MR. MACQUORN RANK1NE ON THERMO-DYNAMICS. 
173 
perature of saturation for its pressure. Having these data, we can solve numerically 
the following problem : — 
Proposition XIX. — Problem. To draw the curve of Free Expansion for vapour 
nascent under a given pressure. 
(Solution.) In fig. 24, let AB, parallel to OX, be the isothermal line of an aggre- 
gate of liquid and vapour at the pressure of evaporation P, corresponding to the tem- 
perature r, : let Ay,, By' be ordinates parallel to OY ; so that y, is the volume of 
Fig. 24. 
unity of weight of the liquid at this temperature, and v\ that of unity of weight of the 
vapour, at saturation. Let DF be a line drawn parallel to OX, at a distance repre- 
senting any lower pressure P 2 corresponding to the temperature r 2 . It is required to 
find the point where the curve of free expansion drawn from B intersects DF. 
Let y 2 be the volume of unity of weight of the liquid at the lower pressure and 
temperature, y 2 D an ordinate parallel to OY, and DA a curve representing the law of 
expansion of the liquid as the pressure and temperature increase. Draw the curves 
of no transmission DN, BL indefinitely prolonged towards X ; ascertain the indefi- 
nitely-prolonged area LBADN ; draw a curve of no transmission MC, cutting DF in 
C, such that the indefinitely-prolonged area MCDN shall be equal to the indefinitely- 
prolonged area LBADN ; then will C be the point required, where the curve of free 
expansion BC intersects the line DF. 
(Demonstration.) Unity of weight of the fluid being raised in the liquid state from 
the temperature 7 2 and corresponding pressure P 2 to the temperature r, and corre- 
sponding pressure P, ; then evaporated completely at the latter pressure and tem- 
perature ; then expanded without performing work, until it falls to the original press- 
ure P 2 ; then cooled at this pressure till it returns to the original temperature r 2 , at 
which it is finally liquefied ; — the area ABCD represents the expansive power deve- 
loped during this cycle of operations, which, as no work is performed, must be wholly 
expended in agitating the fluid, and reproducing, by friction, the heat consumed by 
the free expansion represented by the curve BC ; which heat is measured by the 
indefinitely-prolonged area MCBL ; which area is therefore equal to the area ABCD. 
