90 PROPAGATION OF LONGITUDINAL WAVES—MACGREGOR. 
B, and Q and Q, the masses of the substance passing through A 
and B respectively, in a direction opposite to that of the wave, 
in unit of time. 
Maxwell first shows, in an unexceptionable manner, that 
Qi "Q =O (Say) eae he en 
(we retain Maxwell’s numbering of his equations), and that 
u, = U — Qu, and u, = U — Qy,......() 
He then points out (1) that the substance between the planes 
A and Bis continually acted upon by a resultant force equal to 
p.—p, in the direction of the wave, and (2) that the momentum 
of the substance between these planes, in the direction of the 
wave, increases at a rate equal to Q (wi—wuz) per unit of time. 
“ The only way in which this momentum can be produced,” he 
says, “is by the action of the external pressures p, and p.,” and 
hence in the earlier editions of the Theory of Heat, he puts 
P= p= OG =) ee (6), 
thus applying Newton’s Second Law of Motion. He then sub- 
stitutes in this equation the above values of uw, and w,, and by a 
ship finds that 
P—pP, = Q? (u—m) .. 1.05. @); 
and consequently 
PLO in pA OP aes ne (8). 
It follows that, as A and B are any planes whatever, normal to 
the direction of the wave, and moving with its velocity, if p and 
v are the pressure and specific volume of the substance at any 
such plane, we have 
p + Q’v = const. 
Hence, since for small changes of volume of actual substances 
the increase of pressure is proportional to the decrease of volume, 
the kind of wave under consideration is proved to be possible 
for actual substances, provided the changes of pressure and 
volume involved be small. Expressing, therefore, the elasticity 
(EK) in terms of v and Q, by the aid of equation (7), 
he obtains 
E = 0 Q, 
and deduces at once the important result, 
U’ = Ev. 
