400 
Proceedings of the Royal Society of Edinburgh. [Sess. 
write (15) in the following approximate form, 
dx . x 
dt ' 2 1 
• ( 15 '). 
It 
of 
can easily be verified that when t is very small and x great the velocity 
any zero is given by 
dx m x 
dt = '~2t, 
(16). 
In each case, what we obtain in these equations is also the wave-velocity 
for an infinite train of waves of wave-length equal to that maintaining 
in the immediate neighbourhood of the point x at time t. 
§11. To obtain the velocity of the group of waves of wave-lengths 
approximately equal to A — that maintaining in the neighbourhood of x at 
time t — we put 
xH (x - \)H _ 0 
4/c6T 2 “ 4 k6T 2 “ T 
(17), 
which enables us to write down the group-velocity thus : — 
dx 87 TKbt — Xx 
dt X t 
(18). 
When t is very small and x great, the right-hand side of (18) is approxi- 
mately equal to — If in (17) we take the wave-length A as small 
compared with x, we can obtain the following approximate expression for 
A when t is great : — 
A- 
47 TKbt 
x 
(19). 
With this value for A, the right-hand side of (18) becomes ( 
x 
\ t 
Equations (15)-(19) show that in the two cases, when t is small and x 
great, and when t is great, the group-velocity is twice the wave-velocity ; 
which is in accordance with the theory of group- velocity given by Osborne 
Reynolds and extended by Lord Rayleigh. 
(. Issued separately May 15, 1909.) 
