1904 - 5 .] Lord Kelvin on Deep Water Ship-Waves. 
583 
produced by them, are shown in figs. 22, 23, 24 on different scales, 
of wave-length, and pressure, chosen for the convenience of each 
case. 
§ 60. As most interesting of the three cases take that derived 
from,/ = 9 of our original investigation. By looking at fig. 23 we 
see that a pontoon having its bottom shaped according to the 
D curve from -3 q to +3 q, 1J free wave-lengths, will leave the 
water sensibly flat and at rest if it moves along the canal at the 
velocity for which the free-wave-length is 4 q. And the pressure 
of the water on the bottom of the pontoon is that represented 
hydrostatically by the F curve. 
§ 61. Imagine the scale of abscissas in each of the four diagrams, 
figs. 21-24, to be enlarged tenfold. The greatest steepnesses of the 
D curve in each case are rendered sufficiently moderate to allow it 
to fairly represent a real water-surface under the given forcive. 
The same may be said of figs. 15, 16, 18, 19, 20 ; and of figs. 13, 
1 4 with abscissas enlarged twentyfold. In respect to mathematical 
hydrokinetics generally; it is interesting to remark that a very 
liberal interpretation of the condition of infinitesimality (§ 36 
above) is practically allowable. Inclinations to the horizon of as 
much as 1/10 of a radian (5°-7 ; or, say, 6°), in any real case of 
water-w r aves or disturbances, will not seriously vitiate the mathe- 
matical result. 
§ 62. Fig. 17 represents the calculations of d(0°) and 
(-l) ? d(180°) for twenty-nine integral values of j; 0, 1, 2, 3, 
. ... 19, 20, 30, 40, ... . 90, 100; from the following 
formulas, found by putting 0 = 0° and 0=180°; and with e=’9 
in each case, and c — 1 
d;( (l ”) = (2/ + l)e‘ 
+ 1 + T + T 
-j+l 
2 W 
+ 
+ 1 
9 ," _ 
(92), 
dXi80°) = (-iy'(2y+iy 
* Je 
1 -e 
+ 
p— i+i . p —j 
(93). 
The asymptote of d(0°) shown in the diagram is explained by 
remarking that when j is infinitely great, the travelling velocity of 
