1904-5.] Lord Kelvin on Deep Water Ship-Waves. 577 
without help of hydrokinetic ideas ! Here is the only answer I 
can give at present. 
§ 51. Look at figs. 12-16, and see how, in the forcive de- 
fined by e='9, the pressure is almost wholly confined to the 
spaces #<60° on each side of each of its maximums, and is very 
nearly null from 0=60° to 0 = 300°. It is obvious that if the 
pressure were perfectly annulled in these last-mentioned spaces, 
while in the spaces within 60° on each side of each maximum 
the pressure is that expressed by (74), the resulting motion would 
be sensibly the same as if the pressure were throughout the whole 
space CC (0 = 0° to 0 = 360°), exactly that given by (74). Hence 
we must expect to find through nearly the whole space of 240°, 
from 60° to 300°, an almost exactly sinusoidal displacement of 
water-surface, having the wave-length 360 70* + i) due to the 
translational speed of the forcive. 
§ 52. I confess that I did 'not expect so small a difference from 
sinusoidality through the ivhole 240°, as calculation by {(83), (86), 
(87)} has proved; and as is shown in figs. 18, 19, 20, by the 
D-curve on the right-hand side of C, which represents in each 
case the value of 
D(0) = d(0)-(-iyd(18O°). sin (y + J)0 . . . (89), 
being the difference of d(0) from one continuous sinusoidal curve. 
The exceeding smallness of this difference for distances from 
C exceeding 20° or 30°, and therefore through a range between 
C C of 320°, or 300°, is very remarkable in each case. 
§ 53. The dynamical interpretation of (88), and figs. 18, 19, 20, 
is this : — Superimpose on the solution {(83), (86), (87)} a “free 
wave” solution according to (73), taken as 
- ( - iyd(180°). sin ( j + 1)0 ... . (90). 
This approximately annuls the approximately sinusoidal portion 
between C and C shown in figs. (13), (14), (15); and approxi- 
mately doubles the approximately sinusoidal displacement in the 
corresponding portions of the spaces C C, and C C on the two 
sides of C C. This is a very interesting solution of our problem 
§ 41 ; and, though it is curiously artificial, it leads direct and 
short to the determinate solution of the following general problem 
of canal ship-waves : — 
PEOC. KOY. SOC. EDIN. — YOL. XXV. 
37 
