stion of curvature.
Now that the cylinder and sphere are balanced I shall blow in more air,
making the sphere larger; what will happen to the cylinder? The cylinder
is, as you see, very short; will it become blown out too, or what will
happen? Now that I am blowing in air you see the sphere enlarging, thus
relieving the pressure; the cylinder develops a waist, it is no longer a
cylinder, the sides are curved inwards. As I go on blowing and enlarging
the sphere, they go on falling inwards, but not indefinitely. If I were
to blow the upper bubble till it was of an enormous size the pressure
would become extremely small. Let us make the pressure nothing at all at
once by simply breaking the upper bubble, thus allowing the air a free
passage from the inside to the outside of what was the cylinder. Let me
repeat this experiment on a larger scale. I have two large glass rings,
between which I can draw out a film of the same kind. Not only is the
outline of the soap-film curved inwards, but it is exactly the same as
the smaller one in shape (Fig. 26). As there is now no pressure there
ought to be no curvature, if what I have said is correct. But look at
the soap-film. Who would venture to say that that was not curved? and
yet we had satisfied ourselves that the pressure and the curvature rose
and fell together. We now seem to have come to an absurd conclusion.
Because the pressure is reduced to nothing we say the surface must have
no curvature, and yet a glance is sufficient to show that the film is so
far curved as to have a most elegant waist. Now look at the plaster
model on the table, which is a model of a mathematical figure which also
has a waist.

[Illustration: Fig. 26.]

Let us therefore examine this cast more in detail. I have a disc of card
which has exactly the same diameter as the waist of the cast. I now hold
this edgeways against the waist (Fig. 27), and though you can see that
it does not fit the whole curve, it fits the part close to the waist
perfectly. This then shows that this part of the cast would appear
curved inwards if you looked at it sideways, to the same extent that it
would appear curved outwards if you could see it from above. So
considering the waist only, it is curved both towards the inside and
also away from the inside according to the way you look at it, and to
the same extent. The curvature inwards would make the pressure inside
less, and the curvature outwards would make it more, and as th