194 Proceedings of Royal Society of Edinburgh. [sess. 
factor (col. 6) increases gradually from e - * 2 / 712 at x = 0 , to 1 at 
x=ao. At a? =50 h, the third factor is '99, which is so nearly 
unity that the diminution of amplitude is, for all greater values of 
x, practically given by the first factor alone, which diminishes 
from '2 at x = 50 h, to 0 at x = co . 
§ 9. The diagrams hitherto given, figs. 1, 2, 3, may he 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 = 2 h. It represents a very gradual rise, 
from t — 0 to t= '0, 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 = oo . The same words with altered figures 
describe the changes of water level at any fixed position farther 
from the centre of disturbance than x = 2. The following table 
shows, for the case a =100 h, all the times of zero less than 71//, 
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 approxi- 
mately the greatest in the intervals between the zeros, because the 
third factor (col. 6, § 7) varies hut slowly, as shown in the first 
column of the present table. 
fe=l; * = 100; p= 100005.7t ; 6 = tan-' M = 45° 18'. 
-t2 
€ P 2 
Times of Zero 
and of 
Approximate 
Maximum 
Elevation and 
Depression. 
Approximate 
Maximum 
Elevations and 
Depressions. 
-«2 
€ P 2 
Times of Zero 
and of 
Approximate 
Maximum 
Elevation and 
Depression. 
Approximate 
Maximum 
Elevations and 
Depressions. 
•9922 
8*333 
+ •1403 
•7718 
50-90 
+ •1091 
15-33 
0 
52-42 
0 
•9616 
19*80 
- -1360 
*7478 
53-90 
- -1058 
23*43 
0 
55 -34 
0 
•9317 
26-57 
+ •1317 
•7247 
5674 
+ -1025 
29*38 
0 
58-10 
0 
•9031 
31*94 
- -1277 
7023 
59-45 
- -0993 
34-31 
0 
60*75 
0 
•8750 
36*54 
+ ■1237 
•6806 
62-03 
+ •0962 
38-62 
0 
63-29 
0 
*8480 
40-61 
- *1199 
•6595 
64*51 
- -0933 
42-50 
0 
65-72 
0 
•8219 
44-31 
+ -1162 
•6392 
66-90 
+ -0904 
46*04 
0 
68-07 
0 
•7964 
47-72 
- -1157 
•6195 
69-21 
- -0876 
49*34 
0 
70-34 
0 1 
