366 Prof. Clausius on the Connexion of the Second Proposition 



That the second proposition of the mechanical theory of heat 

 is connected with the principle of least action vfas also stated by 

 me, and, as I subsequently learned, also still earlier by Boltz- 

 mann*. This connexion becomes, as M. Szily quite correctly 

 insists, still more striking when the amplified expression given 

 by Hamilton of the principle of least action is employed. 



But for this purpose Hamilton's equation must not be taken 

 in the form in which it is usually cited in the text-books of me- 

 chanics and is found, for example, in Jacobins ' Lectures on Dy- 

 namics,^ p. 58 : viz., for a system of material points in motion 

 under the influence of forces which have a force-function or 

 ergal, let the vis viva be denoted by T and the ergal by U, in 

 the sense that the sura T + U is constant ; then the usually cited 

 form of Hamilton's equation is 



Sj(T-U)^^=0 (1) 



In order that this equation may be correct, the variation sig- 

 nified by 8 must be so understood that in variating we neglect 

 the alteration of the time — although in reality the altered motion 

 to which the variation refers is different from the original motion 

 not merely in respect of the coordinates and velocities, but also 

 of the time in which it happens. 



If, on the contrary, we understand the variation-symbol h in 

 the sense usual in other cases, as signifying the entire alteration 

 of the variated quantity, the equation must read : — 



Sj(T-U)^^ + (T + XJ)3j^/ = 0, 



or, if we denote by i the duration of time to which the integra- 

 tion refers, and for the sum T + U (the energy of the system) 

 introduce the symbol E, 



1'"- 



U)^/+E5z=0. 



The same equation can, as is readily seen, be also brought 

 into the following still more simple form : — 



2S( \dt = ih^. (2) 



This is the form of Hamilton's equation made use of by Szily, 

 and which expresses the principle of least action with variable 

 energy. It agrees with Boltzmann's equation (23a) in the 

 above-cited memoir, if we understand under the quantity which 

 he denotes by e not merely the vis viva supplied, but, in accord- 



* Sitzungsberichte der Wiener Akademie, vol. liii. 



