252 
Proceedings of the Royal Society of Edinburgh. [Sess. 
by equation (9) that the amplitude of the disturbance is of the order 1/x 2 . 
When t becomes comparable with x, equation (20) shows that the amplitude 
rises to a maximum of the order l/x h , and begins to diminish. When t 
becomes great compared with x, the amplitude becomes vanishingly small. 
§ 15. But for n negative, the rate of rise to the maximum value is much 
quicker than the rate of subsidence from it, for in this case the successive 
waves of the disturbance at the point observed are of smaller and smaller 
wave-length, and take relatively longer to pass, owing to their smaller wave- 
velocity. When we observe the whole disturbance at any particular time, 
however, we find that the greater part of the medium sensibly disturbed lies 
beyond the point where the maximum displacement occurs — that is, towards 
the side where the greater wave-lengths predominate. The general form of 
the amplitude curve at any point x, throughout all time, is shown in fig. 1 ; 
and the general form of the amplitude curve throughout the medium at 
any time, t, is shown in fig. 2. These curves are reproduced from Professor 
Burnside’s paper, referred to in § 2 above, where they were drawn to 
represent the particular case of deep-water waves ( n = — J), but it is clear 
from the equations given below that we can obtain from them, by suitable 
modification of scales used, an indication of the general features of the 
wave-disturbance in any medium for which n is negative. The equations 
to these curves are : 
£= cz€~ d ' z for fig. 1, where !p==^ .... (33), 
£ = -^he - * for fig. 2, where z = x™ .... (34), 
z~r 
c, cl, c\ d' being constants, and the absolute value of n being taken 
throughout. 
When n is positive, the amplitude curve at any point x, throughout all 
time, is given by 
$ = * , where z = ..... (35), 
and the amplitude curve throughout the medium at any fixed time, t, is 
given by 
£ — c\z~y e~ d ' lZ , where z = x™ .... (36). 
These equations, being respectively of the same types as (34) and (33) above, 
show that for n positive and less than unity fig. 1 may be taken as 
representing the general features of the amplitude curve throughout the 
medium at any fixed time, and fig. 2 may be taken as representing the 
general features of the amplitude curve at any fixed place x, throughout all 
