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Proceedings of the Koyal Society of Edinburgh. [Sess. 
every point except the origin ; leaving our assumption to be tested after- 
wards by results. This is equivalent to saying that the main effect at any 
point is produced by a small portion of the total number of trains, whose 
wave-velocities are nearly equal and whose points of equal phase coincide 
with the point of the medium considered. Other trains, whose phases at 
the point considered are nearly equal to that of the trains whose phases 
coincide there, contribute effectively to the resultant disturbance ; but the 
effects of the remaining trains disappear by mutual interference. As we 
pass from point to point of the medium at any time, the mean period of the 
effective trains at each point, and the phase at which their main agreement 
occurs, vary continuously. We can therefore speak of a certain wave-period 
which is the mean period of all the trains whose coincidence of phase at a 
given point determines the displacement of the medium at that point, as 
the predominant period at that point ; and our problem is to determine at 
what point of the medium any specified wave-period will be the predominant 
period at any time. 
§ 10. If k be the mean period of an infinite number of trains of waves 
whose velocities are nearly equal to the wave-velocity corresponding to k, 
we require to know at what point of the medium these trains will agree in 
phase at any time. The equation to any particular phase may be written 
in the form 
k{x - tf(k)} = c . ..... (2), 
where c is a constant ; and the distance Sx between a point of phase c on 
any of the neighbouring trains and the point x is given by the equation 
kBx+ [x-tf(k) — tkf(k))Sk = § . . . (3). 
From this we see that the distance Sx between points of equal phase is zero 
for all values of Sk less than a certain value, provided 
* - tf(k) - tkf(k) = 0 (4). 
This equation therefore determines the place at which k is the predominant 
period at time t ; and it may be written in the form 
x={f(k)+kf\k)}t=m .... (5), 
where U is the group- velocity corresponding to the wave-period k. We 
may define the group-velocity as a function of the wave-length X which 
determines the velocity of a point coinciding at each instant with the point 
of agreement of phase of the infinite number of trains of wave-lengths very 
slightly differing from X. The group-velocity for the mean wave-length of 
a very large number of trains of nearly equal wave-lengths might more 
accurately be termed the velocity of their coincident phase, or simply the 
