564 
Proceedings of Boyal Society of Edinburgh. [sess. 
free surface^ is essentially irrotational. But we need not assume 
this at present : we see immediately that it is proved by our 
equations of motion, when in them we suppose the motion to he 
infinitesimal. The equations of motion, when the density of the 
liquid is taken as unity, are : — 
d^ , _ dp ' 
dt 2 aa? az dx 
dt 2 dx ^ dz dz 
(59), 
where g denotes the force of gravity and p the pressure at (a?, z, t). 
Assuming now the liquid to he incompressible, we have 
d A + rt = o 
dx^ dz 
(60). 
§ 37. The motion being assumed to be infinitesimal, the second 
and third terms of the first members of (59) are negligible, and 
the equations of motion become : — 
d 
dp' 
dt 2 
dx 
d 2 t 
df 2 = (J 
dp 
df. 
• (61). 
This, by taking the difference of two differentiations, gives : — 
dfdl dt\ Q / 62 ) 
dt\dz dx) v ' 
which shows that if at any time the motion is zero or irrotational, 
it remains irrotational for ever. 
§ 38. If at any time there is rotational motion in any part of 
the liquid, it is interesting to know what becomes of it. Leaving 
for a moment our present restriction to canal waves, imagine our* 
selves on a very smooth sea in a ship, kept moving uniformly at 
a good speed by a tow-rope above the water. Looking over the 
ship’s side we see a layer of disturbed motion, showing by dimples 
in the surface innumerable little whirlpools. The thickness of 
this layer increases from nothing perceptible near the bow to 
perhaps 10 or 20 cms. near the 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 
