Determination of the Velocity of Sound. 453 
sake of simplicity, assume the disturbance to be of the type which 
satisfies the equation z,=A, cos (ké), A, being a function of r 
but not of ¢. By substituting this in the preceding differential 
equation, we obtain 
—KA,=mf"( h).(A,-1—2A,+ 4,41) 
+mf'(2h) .(A,-2—2A,+ A,+2) 
+ mf"(3h) . (A,-s—2A,+ A,+3) 
This being a linear equation of partial differences, its solution 
will be of the form 
A,=Ca*" + Cla-2", Sh) S Merah Mpa CE) 
the quantity « being such as to satisfy the equation 
—K=mf'(h) . (a—a-")? + mf"(2h) . (a®—a-2)? +... (2) 
These results may be exhibited in a more simple form by 
writing «—a-!=2 Y —1 sin @, which reduces them to 
$k? = mf" (h) .sin?@ + mf" (2h) sin?26 + mf" (3h) .sin?80+... (3) 
and 
A= DA Con(erO ea) 5 O28 Sn Shae “ys, 7 (4) 
*. &, =2A cos (278 + a) cos (kt) 
=A cos (kt—2r0—a) +A cos (kt +2r0 +a). 
This is the general result for the type of wave which we have 
assumed ; and it indicates that there may be two waves of that 
type travelling in opposite directions. For our purpose it will 
be sufficient to preserve one of them. Hence we have 
a= Acus (ki=278). thee Xe. te sO) 
From this equation it results that if \ be the length of a wave, 
and v the velocity of its transmission, 
ah 
d= >»? . . . . . . . . . . (6) 
and 
Sa — ; — aig =) 
v= atah vin.4 Vf (1). (F") +22. 6120). (FE 
Brent? ; 
48, f(3n). (U2? sha ae MR AT) 
5. Now in the case of all sounds which are audible to human 
ears, \ is immensely larger than 4; and consequently for all 
dibl d sin 6 arb sin 20 
audible sounds, —-=1, —9q 
city of transmission of sounds of every pitch, audible to our 
organs of hearing, though not absolutely the same, is sensibly the 
Phil. Mag, 8, 4, Vol. 19, No, 129, June 1860, 2H 
=1, &e.; and hence the velo- 
