260 M. 0. JSzily on the Dynamical Signification of the 

 Then, taking the mean values, 



ST=0, diJ = 0, SW=0. 



Let us collect the results of the foregoing considerations. 

 It is now easy to perceive when tho thermal state of a body 

 can be regarded as constant : it is not in the least necessary 

 that what are called the variables of state (volume, tempera- 

 ture, tension, &c.) should continually remain constant. Ac- 

 cording to our apprehension, the condition for the constancy 

 of the state of any body whatsoever consists in this, that the 

 period of the oscillation, the mean values of the volume, of the vis 

 viva, and of the potential energy he constant, the mean value of 

 the external work, referred to the duration of a complete oscilla- 

 tion, be nil, and, finally, that the periodical change of the body 

 be an adiabatic one. Therefore, if the state is unchanged, then 



z= const., . . (1) U = const., . . (4) 



v = const., . . (2) rfW=0, ... (5) 

 T= const., . . (3) dQ=0. ... (6) 



So long as the state of the body is unchanged these six 

 equations stand, and vice versa. But these condition-equations 

 are not all independent of one another ; it is possible that by 

 satisfying two or three the rest of them are eo ipso satisfied. 



Having settled these preliminaries, we will now pass to the 

 analytic treatment of state-change, and, in connexion there- 

 with, to the deduction of the Second Proposition. 



2. 



Let us imagine any body, and select any one of its material 

 points. Let m be the mass of this point ; let its rectangular 

 coordinates at the time t = be x , y , z ; the components of 

 its velocity, x f , y' w z f ; and the components of its accelera- 

 tion, x' f , y" , z" . Further, let v Q be the volume of the body, 

 T its vis viva, U its potential energy at the same time. If 

 the state of the body did not vary, the period of an oscillation 

 would remain constant, likewise the mean values referred to 

 it v, T, U, &c, and the point m would come at the end of the 

 time i into a certain position whose coordinates we will desig- 

 nate by x\, . . . . , the velocity-components there appearing 



by x\, . . . , the acceleration-components by x'\, 



But now let the state of the body change infinitesiinally 

 during the time i, so that at the end of it the oscillation-period 

 may be i + hi, the mean value of the vis viva T 4- ST (instead 

 of T), &c. 



At any instant t, between and i, let the coordinates of 

 m be x, , . . . ? the velocity-components x', . . . , and the 



