ON OCR KNOWLEDGE OF THERMODYNAMICS. 95 



Since the external system of conservative forces may be chosen to be 

 any we please, equation (19) mast be true for any cyclical process what- 

 ever, whether or not accompanied by the production or absorption of 

 external work. 



This, then, is the closest connection which exists between Hamilton's 

 principle and the kinetic analogue of the Second Law of Thermodynamics. 



VV"e might avoid the necessity of constructing a different multiply 

 connected field of external force to suit each cyclic process by adopting a 

 generalisation of the principle of Least Action, but this generalisation 

 would no longer belong to the forms given by Hamilton. Thus we might 

 suppose W, and therefore Q, to be a function of the time. This would 

 not affect the form of (17a), but in (176) iBQ, would be replaced by 



SQdt—i.e., iSQ. 



A slightly different method adopted by Helmholtz in his papers on 

 ' Least Action ' (Crelle, 'Journal,' vol. c.) leads to the same result. He 

 supposed the generalised external force components P^ to be functions of 

 the time only; in this case we must write 2(PaPn) instead of W, and, 

 therefore, B + 2(P„F„)=Q. 



18. Under the present section of this Report must be mentioned 

 Prof. J. J. Thomson's theorem that 'when a system consisting of a very 

 great number of molecules is in a steady state, the mean value of the 

 Lagrangian function has a stationary value so long as the velocities of 

 the controllable coordinates are not altered.' ' 



This ' theorem ' is nothing more or less than Hamilton's Principle of 

 Least Action, which is enunciated in a form identical with the above 

 by von Helmholtz in his paper on Least Action.^ In fact, if in equation 

 (18) we write 



H=U-T, 

 and assume the variation to be so chosen that 



8;=0, [2s,Sp,]^=0 .... (19) 



we have at once 



8{'B.dt=0, 



whence 



8(iH)=0, 

 or by (19) _ 



SH = 0; 



so that H has a stationary value. 



The function H, which is merely the Lagrangian function with its 

 sign changed, has been termed by Helmholtz the Kinetic Potential. 



The mean value of this function is the dynamical analogue of the 

 quantity in the theory of heat which is called the Thermodijnamical Poten- 

 tial by Duhem and Massieu, the Force Function of Constant Temperature by 

 J. Willard Gibbs, and the Free Energy by Helmholtz himself. 



The fact that, for a system which undergoes reversible transformations 



' Apjdloatums of Dynamics to Physics and Chemistry, p. Ii2. 

 ^ Crelle, Journal, vol. c. p. 139. 



