ON OUR KNOWLEDGE OF THEBMODTNAMICS. 93 



It should be noted tbat Clausius introduces the conception of a 

 * phase ' in dealing with stationary motions, but this is not an essential 

 feature of the proof, and it only modifies the form of the equations. I 

 have therefore dispensed with it. 



16. Connection with Hamilton's Principle. — Although Thomson and 

 Tait have based their proof of the Principle of Least Action on equa- 

 tion (5), the above investigations do not show more than a very indirect 

 connection between that principle and the equation (14) which corre- 

 sponds to the Second Law of Thermodynamics. Had we used general- 

 ised coordiuates to represent the positions of the molecules, equation (6) 

 would have been replaced by Lagrange's generalised equations of motion, 

 and the connection would hardly have been any closer, depending only, 

 as it would have done, on the fact that Lagrange's equations could be de- 

 duced from the Principle of Least Action, and that equation (14) would 

 have been deduced from Lagrange's equations. 



Clausius recognised at the very outset of his researches the fact that 

 Hamilton's principle could not be applied directly to the case of a 

 system of molecules in which the variation of the motion was accom- 

 panied by the performance of external work through the controllable 

 coordinates of the system. For, as he puts it, Hamilton's principle only 

 holds good when, in the varied motion, the Ergal has the same form as a 

 function of the coordinates as in the original motion.' By the co- 

 ordinates Clausius here means the molecular coordinates only, for he 

 considers the controllable coordinates as variable parameters which enter 

 into and affect the form of the potential energy or ' Ergal.' In consequence 

 of this fact Clausius claimed that his equations involved a new principle 

 which was of more general application than Hamilton's principle. We 

 shall, however, show (i.) that, by means of a certain assumption as to 

 the form taken by the external work, a system can be formed to which 

 Hamilton's principle is directly applicable ; (ii.) that the principle leads 

 immediately to the analogue of the Second Law in the form of equation 

 (14) ; and (iii.) that the assumption made does not really interfere with 

 the generality of the proof. 



17. Our assumption is that the external forces, acting on the control- 

 lable coordinates of the body, belong to a conservative system. This 

 system we may, for convenience, call the ' external system.' When the 

 body performs external work 8W, the potential energy of the external 

 system increases by 8W. Hence we may denote this potential energy by 

 W. The external system and the original body, when taken together, 

 form a complete dynamical system, to which Hamilton's principle can be 

 applied ; for the potential energy of the complete system is a function 

 only of the generalised coordinates of the system. 



Moreover, in the complete system the increment of the total energy is 

 =SE + 8W=8Q by (1). Hence the total energy may be denoted by Q 

 where 



Q=E-|-W=T-fY + W, 



the total potential energy being U where 



U=V + W=Q-T. 



Let P\, Pi . . . denote the generalised coordinates of the complete 

 system, q^, q2 . ■ . the corresponding velocities, so that <7„=p„ ; and let 



» Fhil. Mag. vol. xliv. (1872), p. 365. 



