expression 



N/(x,,a;2, . . . x,^)dxi . . . dx,^ or Jif/du, 



and if the law of distribution of co-ordinates and velocities be given by 



Fda dff' where diy' :=dii . . . d'e^, 



then 8Q/T will be a complete diflferential if either 



(i) F=functionof (r-U)/T=^{(r-U)/T} say . (62) 



(which does not vanish for infinite values of the variables), or 



(ii) ^=K"T-)' -^^K^)- • • • ^^^^ 



where r is the kinetic energy, and XJ the potential energy of a molecule. 



Burbury says : ' And since F and f must vanish for all infinite values 

 of the variables, we are led to 



F=C exp -X^ f=C' exp -X ^ 



whei'e X is some positive numerical quantity. . . . ' 



59. Now this is obviously a solution, but it is not the only solution, 

 and I think the real inferences are slightly different from those he has 

 drawn. They are sufficiently interesting to be treated in detail, and they 

 are intimately connected with another point which at first suggests an 

 objection to the proof, namely, that c^t drr' is the multiple differential 

 of the co-ordinates and velocities, and is therefore not in general an 

 invariant like the multiple differential of the co-ordinates and momenta. 



In § 1 5 of his paper Burbury states that this does not matter. ' If 

 they' (i.e., da and da') 'do vary, that is, in effect, if the limits of integra- 

 tion vary, the assumption F=i^ [(U-fr)/Tj will still make 3Q/T acomplete 

 differential.' 



Now if 2/i, . . . 2/„ denote the generalised momenta con-esponding to 

 the co-ordinates a;, . . . x,„ the Jacobian 



( ■^1, • ■ • ^n ) j(^^ X2 . . . x„) suppose 

 8(2/1, . • • 2/«) 



will in general be a function of the co-ordinates cc,, . . . x,„ and its form, 

 will depend on the choice of co-ordinates. 



Hence, if Burbury' s proof be correct, we have really shown that the- 

 Second Law will be satisfied if the distribution be determined by any 

 expression of the form 



^/-^jj(x-,, . . . x^)dxi . . . dx^dyi . . . dy^ , (64) 



where by suitable choice of co-ordinates the form of J may be varied 

 quite arbitrarily. And by § 22 this expression represents, not necessarily 

 the Boltzmann- Maxwell distribution, but a distribution satisfying the 



' • Analytische Beweise des zweiten Hauptsatzes,' &c., Sitzher. der k. Wiener Akad., 

 Ixiii. (ii.), 1871, p. 712. 



2 Phil. Ma^., January 1876, p. 61. 



1894. H 



