n of a vertical film were
absolutely the same throughout, the middle parts would of necessity fall
with the acceleration of gravity. In reality, the tension adjusts itself
automatically to the weight to be supported at the various levels.

Although throughout a certain range the surface-tension varies rapidly
with the degree of contamination, it is remarkable that, as was first
fully indicated by Miss Pockels, the earlier stages of contamination
have little or no effect upon surface-tension. Lord Rayleigh has shown
that the fall of surface-tension _begins_ when the quantity of oil is
about the half of that required to stop the camphor movements, and he
suggests that this stage may correspond with a complete coating of the
surface with a single layer of molecules.]

Spherical soap-bubble.

_On the Forms of Liquid Films which are Figures of Revolution._--A soap
bubble is simply a small quantity of soap-suds spread out so as to
expose a large surface to the air. The bubble, in fact, has two
surfaces, an outer and an inner surface, both exposed to air. It has,
therefore, a certain amount of surface-energy depending on the area of
these two surfaces. Since in the case of thin films the outer and inner
surfaces are approximately equal, we shall consider the area of the film
as representing either of them, and shall use the symbol T to denote the
energy of unit of area of the film, both surfaces being taken together.
If T' is the energy of a single surface of the liquid, T the energy of
the film is 2T'. When by means of a tube we blow air into the inside of
the bubble we increase its volume and therefore its surface, and at the
same time we do work in forcing air into it, and thus increase the
energy of the bubble.

That the bubble has energy may be shown by leaving the end of the tube
open. The bubble will contract, forcing the air out, and the current of
air blown through the tube may be made to deflect the flame of a candle.
If the bubble is in the form of a sphere of radius r this material
surface will have an area

S = 4[pi]r^2 (1)

If T be the energy corresponding to unit of area of the film the
surface-energy of the whole bubble will be

ST = 4[pi]r^2T (2)

The increment of this energy corresponding to an increase of the radius
from r to r + dr is therefore

TdS = 8[pi]rTdr (3)

Now this increase of energy was obtained by forcing in air at a pressure
greater than the atmos