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 they 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]
Fig. 27.
Fig. 28.
[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 gradually reduced the pressure, namely—
1. Outside the sphere.
2. The sphere.
3. Between the sphere and the cylinder.