pheric pressure, and thus increasing the volume of
the bubble.

Let [Pi] be the atmospheric pressure and [Pi] + p the pressure of the
air within the bubble. The volume of the sphere is

V = 4/3 [pi]r^3, (4)

and the increment of volume is

dV= 4[pi]r^2dr (5)

Now if we suppose a quantity of air already at the pressure [Pi] + p,
the work done in forcing it into the bubble pdV. Hence the equation of
work and energy is

pdV = Tds (6)

or

4[pi]pr^2dr = 8[pi]r drT (7)

or

p = 2T/r (8)

This, therefore, is the excess of the pressure of the air within the
bubble over that of the external air, and it is due to the action of the
inner and outer surfaces of the bubble. We may conceive this pressure to
arise from the tendency which the bubble has to contract, or in other
words from the surface-tension of the bubble.

If to increase the area of the surface requires the expenditure of
work, the surface must resist extension, and if the bubble in
contracting can do work, the surface must tend to contract. The surface
must therefore act like a sheet of india-rubber when extended both in
length and breadth, that is, it must exert surface-tension. The tension
of the sheet of india-rubber, however, depends on the extent to which it
is stretched, and may be different in different directions, whereas the
tension of the surface of a liquid remains the same however much the
film is extended, and the tension at any point is the same in all
directions.

The intensity of this surface-tension is measured by the stress which it
exerts across a line of unit length. Let us measure it in the case of
the spherical soap-bubble by considering the stress exerted by one
hemisphere of the bubble on the other, across the circumference of a
great circle. This stress is balanced by the pressure p acting over the
area of the same great circle: it is therefore equal to [pi]r^2p. To
determine the intensity of the surface-tension we have to divide this
quantity by the length of the line across which it acts, which is in
this case the circumference of a great circle 2[pi]r. Dividing [pi]r^2p
by this length we obtain 1/2pr as the value of the intensity of the
surface-tension, and it is plain from equation 8 that this is equal to
T. Hence the numerical value of the intensity of the surface-tension is
equal to the numerical value of the surface-energy per unit of surface.
We must remember that since th