ick,
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.]

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