282 
PROFESSOR W. J. M. RANKINE ON THE THERMODYNAMIC 
Equation (15) also serves to calculate the pressure j? corresponding to a given velocity 
of disturbance u. It may here be repeated that the linear velocity of advance is «=mS 
(equation 1). 
§ 9. Application to a Perfect Gas . — In a perfect gas, the specific heat at constant 
volume, c s , and the specific heat at constant pressure, c p , are both constant ; and conse- 
quently bear to each other a constant ratio, whose value for air, oxygen, nitrogen, 
Cs 
and hydrogen is nearly T41, and for steam-gas nearly L3. Let this ratio be denoted 
by y. Also, the differential coefficients which appear in equations (13) and (13a) have 
the following values : — 
dr t s s 
dp p~ J(c p — c s )~~ 'J(y — l)c s ’ 
dr t p p 
ds~~ s J(cp — c s ) — J(y — l)c 4 - ’ 
(16) 
dp p J (c p — c s ) J(y—\)c s 
dr r s s 
When these substitutions are made in equation (13), and constant common factors can- 
celled, it is reduced to the following : 
dp . {m 2 s—yp} = 0 (17) 
pi 
But according to the dynamical condition of permanence of type, as expressed in equa- 
tion (8), we have m 2 s=m 2 S + P— p; whence it follows that the value of the integral in 
equation (17) is 
C\lp . { m 2 S +P—(y-j-l)p}= (m 2 S + P)( p 2 —Pi)— ^r 1 ( pi —p\ )=0; 
Jpi 
which, being divided byp 2 —p 1} gives for the square of the mass-velocity of advance the 
following value : 
^{(r+U^-r} (13) 
The square of the linear velocity of advance is 
ff 2 =m 2 S 2 =s|(y+l).^- 1 -P| (19) 
The velocity of disturbance u corresponding to a given pressure p , or, conversely, the 
pressure p corresponding to a given velocity of disturbance, may be found by means of 
equation (15). 
Such are the general equations of the propagation of waves of longitudinal disturbance 
of permanent type along a cylindrical mass of a perfect gas whose undisturbed pressure 
and bulkiness are respectively P and S. In the next two sections particular cases will 
be treated of. 
§ 10. Wave of Oscillation in a Perfect Gas . — Let the mean between the two extreme 
