MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 131 
The equation of a curve of free expansion is 
d(¥+PV) = 0 (17 a.) 
(16.) Corollary . — In fig. 11, the same letters being retained as in the last figure, 
through G draw an isothermal curve QjQ,, which the line P h H produced cuts in h ; 
Fig. 11. 
and from h draw the indefinitely-prolonged curve of no transmission, hn. Then be- 
cause, by the proposition just proved, the areas P g GHP h and MGHN are equal, it 
follows that the indefinitely-prolonged area, MG/m, which represents the latent heat 
of expansion at the constant actual heat Q„ from the volume V G to the volume V A , 
exceeds P g GAP h , by the indefinitely-prolonged area NH/m, which represents the 
heat which the substance would give out, in falling, at the pressure P H , from the 
actual heat Q, to the actual heat corresponding to the point H on the curve of free 
expansion passing through G. Subtracting from this area the excess of the rectangle 
P H V A above the rectangle P G V G , we obtain the excess of the area MG/m above the 
area V g GAV a . 
This conclusion may be thus expressed : — Let Q 2 be the actual heat for the point 
H ; qp the ratio of specific heat at the constant pressure P H to real specific heat ; 
then 
^rfQ-P H V s +P G V 0 = (qA_ i)f v ‘prfV(fo r Q=Q,); 
otherwise: — I Vc/P=Q 1 (F A — F G ) 
r Jp h 
Equation (18.) may be used, either to find points in the curve of free expansion 
which passes through G, when the isothermal curves and the curves of no transmission 
are known ; or to deduce theoretical results from experiments on the form of curves 
of free expansion, such as those which have been for some time carried on by 
Mr. Joule and Professor William Thomson. 
Considered geometrically, these experiments give values of the area NHk. 
The area P G G/iP H =f P WP 
J Ph 
s 2 
1 
( • ( 18 .) 
J 
