Quantities occurring in the Mechanical Theory of Heat. 263 

 To this end let us consider first the signification of 

 ^m(x f dx-\- . . .). 

 This may be written also as follows, 

 l l m(x r2 + ...).dt, 

 or thus, 



2T. dt. 



But this last expression is nothing else but the action of the 

 unchanged body in the time dt. Let that action be dA, then 



%m(x'dx + . . . ) = <£A, 



whence it appears that the sum in question, viz. 



l l m(x / Bx + ...) = 8A, 



signifies the action of the body in the time i, supposing that its 

 material points are not endued with any spontaneous motion, 

 and that the rate of change of state remains the same through- 

 out all the time i as it actually was during the element of time 

 dt. Hence 



SA.dt 



denotes the action of the body in the time dt, supposing that its 

 material points are not endued with any spontaneous motion. 

 If we now add to this the action in the time dt resulting from 

 the spontaneous motion of the material points, we shall get the 

 total action in the time dt equal to 



I 

 Accordingly 



I 



will express the action in the first time-element of the change 

 of state. 



Let us now imagine that the very same infinitely small state- 

 variation does not begin at the time ^=0, but at the time t — i 

 later, and that it also finishes at the same time later ; then 

 will the action in the first time-element of the state-variation be 



^ 



But according to the above-defined conceptions, the thermal 

 properties of the body are periodical functions of the time ; 

 consequently it cannot make any difference whether the change 

 of state starts with the beginning of the nth or (n+ l)th period; 

 and the action in the first time-element of the change must be 



