144 
MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
and let us compare the forms of the indicator-diagrams without and with a regene- 
rator, for a perfect air-engine, working between given limits as to actual heat, defined 
by the isothermal curves QjQj, Q 2 Q 2 in fig 19- 
Fig. 19. 
The amount of expansion at the higher limit of heat being arbitrary, let us suppose 
it to be from the volume V A to the volume V B , corresponding respectively to the 
points A and B, and to be the same in all cases, whether with or without a regene- 
rator. 
The engine being without a regenerator, the diagram corresponding to the maxi- 
mum efficiency has but one form, viz. AVtcd, where Be, Ad are curves of no trans- 
mission. Hence, in this case, there must be an additional expansion, from the volume 
V B to the volume 
v - v * •(!)"> < 32 -) 
for the purpose merely of lowering the actual heat of the air without loss of heat ; 
and the engine must be made large enough to admit of this expansion, otherwise 
heat will be wasted. 
On the other hand, if the engine be provided with a perfect regenerator, any pair 
of curves of equal transmission passing through A and B will complete a diagram of 
maximum efficiency. The property of a pair of these curves being, as shown in Pro- 
position IV., that the difference of their thermo-dynamic functions, 
AF^=j^e?V when Q is constant^, 
is the same for every value of Q, it follows, that for a gas, according to the approxi- 
mate equation (23.), the property of a pair of curves of equal transmission is, that the 
volumes corresponding to the intersections of the two curves by the same isothermal 
curve, are in a ratio which is the same for every isothermal curve. Thus let V a , V* be 
such a pair of volumes, then this equation 
V,_Vb 
v a -v A 
( 33 .) 
