370 Mr. R. 0. Nichols on the Proof of 



supposition no ground whatever has been alleged, the assump- 

 tion that I %edt=t$Q is purely arbitrary, and moreover is 



untrue for any value of t unless it is also true for every value 

 of*. 



The general condition under which the proposition is true 

 which M. Szily has sought to establish, and which has been 

 proved by Mr. Burbury for the case in which the potential 

 energy of the system has the special value deduced from the 

 theory of M. Boltzmann, may be shown in the following 

 manner. 



Let it be observed that the addition of energy dQ is ex- 

 pended (1) upon alteration of kinetic energy, which call dT y 

 (2) upon alteration of potential energy or </U, and (3) upon 

 work done or dX. ; and dQ must be equal to the sum of these 

 quantities, or 



dQ,=dT + dTJ + dX (1) 



Now if v be the volume of the gas and p its elastic force, or 

 the force resisting expansion per unit of surface, dv the change 

 of volume, 



dX=pdv (2) 



And by the equation of Clausius, 



^=f(T+22Q&-)), 



—p x dv +p 2 dv ; 



T 



where Pi= § — is the element of the elastic force resulting from 



the kinetic energy of the particles, and p 2 — ^ 22 ( — ) is the 



element resulting from the internal forces. Now supposing 



the change of volume from v to v + dv were to take place in 



such a manner that the distances of the particles should, 



on the average, preserve the same relative amount, every 



dv 

 dr resulting from the change of volume would equal Jr — , 



or p 2 dv would be equal to 22 ( Ry- J dv, or —j- dv. It is there- 



dU / 

 fore equal to -j— dv, where U 7 denotes a function of (v,T,) 



which varies as the potential of the system would do upon this 

 supposition. Therefore 



