138 
MR. MACQUGRN RANKIN E ON THERMO-DYNAMICS. 
will be the lines of maximum and minimum pressure. Let HE and LG be the 
volumes occupied by the cushion at the maximum and minimum pressures respectively : 
draw the curve EG, such that its co-ordinates shall represent the changes of volume 
and pressure undergone by the cushion during a revolution of the engine. Let 
be any line of equal pressure, intersecting this curve and the apparent indicator- 
diagram ; so that K/o K d shall represent the two volumes assumed by the whole 
elastic body at the pressure OK, and KF the volume of the cushion at the same 
pressure. On this line take 
W>=dX)= KF; 
Fig. 15. 
then it is evident that B and D will be two points in the true indicator-diagram ; and 
in the same manner may any number of points be found. 
The area of the true diagram ABCD is obviously equal to that of the apparent 
diagram abed. 
(23.) Proposition IX. — Problem. The true indicator-diagram of a thermo-dynamic 
engine worked by the expansion and contraction of a substance which does not change its 
condition , and. without a regenerator % being given , it is required to determine the effi- 
ciency of the engine. 
(Solution.) In fig. 15, let A aa' Bb'bA be the 
given true indicator-diagram. Draw two curves 
of no transmission, AM, BN, touching this 
figure at A and B respectively, and indefinitely 
produced towards X. Then during the process 
denoted by the portion Aaa'B of the diagram 
the elastic substance is receiving heat, and the 
mechanical equivalent of the total quantity re- 
ceived is represented by the indefinitely-pro- 
longed area MAaa'BN ; during the process denoted by the portion Bb'bA of the dia- 
gram, the substance is giving out heat, and the mechanical equivalent of the total heat 
given out is represented by the indefinitely-prolonged area MA6//BN ; while the 
difference between those areas, that is, the area of the indicator-diagram itself, re- 
presents at once the heat which permanently disappears and the motive power given 
out. The Efficiency of the engine is the ratio of this last quantity to the total heat 
received by the elastic substance during a revolution ; that is to say, it is denoted 
by the fraction, 
area Aaa'Bb'bA 
area MAao'BN* 
To express this result symbolically, find the limiting points A and B by combining 
the equation of the indicator-diagram with the general equation of curves of no trans- 
mission, viz. — 
d¥=0. 
Then draw two indefinitely-close and indefinitely-prolonged curves of no trans- 
