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also taught us that the numerical relation between the numbers of swings of the two pendulums is shown by the curved figure produced; so that, knowing the figure, we can tell the relative number of vibrations of each pendulum, and, from knowing the latter, we can pre

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dict the curved figure that the double pendulum will draw. But our experiments have taught us that a vibrating rod moves to and fro with the same kind of motion as a swinging pendulum. From this it follows that, if by any means we can combine into one motion the separate

motions of two vibrating rods, we shall make these rods describe the curved figures traced by the double pendulum.

The motions of two vibrating rods may be combined into one motion by means of a beam of light, which, falling on a mirror fastened to the end of one rod, is reflected to a mirror fastened to the end of the other rod, while from this second mirror the beam is reflected to a screen.

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It is absolutely necessary for the success of these experiments that the vibrating rods should be fastened to bodies which are heavy and firm, and do not vibrate when the rods are set in motion. Boxes A and B of Fig. 16, about 14 inches square, half filled with sand, gravel, or dry earth, make such supports. The rods C and D are of clear, white pine, 4 feet (121.9 centimetres) long, 1 inch (25 millimetres) wide, and inch (6.25 millimetres) thick. On the end of each rod is fastened with wax a silvered glass mirror, 1 inch square. The upright rod C is fastened to the side of the box A by two screws, which through the rod and into the box near the edge of its top. Another screw fastens the rod to the box at a distance of several inches below the upper screws. The free end of this rod, above the box, is exactly 30 inches (76.2 centimetres). The length of the horizontal rod D can be changed at will, for it is clamped to the side of the box B by screws, which go through the ends of the two pieces of wood F and G. Two nails are driven into the box under this rod, and serve to guide it in a horizontal direction while we slide it out or in. A piece of paper, 1 inch square, with a hole in its centre of inch in diameter, is pasted on the mirror of the rod C.

EXPERIMENT 17.-To begin the experiment, we place the heliostat in the window. The box A has been made of such a height that, when placed before the window,

the centre of the mirror is opposite the centre of the opening E in the heliostat. We now loosen the screws in the clamps F and G, and slide the rod D under the clamps till it projects beyond the box exactly 20 inches (51.91 centimetres). The boxes A and B are now placed in such positions that the light, falling on the 4-inch circle on the mirror of the rod C, is reflected to the squareinch mirror on the rod D, and thence is reflected to a white screen S at the other end of the room, on which it appears as a little bright circle.

In Fig. 16 the bright lines show the light coming from the heliostat E to the mirror on the rod C, then going to the mirror on the rod D, to be reflected by it to the screen S.

Now pull toward you the rod C, and let it go. At once the bright circle on the screen is drawn out into a vertical line. As the width of the swings of the rod become less and less, the line becomes shorter and shorter, and finally contracts to the little bright circle when the rod has ceased to vibrate. Now pull aside the rod D, and let it go. The little circle on the screen is now drawn out into a horizontal line.

These two motions of the spot of light are at right angles to each other, and are exactly like the motions of the two pendulums of the double pendulum. Hence, if both rods should vibrate at the same time, we should see the circle of light thrown into one of the familiar curved traces of the double pendulum. Let us try the experiment. Pull both rods aside, and let them go at the same instant. At once the little circle vanishes from the screen, and there appears in its place this figure (Fig. 17). We at once recognize it as the same figure which the double pendulum drew in sand when one of its

pendulums made two swings while its other pendulum made one. Therefore, one of these rods swings twice while the other swings once.

It may be that the figure on the screen is not stationary, but appears to twist and untwist itself with a sort of

FIG. 17.

revolving motion. If it does so, it will go through curious changes. The horns of the above figure will split open at their ends, as shown at B in Fig. 18, and, while they open more and more, they bend more and more into a line with each other, until the figure is like an 8, as at

B

FIG. 18.

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C. Then the 8 bends in its middle to the right, as in D, while its openings close up more and more, until the figure A again appears at E, but with the horns pointing to the right instead of to the left. Thus the figure changes, becoming smaller and smaller, till it vanishes into the circle of light from which it sprang.

By drawing the rod D in or out, or by loading it, or the rod C, with a lump of wax, the figure (17) may be made stationary as long as the rods vibrate; and, when this has been done, we know that one of the rods makes one vibration while the other makes exactly two; for the twisting and untwisting of the figure are caused by one of the rods making slightly more or less than one vibration while the other makes two.

Indeed, so delicate is this method of tuning one vibrating body with another, that it is used as the most precise one known to bring two tuning-forks to any required ratio in their vibrations. Hence these figures are sometimes called "the acoustic curves." In testing the forks, they are placed, like the rods, with their prongs at right angles, and the light is reflected from their polished prongs, as is shown in Fig. 16. Then, with a file, some of the metal of one of the forks is removed, either from the ends or base of its prongs, till the figure on the screen remains stationary.

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EXPERIMENT 18.-The rod D is now unclamped, and slid out till 24 inches (61.38 centimetres) of its end project beyond the edges of the clamps and box. The clamps are again screwed tightly against the rod. Now, on vibrating the rods together, we have on the screen

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