244 
Proceedings of the Royal Society of Edinburgh. [Sess. 
which indicates that each particle of the medium performs a simple 
harmonic motion, of period Z-rr/b, and of amplitude equal to the initial 
displacement of the particle ; in exactly the same way as a row of Reynold’s 
disconnected pendulums would swing after being held at rest in a displaced 
position and then let go. The form of equations (5) and (6) shows that 
there is no resolution of the initial disturbances by the medium according 
to wave-length whereby effects due to the separate constituent trains are 
observable at distinct parts of the medium. 
§ 4. Taking now the particular initial conditions 
! = ° ' • • K 1 ) repeated], 
where a is a constant, in the general case where V = A k n , we shall consider 
first the solution corresponding to equation (2) above : 
OO +00 
^ — [dkfdx ,—^ — - cos k(x - x,) cos Ak n+1 t . . . (7). 
7 T J J a~ + X , 2 
0-00 
The integration with respect to x 1 can be performed by means of the well- 
known integrals 
+ 00 +00 
dx 1 cos /foq 
a 2 + X j 2 
-00 -00 
the result being given by the equation 
£ ~ ak # j dx Y sin Jcx l _ q _ 
a ’ I a 2 + X x 2 
oo 
£=- fd/ce- ak cos /cx cos AJc n+1 t .... (8). 
a I 
o 
By expanding cos A k n+1 t, and integrating the series term by term, we obtain 
finally the following value for f : 
where 
(-i) r (A tr 
a 2 rl 
j _ i x 
x = tan x — 
a 
T{2r(n + 1) + 1 } 
( a 2 + x 2 ) nr+r+i 
cos [{2 r(n + 1) 4- 1 }x] 
(9). 
This expression can be used to calculate the displacement at any point x, 
with moderate labour so long as t is small compared with x. As t increases, 
however, the series becomes sluggishly convergent, and even ultimately 
divergent * when n>0, so that it is necessary to adopt another method of 
evaluating the right-hand side of (7) when t is of the same order of 
magnitude as, or greater than, x. 
* See Lord Rayleigh, Phil. Mag., July 1909. 
