152 
MR. MACQUORN RANKINE ON THERMO-DYNAMICS, 
Suppose, for example, that the form assumed for AaD is a hyperbola, concave 
towards OY, and having the following equation, — 
P„= 
(4 7 a.) 
V 
-/3-V a ’ 
in which a and (3 are two arbitrary constants ; and let the ratio ^-=r. 
Then must the curve B&C be another hyperbola concave towards OY, having for 
its equation 
7 (47 b.) 
b ~r(3-V b 
The total expenditure of heat, per pound of air per stroke, in a perfect air-engine, 
is the latent heat of expansion from V A to V B , given by equation (45.). 
The heat to be abstracted by refrigeration is the latent heat of compression from 
V c to V D ; and is found by substituting in the same equation, the lower temperature 
T 2 for the higher temperature T x . 
The indicated work, per pound of air per stroke, being the difference between those 
two quantities, is found by multiplying the range of temperature by the difference of 
the thermo-dynamic functions <E> for the curves AD, BC, or by multiplying the latent 
heat of expansion by the efficiency, and has the following value : — 
T — T 
1 1 1 <2 
V 
E=H 1 -H 2 =(T 1 -T 2 ).(O B -a> A )=P 0 V 0 .^ 1 r^.hyp.log^. . . . (48.) 
- 1 n v A 
The heat alternately stored up and given out by the regenerator (supposing it to 
work perfectly) is to be computed as follows. Let the arbitrary manner in which 
volume is made to vary with temperature, on either of the curves DaA, C6B, be ex 
pressed by an equation 
V=¥.T, 
then the thermo-dynamic function O takes the form 
0=K V hyp. log (T+T.)+5* hyp. log Y.T ; 
and the total heat stored up and given out, per pound of air per stroke, is 
For example, if, as before, 
P„=: 
be the equation of the curve DA, then 
V„=, 
■'k.T 
dT. 
(49.) 
'/3-V a 
/3(T + T 0 ) 
■T + T, 
»( 1 + p 0 v) 
