Soap Bubbles
and the Forces which Mould Them

the forces which mould them. 67 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 cate- noid. 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. 4. The cylinder. 5. Between the cylinder and the catenoid. 6. The catenoid. 7. Inside the catenoid. 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