For the upper harmonies, 1:4 down to 3:8, the two pendulums may be almost equally weighted, the top one somewhat more heavily than the other. The upper weight is brought down the rod as the ratio is lowered.
To continue the harmonies beyond, say, 2:5, it is necessary to load the upper pendulum more heavily, and to lighten the lower one so that the proportionate weights are 5 or 6:1. Starting again with the upper weight high on the rod, several more harmonies may be established, perhaps down to 4:7. Then a third alteration of the weights is needed, the lower being reduced to about one-twentieth of the upper, and the upper weight is once more gradually brought down the rod.
Exact figures are not given, as much depends on the proportions of the apparatus, and the experimenter must find out for himself the exact position of the main weight which gives any desired harmonic. A few general remarks on the action and working of the Twin Elliptic will, however, be useful.
1. Every ratio has two forms.
(a) If the pendulums are working against each other— antagonistically—there will be loops or points on the outside of the figure equal in number to the sum of the figures in the ratio.
(b) If the pendulums are working with each other—concurrently—the loops form inside the figure, and are equal in number to the difference between the figures of the ratio. To take the 1:3 ratio as an example. If the tracing has 3+1=4 loops on the outside, it is a specimen of antagonistic rotation. If, on the other hand, there are 3-1=2 loops on the inside, it is a case of concurrent rotation. (Fig. 176, A.)
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.