Part 3

I have now blown a bubble upon a ring of wire. I shall hang a small ring upon it, and to show more clearly what is happening, I shall blow a little smoke into the bubble. Now that I have broken the film inside the lower ring, you will see the smoke being driven out and the ring lifted up, both of which show the elastic nature of the film. Or again, I have blown a bubble on the end of a wide pipe; on holding the open end of the pipe to a candle flame, the outrushing air blows out the flame at once, which shows that the soap-bubble is acting like an elastic bag (Fig. 21). You now see that, owing to the elastic skin of a soap-bubble, the air inside is under pressure and will get out if it can. Which would you think would squeeze the air inside it most, a large or a small bubble? We will find out by trying, and then see if we can tell why. You now see two pipes each with a tap. These are joined together by a third pipe in which there is a third tap. I will first blow one bubble and shut it off with the tap 1 (Fig. 22), and then the other, and shut it off with the tap 2. They are now nearly equal in size, but the air cannot yet pass from one to the other because the tap 3 is turned off. Now if the pressure in the largest one is greatest it will blow air into the other when I open this tap, until they are equal in size; if, on the other hand, the pressure in the small one is greatest, it will blow air into the large one, and will itself get smaller until it has quite disappeared. We will now try the experiment. You see immediately that I open the tap 3 the small bubble shuts up and blows out the large one, thus showing that there is a greater pressure in a small than in a large bubble. The directions in which the air and the bubble move is indicated in the figure by arrows. I want you particularly to notice and remember this, because this is an experiment on which a great deal depends. To impress this upon your memory I shall show the same thing in another way. There is in front of the lantern a little tube shaped like a U half filled with water. One end of the U is joined to a pipe on which a bubble can be blown (Fig. 23). You will now be able to see how the pressure changes as the bubble increases in size, because the water will be displaced more when the pressure is more, and less when it is less. Now that there is a very small bubble, the pressure as measured by the water is about one quarter of an inch on the scale. The bubble is growing and the pressure indicated by the water in the gauge is falling, until, when the bubble is double its former size, the pressure is only half what it was; and this is always true, the smaller the bubble the greater the pressure. As the film is always stretched with the same force, whatever size the bubble is, it is clear that the pressure inside can only depend upon the curvature of a bubble. In the case of lines, our ordinary language tells us, that the larger a circle is the less is its curvature; a piece of a small circle is said to be a quick or a sharp curve, while a piece of a great circle is only slightly curved; and if you take a piece of a very large circle indeed, then you cannot tell it from a straight line, and you say it is not curved at all. With a part of the surface of a ball it is just the same--the larger the ball the less it is curved; and if the ball is very large indeed, say 8000 miles across, you cannot tell a small piece of it from a true plane. Level water is part of such a surface, and you know that still water in a basin appears perfectly flat, though in a very large lake or the sea you can see that it is curved. We have seen that in large bubbles the pressure is little and the curvature is little, while in small bubbles the pressure is great and the curvature is great. The pressure and the curvature rise and fall together. We have now learnt the lesson which the experiment of the two bubbles, one blown out by the other, teaches us.

A ball or sphere is not the only form which you can give to a soap-bubble. If you take a bubble between two rings, you can pull it out until at last it has the shape of a round straight tube or cylinder as it is called. We have spoken of the curvature of a ball or sphere; now what is the curvature of a cylinder? Looked at sideways, the edge of the wooden cylinder upon the table appears straight, _i. e._ not curved at all; but looked at from above it appears round, and is seen to have a definite curvature (Fig. 24). What then is the curvature of the surface of a cylinder? We have seen that the pressure in a bubble depends upon the curvature when they are spheres, and this is true whatever shape they have. If, then, we find what sized sphere will produce the same pressure upon the air inside that a cylinder does, then we shall know that the curvature of the cylinder is the same as that of the sphere which balances it. Now at each end of a short tube I shall blow an ordinary bubble, but I shall pull the lower bubble by means of another tube into the cylindrical form, and finally blow in more or less air until the sides of the cylinder are perfectly straight. That is now done (Fig. 25), and the pressure in the two bubbles must be exactly the same, as there is a free passage of air between the two. On measuring them you see that the sphere is exactly double the cylinder in diameter. But this sphere has only half the curvature that a sphere half its diameter would have. Therefore the cylinder, which we know has the same curvature that the large sphere has, because the two balance, has only half the curvature of a sphere of its own diameter, and the pressure in it is only half that in a sphere of its own diameter.

I must now make one more step in explaining this question of curvature. Now that the cylinder and sphere are balanced I shall blow in more air, making the sphere larger; what will happen to the cylinder? The cylinder is, as you see, very short; will it become blown out too, or what will happen? Now that I am blowing in air you see the sphere enlarging, thus relieving the pressure; the cylinder develops a waist, it is no longer a cylinder, the sides are curved inwards. As I go on blowing and enlarging the sphere, they go on falling inwards, but not indefinitely. If I were to blow the upper bubble till it was of an enormous size the pressure would become extremely small. Let us make the pressure nothing at all at once by simply breaking the upper bubble, thus allowing the air a free passage from the inside to the outside of what was the cylinder. Let me repeat this experiment on a larger scale. I have two large glass rings, between which I can draw out a film of the same kind. Not only is the outline of the soap-film curved inwards, but it is exactly the same as the smaller one in shape (Fig. 26). As there is now no pressure there ought to be no curvature, if what I have said is correct. But look at the soap-film. Who would venture to say that that was not curved? and yet we had satisfied ourselves that the pressure and the curvature rose and fell together. We now seem to have come to an absurd conclusion. Because the pressure is reduced to nothing we say the surface must have no curvature, and yet a glance is sufficient to show that the film is so far curved as to have a most elegant waist. Now look at the plaster model on the table, which is a model of a mathematical figure which also has a waist.

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]

[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 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 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.

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.

You may well ask what all this has to do with a soap-bubble. You will see in a moment. When you light a candle, the base of the candlestick throws the space behind it into darkness, and the form of this dark space, which is everywhere round like the base, and gets larger as you get further from the flame, is a cone, like the wooden model on the table. The shadow cast on the wall is of course the part of the wall which is within this cone. It is the same shape that you would find if you were to cut a cone through with a saw, and so these curves which I have shown you are called conic sections. You can see some of them already made in the wooden model on the table. If you look at the diagram on the wall (Fig. 31), you will see a complete cone at first upright (A), then being gradually tilted over into the positions that I have specified. The black line in the upper part of the diagram shows where the cone is cut through, and the shaded area below shows the true shape of these shadows, or pieces cut off, which are called sections. Now in each of these sections there are either one or two points, each of which is called a focus, and these are indicated by conspicuous dots. In the case of the circle (D Fig. 31), this point is also the centre. Now if this circle is made to roll like a wheel along the straight line drawn just below it, a pencil at the centre will rule the straight line which is dotted in the lower part of the figure; but if we were to make wheels of the shapes of any of the other sections, a pencil at the focus would certainly not draw a straight line. What shape it would draw is not at once evident. First consider any of the elliptic sections (C, E, or F) which you see on either side of the circle. If these were wheels, and were made to roll, the pencil as it moved along would also move up and down, and the line it would draw is shown dotted as before in the lower part of the figure. In the same way the other curves, if made to roll along a straight line, would cause pencils at their focal points to draw the other dotted lines.

We are now almost able to see what the conic section has to do with a soap-bubble. When a soap-bubble was blown between two rings, and the pressure inside was varied, its outline went through a series of forms, some of which are represented by the dotted lines in the lower part of the figure, but in every case they could have been accurately drawn by a pencil at the focus of a suitable conic section made to roll on a straight line. I called one of the bubble forms, if you remember, by its name, catenoid; this is produced when there is no pressure. The dotted curve in the second figure B is this one; and to show that this catenary can be so drawn, I shall roll upon a straight edge a board made into the form of the corresponding section, which is called a parabola, and let the chalk at its focus draw its curve upon the black board. There is the curve, and it is as I said, exactly the curve that a chain makes when hung by its two ends. Now that a chain is so hung you see that it exactly lies over the chalk line.

All this is rather difficult to understand, but as these forms which a soap-bubble takes afford a beautiful example of the most important principle of continuity, I thought it would be a pity to pass it by. It may be put in this way. A series of bubbles may be blown between a pair of rings. If the pressures are different the curves must be different. In blowing them the pressures slowly and _continuously_ change, and so the curves cannot be altogether different in kind. Though they may be different curves, they also must pass slowly and continuously one into the other. We find the bubble curves can be drawn by rolling wheels made in the shape of the conic sections on a straight line, and so the conic sections, though distinct curves, must pass slowly and continuously one into the other. This we saw was the case, because as the candle was slowly tilted the curves did as a fact slowly and insensibly change from one to the other. There was only one parabola, and that was formed when the side of the cone was parallel to the plane of section, that is when the falling grease just touched the edge of the candlestick; there is only one bubble with no pressure, the catenoid, and this is drawn by rolling the parabola. As the cone is gradually inclined more, so the sections become at first long ellipses, which gradually become more and more round until a circle is reached, after which they become more and more narrow until a line is reached. The corresponding bubble curves are produced by a gradually increasing pressure, and, as the diagram shows, these bubble curves are at first wavy (C), then they become straight when a cylinder is formed (D), then they become wavy again (E and F), and at last, when the cutting plane, _i. e._ the black line in the upper figure, passes through the vertex of the cone the waves become a series of semicircles, indicating the ordinary spherical soap-bubble. Now if the cone is inclined ever so little more a new shape of section is seen (G), and this being rolled, draws a curious curve with a loop in it; but how this is so it would take too long to explain. It would also take too long to trace the further positions of the cone, and to trace the corresponding sections and bubble curves got by rolling them. Careful inspection of the diagram may be sufficient to enable you to work out for yourselves what will happen in all cases. I should explain that the bubble surfaces are obtained by spinning the dotted lines about the straight line in the lower part of Fig. 31 as an axis.

As you will soon find out if you try, you cannot make with a soap-bubble a great length of any of these curves at one time, but you may get pieces of any of them with no more apparatus than a few wire rings, a pipe, and a little soap and water. You can even see the whole of one of the loops of the dotted curve of the first figure (A), which is called a nodoid, not a complete ring, for that is unstable, but a part of such a ring. Take a piece of wire or a match, and fasten one end to a piece of lead, so that it will stand upright in a dish of soap water, and project half an inch or so. Hold with one hand a sheet of glass resting on the match in middle, and blow a bubble in the water against the match. As soon as it touches the glass plate, which should be wetted with the soap solution, it will become a cylinder, which will meet the glass plate in a true circle. Now very slowly incline the plate. The bubble will at once work round to the lowest side, and try to pull itself away from the match stick, and in doing so it will develop a loop of the nodoid, which would be exactly true in form if the match or wire were slightly bent, so as to meet both the glass and the surface of the soap water at a right angle. I have described this in detail, because it is not generally known that a complete loop of the nodoid can be made with a soap-bubble.

We have found that the pressure in a short cylinder gets less if it begins to develop a waist, and greater if it begins to bulge. Let us therefore try and balance one with a bulge against another with a waist. Immediately that I open the tap and let the air pass, the one with a bulge blows air round to the one with a waist and they both become straight. In Fig. 32 the direction of the movement of the air and of the sides of the bubble is indicated by arrows. Let us next try the same experiment with a pair of rather longer cylinders, say about twice as long as they are wide. They are now ready, one with a bulge and one with a waist. Directly I open the tap, and let the air pass from one to the other, the one with a waist blows out the other still more (Fig. 33), until at last it has shut itself up. It therefore behaves exactly in the opposite way that the short cylinder did. If you try pairs of cylinders of different lengths you will find that the change occurs when they are just over one and a half times as long as they are wide. Now if you imagine one of these tubes joined on to the end of the other, you will see that a cylinder more than about three times as long as it is wide cannot last more than a moment; because if one end were to contract ever so little the pressure there would increase, and the narrow end would blow air into the wider end (Fig. 34), until the sides of the narrow end met one another. The exact length of the longest cylinder that is stable, is a little more than three diameters. The cylinder just becomes unstable when its length is equal to its circumference, and this is 3-1/7 diameters almost exactly.

I will gradually separate these rings, keeping up a supply of air, and you will see that when the tube gets nearly three times as long as it is wide it is getting very difficult to manage, and then suddenly it grows a waist nearer one end than the other, and breaks off forming a pair of separate and unequal bubbles.

If now you have a cylinder of liquid of great length suddenly formed and left to itself, it clearly cannot retain that form. It must break up into a series of drops. Unfortunately the changes go on so quickly in a falling stream of water that no one by merely looking at it could follow the movements of the separate drops, but I hope to be able to show to you in two or three ways exactly what is happening. You may remember that we were able to make a large drop of one liquid in another, because in this way the effect of the weight was neutralized, and as large drops oscillate or change their shape much more slowly than small, it is more easy to see what is happening. I have in this glass box water coloured blue on which is floating paraffin, made heavier by mixing with it a bad-smelling and dangerous liquid called bisulphide of carbon.

[Sidenote: _See Diagram at the end of the Book._

Fig. 35.]