Quantities occurring in the Mechanical Tlteovy of Heat. ^61 



acceleration-components x" , . . . . ; at the same time let the 

 vis viva of the body be T, its potential energy U. In the in- 

 definitely short time clt subsequent to t the position of the 



. , . , ,, Sx.dt ., n ., , 8a/. dt 



point is changed by — — ,...., its velocity by — -. — , . . . . , 



z i 



the vis viva of the body by — r — , its potential energy by 

 — ^ — . Meanwhile the body does the work — ^— , and receives 



from without the energy . . According to this notation, 



the variation 8 signifies that indefinitely small change of the 

 quantity concerned which would ensue during the time i if 

 the rate of the change remained the same throughout the 

 time i as it was during the element of time dt. From the 

 principle of the conservation of energy we have then 



SQ=ST + SU + SW. 

 Since 



and 



8TJ + 8W = -t(a/ f 8x+y"8y+z"8z) 



(that is, when the work is reckoned positive which the forces 

 do against the body), therefore 



SQ=Zm(V&*' + . . .-x"8x-. . .). 



Now, in order that we may form SQ, we multiply the 

 equation by dt and take into consideration that 



a/dt — dx, 



x // dt = dx / . 

 Then 



8Q.dt=Z?n(dx.8x' + . . . . -dx / 8x-. . . .). 

 But now 



dx / 8x = d{x f 8x) — x'd8x 

 and 



a/dSai = x'8dx — 8(x'dx) — dx8x f ; 

 so that 



8Q.dt=-I,?nd(x'8x + . . . .)+tm§(a/dx + . . . .), 



or, writing the symbols of differentiation and variation before 

 that of summation, and restoring in the last sum the previous 

 value of dx, taking into account the expression of the vis viva, 

 and integrating between and i, 



