1904 - 5 .] Lord Kelvin on Deep Water Ship-Waves. 
567 
II = ^Bcos m(x-vt + /3) (72), 
d = 3 — ^ — cos mix -vt + B) + A cos -vt + y) . (7 3), 
g - mv 1 v 2 
where B, m, /3 are given constants having different values in the 
different terms of the sums ; and v is a given constant velocity. 
The last term of (73) expresses, with two arbitrary constants 
(A, y), a train of free waves which we may superimpose on any 
solution of our problem. 
§ 41. It is very interesting and instructive in respect to the 
dynamics of water-waves, to apply (72) to a particular case of 
Fourier’s expansion of periodic arbitrary functions such as a dis- 
tribution of alternate constant pressures, and zeros, on equal 
successive spaces, travelling with velocity v. But this must be 
left undone for the present, to let us get on with ship-waves ; and 
for this purpose we may take as a case of (72), (73), 
II = + ec°s0 + e 2 cos 20 4- etc.) = gc — ^ e ) - (74), 
' 1 - 2ecos 0 + e 2 v ' 
d= Je { ^ + jL cos0+ r?2 cos2<9+etc ’ } ■ ■ ■ 0$), 
where 
6=—(x 
a 
vt + /I) 
2 T 
a ga 
(76) ; 
(77) ; 
and e may he any numeric < 1. Remark that when v — 0 , J = go , 
and we have by (75) and (74), d = n/^, which explains our unit 
of pressure. 
§ 42. To understand the dynamical conditions thus prescribed, 
and the resulting motion : — remark first that (7 4), with (7 6), 
represents a space-periodic distribution of pressure on the surface, 
travelling with velocity v ; and (7 5) represents the displacement 
of the water-surface in the resulting motion, when space-periodic 
of the same space-period as the surface-pressure. Any motion 
whatever; consequent on any initial disturbance and no subse- 
quent application of surface-pressure ; may he superimposed on the 
solution represented by (75), to constitute the complete solution 
