168 Prof. Barnard on the comparative Expenditure of Heat 
ential area cannot be enlarged unless the curve F G rise in some 
point above, or the curve K H descend in some point below, the 
logarithmic curve; which neither can do without transcending 
the limit 7 or z,. 
The practical difficulty of maintaining a body of air, while 
undergoing dilatation or compression, at an unvarying tempera- 
ture, will render it always, probably, an impossibility to work an 
engine on this principle, and therefore to realize so large an ad- 
vantage from the heat expended, as the extremes of temperature 
. in furnace and refrigerator might lead us theoretically to antic _ 
pate. It is easy, however, to secure a constant pressure, and this 
presents the question of economy under a new form 
Suppose the air to expand from B to C, with rising tempera- 
ture, and a constant pressure represented 
B=GC, then to expand, without 
receiving or imparting heat, to D, at the 
lower pressure H D, then to be compressed 
to E, with loss of heat and constant pres- 
sure EK = HD, and finally to be com- 
pressed to its original bulk, acquiring at “3% ele 
the same time by the compression, and without receiving or im- 
parting heat, the original temperature and pressure. The differ- : 
ential area, F'G H K, bounded by the parallel straight lines FG, 


fect. If we draw a line, as vv’, parallel to FG, it will represent 
the difference of volume of the air when at equal pressure in the 
two opposite processes of expansion and compression. Calling 
these volumes v and wv’, and the corresponding pressure p, We 
shall have (using the other symbols as before), 
Vi 
Or, as P’=P”, —— —. and 
v v 

v'(V” —V’) 
fy fee "Ed an 19 a pe ee, 
W432 VV i: 0s == z 
Pp” ee Ae & 
But v=v(—/) ; whence '—o=(Wevy(— 
i a / ye Y i bai 
Now area PGuK=/ (v —ldp= J (W =¥) (=) 
Or, observing that dp is negative, ; 
= (wiv (prope, 7) ewe \( -(F)") 
w= (v"—V9(P —P'TP, 7 |e qr) I= (7 
Where P, represents the pressure at D or E, and is found in terms 
/ * . 
of P” by the formula pi= y_/ > after which P’ is put for us 
equal Pp”, ; 
