190 R. Clausius on the Application of the 
creased by d7, the quantity of heat necessary, will be repre- 
sented generally by 
dQ 
a? dT. 
This quantity of heat consists of three portions—1. The fluid 
portion, 4—m of the whole mass, must be warmed by d 7, for 
which purpose, if c denotes the specific heat of the liquid, the 
ares of heat (M—m)cd T is necessary. 
The portion m in the state of vapor must in like manner 
be ‘amen by dT, but will thereby at the same time be so much 
mpressed, th at for the increased temperature T+d 7, it is 
arte at a maximum density. The quantity of heat ‘which 
must be communicated toa unit of mass of vapor during its 
compression, in order that it shall have at every density pre- 
cisely the temperature for which this density is a maximum, we 
shall denote for an increase of temperature of d 7, in general by 
T in which h is a magnitude which is previously unknown as 
to its value, and even as to itssign. The quantity of heat neces- 
sary for our case, will hence be en mhd % 
3. In the process of heating, a small ciiasttiey of the previously - 
fluid portion, passes into the state of vapor, whith is Tep 
generally by a7 Re pe J, and which consumes the quantity of heat 
PES Ss AY ae 
m du _M do 
—s - ‘ av ee . “ale ¥. 
If we add these three quantities of heat together, and Pat their 
sum equal to aoa T we have 
"3 Q r r du 
(9) wa (5 a) m(h— o-<. 77 
+ d dQ a 
13. The first of these expressions for ic and —~, must now 
aT” : 
also, as is signified in equation (rI1), be y ee the first 4 
with respect to 7; and the last with respect tov. If we co : 
moreover that the quantity J is constant, the quantities u,% 7, 
