Soap-Bubbles and the Forces Which Mould Them

by C. V. Boys

4. The cylinder.

5. Between the cylinder and the catenoid.

6. The catenoid.

7. Inside the catenoid.

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.

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 candlestick, then when the candle is just level, and the grease pouring away, the shadow will be almost a circle; it would be an exact circle if the flame did not flare up. Now if you go on tilting the candle, until at last the candlestick is upside down, the curves already obtained will be reproduced in the reverse order, but above instead of below you.