dually reduced the
pressure, namely--

1. Outside the sphere.

2. The sphere.

3. Between the sphere and the cylinder.

4. The cylinder.

5. Between the cylinder and the catenoid.

6. The catenoid.

7. Inside the catenoid.

[Illustration: Fig. 29.]

Now I am not going to say much more about all these curves, but I must
refer to the very curious properties that they possess. In the first
place, they must all of them have the same curvature in every part as
the portion of the sphere which forms the cap; in the second place, they
must all be the curves of the least possible surface which can enclose
the air and join the rings as well. And finally, since they pass
insensibly from one to the other as the pressure gradually changes,
though they are distinct curves there must be some curious and intimate
relation between them. This though it is a little difficult, I shall
explain. If I were to say that these curves are the roulettes of the
conic sections I suppose I should alarm you, and at the same time
explain nothing, so I shall not put it in that way; but instead, I shall
show you a simple experiment which will throw some light upon the
subject, which you can try for yourselves at home.

[Illustration: Fig. 30.]

I have here a common bedroom candlestick with a flat round base. Hold
the candlestick exactly upright near to a white wall, then you will see
the shadow of the base on the wall below, and the outline of the shadow
is a symmetrical curve, called a hyperbola. Gradually tilt the candle
away from the wall, you will then notice the sides of the shadow
gradually branch away less and less, and when you have so far tilted the
candle away from the wall that the flame is exactly above the edge of
the base,--and you will know when this is the case, because then the
falling grease will just fall on the edge of the candlestick and splash
on to the carpet,--I have it so now,--the sides of the shadow near the
floor will be almost parallel (Fig. 30), and the shape of the shadow
will have become a curve, known as a parabola; and now when the
candlestick is still more tilted, so that the grease misses the base
altogether and falls in a gentle stream upon the carpet, you will see
that the sides of the shadow have curled round and met on the wall, and
you now have a curve like an oval, except that the two ends are alike,
and this is called an ellipse. If you go on tilting the candlest