Suppose that the pendulum goes from a to b, or from b to a, in one second, and that, while the point of the funnel is just over L, we slide the board so that, in two seconds, the end M of the line L M comes under the point of the funnel. In this case, the sand will be strewed by the pendulum to and fro, while the paper moves under it through the distance L M. The result is, that the sand appears on the paper in a beautiful curve, LC ND M. Half of this curve is on one side of L M, the other half on the opposite side of this line.

The experimenter may find it difficult to begin moving the paper at the very instant that the mouth of the funnel is over L; but, after several trials, he will succeed in doing this. Also, he need not keep the two sand-lines, L M and a b, on paper during these trials; he may as well use their traces, made by drawing a sharply-pointed pencil through them on to the paper.

By having a longer board, or by sliding the board slowly under the pendulum, a trace with many waves in it may be formed, as in Fig. 8.

As the sand-pendulum swung just like an ordinary pendulum when it made the wavy lines of Figs. 7 and 8, it follows that these lines must be peculiar to the motion of a pendulum, and may serve to distinguish it. If so, this curve must have some sort of connection with the motion of the conical pendulum, described in Experiment 6. This is so, and this connection will be found out by an attentive study of Fig. 9.

In this figure we again see a wavy curve, under the same circular figure which we used in explaining how the motion of an ordinary pendulum may be obtained from the motion of a conical pendulum. This wavy curve is made directly from measures on the circular figure, and certainly bears a striking resemblance to the wavy trace made by the sand-pendulum in Experiment 7. You will soon see that to prove that these two curves are precisely the same, is to prove that the apparent motion of the conical pendulum is exactly like the motion of the ordinary pendulum.

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The wavy line of Fig. 9 is thus formed: The dots on A B, as already explained, show the apparent places of the ball on this line, when the ball really is at the points correspondingly numbered on the circumference of the circle. Without proof, we stated that this apparent motion on the line AB was exactly like the motion of a pendulum. This we must now prove. The straight line L M is equal to the circumference of the circle stretched out. It is made thus: We take in a pair of dividers the distance 1 to 2, or 2 to 3, etc., from the circle, and step this

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M

FIG. 9.

distance off 16 times on the line L M; hence L M equals the length of the circumference of the circle. In time this length stands for two seconds, for the ball in Experiment 6 took two seconds to go round the circle. This same length, you will also observe, was made in the same time as the sand-line L M was made in Experiment 7. In Fig. 9 the length L M, of two seconds, is divided into 16 parts; hence each of them equals one-eighth of a second, just as the same lengths in the circle equal eighths of a second. Thus the line L M of Fig. 9, as far as a record of time is concerned, is exactly like the sand-line L M of Experiment 7, and the line A B of Fig. 9, in which the ball appeared to move, is like the line a b of Fig. 7, along which the sand-pendulum swung.

Now take the lengths from 1 to 2, 1 to 3, 1 to 4, 1 to 5, and so on, from the line A B of Fig. 9, and place these lengths at right angles to the line L M at the points 1, 2, 3, 4, 5, and so on; by doing so, we actually take the distances at which the ball appeared from 1 (its place of greatest velocity), and transfer them to LM; therefore, these distances correspond to the distances from L M, Fig. 7, to which the sand-pendulum had swung at the end of the times marked on L M of Fig. 9.

Join the ends of all these lines, 2 2′, 3 3′, 4 4′, etc., by drawing a curve through them, and we have the wavy line of Fig. 9.

This curve evidently corresponds to the curve LCN DM of Fig. 7 made by the sand-pendulum; and it must be evident that, if this curve of Fig. 9 is exactly like the curve traced by the sand-pendulum in Experiment 7, it follows that the apparent motion of the conical pendulum, as seen in the plane in which it revolves, is exactly like the real motion of an ordinary pendulum.

EXPERIMENT 8.—To test this, we make on a piece of paper one of the wavy curves exactly as we made the one in Fig. 9, and we tack this paper on the board L M of the sand-pendulum, being careful that when the board is slid under the stationary pendulum the point of the funnel goes precisely over the centre line L M (Fig. 9) of the curve.

Now draw the point of the funnel aside to a distance from the line L M equal to one-half of A B, or, what is the same, from 5 to 5' of Fig. 9. Pour sand in the funnel, and let the bob go. At the moment the point of the funnel is over L, slide the board along so that, when the point of the funnel comes the third time to the line L M, it is at the end M of this line. This you may not succeed in doing at first, but after several trials you will succeed, and then you will have an answer from the pendulum as to the kind of motion it has, for you will see the sand from the swinging pendulum strewed precisely over the curve you placed under it. Thus you have conclusively proved that the apparent motion of the conical pendulum, along the line A B, is exactly like the swinging motion of an ordinary pendulum.

As it is difficult to start the board with a uniform motion at the very moment the pendulum is over the line L M, it may be as well to tack a piece of paper on the board with no curve drawn on it, and then practise till you succeed in sliding the board under the pendulum, through the distance L M, in exactly the time that it takes the pendulum to make two swings. Now, if you have been careful to have had the swing of your pendulum just equal to A B, or from 5 to 5' on the drawing of the curve, you will have made a curve in sand which is precisely like the curve you have drawn ; for, if you trace

the sand-curve on the paper by carefully drawing through it the sharp point of a pencil, and then place this trace against a window-pane with the drawing of the curve, Fig. 9, directly over it, you will see that one curve lies directly over the other throughout all their lengths.

This curve, which we have made from the circle in Fig. 9, and have traced in sand by the pendulum, is called the curve of sines, or the sinusoid. It is so called because it is formed by stretching the circumference of a circle out into a line, and then dividing this line, L M of Fig. 9, into any number of equal parts. From the points of these divisions, 1, 2, 3, 4, 5, etc., of LM, we erect perpendiculars, 2 2', 3 3', 4 4', 5 5', etc., equal to the lines a 2, b 3, c 4, d 5, etc., in the circle. These lines in the circle are called sines; so, when we join the ends of these lines, erected to the straightened circumference, by a curve, we form the curve of sines, or the sinusoid.

The sinusoid occurs often during the study of natural philosophy. We may meet with it again in our book on the nature of light, and it certainly will occur in our book on heat.

AN EXPERIMENT WHICH GIVES US THE TRACE OF A

VIBRATING PINE ROD.

A in Fig. 10 represents a rod 4 feet (121.9 centimetres) long, 1 inch (25 millimetres) wide, and inch (6) millimetres) thick, made of clear, well-seasoned pine. This is fastened by means of small screws to the wooden box B standing on a table. This box may be of any convenient size; but, as it is to be used for another experiment, it may be made about 14 inches (35.5 centimetres) square and 30 inches (76.2 centimetres) high. A shoe-box will answer for the purpose. This