resulting from Hamilton's Theory of Motion. 293 



motion designated heat, making use at the same time of the hy- 

 pothesis that the temperature is proportional to the vis viva of 

 the motion, we arrive (as Boltzmann, Clausius, and Ledieu have 

 shown) easily at the second proposition of the mechanical theory 

 of heat. In general, however, the motion of the molecules of a 

 body does not take place in closed paths. With respect to fluids, 

 for example, we are not even justified in assuming for them a 

 fixed mean position ; and in the case of solids, where such an 

 assumption is indeed necessary, the actual motion will yet be 

 distributed along all the dimensions. Now, for such cases Clau- 

 sius has recently called attention to a second, analogous equation, 

 which substitutes another hypothesis for that of closed paths. 

 A more direct derivation of the second main proposition from 

 the theorem of the action shall here be given. 



The suppositions which have been made respecting the system 

 of points representing the body are simply that the motion is a 

 stationary one, and that it is infinitesimally changed by the com- 

 munication of an elementary quantity of heat. The subject of 

 investigation is the quantity 



Px dt Fl dt' 

 which refers to the variation mentioned. Since infinitesimal 

 alterations of the velocities in the time-particle dt produce only 

 infinitesimal path-changes of the second order, this makes 



Further, the system of variations ~ can be split into two. 



Let the first be the distances q t dt which are traversed in the ori- 

 ginal motion during the time-element dt from the points q { . 

 This portion furnishes the sum 



Let the second partial system be the distances e L dt which lead 

 from the above-mentioned last positions in the original motion 

 to the final positions in the changed motion. In it, under the 

 suppositions made, to every value of;? there come just as many 

 positive as negative e ; this portion therefore furnishes the sum 



2^=0. 



Accordingly, for the infinitesimal variation which in the sta- 

 tionary motion of the point-system is conditioned by an infinitely 

 small quantity of heat, 



v ?J d( u_ Sv AA _ {) 



