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.
[Illustration: Fig. 26.]
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]
[Illustration: Fig. 27.]
[Illustration: Fig. 28.]
[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.
[Illustration: 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.
[Illustration: 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.
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.
[Illustration: Fig. 31.]
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.
[Illustration: Fig. 32.]
[Illustration: Fig. 33.]
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.
[Illustration: Fig. 34.]
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.
[Illustration: Fig. 36.]
[Illustration: Fig. 37.]
[Illustration: Fig. 38.]
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.)
[Illustration: Fig. 39.]
[Illustration: Fig. 40.]
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.
[Illustration: Fig. 41.]
[Illustration: Fig. 42.]
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.