464 Proceedings of the Royal Society of Edinburgh. [Sess. 
and velocity 2jjnr. The zeros are numbered in the order of their coming 
into existence, so that we can study the history of each perceptible wave, 
or of each group defined as the part of the whole disturbance which has a 
given wave-length. According to the argument of § § 20, 21 above, the 
wave-trains are initially in agreement of phase near the origin, and, as 
their points of predominance separate from each other, the wave-length of 
the resultant curve in this case increases continuously as we pass from the 
middle outwards. It is clear from the diagrams that we can take twice 
the perceptible half-wave between ^any two zeros as the wave-length pre- 
dominating somewhere near the point of maximum displacement between 
the zeros, this being a very close approximation when the wave-length does 
not vary quickly in the neighbourhood. The following table of numbers, 
calculated from the numerical results from which the diagrams were drawn 
wherever convenient, or taken direct from the diagrams themselves, shows 
that each perceptible wave-length of the disturbance appears at each time 
of observation at a distance from the place of the original disturbance 
corresponding to its group-velocity. Column 1 gives the diagram observed 
in each case ; column 2 gives the zeros twice whose distance apart is used 
as the wave-length ; column 3 gives the actual position of the maximum 
or minimum between the zeros, from the original calculations or from the 
diagrams ; and column 4 gives its position approximately by the group- 
velocity theory of § 10, i.e. column 4 gives group- velocity multiplied by 
the time. 
Diagram. 
Zeros. 
Position of Maximum. 
On the Diagram. 
By Group- Velocity. 
2 
1 and 2 
1-3 
1 
3 
1 „ 2 
3 
3-4 
4 
1 „ 2 
5 
7*3 
4 
2 „ 3 
2 
2*3 
6 
4 „ 5 
19-8 
19-9 
6 
5 „ 6 
15-1 
15-2 
6 
6 „ 7 
12-2 
12-2 
6 
7 „ 8 
10-3 
10-3 
6 
8 „ 9 
8'8 
8-8 
6 
9 „ 10 
8-0 
8*0 
An examination of the curves makes it clear that the agreement is practi- 
cally perfect where the change of wave-length between the zeros is slight ; 
and when the change of wave-length is considerable, as in the foremost 
