250 Proceedings of the Royal Society of Edinburgh. [Sess. 
tained in equations (25)-(31), for our initial form of disturbance l/(a 2 -{-x 2 ), 
as follows. Equation (28) shows that the wave-length associated with the 
maximum disturbance at any point x of the medium depends only on the 
form of the initial disturbance, and is independent of the dispersive nature 
of the medium or of the part of the medium considered. The greatest 
amplitude of disturbance, however, at any point of a particular medium falls 
off according to the inverse square root of the distance of the point from 
the centre of the. initial disturbance, as we see by equation (26). 
It was shown in my former paper that the position of any given wave- 
length in a wave-system changes uniformly at the group-velocity corre- 
sponding to that wave-length. Hence, or by equation (25) above, we see 
that in any one medium the time which elapses before the disturbance at 
any point reaches its maximum increases uniformly according to the dis- 
tance of the point observed from the place of the original disturbance. If 
we observe the displacements at points corresponding to a fixed value of x 
in several dispersive media, the times of the maximum displacements vary 
from one medium to another according to equation (25) ; and, by (28), the 
wave-length, and therefore the velocity of this maximum in each medium 
depends only on the form of the initial disturbance. Its amplitude in each 
medium varies as x~K If we observe all the media at a fixed time, the 
places of maximum displacement, and the amounts, vary from one medium 
to another according to equation (30). 
As to the beginning of wave-disturbance, we see by § 1 2 of my former 
paper that there is practically instantaneous propagation of disturbance 
from the middle to each point of the medium, by wave-trains whose group- 
velocities are very great. 
§ 13. The association of a certain wave-length with the maximum 
disturbance at any point in any medium, and also of a distinct wave-length 
with the maximum disturbance at any time for each medium, is in accord- 
ance with the general views of wave-propagation expressed in § 21 of my 
former paper, and illustrates the fact that the amplitude associated with 
any chosen wave-length at any time depends on the manner in which the 
energy is distributed initially among the effective Fourier trains. This is 
made clearer by an examination of the law of falling off of the amplitude 
of disturbance corresponding to a definite wave-length, by equation (20). 
The position in the wave-system of the wave-length X, at time t, is given 
by equation (28), and from this equation combined with equation (20) we 
find that the amplitude associated with wave-length X is given by 
a _ J 2 ? T?l \ * 1 c ~Y a 
a ( {n+ \ )BA j x l A a ( nXx ) j€ 
(32), 
