MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
133 
definitely to the sum of the areas MA am and raaBN ; the ultimate ratio of which 
sum to the area MAara is therefore the required ratio of the specific heats. Now 
w«BN, as qq approaches QQ, approximates indefinitely to the latent heat of the small 
expansion V B — V A at the actual heat Q, and this small expansion bears ultimately to 
the increment of pressure P a — P A , the ratio of the subtangent of the isothermal curve 
QQ to its ordinate at the point A. 
The symbolical expression of this proposition is as follows : — Let c$Q denote the 
indefinitely small difference of actual heat between the isothermal curves QQ, qq ; 
W the indefinitely small variation of volume V B — V A ; &P the indefinitely small 
K K 
variation of pressure P a — P A ; -y^Q, y- the quantities of heat required to produce 
the variation &Q, at the constant volume V A , and at the constant pressure P A re- 
spectively. 
d P 
Then SV=-^p=-^p.SQ; 
~dV ~dV 
and 
consequently 
Kp 
k • 
5Q=Y v -»Q+qS- 8v ={t 
(19.) 
equations agreeing with equation 31 of a paper on the Centrifugal Theory of Elasti- 
city before referred to. 
(18.) First Corollary . — -As the curves AM, am, BN approximate indefinitely to- 
wards parallelism, and the point a towards C, where am intersects AB, the ratio of 
the areas MABN : MAam, approximates indefinitely to that of the lines AB : AC, 
which are ultimately proportional, respectively, to the subtangents of the isothermal 
curve and the curve of no transmission passing through A. Therefore, 
K P Subtangent of Isothermal Curve . 
K v Subtangent of Curve of No Transmission \ v 
(19.) Second Corollary. — Velocity of Sound. The subtangents of different curves 
at a given point on a diagram of energy being inversely proportional to the increase of 
pressure produced by a given diminution of volume according to the respective curves, 
are inversely proportional to the squares of the respective velocities with which waves 
of condensation and rarefaction will travel when the relations of pressure to volume 
are expressed by the different curves. Therefore, if there be no sensible transmission of 
heat between the particles of a fluid during the passage of sound, the square of the 
velocity of sound must be greater than it would have been had the transmission of 
heat been instantaneous in the ratio of the subtangent of an isothermal curve to that 
