142 
MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
the points of contact, BandD, draw the isothermal carves, CLQi cutting the diagram 
in A and B, and Q 2 Q 2 cutting it in C and D. Then because, during the whole of the 
change from D through A to B, the working substance is receiving heat, and during 
the whole of the change from B through C to D, emitting heat, the regenerator can 
have no action above the isothermal curve CLQ 15 nor below the isothermal curve 
QoQa 1 
The whole of the diagram between these curves is to be divided by indefinitely- 
close isothermal curves into stripes like abed-, and the saving of heat effected by the 
layer of the regenerator corresponding to each stripe ascertained in the manner de- 
scribed, when the whole saving may be found by summation or integration. 
The symbolical expression of this result is as follows. Let the points of contact, 
B, D, which limit the action of the regenerator, and the corresponding quantities of 
actual heat, Q 1} CL, be found, as in Proposition IX., by means of the equation dF=0. 
Then 
/7 Ti 1 />Qi /K ,/p ,7 V\ 
the saving of heat =j Q^QA (t+Q^q- m) d Q ( 30 -) 
v \aC2 W 2 V 
d. F 
care being taken, when has different values for the same value of Q, correspond- 
ing respectively to the two sides of the diagram, to choose the smaller in performing 
the integration. 
(28.) Corollary. — It is evident that the regenerator acts most effectually, when the 
outlines of the indicator-diagram from A to D, and from B to C, are portions of a 
pair of curves of equal transmission (determined as in Proposition IV.) ; for then, if 
the operation of the regenerator is perfect, the changes from B to C and from D to A 
will be effected without expenditure of heat ; the heat transmitted from the working 
substance to a given stratum of the regenerator, during any part such as be, of the 
operation BC, being exactly sufficient for the corresponding part, da, of the operation 
d,Y 
DA. In this case for each value of Q between CL and Q 2 , has the same value at 
either side of the diagram. 
In fact, the effect of a perfect regenerator is, to confer upon any pair of curves of 
equal transmission the properties of a pair of curves of no transmission. 
(29.) Proposition XL— Theorem. The greatest efficiency of a thermo-dynamic en- 
gine, working between given limits of actual heat, with a perfect regenerator, is equal 
to the greatest efficiency of a thermo-dynamic engine, working between the same limits 
of actual heat , without a regenerator. 
(Demonstration.) In fig. 18, let Q,CL, Q 2 Q 2 be the isothermal curves denoting the 
given limits of actual heat. Let AD, BC be a pair of curves of equal transmission 
of any form. Then by the aid of a perfect regenerator, the whole of the heat given 
out by the elastic substance during the operation BC may be stored up, and given 
