et 1853_), a vessel
containing olive oil is placed with its mouth downwards in a vessel
containing a mixture of alcohol and water, the mixture being denser than
the oil. The surface of separation is in this case horizontal and
stable, so that the equilibrium is established of itself. Alcohol is
then added very gradually to the mixture till it becomes lighter than
the oil. The equilibrium of the fluids would now be unstable if it were
not for the tension of the surface which separates them, and which, when
the orifice of the vessel is not too large, continues to preserve the
stability of the equilibrium.

When the equilibrium at last becomes unstable, the destruction of
equilibrium takes place by the lighter fluid ascending in one part of
the orifice and the heavier descending in the other. Hence the
displacement of the surface to which we must direct our attention is one
which does not alter the volume of the liquid in the vessel, and which
therefore is upward in one part of the surface and downward in another.
The simplest case is that of a rectangular orifice in a horizontal
plane, the sides being a and b.

Let the surface of separation be originally in the plane of the
orifice, and let the co-ordinates x and y be measured from one corner
parallel to the sides a and b respectively, and let z be measured
upwards. Then if [rho] be the density of the upper liquid, and [sigma]
that of the lower liquid, and P the original pressure at the surface
of separation, then when the surface receives an upward displacement
z, the pressure above it will be P - [rho]gz, and that below it will
be P - [sigma]gz, so that the surface will be acted on by an upward
pressure ([rho] - [sigma])gz. Now if the displacement z be everywhere
very small, the curvature in the planes parallel to xz and yz will be
d^2z/dx^2 and d^2z/dy^2 respectively, and if T is the surface-tension
the whole upward force will be

/ d^2z d^2z \
T ( ---- + ---- ) + ([rho] - [sigma])gz.
\ dx^2 dy^2 /

If this quantity is of the same sign as z, the displacement will be
increased, and the equilibrium will be unstable. If it is of the
opposite sign from z, the equilibrium will be stable. The limiting
condition may be found by putting it equal to zero. One form of the
solution of the equation, and that which is applicable to the case of
a rectangular orifice, is

z = C sin px sin qy.

Substituting in