126 Prof. R. Clausius on a Mechanical Theorem 



very simply ; for, p signifying the pressure, and v the volume of 

 the body, it is represented by 



Denoting, further, by h the vis viva of the internal motions 

 (which we call heat), we can form the following equation : — 



h=±2r<l>(r) +%pv. 



We have still to adduce the proof of our theorem of the rela- 

 tion between the vis viva and the virial, which can be done very 

 easily. 



The equations of the motion of a material point are : — 



d 2 w „ d 2 y , r d*z „ 



m -df =X ' m w =Y > m w =z - 



But we have 



d 2 (x 2 ) n d( dx\ n (dooY d*x 



or, differently arranged, 



/^\ 2 d?x d\x*) 



771 



Multiplying this equation by — , and putting the magnitude X 



for m-r^-y we obtain 

 dr 



m(dxy__ _ m d*{x*) 



2\dt/~ 2 X+ 4>'~aW 



The terms of this equation may now be integrated for the time 

 from to t, and the integral divided by I ; we thereby obtain 



.2 



tfdvYjt 1 TV j*, mVd(^) /d(z*)\l 



in which { — tr- ) denotes the initial value of -— — 

 V at / dt 



The formulae 



f£(S) * and \& dt > 



occurring in the above equation, represent, if the duration of time t 

 is properly chosen, the mean values ofii-J and X#, which were 



/^\? 



denoted above by ( -57 J and Xa?. For a periodic motion the 



