312 
NATURE 
[ Aug. 17, 1871 

I procured three or four path-tracks superposed, including 
vibrations in the planes of obedience, by which it was 
easy and instructive to examine at leisure the geometrical 
relations of the various planes. Fig. 14 is a specimen of 
these twig-tracings with cross-vibrations to show the 

Fic. 14.—Specimen (obtained from nature) of the path of a stiff vibrating 
hazel shoot. Much friction. a, the starting-point. 3s, the end of the first 
cycle xx’, the path of the twig set vibrating exactly in the plane of slow 
vibration. yy’, nearly in the plane of quick vibration. 

planes of obedience (Xx’, yY’.) Ais the starting point, 
and B the point where the first retrograde step begins in 
the secondary plane. It will be seenon examination that 
from A to B the twig has accomplished exactly six quasi- 
elliptic journeys resolved parallel to the plane xx’, and 
six and a half resolved parallel to yy’. So xx’ is the 
plane of slow vibration, and yy’ is the plane of quick 
vibration, and the periods of vibration in those two planes 
respectively are in the proportion of 13 to 12. 
While considering these points, it occurred to me that 
similar results would be given by the oscillation of a 
pendulum jointed in such a manner as to swing in one 
plane only by one joint, and in the cross-plane only by a 
second joint at a different level from the first. The oscil- 
lation from the lower joint would be more rapid than 
that from the higher, and we should have exactly the 
same conditions of simultaneous motion in two planes 
with unequal periods as we had in the case of the acacia- 
twig. This was easily tried. From a cross-bar on an 
extempore tripod-stand I hung a rod by string-hinges, 
with an intermediate piece having its joint-edges at 
right angles, so that the rod was swinging in one plane 
by the joint between the cross-bar and the intermediary, 
and in the cross-plane by the joint between the inter- 
mediary and the rod. In any intermediate plane the 
rod could only swing by do¢/ these joints; its motion 
being really and veritably resolved in those two planes 
at right angles ; with a longer period for the part resolved 
in the plane allowed by the upper joint than for the part 
resolved in the plane allowed by the lower. With the 
help of a weight of lead at the bottom of the rod, my 
make-shift pendulum gave a capital illustration of the 
problem, and the gravity and deliberation of its behaviour 
afforded better opportunity for study than was given by 
the more brilliant but less persistent energy of the acacia- 
twig. 
The next step that naturally suggested itself was to 
obtain a permanent authentic record of the grave gyra- 


tions of my pendulum. I wanted something more per- 
manent than pencil-marks, and more delicate than the 
daubs produced by a paint-brush full of colour. Clearly 
I wanted a pen that would deliver its ink in any direction 
all round universally. Such a pen I obtained by taking a 
small piece of glass tube four or five inches long and 
about a quarter of an inch in diameter,and melting one 
end in the flame of a Bunsen’s burner, and drawing it 
out to a capillary tube, then breaking the point off square, 
and smoothing the broken edges of the pore in the flame, 
to run smoothly onthe paper. By suction 1 drew up a 
small quantity of ink into the tube through the micro- 
scopic pore at the point, and then fastened my pen in a 
groove at the end of the pendulum-red by elastic bands, 
so that it could be raised or lowered within shout limits 
at pleasure. Then having adjusted the elastic suspension 
of the paper so that it hung evenly beneath the pen with 
a slight concavity to accommedate the nearly spherical 
“locus” of the pen-poist, I drew the pendulum aside, 
and lowered the pen till it was on the point of touching 
the paper, then let the pendulum recede till the pen 
actually touched the paper, and then let go. It was 
beautiful to see the unerring certainty with which the 
pen-point struck its curves in obedience to the law im- 
posed by its two-fold suspension. The yery first back- 
stroke began the deviation from the primary plane, and 
every successive stroke made the ellipse wider and shorter 
by steps whose regularity was marvellous to watch. 
Slowly and surely the figure was filled up, line within 
line, line across line, as the ever-changing ellipse oscil- 
lated slowly from one side to the other of the plane of 
slow vibration, 

Fic, 15 
unity, about 50 : 51. 
Curve traced by double jointed pendulum, with proportion near 
Fig. 15 illustrates this stage of experiment. Contact in 
this case was purposely broken at the moment when the 
oscillation had reached the secondary plane. The tubular 
glass pen did its work very well, delivering the ink with ease 
and regularity, and gliding almost noiselessly over the 
paper with very little friction. But even that small amount 
of friction, added to hinge-rub and air-resistance, required 
great weight in the pendulum to overcome it. I kept 
adding one lump after another till there were forty or fifty 
pounds of lead lashed to the rod immediately above the pen. 
The elastic suspension of the paper, by india-rubber bands 
attached to the four corners, was very serviceable, and 
