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.

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