730 Lord Kelvin on 



stern ; more or less according to the length and speed of the 

 ship. If now the water suddenly loses viscosity and becomes 

 a perfect fluid, the dynamics of vortex 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 in- 

 creasing assemblage of regular waves extending farther and 



farther right astern, and over an area of 19° 28' ( tan _1 A / -x J 



on each side of mid-wake, as we shall see in about § 80 below. 

 Returning now to two-dimensional motion and canal waves : 

 we, in virtue of (62), put 



f«£. *-$ ^ 



where <f> denotes what is commonly called the " velocity- 

 potential " ; which, when convenient, we shall write in full 

 <£(#, z, t). With this notation (61) gives by integration with 

 respect to x and z, 



d<j> 



And (60) gives 



, t =-p + <t(s+C) .... (64). 

 ^ + ^=0 ...... (65). 



da? T dz? 

 Following Fourier's method, take^now 



<j)(x,z,t) = -k€- mz sinr)i(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 intinite 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 Stabilitv of Steady and Periodic Fluid Motion," in the Phil. Mag. 

 for May 1887. 



