1904-5.] Lord Kelvin on Deep Water Ship- Waves. 
565 
motion tells us that the rotationally moving water gets left behind 
by the ship, and spreads out in the more and more distant wake 
and becomes lost ; * without, however, losing its kinetic energy, 
which becomes reduced to infinitely small velocities in an 
infinitely large portion of liquid. The ship now goes on through 
the calm sea without producing any more eddies along its sides 
and stern, but leaving within an acute angle on each side of its 
wake, smooth ship-waves with no eddies or turbulence of any 
kind. The ideal annulment of the water’s viscosity diminishes 
considerably the tension of the tow-rope, but by no means annuls 
it ; it has still work to do on an ever increasing assemblage of 
regular waves extending farther and farther right astern, and 
over an area of 19° 28' 
on each side of mid-wake, as 
we shall see in about § 80 below. Returning now to two-dimen- 
sional motion and canal waves : we, in virtue of (62), put 
dx ’ 
(63), 
where </> denotes what is commonly called the “velocity- 
potential ” ; which, when convenient, we shall write in full 
<f>(x,z,t). With this notation (61) gives by integration with 
respect to x and z, 
And (60) gives 
dJ4> d^ = 
dx 2 dr? 
(64) . 
(65) . 
Following Fourier’s method, take now 
<f)(x,z,t)= - ~ke~ mz sin m(x - vt) . . . . (66), 
* It now seems to me certain that if any motion be given within a finite 
portion of an infinite incompressible liquid originally at rest, its fate is 
necessarily dissipation to infinite distances with infinitely small velocities 
everywhere; while the total kinetic energy remains constant. After 
many years of failure to prove that the motion in the ordinary Helmholtz 
circular ring is stable, I came to the conclusion that it is essentially unstable, 
and that its fate must be to become dissipated as now described. I came 
to this conclusion by extensions not hitherto published of the considerations 
described in a short paper entitled : “On the stability of steady and periodic 
fluid motion,” in the Phil. Mag. for May 1887. 
