By changing the relation and phases of the oscillations we obtain curves of infinitely varied aspect. M. Tisley has a collection of more than three thousand curves, which we have had occasion to glance over, in which we failed to meet with two corresponding figures. The ratio between these curves corresponds with some special class, of which the analyst may define the general characters, but which is outside our present subject. By giving the plate P a rotatory movement, we obtain spiral curves of a very curious effect, but the apparatus is more complicated. Considered from this point of view it constitutes an interesting mechanical apparatus, showing the combination of oscillations, and resolving certain questions of pure mechanics. From the point of view of acoustics it constitutes a less curious object of study. The experiments of M. Lissajons have proved that the vibrations of diapasons are oscillations similar to, though much more rapid than those of the pendulum. We can therefore with this apparatus reproduce all the experiments of M. Lissajons, with this difference, that the movements being slower are easier to study. When the ratio between the _number of vibrations_—we purposely use the term vibration instead of the term oscillation—is a whole number, we obtain figs. 177 and 178. If the ratio is not exact, we obtain fig. 179, which is rather irregular in appearance, corresponding to the distortions noticeable in M. Lissajon’s experiments. Fig. 178 has been traced in the exact ratio 2:3; fig. 177 in the ratio 1:2; and fig. 179 corresponds to the ratio 1:2 and a small fraction, which causes the irregularity of the figure. [Illustration: Fig. 177.—Ratio 1:2. Fig. 178.—Ratio 2:3.] [Illustration: Fig. 179.—Ratio 1:2 and a fraction.] [Illustration: Fig. 180. Construction of the Harmonograph. Fig. 181.] [Illustration: Fig. 182.—Method of constructing an Harmonograph.] [Illustration: Fig. 183.—The apparatus completed.]