of the Mechanical Theory of Heat with Hamilton's Principle, 367 



ance with his later explanation, the augmentation of vis viva and 

 work, so as to attribute to it the same signification as to the va- 

 riation 3E in equation (2) above. 



Szily derives, in the simplest manner possible, from equation 

 (2) another equation, which he regards as synonymous with the 

 second proposition of the mechanical theory of heat. In relation 

 to this I cannot help saying that his development appears to me 

 too simple, because in it important difficulties remain unnoticed 

 and unsolved. 



Of that mechanical equation which I produced and employed 

 in my memoir mentioned at the commencement he speaks as if 

 it were contained in Hamilton's equation. But that is not pos- 

 sible; for my equation possesses a more general applicability 

 than Hamilton's. The latter, namely, presupposes as a neces- 

 sary condition that with the altered motion the ergal is to be 

 expressed by the same function of the space-coordinates as with 

 the original motion ; while my equation remains valid even when 

 the function of coordinates which represents the ergal undergoes 

 an alteration. As the simplest case of this kind, we may assume 

 that the function contains, besides the coordinates, also a quan- 

 tity which with each motion remains constant. Hamilton's then 

 presupposes that this quantity with the altered motion has the 

 same value as with the original motion, while mine permits an 

 alteration of the value of this quantity. 



I have already, in a memoir^ relative to Boltzmann's, ex- 

 plained this difi^erence between the two equations, and have there 

 shown more particularly how far Hamilton's equation would be 

 incorrect for the case in which the ergal undergoes an alteration 

 independent of the alteration of the coordinates. 



The erroneous supposition that my equation can be derived 

 from Hamilton's appears to me to have arisen from this, that 

 M. Szily has not quite correctly understood my notation. 



For the more convenient elucidation of the matter, we will 

 here confine ourselves to the consideration of a single moveable 

 point. Given, then, a material point of mass m, which with the 

 original motion describes a closed path or moves between two 

 given points. Let it likewise with the altered motion describe 

 a closed path or move between two points, which latter may 

 either be identical with those previously given, or, when that is 

 not the case^ fulfil the condition that the quantity 



dx ^ dy ^ , dz ^ 



has the same value at the final point of the motion as at the 

 * Pogg. Ann. vol. cxliv. p. 268. 



