270 PROFESSOR WILLIAM THOMSON ON THE 
they were at the beginning, however arbitrarily they may have been made to 
vary in the period, the total external effect must, according to Prop. I., amount 
to nothing; and hence 
(p—-IM)dv-—JNdat 
_ Inust be the differential of a function of two independent variables, or we must have 
d(p—JM) d(-JN 
RE DN chi DN aT hee esse a 
this being merely the analytical expression of the condition, that the preceding 
integral may vanish in every case in which the initial and final values of v7 and 
t are the same, respectively. Observing that J is an absolute constant, we may 
put the result into the form 


d dM aN 
1 (a - 7s) oo a 
This equation expresses, in a perfectly comprehensive manner, the application of 
the first fundamental proposition to the thermal and mechanical circumstances 
of any substance whatever, under uniform pressure in all directions, when sub- 
jected to any possible variations of temperature, volume, and pressure. 
21. The corresponding application of the second fundamental proposition is 
completely expressed by the equation 
dp _ , 
Fi ee set ia ile re Seabee ay oo en 
where , denotes what is called “ Carnor'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 temperature of the 
refrigeration differs infinitely little from that of the source, without being true 
universally. Now, if a substance be allowed first to expand from v to v + dz, its 
temperature being kept constantly ¢; if, secondly, it be allowed to expand farther, 
without either emitting or absorbing heat till its temperature goes down through 
an infinitely small range, to ¢ — 7; if, thirdly, it be compressed at the constant 
temperature ¢ — 7, so much (actually by an amount differing from dv by only an 
infinitely small quantity of the second order), that when, fourthly, the volume 
is further diminished to » without the medium’s being allowed to either emit or 
absorb heat, its temperature may be exactly ¢; it may be considered as consti- 
tuting a thermo-dynamic engine which fulfils Carnot’s condition of complete 
reversibility. Hence, by Prop IL, it must produce the same amount of work for 
the same quantity of heat absorbed in the first operation, as any other substance 
similarly operated upon through the same range of temperatures. But as tT. dv 
is obviously the whole work done in the complete cycle, and (by the definition of 

