374 Propagation of Pulses in Elastic Media. 
@ point in the line AB, in advance of C, where the particles are 
at rest, not having yet ‘felt the influence of the approaching pulse ; 
and let the two imaginary points C and D be conceived to move 
with the same velocity as the pulse. Then each travels over the 
same space and through the same number of particles in a given 
time. Consequently while C moves over a space equal to that 
occupied by a particle in its natural state, it only moves through 
one particle in its most condensed state; that is to say, while C 
moves over the space mn, it moves, relatively to the particle 
through which it passes, only over sm. Consequently the parti- 
cle itself moves in the same time over a space equal to ms. 
Hence when mm represents the velocity of the pulse, ms repre-- 
sents the final er which the pulse gives to every particle 
through which it passe 
Let H be the athena subtangent ; or the length of the 
column of particles of the natural density whose weight is equal 
to the elastic force of a particle in its natural state, and let H+A 
be the length of a similar column whose weight is equal to the 
elastic force of a particle at the maximum density.—Since the 
_ we te by a particle is inversely as the compressing force, 
lies suiitH+A: Hor ms+snisn::H+h: Hi 
consequently msisn:ih:H. Whence H=-"" -: 
The force which accelerates all the particles in Paves of €, 
which have felt the influence of the pulse but have not yet 
reached their maximum velocity, is the difference of the elastic 
forces which correspond to the natural and the maximum densities. 
This is a constant force, and must evidently be that force which 
is competent to give the velocity ms to all the sebeeel in any 
in the time in which the pulse runs over that s 
us suppose the pulse runs over the space h. ‘Then the en runs 
over / in the time in which the difference of those elastic forces 
will give to all the particles in A the velocity ms. But the dif- 
ference of the elastic forces is equal to oa weight of all the par- 
ticles in h. Therefore the pulse runs over & in the time in 
which those Ag would in falling by their own weigmt ~ 
quire the velocity m 
e time in which a falling body acquires the velocity ms 
is to the time in whieh it would acquire the velocity of the 
or mn, as ms to mn, and the spaces over which the pulse would 
rin in Ay times are as the times and therefore as ms to ™ 
re putting S for the space which the pulse would ran 
over while a falling ao @, would oie the velocity of the pulse; 
have ms imn::h:S. Whe 
oO ads ae pn _msss ms+snXh- _sn smh pg me sit ee 
ms 
