glass may be laid and drawn under the brass point or style by a cord passing over the pulley C. Two weights, D and E, are attached below, the upper one, D, being just equal to the friction of the carriage. Some pieces of glass about three inches by four, are needed, and a gas-burner, by which they may be covered with lampblack. By using the size of glass employed in the lantern for projections, the curves may be thrown on the screen on a greatly enlarged scale.

B

Experiment. Cover one of the plates of glass with a layer of lampblack by holding it by one corner over the gas-flame, and moving it about so that the coating shall be uniform, and very thin. Instead of lampblack, collodion may be used, pouring it on in the usual way, as when taking a photograph. Care must be taken to select such collodion as

Fig. 47.

will give an opaque and very tender film, when results of extreme beauty and delicacy will be obtained. Lay the glass down on the carriage, and raise it so that when passed under the style, the latter will just touch its surface. This may be accomplished by wedges or levelling screws under the glass. Draw B back a short distance beyond the style, and release it, when it will begin to move under the action of the two weights D and E. The length of the cord should be such that when the wagon reaches the style, E will touch the floor so that the carriage will move with a uniform motion by its inertia, the friction being just compensated by D. The style will accordingly draw a fine unbroken straight line over the glass. Now sound the fork by the violin bow (see Experiment 61), and again pass the carriage under, when the line, instead of being straight, will be marked by sinuosities, one corresponding to each vibration of the fork.

Next sound the harmonic, by drawing the bow somewhat more rapidly, and with less pressure than before, at a point about twothirds of the distance from the end of the prong to the handle. The sound sometimes comes out more readily by lightly touching the intermediate one-third point or node with the finger. A high, clear note is thus produced, and on drawing the carriage back the

same distance as before, and letting it again pass under, another curve is obtained, with indentations much nearer together, owing to the greater rapidity of the undulations. Of course the plate is moved sideways a short distance each time, to prevent the curves from overlapping. Produce the fundamental note, and while it is sounding draw the bow so as to give the harmonic, and immediately let go the carriage. A curve is thus obtained, resulting from these two systems of vibrations, and consisting of small sinuosities superimposed on larger ones. Determine their ratio by seeing how many of the former correspond to an exact number of the latter. Write on the lampblack your name and the date, and if all the curves are good, varnish the plates to render them permanent. For this purpose expose the blackened surface to the vapor of boiling alcohol to remove the grease, then holding it by one corner pour amber varnish over it precisely as when varnishing a photographic negative.

To compare the lines with theory, place the glass in a magic lantern, and project an image of it on the screen. If the sun is used as a source of light, it is scarcely necessary to darken the room. Place a sheet of paper so that three or four undulations of the curve of the fundamental note shall fall on it. Trace them carefully with a pencil and an enlarged reproduction of the original is obtained. Draw lines tangent to the waves above and below, and bisect the space between them by a line. It will intersect the curve in points at regular intervals b, any one of which may be taken as the origin of coördinates. If a is the height of the wave, or one half the distance between the two tangent lines, the theoretical equation will be y = a sin7 Construct points of this curve by dividing the space between two consecutive intersections of the curve into six equal parts, and lay off vertical distances equal to a multiplied by sin 15°, 30°, 45°, etc., to 180°. These sines have the following values: sin 15° .259, sin 30°.500, sin 45° = .707, sin 60°.866, sin 75° = .966. Draw a smooth curve through the points thus obtained, and compare it with that given by the forks. To test the combination of the two systems of vibrations is more difficult, but it may be done

που

που

and y"

a' sin and adding them so that y=y'+y" = a sin +

a' sin

πο

b

An immense variety of curves may be obtained by mounting the plate on a second tuning-fork, which is also set vibrating. Different curves are thus obtained, according as the motion of the style is parallel or perpendicular to the vibrations of the plate, also with every change in the interval between the two forks. With this arrangement it is much better to maintain the vibrations of one or both forks continuously by electricity. Better effects are also obtained in this way in Melde's and Lissajous' experiments.

Instead of projecting the curve on the screen it may be measured by the Dividing Engine, Experiment 21, or enlarged by a microscope and drawn by a camera lucida. The length of the waves gives a very delicate test of the uniformity of the motion of the car, a difference of a ten thousandth of a second being easily perceived.

65. LISSAJOUS' EXPERIMENT.

Apparatus. Mirrors are attached to the ends of the prongs of two tuning-forks, and the image of a spot of light reflected in them is viewed in a telescope. The planes of the tuning-forks must be perpendicular, that is, one must vibrate in a vertical, the other in a horizontal plane. It is best to have a series of forks with sliding weights, so that all the intervals in the octave may be obtained. A good spot of light is produced by a gas flame shining through a small aperture in a metallic plate, or a mirror may be used to reflect the light of the sky.

Experiment. On looking through the telescope a minute spot of light should be visible. When one of the tuning-forks is sounded the mirror is moved from side to side, carrying the image. of the spot with it so rapidly as to make it appear like a horizontal line of light. In the same way the motion of the other fork produces a vertical line. When both sound, a curve is formed, which remains unchanged if the concord is exact, but continually alters if the forks are not a perfect tune. Bring the forks in uni

son by placing the weights on the corresponding points of each. They are best sounded by a bass-viol bow, drawing it slowly and with pressure over the end of one prong nearly parallel to, but not touching, the other. As the bows soon wear out by the horsehair giving way, a convenient and cheap substitute is made by covering a strip of wood of proper shape with leather, which when rubbed with resin, answers very well.

On sounding both forks, having brought them in unison as above, the point of light is in general converted into an ellipse which, as it is impossible to tune them exactly by the ear, gradually changes into a straight line, then into an ellipse, a circle, an ellipse turned the other way, a straight line and so on. Raise the pitch of one of the forks slightly, by moving the weight towards the handle, and if the changes take place more slowly the unison is more perfect. By trial, first moving the weights one way and then the other, they may be brought in tune with any desired degree of exactness, and far nearer than is possible by the ear alone, as the complete change of the curve from one line to the other denotes that one fork has advanced only a single vibration.

Next make one fork the octave of the other, and a curve is obtained, changing from the parabola to the lemnescata, or figure 8. A simple rule serves to determine the interval in all cases from the curve. Count the number of points where the latter touches the sides of the rectangle bounding it, also the number of points where it touches its top or bottom; the ratio of these two is the interval between the forks. When the curve terminates in either corner this point must be counted as one half on the horizontal, and half on the vertical, bounding line. Thus in Figure 48, both A and B correspond to the ratio of 2 3, or the interval of the fifth.

Fig. 48.

B

The more perfect the concord the more slowly will the curves alter their form, and the simpler the ratio of the number of vibra tions the simpler the curve. When the forks are not quite in unison, beats will be heard, and the curve will then be seen to alter its form so as to keep time with them. Next try some other ratios, as 2, %, t, §, 1, 4; also some more complex curves, as §, §,

and 71.

3 3

9

66. CHLADNI'S EXPERIMENT.

Apparatus. A number of brass plates attached to a stand, a violin bow and some sand. A good series of plates consists of three circles, whose diameters are as 22: 1, and their thickness as 1:2:1. Also three square plates, similarly proportioned.

Experiment. The plates are sounded by touching them at certain points and drawing the bow across their edges, holding it nearly vertical, and moving it slowly and with considerable pressure. A sound is thus produced, and certain lines are formed on the plate called nodal lines, which remain at rest, the other parts vibrating. If sand is sprinkled uniformly over the plate, that on the nodal lines will remain there, the rest being thrown up and down, so that finally it will all collect on these lines. The higher the note the more complex the nodal lines, and the nearer they are together.

Taking first the largest and thinnest circular plate, touch it at any point of the circumference, and bow it at a point about 45° distant. The sand will collect on two lines at right angles. Next bow it at a point 90° distant, and it will divide into six parts. By touching the plate at two points distant 45° with the thumb and middle finger of the left hand, and bowing the point midway between them, a division into eight equal parts is obtained. In the same way 10, 12, or any even number of parts are formed, until the divisions become so small that they cannot be sounded.

Next try the first square plate. The lowest sound this will give is obtained by touching the centre of one side, and bowing the corner. The next note, a fifth above, is produced, when the corner is held and the centre bowed. By altering the position of the fingers and bow, a great variety of figures may be obtained, which may be still further extended by changing the points of support, or the form of the plate. Moreover, among plates of the same shape some seem to give out certain curves more easily than others, owing probably to peculiarities in their internal structure. A square plate generally gives readily, besides the curves described above, one formed of two diagonal lines, and four half ovals on its edges. The pitch is three octaves above that of the diagonal lines alone. Another curve of extreme beauty consists of a circle with eight radial lines, and the intermediate spaces