352 On the Expenditure of Heat in the Hot-air Engine. 
in which p’ is the working pressure, ¢ the density of the compres- 
sed air, p the atmospheric pressure, and ¢ the density of the atmos- 
phere, assumed =1. Now, if ¢” represent the density of the air 
in the working cylinder at the moment of cut-off, and o” the den- 
sity at that of final escape, we shall have, by virtue of the same 
equation, for the pressure at escape (represented by p’’) 
pil=p (5) 
In this, if we substitute the foregoing value of p’, we obtain 
g! ¥ pe x 
p’=7() F) 
But, as in the nature of things, p” must equal p, this equation 
becomes, 
o! ¥ ov y o ¥ oll! % 
1 ()') > (9) = (e): 
Now, by Poisson’s second equation, 
6"=(O+4T) (5 } a nO 
ol\ 7 
Whence, also, 
lo q-1 
G'==( O+T) (F — 0, 
And we have no occasion to determine o’” and 9 
For o’ we proceed thus. In the last number of the Journal we 
have these two — 
7 sow. 
. T+9 Pao of mn 1 Q l 
== (S45)(Q)", st eA eat) 
+ aelehagae the value of m in the last expression, we obtain, 
after reduction, 
t=(1( evan 
ait) 
- by iain again in the preceding formula, we have 
finally, a, i 
w=(0+T)(s(arn)) — 2, 
Or, o=| (= moo)" (o4t) | i 
As the factors (9+4) 7 , and (O17), may be regarded as 
constant, we may represent their product by a single apeabole ol, as 
N, which being substituted will ine i the expression thus 
1- 
a"=N(<) 7 - 0, i 
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