in different forms of the Air-Engine. 171 
In the engine of Stirling, and all others resting on the same 
principle, we arrive at a similar form of expression for the eco- 
nomical ratio, though the expression for the work done assumes 
asimpler shape. In order to fix our ideas, we must first examine 
a theoretic case, which, in practice, it is somewhat difficult to 
realize. 
Let AB be an air-tight cylinder, in which ee 
a piston P, capable of moving without fric- + i 
tion, separates completely two equal masses al c lp c’ 
ra 

8 
of air, C and C’. Suppose that, by the al- 
ternate and instantaneous changes of temperature of these mass- 
es, the piston traverses the space LL’. When it reaches one of 
these limits, let the temperature of the air before it be v’, and of 
that behind it t,, Then, by an instantaneous change, let the for- 
mer temperature rise to 7”, the maximum, and the latter fall to 
t,, Which is the minimum. ‘There are but two volumes here to 
be considered, and we may represent the minimum by V and the 
maximum by V’. Put also P = pressure at minimum volume, 
and P’ == pressure at maximum volume, both being taken at the 
end of the stroke, and before the change of temperature. Then 


: +5 ‘ 5 
the maximum pressure will be P—; and in accordance with what 
b 
moh already seen, the positive power exerted at each stroke 
Wil be 
PV qi 7) 
y-ir L (= ) 
“4 
And the resistance a (.- (5) ) 
y-1 ie 
Whence =P¥ (Pa) (1-(5)" ) 
7-1\" bas 
But, in order to eliminate t’, so that the expression may con- 
tain only the maximum and minimum temperatures, which, being 
those of the source of heat and the refrigerator, may be suppose 
to be controllable, or known, we may take ; 
/ 
-1 
Pant; (=) Z whence 
i - WAT 
PH) =) (0 
y-1 Ty ¥ : 
Also, as in the former case, the mechanical equivalent of the 
heat imparted is, (vol. being constant during the heating, 
Pv (ot PW (envy a) ot 
~y-l vi 
Ty, 
ee | 
tT! — 7 
Whence Weou(1—(¥)"}=H(1-5)=8( “= ) 

M 7 
