172 Prof. Barnard on the comparative Expenditure of Heat 
If, in the expression for W, above, we make V, variable, we 
shall find that the maximum effect is obtained from a body o 
air, of which the minimum bulk and pressure are expressed by 
Sic 
Qui tet. 
sd at ") 
An expansion to a volume approaching or exceeding two-fold 
is therefore usually necessary to obtain the greatest effect; and 
the ratio rises rapidly as the maximum temperature is increased 
while the minimum is constant ; the index s1; being about 2-44 
for air. Fora maximum temperature of 480° F., and a minimum 
of 60°, V’ will be 1:8 V. For 750° maximum and the same mini- 
mum, V’ will be 2:25 V. 
=] 
As the expression (1- bay } is the measure of the heat 
made available, it is evident that the economy will increase with 
the expansion; but there occurs here, as in the other form, 4 
negative pressure at the end of the stroke, after passing a certain 
limit. P’, which has been taken for the final pressure, will be 
expressed thus, 
vev| 
TVW \ 27-1 
7/— P| _ 
: =Po ly) 
In the two cases above supposed, P’=-59P for the first, and 52P 
for the second. ; 
If we would impose a condition that there shall be no negative 
pressure at the end of the stroke, or that the final pressure shall 
ear any ratio, expressed by n, to the resistance, we shall have 
ee 2y-1 F 2y-1 
P’=nP=P= (7) , OF n=— (7) a 
ae sj Ly, 
and if we make n=1, we shall have 
vy’ qi’! soe ; as 
a (=) ‘ which limits the expansion to about once and 
a half the minimum volume, in the cases foregoing; V’ beilg 
equal to 1:4 V and to 1-6 V in those cases respectively. 
By substituting 2V and 1-5V successively, for V’, in the eX 
Vi 7-3 
pression (1 - (7) )H, we find that the fraction of heat col 
by the use of the regenerator, there may be theoretically a larg? 
saving. All the heat absorbed, which is not expended in work 
