246 
Proceedings of the Royal Society of Edinburgh. [Sess. 
§ 6. The evaluation of (12) given by Lord Kelvin and repeated in § 14 
of my former paper is 
cos |^,'{ar -£/(&„)} ± | 
J= 2W(WWM]'' • • ' (13) ’ 
where k 0 determines the particular group of wave-trains which predominate 
at point x, at time t, t being great, though probably not very great. The 
ambiguous sign of the expression in the denominator is to be chosen so as 
to make the expression positive, and in the numerator the opposite sign 
is required. The relation between k 0 and x at time t is given by the 
equation 
a = { AK) + V(&o) P = m .... (14), 
where U is the group-velocity for waves of wave-length 2i r/k 0 . Putting 
now f(k 0 ) = Ak 0 n , and eliminating k 0 from (13) by means of (14), we obtain 
finally, as the value of the integral in (12), 
where 
t — 1 — — cos 
£2n 
B 
ii+i 
X n 
fa 
27 m{{n + l)Ap 
; B = 
n + 1 1 
(1 + n) n A » 
(15). 
The condition to which this evaluation is subject is given in § 14 of my 
former paper: — that the process of dispersion described in § 12 is exceed- 
ingly far advanced, and t therefore very great. The further condition 
given in Lord Kelvin’s paper of 1887, that the denominator of (13) must 
be very great, is also stated. But this can be proved to be inconsistent 
with the well-known condition to which (13) is subject in the case of 
deep-water waves (equation (14) of Lord Kelvin’s paper), namely, that 
2^- is very great. 
4a? J s 
When this expression for the effect of a single initial impulse is inserted 
in equation (10), the displacement f at any point x, at time t, due to the 
initial disturbance f(x) can be written in the form 
+ 00 
r 
r n+i ~i 
1 1 - 1 
- I dx 1 f(x 1 )(x - aq) 2w ^cos 
bd^ 
i 
1 + 
V 
-00 
L i n J 
( 16 ), 
the positive sign being taken when n lies between 0 and — 1. 
