MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
163 
(Solution.) In fig. 22, through the given point B draw the straight isothermal line 
AB corresponding to the absolute temperature r„ and cutting the ordinate corre- 
sponding to the volume of total liquefaction in A. Through A, according to the last 
proposition, draw the curve of no transmission, ADM. Let EDO be any other 
isothermal line, corresponding to the absolute temperature r 2 , and cutting the curve 
AM in D. Draw isothermal lines ab, edc at indefinitely small distances from AB, 
Fig. 22. 
EDC respectively, corresponding to the same indefinitely small difference of tempe- 
rature Sr. Draw the ordinates V d gD, V b 6B ; then draw the ordinate V c cC at such a 
distance from V d g(D, that the indefinitely small rectangles DCcd, AB&a shall be equal. 
Then as the difference Sr is indefinitely diminished, C approximates indefinitely to a 
point on the required curve of no transmission, BN. 
This is Proposition III. applied to aggregates, mutatis mutandis. 
The symbolical representation of this proposition is as follows : — let P, and P 2 be 
the pressures of the aggregate of liquid and vapour corresponding respectively to the 
temperatures r l and r 2 ; then the following expressions for the difference between the 
thermo-dynamic functions of the curves AM, BN are equal, 
A®=^ ! (V c -V d )=^(V b -») (66.) 
(43.) Corollary. (. Absolute Maximum Efficiency of Vapour-Engines.) 
If the volume V B be that corresponding to complete evaporation at the tempera- 
ture r 15 that is to say, if 
V B =t/, 
then the curve BCN will represent the mode of expansion under pressure, of vapour 
of saturation in working an engine, and will be defined by the equation 
V C -V D = 
rfP, 
dr 
(v 1 — v) 
dr 
Y 2 
(67.) 
