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Proceedings of the Royal Society of Edinburgh. [Sess. 
wave-length and period as the effective Fourier trains which predominate in 
the neighbourhood. It is clear also, from our equation, or from the considera- 
tions of § 12, that the correspondence of wave-lengths referred to becomes 
more and more perceptible and the approach to a regular sine curve in each 
neighbourhood more and more close as time goes on ; for the predominant 
period of the Fourier trains varies more and more slowly as we pass along 
the £ curve at later and later times. In this case therefore the demonstra- 
tion of a coincident-phase-velocity for the constituent wave-trains carries 
with it a demonstration of group-velocity for each part of the resultant wave- 
curve where a definite wave-length is observed. If, following Professor 
Lamb, we understand by “ group ” in this connection “ a long succession of 
waves in which the distance between successive crests and the amplitude 
vary very slightly,” our result would mean that, whereas a certain wave- 
length X is observable in the group at place x at time t , this particular wave- 
length will be found at any later time t' at a place x' given by the equation 
x=x + V(£ - t) ..... (15), 
where U is the group- velocity corresponding to the wave-length X. In 
this sense we can speak of a group-velocity with reference to each small 
part of the original “ group,” each part having a slightly different group- 
velocity from contiguous parts ; but the “ group ” as a whole has not any 
definite “ group-velocity,” unless we define its group-velocity to be the mean 
value of the group- velocities of its parts. Such a “ group,” however, quickly 
loses definite marks by which it could be recognised on account of its con- 
tinual extension by the process of § 12. 
§ 16. It is important to observe from equation (14) that each wave of 
the resultant displacement curve moves at each instant with the velocity 
corresponding to its length. Thus in his paper of 1887, referred to in § 6 
above, Lord Kelvin remarks : “ The result of our work will show T us that 
the velocity of progress of a zero, or maximum, or minimum, in any part of 
a varying group of waves is equal to the velocity of progress of periodic 
waves of wave-length equal to a certain length, which may be defined as 
the wave-length in the neighbourhood of the particular point looked to in 
the group (a length which will generally be intermediate between the 
distances from the point considered to its next neighbour corresponding 
points on the preceding and following waves).” To illustrate this for any 
medium in which the group-velocity is positive, let us take V = A k n , where 
n may be positive or negative, but not less than — 1. Equation (8) becomes 
in this case 
x = (1 + n)A k Q t = \Jt 
. ( 16 ); 
