ey are
equal they just balance, and there is no pressure at all. If we could in
the same way examine the bubble with the waist, we should find that this
was true not only at the waist but at every part of it. Any curved
surface like this which at every point is equally curved opposite ways,
is called a surface of no curvature, and so what seemed an absurdity is
now explained. Now this surface, which is the only one of the kind
symmetrical about an axis, except a flat surface, is called a catenoid,
because it is like a chain, as you will see directly, and, as you know,
_catena_ is the Latin for a chain. I shall now hang a chain in a loop
from a level stick, and throw a strong light upon it, so that you can
see it well (Fig. 28). This is exactly the same shape as the side of a
bubble drawn out between two rings, and open at the end to the air.[1]

[Illustration: Fig. 27.]

[Illustration: Fig. 28.]

[Footnote 1: If the reader finds these geometrical relations too
difficult to follow, he or she should skip the next pages, and go on
again at "We have found...." p. 77.]

Let us now take two rings, and having placed a bubble between them,
gradually alter the pressure. You can tell what the pressure is by
looking at the part of the film which covers either ring, which I shall
call the cap. This must be part of a sphere, and we know that the
curvature of this and the pressure inside rise and fall together. I have
now adjusted the bubble so that it is a nearly perfect sphere. If I blow
in more air the caps become more curved, showing an increased pressure,
and the sides bulge out even more than those of a sphere (Fig. 29). I
have now brought the whole bubble back to the spherical form. A little
increased pressure, as shown by the increased curvature of the cap,
makes the sides bulge more; a little less pressure, as shown by the
flattening of the caps, makes the sides bulge less. Now the sides are
straight, and the cap, as we have already seen, forms part of a sphere
of twice the diameter of the cylinder. I am still further reducing the
pressure until the caps are plane, that is, not curved at all. There is
now no pressure inside, and therefore the sides have, as we have
already seen, taken the form of a hanging chain; and now, finally, the
pressure inside is less than that outside, as you can see by the caps
being drawn inwards, and the sides have even a smaller waist than the
catenoid. We have now seen seven curves as we gra