Principle in Thermodynamics. 431 



This is Clausius' s equation. 



Now I affirmed that Mi's equation, and not Thomson's, is iden- 

 tical with Hamilton's. This assertion of mine I still maintain 

 in its full extent ; and I believe that after the preceding state- 

 ments Professor Clausius will urge no objection to it. 



Still, although Clausius' s mathematical contemplations on this 

 subject have not led to an altogether new equation, but have for 

 their result merely Thomson's and Hamilton's equations some- 

 what transformed, it would be a great mistake to suppose that 

 those highly important disquisitions had remained unfruitful. As 

 Boltzmann on his part, so I also most willingly acknowledge that 

 the portion which chiefly refers to the change of form of the force- 

 function was absolutely necessary — nay, that only with their aid 

 could Hamilton's equation have been, with any result, introduced 

 into the mechanical theory of heat. Hamilton has proved his 

 equation only for the case in which the form of the force-func- 

 tion undergoes no change when the motion is changed; but 

 from the researches of Clausius it follows that the same equation 

 (Hamilton's) can be made use of also when the form of the force- 

 function changes, provided that the mean value of the resulting 

 variation be =0. Such an extension of the validity of Hamil- 

 ton's equation was at all events requisite, because in the thermo- 

 dynamic theory changes of state of bodies come into consideration 

 which are not dependent merely on space-alterations, but also 

 on the changed form of the force-function. 



Let us now examine how far Hamilton's equation must be 

 modified to make it applicable to the case in which we have to 

 do not merely with a single point, but with the change of mo- 

 tion of an entire system of points. And indeed let us just take 

 a system the individual points of which do not travel their paths 

 with equal vires viva or in equal times. Professor Clausius 

 remarks, quite correctly, that in this case Hamilton's equation 

 is not immediately applicable, and consequently special investi- 

 gations were necessary. 



"When, however, we consider the matter a little more closely, 

 we find that in the form (8) of Hamilton's equation no other 

 modification is required than that which, relative to the energy, 

 naturally results from the notion of a heterogeneous system of 

 points ; that is, we have only to prefix the sign of summation to 

 the right- and left-hand terms, and to extend this summation to 

 all the points of the system. Thus, whilst Hamilton's equation 

 expresses the variation of the energy for one point as follows, 



i 



we obtain the variation of the energy relative to an entire system 



