y y = a + c sin(x/a) where c is small. This is
a simple harmonic wave-line, whose mean distance from the axis is a,
whose wave-length is 2[pi]a and whose amplitude is c. The internal
pressure corresponding to this unduloid is as before p = T/a. Now
consider a portion of a cylindric film of length x terminated by two
equal disks of radius r and containing a certain volume of air. Let one
of these disks be made to approach the other by a small quantity dx. The
film will swell out into the convex part of an unduloid, having its
largest section midway between the disks, and we have to determine
whether the internal pressure will be greater or less than before. If A
and C (fig. 13) are the disks, and if x the distance between the disks
is equal to [pi]r half the wave-length of the harmonic curve, the disks
will be at the points where the curve is at its mean distance from the
axis, and the pressure will therefore be T/r as before. If A1, C1 are
the disks, so that the distance between them is less than [pi]r, the
curve must be produced beyond the disks before it is at its mean
distance from the axis. Hence in this case the mean distance is less
than r, and the pressure will be greater than T/r. If, on the other
hand, the disks are at A2 and C2, so that the distance between them is
greater than [pi]r, the curve will reach its mean distance from the axis
before it reaches the disks. The mean distance will therefore be greater
than r, and the pressure will be less than T/r. Hence if one of the
disks be made to approach the other, the internal pressure will be
increased if the distance between the disks is less than half the
circumference of either, and the pressure will be diminished if the
distance is greater than this quantity. In the same way we may show that
if the distance between the disks is increased, the pressure will be
diminished or increased according as the distance is less or more than
half the circumference of either.

Now let us consider a cylindric film contained between two equal fixed
disks. A and B, and let a third disk, C, be placed midway between. Let C
be slightly displaced towards A. If AC and CB are each less than half
the circumference of a disk the pressure on C will increase on the side
of A and diminish on the side of B. The resultant force on C will
therefore tend to oppose the displacement and to bring C back to its
original position. The equilibrium of C is therefore stable. It is easy
to show tha