Prof. Thomson on the Dynamical Theory of Heat. 19 



accordiug to Prop. I., amount to nothing ; and hence 



{p-:iM.)dv-mdt^ 

 must be the differential of a function of two independent vari- 

 ables, or we must have 



d{p-m) _ d{-m) 



dt ~ dv ' ' ' ' ' ^^' 



this being merely the analytical expression of the condition, that 

 the preceding integral may vanish in eveiy case in which the 

 initial and final values of v and t are the same, respectively. 

 Observing that J is an absolute constant, we may put the result 

 into the form 



dp (m d^\ 

 dt-\~dt~lh) ^'^> 



This equation expresses, in a perfectly comprehensive manner, 

 the application of the fii'st fundamental proposition to the ther- 

 mal and mechanical circumstances of any substance whatever;^ 

 under uniform pressure in all directions, when subjected to any 

 possible variations of temperature, volume and pressui'e. 



21. The corresponding application of the second fundamental 

 proposition is completely expressed by the equation 



|=''M. (3) 



where j«. denotes what is called " Carnot's function," a quantity 

 which has an absolute value, the same for all substances for any 

 given temperature, but which may vary with the temperature in 

 a manner that can only be determined by experiment. To prove 

 this proposition, it may be remarked in the first place that 

 Prop. II. could not be true for every case in which the tempera- 

 ture of the refrigerator differs infinitely little from that of the 

 soiu-ce, without being true universally. Now, if a substance be 

 allowed first to expand from v to v + dv, its temperatui'e being 

 kept constantly t ; if, secondly, it be allowed to expand further, 

 without either emitting or absoi'bing heat till its temperature 

 goes down through an infinitely small range, to t — r; if, thirdly, 

 it be compressed at the constant temperature t—r, so much 

 (actually by an amount differing from dv by only an infinitely 

 small quantity of the second order), that when, fomihly, the 

 volume is further diminished to v without the medium's being 

 allowed to either emit or absorb heat, its temperatui'e may be 

 exactly t ; it may be considered as constituting a thermo-dynamic 

 [* The integial functiony{(JM— /;)f/y + JNJ/} may obviously be called 

 the mechanical mcryy of the fluid mass ; iis (when the eoustant of integration 

 is properly assigned) it expresses the whole work the fluid has in it to 

 produce. The consideration of this function is the snl)jectof a short pajjcr 

 communicated to the Royal Society of Edinburgh, Dec. 16, 1851, as an ap- 

 pendix to the i)aper at present repubUshed.] 



C2 



