278 Prof. J. J. Muller on a Mechanical Principle 



dinates ; and this function is nothing else but the above-defined 

 primary function of the system. This peculiarity gives to the 

 primary function a real signification similar to that obtained by 

 the force- function in the potential energy, and makes the coor- 

 dination between the principle of energy and the new proposi- 

 tion still more evident. 



If this proposition was the general principle at which those 

 investigations of the theory of heat aimed, it must have included 

 as a special case the second proposition of that doctrine, in the 

 same way as the principle of energy included the first. In this 

 relation it is remarkable that, applied to the mechanical theory 

 of heat, it leads direct to the second main proposition as soon as 

 we make the apparently indispensable supposition that the tem- 

 perature of bodies is proportional to the vis viva of their mole- 

 cular motion. Corresponding to this, the principle seems also 

 capable of a series of further applications like those of the prin- 

 ciple of energy ; those which will be here given, however, are 

 limited to the case belonging to the theory of heat. 



The proposition cited resulted from the combination of two 

 long-known mechanical equations. Hamilton, namely, had 

 given his equations of motion both in reference to the function 

 V and in reference to the function W ; and the separate results 

 needed only to be combined, in order at once to furnish the new 

 one. It would be obtained in the most general form by intro- 

 ducing the integral elements generalized in the sense of Lip- 

 schitz and Schering. As, however, the essential point was its 

 application to real physical motions, and it had to be presented 

 first in its simplest form, I have preferred to give it in connexion 

 with the older method of Hamilton and Jacobi, which moves en- 

 tirely on this ground. But then this process must in another 

 respect be conceived more generally ; for, in every form in which 

 it has hitherto been carried out, it presupposes the force-func- 

 tion unaltered in form with the variation of the motion, while 

 such an alteration of form is sometimes essential in physical 

 considerations. This is the case, for instance, with the mole- 

 cular motions designated as heat, as soon as the bodies are sub- 

 jected to changes of volume and pressure. In regard to the 

 quantities accentuated especially by Clausius, which occur toge- 

 ther with the coordinates in the force-function, and vary with 

 the variation of the motion, while they remain constant within a 

 given motion, it was therefore needful that the method should 

 be amplified; and this has led to a somewhat more general form 

 of the equations of motion. 



§1. 



Given a system of n material points reciprocally attracting and 



