Generally speaking, great comets come to us once and are then never
seen again. Such bodies do not move in closed ovals or ellipses, they
follow another kind of curve, like that represented in Fig. 74. It
is one that every boy ought to know. In fact, in one of his earliest
accomplishments he learned how to make a parabola. It is true he did
not call it by any name so fine as this, but every time a ball is
thrown into the air it describes a part of the beautiful curve which
geometers know by this word (Fig. 74). In fact, you could not throw
a ball so that it should describe any other curve except a parabola.
No boy could throw a stone in a truly horizontal line. It will always
curve down a little, will always, in fact, be a portion of a parabola.
[Illustration: FIG. 74.--The Path of a Projectile is a Parabola.]
There are big parabolas and there are small ones. One of the shells
which are thrown into a town when bombarded from a distance describes,
as it rises and then slopes down again, part of a mighty parabola. So
does a tennis ball thrown by the hand or struck by the racket; though
here, indeed, I admit that a spin may be given to the ball which will
somewhat detract from the simplicity of its movement. In playing
baseball, a large part of the skill of the pitcher consists in throwing
the ball in such a way that it shall not move in a parabola, but in
some twisting curve by which he hopes to baffle his adversary. Setting
aside these exceptions, and such another as the case of a body tossed
straight up or dropped straight down, we may assert that the path of a
projectile is a parabola.
[Illustration: FIG. 75.--The Lighthouse Reflector.]
There are some remarkable applications of the same curve for practical
purposes. From our lighthouses we want to send beams off to sea, so
as to guide ships into port. If we merely employed a lamp without
concentrating its rays, we should have a very imperfect lighthouse,
for the lamp scatters light about in all directions. Much of it goes
straight up into the air, much of it would be directed inland; in fact,
there is only an extremely small part of the entire number of rays
that will naturally take the useful direction. We therefore require
something round the lamp which shall catch the truant rays that are
running away to idleness and loss, and shall concentrate them into the
direction in which they will be useful to the mariner. An effective
way of doing this is to furnish the lamp with a reflector. On its
bright surface (Fig. 75) all the rays fall which would otherwise
have gone astray, and each of them is properly redirected, where the
sailors can see it. It is essential that the mirror shall do this work
accurately, and this it will only do when it has been truly shaped so
as to be a parabola.
You will remember, also, how I described to you the reflector which
Herschel made for his great telescope. The shape of the mirror must
be most accurately worked, and it, too, must have a parabola for its
section; so that you see this curve is one of importance in a variety
of ways.
But the grandest of all parabolas are those which the comets pursue.
Unlike the ellipse, the parabola is an open curve; it has two branches
stretching away and away forever, and always getting further apart. Of
course, in the examples of this curve that I have given it is only a
small part of the figure that is concerned. When you throw a stone it
only describes that part of the parabola that lies between your hand
and the spot where the stone hits the ground. It is just a part of the
curve in the same way that a crescent may be a bit of a circle. It is
to comets that we must look for the most complete illustration of the
ample extent of a parabola.
The shape of this grand curve will explain why so many comets only
appear to us once. It is quite clear that if you begin to run round a
closed racecourse, you may, if you continue your career long enough,
pass and repass the starting-post thousands of times. Thus, comets
which move in ellipses, and are consequently tracing closed curves,
will pass the earth times without number. For this reason we may see
them over and over again, as we do Encke’s comet or Halley’s comet. But
suppose you were travelling along a road which, no matter how it may
turn, never leads again into itself, then it is quite plain that, even
if you were to continue your journey forever, you can never twice pass
the same house on the roadside. That is exactly the condition in which
most of the comets are moving. Their orbits are parabolas which bend
round the sun; and, generally speaking, the sun is very close to the
turning-point. The earth is also, comparatively speaking, close to the
sun; so that while the comet is in that neighborhood we can sometimes
see it. We do not see the comet for a long time before it approaches
the sun, or for a long time after it has passed the sun. All we know,
therefore, of its journey is that the two ends of the parabola stretch
on and on forever into space. The comet is first perceived coming in
along one of these branches to whirl round the sun; and after doing so,
it retreats along the other branch, and gradually sinks into the depths
of space.
Why one of these mysterious wanderers should approach in such a hurry,
and then why it should fly back again, can be partially explained
without the aid of mathematics.
Let us suppose that, at a distance of thousands of millions of miles,
there floated a mass of flimsy material resembling that from which
comets are made. Notwithstanding its vast distance from the sun, the
attraction of that great body will extend thither. It is true the
pull of the sun on the comet will be of the feeblest and slightest
description, on account of the enormously great distance. Still, the
comet will respond in some degree, and will commence gradually to move
in the direction in which the sun invites it. Perhaps centuries, or
perhaps thousands, or even tens of thousands, of years will elapse
before the object has gained the solar system. By that time its speed
will be augmented to such a degree, that after a terrific whirl around
the sun, it will at once fly off again along the other branch of the
parabola. Perhaps you will wonder why it does not tumble straight
into the sun. It would do so, no doubt, if it started at first from a
position of rest; generally, however, the comet has a motion to begin
with which would not be directed exactly to the luminary. This it is
which causes the comet to miss actually hitting the sun.
It may also be difficult to understand why the sun does not keep the
comet when at last it has arrived. Why should the wandering body be in
such a hurry to recede? Surely it might be expected that the attraction
of the sun ought to hold it. If something were to check the pace of
the comet in its terrific dash round the sun, then, no doubt, the
object would simply tumble down into the sun and be lost. The sun has,
however, not time to pull in the comet when it comes up with a speed
20,000 times that of an express train. But the sun does succeed in
altering the _direction_ of the motion of the comet, and the attraction
has shown itself in that way.
I can illustrate what happens in this manner. Here is a heavy weight
suspended from the ceiling by a wire; it hangs straight down, of
course, and there it is kept by the pull of the earth. Supposing I draw
the weight aside and allow it to swing to and fro, then the motion
continues like the beat of a pendulum. The weight is always pulled
down as near to the earth as possible, but when it gets to the lowest
point, it does not stay there, it goes through that point, and rises
up at the other side. The reason is that the weight has acquired speed
by the time it reaches the lowest point; and that, in virtue of its
speed, it passes through the position in which it would naturally rest,
and actually ascends the other side in opposition to the earth’s pull,
which is dragging it back all the time. This will illustrate how the
comet can pass by and even recede from the body which is continually
attracting it.
Just a few words of caution must be added. Suppose you had an ellipse
so long that the comet would take thousands and thousands of years to
complete a circuit, then the part of the ellipse in which the comet
moves during the time when we can see it is so like a parabola, that
we might possibly be mistaken in the matter. In fact, a geometer will
tell us that if one end of an ellipse was to go further and further
away, the end that stayed with us would gradually become more and more
like this curve. Therefore, some of those comets which seem to move in
parabolas may really be moving in extremely elongated ellipses, and
thus, after excessively long periods of time, may come back to revisit
us.