ha$, we find

p = T(1/R1 + 1/R2) (14).

This equation, which gives the pressure in terms of the principal radii
of curvature, though here proved only in the case of a surface of
revolution, must be true of all surfaces. For the curvature of any
surface at a given point may be completely defined in terms of the
positions of its principal normal sections and their radii of curvature.

Before going further we may deduce from equation 9 the nature of all the
figures of revolution which a liquid film can assume. Let us first
determine the nature of a curve, such that if it is rolled on the axis
its origin will trace out the meridian section of the bubble. Since at
any instant the rolling curve is rotating about the point of contact
with the axis, the line drawn from this point of contact to the tracing
point must be normal to the direction of motion of the tracing point.
Hence if N is the point of contact, NP must be normal to the traced
curve. Also, since the axis is a tangent to the rolling curve, the
ordinate PR is the perpendicular from the tracing point P on the
tangent. Hence the relation between the radius vector and the
perpendicular on the tangent of the rolling curve must be identical with
the relation between the normal PN and the ordinate PR of the traced
curve. If we write r for PN, then y = r cos[alpha], and equation 9
becomes

/ T \ F
y^2( 2 -- -1 ) = -----.
\ pr / [pi]p

This relation between y and r is identical with the relation between the
perpendicular from the focus of a conic section on the tangent at a
given point and the focal distance of that point, provided the
transverse and conjugate axes of the conic are 2a and 2b respectively,
where

T F
a = ---, and b^2 = -----.
p [pi]p

Hence the meridian section of the film may be traced by the focus of
such a conic, if the conic is made to roll on the axis.

_On the different Forms of the Meridian Line._--1. When the conic is an
ellipse the meridian line is in the form of a series of waves, and the
film itself has a series of alternate swellings and contractions as
represented in figs. 9 and 10. This form of the film is called the
unduloid.

1a. When the ellipse becomes a circle, the meridian line becomes a
straight line parallel to the axis, and the film passes into the form of
a cylinder of revolution.

1b. As the ellipse degenerates into the straight line join