PROPAGATION OF LONGITUDINAL WAVES—MACGREGOR. 89 
Art. IIL—On THE ELEMENTARY TREATMENT OF THE PROPAGA- 
TION OF LONGITUDINAL Wavers.—By Pror. J. G. 
Macerecor, D. Sc. 
Read Jenuary 9th, 1888. 
The best elementary treatment of the propagation of longitu- 
dinal waves known to me is that contained in Maxwell’s Theory 
of Heat, chapter XV. It is based on Rankine’s more difficult 
treatment of the same subject in his paper “ On the thermodyna- 
mie theory of waves of finite longitudinal disturbance,” pub- 
lished in the Philosophical Transactions of 1870, and in 
Rankine’s Miscellaneous Scientific Papers, page 530. “The kind 
of waves to which the investigation applies are those in which 
the motion of the parts of the substance is along straight lines 
parallel to the direction in which the wave is propagated, and the 
wave is defined to be one which is propagated with constant 
velocity, and the type of which does not alter during its propa- 
gation.” Maxwell’s investigation involves an error which vitiates 
his result, and it is the object of this paper to point out the error 
and to obtain the same result in a legitimate manner, 
Maxwell imagines a plane of unit area, which he calls the plane 
A, perpendicular to the direction of propagation of the wave, to 
move with the velocity of the wave. Behind it another plane, 
B, of the same area, moves with the same velocity, thus main- 
taining its distance from A. As both these planes move with the 
velocity of the wave, the velocity, the specific volume (7. e., volume 
of unit of mass), and the pressure, of the substance, at each of them, 
must remain the same. The same is true of any plane normal to 
the direction of propagation of the wave and moving with its 
velocity. During the passage of the wave portions of the sub- 
stance are continually passing through the planes A and B. 
Let V be the velocity of the wave, w, %, and p,, the velocity, 
specific volume, and pressure, of the substance at P, w., v, and 
px, the velocity, specific volume and pressure of the substance at 
