the figure corresponding to the ratio 2: 3, given in Fig. 19.

EXPERIMENT 19.—Making the experiment with the free length of the rod D, of 25ğ inches (65.09 centimetres), the figure on the screen is like that opposite the ratio 3:4 of Fig. 15.

EXPERIMENT 20.-Giving the rod D 26 inches (66.7 centimetres) of free vibrating end, it makes with the rod C the figure opposite the ratio 4:5 in Fig. 15, showing that one rod makes 4 vibrations in the time that the other makes 5.

You may not succeed the first time in getting these figures from the directions I have given. This is because the pine rods which you use may have different elasticities and weights from those which gave us the lengths we have put in this book. But, by drawing out or pushing in the rod D, you will, after the expenditure of a little patience, find the lengths of the rod D which give the desired figures. When found, these lengths should be preserved by drawing a lead-pencil across the rod along the outer edge of the clamp F. These marks will serve you when you wish to repeat these experiments before your friends. A piece of wax will assist you in getting the right lengths of the rod D. Stick the piece of wax on one of the rods. If the figure on the screen becomes more quiet, this shows that this rod which carries the wax should be lengthened. If the wax shows that the rod should be lengthened, then we shorten the rod D, because the rod C always remains of the same length.

THE WAY TO DRAW THE ACOUSTIC CURVES.

By the aid of Fig. 5 we explained how one can plot on a line, A B, the distances that a pendulum goes

through in equal small portions of time, by drawing perpendiculars to the line A B from the points of equal divisions of the circumference of the circle above it. In Fig. 5, if we assume that the pendulum goes from A to B in one second, then it goes through each of the divisions on the line AB in one-eighth of a second.

The above mode of getting the distances the pendulum goes over in successive small portions of time will serve us to draw at our leisure all the acoustic curves given by Blackburn's double pendulum, or by the two vibrating rods.

For example, suppose we wish to draw the figure which is made when the two pendulums of the double pendulum, or the two rods in our last experiment, vibrate in the ratio of 4 to 5; that is, when one pendulum or rod makes 4 vibrations while the other pendulum or rod makes 5.

Draw the circle A C B D, Fig. 20, and inclose it in a square. Then draw the two diameters A B and CD parallel to the sides of the square. Divide the half-circumference ACB into 8—that is, twice 4-equal parts, and through the points of these divisions, 1, 2, 3, 4, 5, 6, 7, draw lines parallel to the diameter CD. Now turn the square round so that the diameter C D runs right and left. Then divide the half-circumference DA C into 10-that is, twice 5-equal parts, and through the points of these divisions, a, b, c, d, e, f, g, h, i, draw lines parallel to the diameter A B. By these operations we have drawn a network of lines in the square E F G H. The spaces on the line E H, or on any line parallel to it, show how far one pendulum or vibrating rod moves in equal times. Let us suppose these times eighths of a second. The spaces

on the line EF, or on any line parallel to it, show how far the other pendulum or rod moves in successive eighths of a second. Now let us begin at the corner E, and suppose that the point of the funnel of the double pendulum is over this corner. Where will the point of the funnel be

at the end of the first eighth of a second? By the motion of one pendulum it will have moved from E to I, and by the motion of the other pendulum it will have moved from E to J. Therefore, at the end of the first eighth of a second, the point of the funnel will be where the lines

drawn through the points I and J meet.

For a like reason, at the end of the second eighth of a second, the pendulum-bob is at the point of meeting of the two lines drawn through the points K and L. It now at once appears how to draw the figure. Begin at the corner E, and draw a line to the opposite corner of the little parallelogram JEI; then continue the line to the next diagonally opposite corner, always passing diagonally from corner to corner of the successive parallelograms. Never leave any parallelogram, save at the corner, and you will end by tracing the complete figure, and then you will find the point of your pencil in the corner H.

In like manner, the curves corresponding to any given ratio of vibrations may be drawn. The formation of these curves is a very fascinating occupation. After you have gone over one with a lead-pencil, you may widen the line with a drawing-pen, or a camel's-hair pencil dipped in Indian-ink. If you should hang up in your room these evidences of your progress in the art of experimenting, no one will call you vain.

EXPERIMENT 21.-One more experiment, and we will begin the study of vibrations giving sound.

Draw one of the acoustic curves in a square of 3 inches on a side, and place the figure so that its centre is directly under the point of the funnel of the double pendulum when this is at rest, and see that the sides of the square are parallel to the edges of the base-board of the pendulum.

The double pendulum having been accurately adjusted to trace the figure under it, draw the bob aside so that the point of its funnel is exactly over the corner of the square containing the figure. Pour sand in the funnel, and burn the thread. The pendulum starts on its journey, and as it

goes it really seems guided in its motion by the figure under it, for it strews the sand over its lines in the most precise manner; showing again, very neatly, that the motion of a pendulum is indeed very accurately reproduced by looking at a ball in the plane of the circle around which it uniformly revolves.