2.  Figures with a ratio of which the sum of the numbers composing it is an even number (examples, 1:3, 3:5, 3:7) are symmetrical, one half of the figure reproducing the other.  If the sum is Uneven, as in 1:2, 2:3, 2:7, the figure is unsymmetrical. (Fig. 177, A.)

3.  The ratio 1:3 is the easiest to begin upon, so the experimenter’s first efforts may be directed to it.  He should watch the growth of the figure closely, and note whether the repeat line is made in front of or behind the previous line of the same loop.  In the first case the figure is too flat, and the weight of the upper pendulum must be raised; in the second case the weight must be lowered.  Immediately an exact harmonic is found, the position of the weight should be recorded.

Interesting effects are obtained by removing the lower pendulum and allowing the apparatus to describe two elliptical figures successively, one on the top of the other, on the same card.  The crossing of the lines gives a “watered silk” appearance to the design, which, if the pen is a very fine one and the lines very close together, is in many cases very beautiful.

Readers who wish for further information on this fascinating subject are recommended to purchase “Harmonic Vibrations,” published by Messrs. Newton and Co., 72 Wigmore Street, London, W. This book, to which I am much indebted, contains, besides much practical instruction, a number of charming reproductions of harmonograms.

Before closing this chapter I should like to acknowledge the kind assistance given me by Mr. C. E. Benham, who has made a long and careful study of the harmonograph.

XXXII.  A SELF-SUPPLYING MATCHBOX.

This useful little article can be constructed in a couple of hours by a handy person.  In general idea it consists of a diamond-shaped box to hold vestas, working up and down diagonally on a vertical member (A in Fig. 179 (1)), which passes through slits at the top and bottom, and runs in grooves cut in the sides of the box.  The top of A is grooved to allow a match to rest on it.  When the box is drawn up to the full extent allowed by a transverse pin in the slot shown in Fig. 179 (2), the groove is at the lowest point of the box, and is covered by the matches.  When the box is lowered, A catches a vesta and takes it up through the top, as seen in Fig. 178, for removal by the fingers.

The only materials required are a cigar-box, some pins, and a supply of glue.  The box should be carefully taken to pieces, and the parts soaked in hot water till freed of all paper, and then allowed to dry under pressure, small slips of wood being interposed across the grain to keep them separate and permit the passage of air.

[Illustration:  Fig. 178.—­Self-supplying matchbox, with match in position for removal by fingers.]

When the wood is dry, cut out with a fret saw two pieces shaped like Fig. 179 (3), to form the ends of the box.  Allow a little surplus, so that the edges may be finished off neatly with chisel and plane.  The two ends should match exactly, or there will be trouble at a later stage.