§8. Look at the values shown in the previous table for the three factors which constitute ;-we see that the first factor (col. 2) decreases slowly from a=0 to a=; the second factor (col. 5) alternates between +1 and -1 with increasing distances (semi-wave-lengths) from zero to zero as a increases.

=

The third factor (col. 6) increases gradually from €-f2** at x=0, to 1 at x=∞. At a 50h, the third factor is 99. which is so nearly unity that the diminution of amplitude is. for all greater values of a, practically given by the first factor alone, which diminishes from 2 at x=50h, to 0 at x=∞.

§ 9. The diagrams hitherto given, figs. 1, 2, 3, may be called space-curves, as on each of them abscissas represent distance from the centre of the disturbance. Fig. 4 is a time-curve (abscissas representing time) for x=2h. It represents a very gradual rise, from t=0 to t='6, followed by a fall to a minimum at t=2.8, and a succession of alternations, with smaller and smaller maximum elevations and depressions, and shorter and shorter times from zero to zero, on to t∞. The same words with altered figures describe the changes of water level at any fixed position farther from the centre of disturbance than 2. The following table shows, for the case = 100h, all the times of zero less than 71h, and the elevations and depressions at the intermediate times when the second factor (col. 5 of §7) has its maximum and minimum values (+1). These elevations and depressions are very approximately the greatest in the intervals between the zeros, because the third factor (col. 6, §7) varies but slowly, as shown in the first column of the present table.

h=1; a=100; p= =100·005.h; 0=tan-1,

Times of Zero
and of

101

-45° 18'.

99

Approximate

Approximate
Maximum

Times of Zero
and of

-42

Approximate
Maximum

Maximum

Elevations and

Depressions.

Depression.

Elevation and Depressions.
Depression.

Approximate
Maximum

Elevations and