Chapter II: Part 2

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.]

The water is only a very little heavier than the mixture. If I now dip a pipe into the water and let it fill, I can then raise it and allow drops to slowly form. Drops as large as a shilling are now forming, and when each one has reached its full size, a neck forms above it, which is drawn out by the falling drop into a little cylinder. You will notice that the liquid of the neck has gathered itself into a little drop which falls away just after the large drop. The action is now going on so slowly that you can follow it. Fig. 35 contains forty-three consecutive views of the growth and fall of the drop taken photographically at intervals of one-twentieth of a second. For the use to which this figure is to be put, see page 149. If I again fill the pipe with water, and this time draw it rapidly out of the liquid, I shall leave behind a cylinder which will break up into balls, as you can easily see (Fig. 36). I should like now to show you, as I have this apparatus in its place, that you can blow bubbles of water containing paraffin in the paraffin mixture, and you will see some which have other bubbles and drops of one or other liquid inside again. One of these compound bubble drops is now resting stationary on a heavier layer of liquid, so that you can see it all the better (Fig. 37). If I rapidly draw the pipe out of the box I shall leave a long cylindrical bubble of water containing paraffin, and this, as was the case with the water-cylinder, slowly breaks up into spherical bubbles.

Having now shown that a very large liquid cylinder breaks up regularly into drops, I shall next go the other extreme, and take as an example an excessively fine cylinder. You see a photograph of a spider on her geometrical web (Fig. 38). If I had time I should like to tell you how the spider goes to work to make this beautiful structure, and a great deal about these wonderful creatures, but I must do no more than show you that there are two kinds of web--those that point outwards, which are hard and smooth, and those that go round and round, which are very elastic, and which are covered with beads of a sticky liquid. Now there are in a good web over a quarter of a million of these beads which catch the flies for the spider's dinner. A spider makes a whole web in an hour, and generally has to make a new one every day. She would not be able to go round and stick all these in place, even if she knew how, because she would not have time. Instead of this she makes use of the way that a liquid cylinder breaks up into beads as follows. She spins a thread, and at the same time wets it with a sticky liquid, which of course is at first a cylinder. This cannot remain a cylinder, but breaks up into beads, as the photograph taken with a microscope from a real web beautifully shows (Fig. 39). You see the alternate large and small drops, and sometimes you even see extra small drops between these again. In order that you may see exactly how large these beads really are, I have placed alongside a scale of thousandths of an inch, which was photographed at the same time. To prove to you that this is what happens, I shall now show you a web that I have made myself by stroking a quartz fibre with a straw dipped in castor-oil. The same alternate large and small beads are again visible just as perfect as they were in the spider's web. In fact it is impossible to distinguish between one of my beaded webs and a spider's by looking at them. And there is this additional similarity--my webs are just as good as a spider's for catching flies. You might say that a large cylinder of water in oil, or a microscopic cylinder on a thread, is not the same as an ordinary jet of water, and that you would like to see if it behaves as I have described. The next photograph (Fig. 40), taken by the light of an instantaneous electric spark, and magnified three and a quarter times, shows a fine column of water falling from a jet. You will now see that it is at first a cylinder, that as it goes down necks and bulges begin to form, and at last beads separate, and you can see the little drops as well. The beads also vibrate, becoming alternately long and wide, and there can be no doubt that the sparkling portion of a jet, though it appears continuous, is really made up of beads which pass so rapidly before the eye that it is impossible to follow them. (I should explain that for a reason which will appear later, I made a loud note by whistling into a key at the time that this photograph was taken.)

Lord Rayleigh has shown that in a stream of water one twenty-fifth of an inch in diameter, necks impressed upon the stream, even though imperceptible, develop a thousandfold in depth every fortieth of a second, and thus it is not difficult to understand that in such a stream the water is already broken through before it has fallen many inches. He has also shown that free water drops vibrate at a rate which may be found as follows. A drop two inches in diameter makes one complete vibration in one second. If the diameter is reduced to one quarter of its amount, the time of vibration will be reduced to one-eighth, or if the diameter is reduced to one-hundredth, the time will be reduced to one-thousandth, and so on. The same relation between the diameter and the time of breaking up applies also to cylinders. We can at once see how fast a bead of water the size of one of those in the spider's web would vibrate if pulled out of shape, and let go suddenly. If we take the diameter as being one eight-hundredth of an inch, and it is really even finer, then the bead would have a diameter of one sixteen-hundredth of a two-inch bead, which makes one vibration in one second. It will therefore vibrate sixty-four thousand times as fast, or sixty-four thousand times a second. Water-drops the size of the little beads, with a diameter of rather less than one three-thousandth of an inch, would vibrate half a million times a second, under the sole influence of the feebly elastic skin of water! We thus see how powerful is the influence of the feebly elastic water-skin on drops of water that are sufficiently small.

I shall now cause a small fountain to play, and shall allow the water as it falls to patter upon a sheet of paper. You can see both the fountain itself and its shadow upon the screen. You will notice that the water comes out of the nozzle as a smooth cylinder, that it presently begins to glitter, and that the separate drops scatter over a great space (Fig. 41). Now why should the drops scatter? All the water comes out of the jet at the same rate and starts in the same direction, and yet after a short way the separate drops by no means follow the same paths. Now instead of explaining this, and then showing experiments to test the truth of the explanation, I shall reverse the usual order, and show one or two experiments first, which I think you will all agree are so like magic, so wonderful are they and yet so simple, that if they had been performed a few hundred years ago, the rash person who showed them might have run a serious risk of being burnt alive.

You now see the water of the jet scattering in all directions, and you hear it making a pattering sound on the paper on which it falls. I take out of my pocket a stick of sealing-wax and instantly all is changed, even though I am some way off and can touch nothing. The water ceases to scatter; it travels in one continuous line (Fig. 42), and falls upon the paper making a loud rattling noise which must remind you of the rain of a thunder-storm. I come a little nearer to the fountain and the water scatters again, but this time in quite a different way. The falling drops are much larger than they were before. Directly I hide the sealing-wax the jet of water recovers its old appearance, and as soon as the sealing-wax is taken out it travels in a single line again.

Now instead of the sealing-wax I shall take a smoky flame easily made by dipping some cotton-wool on the end of a stick into benzine, and lighting it. As long as the flame is held away from the fountain it produces no effect, but the instant that I bring it near so that the water passes through the flame, the fountain ceases to scatter; it all runs in one line and falls in a dirty black stream upon the paper. Ever so little oil fed into the jet from a tube as fine as a hair does exactly the same thing.

I shall now set a tuning-fork sounding at the other side of the table. The fountain has not altered in appearance. I now touch the stand of the tuning-fork with a long stick which rests against the nozzle. Again the water gathers itself together even more perfectly than before, and the paper upon which it falls is humming out a note which is the same as that produced by the tuning-fork. If I alter the rate at which the water flows you will see that the appearance is changed again, but it is never like a jet which is not acted upon by a musical sound. Sometimes the fountain breaks up into two or three and sometimes many more distinct lines, as though it came out of as many tubes of different sizes and pointing in slightly different directions (Fig. 43). The effect of different notes could be very easily shown if any one were to sing to the piece of wood by which the jet is held. I can make noises of different pitches, which for this purpose are perhaps better than musical notes, and you can see that with every new noise the fountain puts on a different appearance. You may well wonder how these trifling influences--sealing-wax, the smoky flame, or the more or less musical noise--should produce this mysterious result, but the explanation is not so difficult as you might expect.

I hope to make this clear when we meet again.

LECTURE III.

At the conclusion of the last lecture I showed you some curious experiments with a fountain of water, which I have now to explain. Consider what I have said about a liquid cylinder. If it is a little more than three times as long as it is wide, it cannot retain its form; if it is made very much more than three times as long, it will break up into a series of beads. Now, if in any way a series of necks could be developed upon a cylinder which were less than three diameters apart, some of them would tend to heal up, because a piece of a cylinder less than three diameters long is stable. If they were about three diameters apart, the form being then unstable, the necks would get more pronounced in time, and would at last break through, so that beads would be formed. If necks were made at distances more than three diameters apart, then the cylinder would go on breaking up by the narrowing of these necks, and it would most easily break up into drops when the necks were just four and a half diameters apart. In other words, if a fountain were to issue from a nozzle held perfectly still, the water would most easily break into beads at the distance of four and a half diameters apart, but it would break up into a greater number closer together, or a smaller number further apart, if by slight disturbances of the jet very slight waists were impressed upon the issuing cylinder of water. When you make a fountain play from a jet which you hold as still as possible, there are still accidental tremors of all kinds, which impress upon the issuing cylinder slightly narrow and wide places at irregular distances, and so the cylinder breaks up irregularly into drops of different sizes and at different distances apart. Now these drops, as they are in the act of separating from one another, and are drawing out the waist, as you have seen, are being pulled for the moment towards one another by the elasticity of the skin of the waist; and, as they are free in the air to move as they will, this will cause the hinder one to hurry on, and the more forward one to lag behind, so that unless they are all exactly alike both in size and distance apart they will many of them bounce together before long. You would expect when they hit one another afterwards that they would join, but I shall be able to show you in a moment that they do not; they act like two india-rubber balls, and bounce away again. Now it is not difficult to see that if you have a series of drops of different sizes and at irregular distances bouncing against one another frequently, they will tend to separate and to fall, as we have seen, on all parts of the paper down below. What did the sealing-wax or the smoky flame do? and what can the musical sound do to stop this from happening? Let me first take the sealing-wax. A piece of sealing-wax rubbed on your coat is electrified, and will attract light bits of paper up to it. The sealing-wax acts electrically on the different water-drops, causing them to attract one another, feebly, it is true, but with sufficient power where they meet to make them break through the air-film between them and join. To show that this is no fancy, I have now in front of the lantern two fountains of clean water coming from separate bottles, and you can see that they bounce apart perfectly (Fig. 44). To show that they do really bounce, I have coloured the water in the two bottles differently. The sealing-wax is now in my pocket; I shall retire to the other side of the room, and the instant it appears the jets of water coalesce (Fig. 45). This may be repeated as often as you like, and it never fails. These two bouncing jets are in fact one of the most delicate tests for the presence of electricity that exist. You are now able to understand the first experiment. The separate drops which bounced away from one another, and scattered in all directions, are unable to bounce when the sealing-wax is held up, because of its electrical action. They therefore unite, and the result is, that instead of a great number of little drops falling all over the paper, the stream pours in a single line, and great drops, such as you see in a thunder-storm, fall on the top of one another. There can be no doubt that it is for this reason that the drops of rain in a thunder-storm are so large. This experiment and its explanation are due to Lord Rayleigh.

The smoky flame, as lately shown by Mr. Bidwell, does the same thing. The reason probably is that the dirt breaks through the air-film, just as dust in the air will make the two fountains join as they did when they were electrified. However, it is possible that oily matter condensed on the water may have something to do with the effect observed, because oil alone acts quite as well as a flame, but the action of oil in this case, as when it smooths a stormy sea, is not by any means so easily understood.

When I held the sealing-wax closer, the drops coalesced in the same way; but they were then so much more electrified that they repelled one another as similarly electrified bodies are known to do, and so the electrical scattering was produced.

You possibly already see why the tuning-fork made the drops follow in one line, but I shall explain. A musical note is, as is well known, caused by a rapid vibration; the more rapid the vibration the higher is the pitch of the note. For instance, I have a tooth-wheel which I can turn round very rapidly if I wish. Now that it is turning slowly you can hear the separate teeth knocking against a card that I am holding in the other hand. I am now turning faster, and the card is giving out a note of a low pitch. As I make the wheel turn faster and faster, the pitch of the note gradually rises, and it would, if I could only turn fast enough, give so high a note that we should not be able to hear it. A tuning-fork vibrates at a certain definite rate, and therefore gives a definite note. The fork now sounding vibrates 128 times in every second. The nozzle, therefore, is made to vibrate, but almost imperceptibly, 128 times a second, and to impress upon the issuing cylinder of water 128 imperceptible waists every second. Now it just depends what size the jet is, and how fast the water is issuing, whether these waists are about four and a half diameters apart in the cylinder. If the jet is larger, the water must pass more quickly, or under a greater pressure, for this to be the case; if the jet is finer, a smaller speed will be sufficient. If it should happen that the waists so made are anywhere about four diameters apart, then even though they are so slightly developed that if you had an exact drawing of them, you would not be able to detect the slightest change of diameter, they will grow at a great speed, and therefore the water column will break up regularly, every drop will be like the one behind it, and like the one in front of it, and not all different, as is the case when the breaking of the water merely depends upon accidental tremors. If the drops then are all alike in every respect, of course they all follow the same path, and so appear to fall in a continuous stream. If the waists are about four and a half diameters apart, then the jet will break up most easily; but it will, as I have said, break up under the influence of a considerable range of notes, which cause the waists to be formed at other distances, provided they are more than three diameters apart. If two notes are sounded at the same time, then very often each will produce its own effect, and the result is the alternate formation of drops of different sizes, which then make the jet divide into two separate streams. In this way, three, four, or even many more distinct streams may be produced.

I can now show you photographs of some of these musical fountains, taken by the instantaneous flash of an electric spark, and you can see the separate paths described by the drops of different sizes (Fig. 46). In one photograph there are eight distinct fountains all breaking from the same jet, but following quite distinct paths, each of which is clearly marked out by a perfectly regular series of drops. You can also in these photographs see drops actually in the act of bouncing against one another, and flattened when they meet, as if they were india-rubber balls. In the photograph now upon the screen the effect of this rebound, which occurs at the place marked with a cross, is to hurry on the upper and more forward drop, and to retard the other one, and so to make them travel with slightly different velocities and directions. It is for this reason that they afterwards follow distinct paths. The smaller drops had no doubt been acted on in a similar way, but the part of the fountain where this happened was just outside the photographic plate, and so there is no record of what occurred. The very little drops of which I have so often spoken are generally thrown out from the side of a fountain of water under the influence of a musical sound, after which they describe regular little curves of their own, quite distinct from the main stream. They, of course, can only get out sideways after one or two bouncings from the regular drops in front and behind. You can easily show that they are really formed below the place where they first appear, by taking a piece of electrified sealing-wax and holding it near the stream close to the nozzle and gradually raising it. When it comes opposite to the place where the little drops are really formed, it will act on them more powerfully than on the large drops, and immediately pull them out from a place where the moment before none seemed to exist. They will then circulate in perfect little orbits round the sealing-wax, just as the planets do round the sun; but in this case, being met by the resistance of the air, the orbits are spirals, and the little drops after many revolutions ultimately fall upon the wax, just as the planets would fall into the sun after many revolutions, if their motion through space were interfered with by friction of any kind.

There is only one thing needed to make the demonstration of the behaviour of a musical jet complete, and that is, that you should yourselves see these drops in their different positions in an actual fountain of water. Now if I were to produce a powerful electric spark, then it is true that some of you might for an instant catch sight of the drops, but I do not think that most would see anything at all. But if, instead of making merely one flash, I were to make another when each drop had just travelled to the position which the one in front of it occupied before, and then another when each drop had moved on one place again, and so on, then all the drops, at the moments that the flashes of light fell upon them, would occupy the same positions, and thus all these drops would appear fixed in the air, though of course they really are travelling fast enough. If, however, I do not quite succeed in keeping exact time with my flashes of light, then a curious appearance will be produced. Suppose, for instance, that the flashes of light follow one another rather too quickly, then each drop will not have had quite time enough to get to its proper place at each flash, and thus at the second flash all the drops will be seen in positions which are just behind those which they occupied at the first flash, and in the same way at the third flash they will be seen still further behind their former places, and so on, and therefore they will appear to be moving slowly backwards; whereas if my flashes do not follow quite quickly enough, then the drops will, every time that there is a flash, have travelled just a little too far, and so they will all appear to be moving slowly forwards. Now let us try the experiment. There is the electric lantern sending a powerful beam of light on to the screen. This I bring to a focus with a lens, and then let it pass through a small hole in a piece of card. The light then spreads out and falls upon the screen. The fountain of water is between the card and the screen, and so a shadow is cast which is conspicuous enough. Now I place just behind the card a little electric motor, which will make a disc of card which has six holes near the edge spin round very fast. The holes come one after the other opposite the hole in the fixed card, and so at every turn six flashes of light are produced. When the card is turning about 21-1/2 times a second, then the flashes will follow one another at the right rate. I have now started the motor, and after a moment or two I shall have obtained the right speed, and this I know by blowing through the holes, when a musical note will be produced, higher than the fork if the speed is too high, and lower than the fork if the speed is too low, and exactly the same as the fork if it is right.

To make it still more evident when the speed is exactly right, I have placed the tuning-fork also between the light and the screen, so that you may see it illuminated, and its shadow upon the screen. I have not yet allowed the water to flow, but I want you to look at the fork. For a moment I have stopped the motor, so that the light may be steady, and you can see that the fork is in motion because its legs appear blurred at the ends, where of course the motion is most rapid. Now the motor is started, and almost at once the fork appears quite different. It now looks like a piece of india-rubber, slowly opening and shutting, and now it appears quite still, but the noise it is making shows that it is not still by any means. The legs of the fork are vibrating, but the light only falls upon them at regular intervals, which correspond with their movement, and so, as I explained in the case of the water-drops, the fork appears perfectly still. Now the speed is slightly altered, and, as I have explained, each new flash of light, coming just too soon or just too late, shows the fork in a position which is just before or just behind that made visible by the previous flash. You thus see the fork slowly going through its evolutions, though of course in reality the legs are moving backwards and forwards 128 times a second. By looking at the fork or its shadow, you will therefore be able to tell whether the light is keeping exact time with the vibrations, and therefore with the water-drops.

Now the water is running, and you see all the separate drops apparently stationary, strung like pearls or beads of silver upon an invisible wire (_see_ Frontispiece). If I make the card turn ever so little more slowly, then all the drops will appear to slowly march onwards, and what is so beautiful,--but I am afraid few will see this,--each little drop may be seen to gradually break off, pulling out a waist which becomes a little drop, and then when the main drop is free it slowly oscillates, becoming wide and long, or turning over and over, as it goes on its way. If it so happens that a double or multiple jet is being produced, then you can see the little drops moving up to one another, squeezing each other where they meet and bouncing away again. Now the card is turning a little too fast and the drops appear to be moving backwards, so that it seems as if the water is coming up out of the tank on the floor, quietly going over my head, down into the nozzle, and so back to the water-supply of the place. Of course this is not happening at all, as you know very well, and as you will see if I simply try and place my finger between two of these drops. The splashing of the water in all directions shows that it is not moving quite so quietly as it appears. There is one more thing that I would mention about this experiment. Every time that the flashing light gains or loses one complete flash, upon the motion of the tuning-fork, it will appear to make one complete oscillation, and the water-drops will appear to move back or on one place.

I must now come to one of the most beautiful applications of these musical jets to practical purposes which it is possible to imagine, and what I shall now show are a few out of a great number of the experiments of Mr. Chichester Bell, cousin of Mr. Graham Bell, the inventor of the telephone.

To begin with I have a very small jet of water forced through the nozzle at a great pressure, as you can see if I point it towards the ceiling, as the water rises eight or ten feet. If I allow this stream of water to fall upon an india-rubber sheet, stretched over the end of a tube as big as my little finger, then the little sheet will be depressed by the water, and the more so if the stream is strong. Now if I hold the jet close to the sheet the smooth column of liquid will press the sheet steadily, and it will remain quiet; but if I gradually take the jet further away from the sheet, then any waists that may have been formed in the liquid column, which grow as they travel, will make their existence perfectly evident. When a wide part of the column strikes the sheet it will be depressed rather more than usual, and when a narrow part follows, the depression will be less. In other words, any very slight vibration imparted to the jet will be magnified by the growth of waists, and the sheet of india-rubber will reproduce the vibration, but on a magnified scale. Now if you remember that sound consists of vibrations, then you will understand that a jet is a machine for magnifying sound. To show that this is the case I am now directing the jet on to the sheet, and you can hear nothing; but I shall hold a piece of wood against the nozzle, and now, if on the whole the jet tends to break up at any one rate rather than at any other, or if the wood or the sheet of rubber will vibrate at any rate most easily, then the first few vibrations which correspond to this rate will be imparted to the wood, which will impress them upon the nozzle and so upon the cylinder of liquid, where they will become magnified; the result is that the jet immediately begins to sing of its own accord, giving out a loud note (Fig. 47).