
Sept. 7, 1871] 
NATURE 
371 

arranged, they conspire to establish a virtual second joint, 
at right angles to the topmost joint, and at some point 
below it not necessarily coinciding with the position of 
any one of the actual joints, and apparently varying for 
different positions of the plane of vibration. 
At this stage I showed some of my pendulum’s tracery 
to my brother-in-law, Mr. E. J. Routh, M.A., of St. 
Peter’s College, Cambridge, and received some hints from 
him which led me to adopt an arrangement whereby the 
position of the lower joint could be varied, so as to bring 
the two periods of vibration into any simple proportion, 
as-1: 2,2:3, &c, This was done by a very simple mode 
of suspension,—two cords hanging from two fixed points 
some distance apart, and passing through a small ring 
that could slide up and down the concurrent cords and be 
fixed at any height. In this arrangement I found the 
germs of an infinite variety of curious and elegant curves, 
that gave a new lease of life to my course of experiment. 
Before attempting anything, I drew in my pocket-book 
the skeletons of the curves that would be described corre- 
sponding to the proportions, 1:2, 2:3, 3:4, 4:5, 5:6, 
6:7, 7:8, 8:9, in the periods of quick and slow vibra- 
tion, Then I hung a bullet by two threads from two 
points about ten inches apart on the circumference of a 
child’s wooden hoop, fixing the hoop upright with the 
bullet swinging inside the circle. Another bit of thread 
furnished the sliding-ring to nip the two convergent 
threads at any desired point, anda few trials enabled me 
to fix it pretty accurately for the proportion 1 : 2, so that 
the bullet would swing in the plane of the hoop in half 
the time it took to swing at right angles to the plane of 
the hoop, making two vibrations in the plane of the hoop 
to one vibration athwart. (The length of the pendulum 
varies as the square of the period of vibration, so the 
slider was at a point one quarter of the distance from the 
bullet to the horizontal line between the two points of 
suspension.) It was with great interest that I watched 
the motion of my pensile bullet, and greeted the verifica- 
tion of my pocket-bo>k sketch. Pull the bullet aside and 
start it obliquely, and it describes a crescent-moon, the 
two horns being formed by the double s ving in the plane 
of the hoop, while the length of the crescent measured 
from tip to tip is given bya single swing athwart the 
plane of the hoop. Start the bullet from rest at the centre 
by a sudden blow obliquely given, and it describes a figure 
of eight with its length athwart the plane of the hoop. 
(See Figs. 5 and 6.) Other proportions gave still more 
curious results in accordance with my skeleton sketches, 
and made me impatient to try them on a larger scale. 
The lead at my command I packed into a long-shaped 
zinc box, with a tubular orifice at the top, which in my 
service became the bottom, To the lead I added a stone 
jar containing some 1olbs. of mercury, and made every- 
thing secure with cordage. The dependent tubular orifice 
seemed made on purpose to accommodate a few inches 
of wooden roller which carried the glass pen, and a 
diagonal beam in the ceiling of my bedroom offered a 
capital fixture for two rings about four feet apart, giving 
suspension to two iron chains by which my incongruous 
pendulum-bob was doomed to swing. To nip the chains 
together at the requisite height, I used a loose link which 
hooked into the corresponding links in the two conver- 
gents, and made a very rudeand coarse adjustment, which 
left all accuracy to chance. It chanced, however, that 
the adjustment for the proportion 2:3 was beautifully 
accurate, and I shall never forget the feeling of delight 
which I experienced while watching the marvellous 
fidelity with which the pen-point traced the curve appro- 
priate to that proportion. The pendulum was drawn 
aside and started obliquely on one side of the plane of 
slow vibration, and having to make three vibrations across 
that plane to two vibrations to and fro, it compounds these 
into a curve like a capital Q with two tails, one on each 
side, looking like a swallow-tailed balloon. (See Fig. 9.) 



At the end of the second to-and-fro vibration the pen 
returns to the point whence it started, except that friction 
compels it to fall short little by little at every stroke ; but 
if the adjustment is accurate, as it was in this particular 
case, the shape of the curve remains the same from first 
to last, and the figure is filled up to the very centre by the 
orderly description of curve within curve conspiring to 
produce a web of lines of astonishing regularity. If the 
adjustment of the connecting link is very slightly inaccu- 
rate, the curve begins to change its shape little by little at 
every stroke, in one way or another, according as the link 
is too high or too low; and wonderfully intricate is the 
result, for after a due series of intermediate stages, the 
original figure reappears, but reversed ; and after ano ter 
series of changes it presents itsclf again to view in its 
original posture, but much diminished by the friction thathes 
been in operation throughout all these changeful phases. 
It may be imagined how intricate the web becomes, 
though the limits of illustration do not.allow me to give a 
specimen here. It was easy to eliminate all the transi- 
tion-curves from the tracery, by depressing the paper for 
the proper interval, and allowing it to return to contact 
with the pen only at: those distinctive phases when the 
original figure was reproduced either ere:t or reversed. 
I obtained a very curious specimen by applying this 
selective method to the case of the proportion I : 2, allow- 
ing the pen to mark only the crescents and the figures of 
8 in alternate series, converging orderly to the centre. 
For the suspension of the paper, [ fixed four pairs of 
upright rods at the four corners of a shallow tray, which 
could be slipped under the pendulum, and each rod gave 
support to an india-rudber band, which, with its fellow at 
right angles, was attached by a small hook to the corre- 
sponding angle of the paper. Each bind could be slid up 
or down its rod, to allow of nice adjustment of the level of 
the paper, and the whole tray could be raised on a foot- 
stool or chair, to suit the elevation of the pendu um when 
the slider was run aloft in attaining the proportions nearer 
unity, such as 7 : 8, or 8:9. 
The iron chain was soon exchanged: for strong cords, 
passing through a narrow wire ring, which could be ar- 
rested at any point bya needle driven through both cords 
below the ring. This was a small improvement, allowing 
more accuracy in the adjustment of the slider, and there- 
fore more accuracy in the proportion between the two 
periods of quick and slow vibration. But it was. still 
very far from satisfactory. Meanwhile I had ordered 
a cylinder of lead, weighing half a cwt, to be cast, 
with a hole through the axis; for my zinc box full 
of “notions” was so tall that I could not bring 
the slider near enough to the centre of gravity to obtain 
any proportion lower than 1 : 3, and that only with great 
trouble. When the cylinder of lead appeared, I sawed it 
into two unequal portions, so that [ could use either or 
both ; and instead of simple cords, which twisted in a 
most troublesome manner below the sliding ring I intro- 
duced a stiff rod of fir to carry the lead by a cross pin, 
and I used two pairs of cords passing throvgh holes in a 
slider on either side of the central rod. This slider wasa 
small block of wood pierced to fit the rod, and provided 
with a lateral screw to fix it at any required height. This 
arrangement ensured admirable steadiness and freedom 
from torsion, and a great many sheets were filled with the 
improved performances of the machine ; but there still 
reinained an important defect to remedy. The coarse 
cords, at the point where they entered the holes in the 
slider, made a very rough hinge for the cross vibration to 
rely upon, and it was manifest in the tell-tale records of 
the curves described that considerable change of period 
took place between the beginning and the end of the web ; 
and the change was always such as to increase the dis- 
parity between the two periods, which could only mean 
that the level of the centre of quick vibration in the cords 
immediately above the slider was lowered when the range 
