284 
PROFESSOR W. J. M. RANKINE ON THE THERMODYNAMIC 
It is to be borne in mind that the last three equations represent a state of matters which 
may be approximated to, but not absolutely realized. 
Equation (25) gives the velocity with which a piston in a tube is to be moved inwards 
or outwards as the case may be, in order to produce a change of pressure from P to p, 
travelling along the tube from the piston towards the further end. Equation (25) may 
be converted into a quadratic equation, for finding p in terms of u; in other words, for 
finding what pressure must be applied to a piston in order to make it move at a given 
speed along a tube filled with a perfect gas whose undisturbed pressure and bulkiness 
are P and S. The quadratic equation is as follows : 
f- 2P 
y+ 1 2 
vr 
2S 
^-.^- 2 + P *=: 0 ; 
7-1 
P—2 
and its alternative roots are given by the following formula : 
— p i ZJll 
i>=P-i 
4S 
u 
M 7 " 
(y + P 
16S 2 
(29) 
The sign + or — is to be used, according as the piston moves inwards so as to produce 
condensation, or outwards so as to produce rarefaction. Suppose, now, that in a tube of 
unit area, filled with a perfect gas whose undisturbed pressure and volume are P and S, 
there is a piston dividing the space within that tube into two parts, and moving at the 
uniform velocity u : condensation will be propagated from one side of the piston, and 
rarefaction from the other; the pressures on the two sides of the piston will be 
expressed by the two values of p in equation (29) ; and the force required in order to 
keep the piston in motion will be the difference of these values ; that is to say, 
A iJ= 2«Vjk P +^ 
(30) 
Two limiting cases of the last equation may be noted : first, if the velocity of the piston 
Sm 2 . 
is very small compared with the velocity of sound, that is if is very small, we have 
A p nearly > • * (30 a) 
secondly, if the velocity of the piston is very great compared with the velocity of sound, 
7 p 
that is if 4^2 is very small, we have 
Ap nearly = 
(30 b) 
§ 12. Absolute Temperature . — The absolute temperature of a given particle of a given 
substance, being a function of the pressure p and bulkiness s, can be calculated for a point 
in a wave of disturbance for which p and s are given. In particular, the absolute tempe- 
rature in a perfect gas is given by the following well-known thermodynamic formula : 
r 
ps 
(J Cp — c,) ’ 
(31) 
and if, in that formula, there be substituted the value of s in terms of p, given by equa- 
