269 
and that for all other values of # the functions y, , and y’, , 
vanish: which is equivalent to supposing that at the origin 
of ¢, and for a large number ¢ of wave-lengths (each = 2) be- 
hind the origin of z, the displacements and velocities of the 
particles are such as to agree with the following law of un- 
dulatory vibration, 
Y,, = 0 vers (22 — 2 at sin ), (3)’ 
but that all the other particles are, at that moment, at rest: 
it is required to determine the motion which will ensue, as 
a consequence of these initial conditions. The solution is 
expressed by the following formula, which is a rigorous de- 
duction from the equation in mixed differences (1) : 
ant sinin® explant pinto enf) dO; (4) 
nf{. 
¥.,=- (sin = : 
as 55 sin cos #—cos — 
an expression which tends indefinitely to become 
Mee 2 vers (2 x — 2atsin e 
2 ¢ sin (2x0—2at sind 
Patna yal aay Case 
s = ) 
as the number j increases without limit. The approximate 
values are discussed, which these rigorous integrals acquire, 
when the value of ¢ is large. It is found that a vibration, of 
which the phase and the amplitude agree with the law (3)’, 
is propagated forward, but not backward, so as to agitate 
successively new and more distant particles, (and to leave 
successively others at rest, if é be finite,) with a velocity of 
progress which is expressed by a cos = and which is there- 
fore less, by a finite though small amount, than the velocity 
Q 07 ‘ hacia’ 
of passage a — sin — of any given phase, from one vibrating 
; rn 
2a 
