* 
Aug. 17, 1871] 
NATURE 
311 

at last became a straight line in a second plane at right 
angles (roughly speaking) to the first. The vibration 
continuing, the twig began to retrace its path, and re- 
turned to the plane in which it started, by a complete 
recantation of its former errors, though the gradually fail- 
ing strength of its oscillation was gradually diminishing 
the range of its orbit. No sooner was the original primary 
plane regained, than it was again forsaken for the 
secondary, the errant twig repeating its delirious maze of 
elliptic gyration, but always with a method in its madness, 
across and across, again and again, till it finally came to 
rest in the centre of its web, still striving to the very last 
perceptible tremor to persevere in its life-long career of 
consistent vacillation. 
Repeating the experiment again and again, I found that 
there were two planes, at right angles, in either of which 
tke twig would vibrate obediently, without deviation to 
one side or the other, and that the primary and secondary 
planes of the first experiment made equal angles with 
either plane of obedient vibration. When the twig was 
started only a few degrees on one side of either plane 
of obedience, its elliptic error carried it into a secondary 
plane only a few degrees on the other side, and then 
back again and again; while if the primary plane was 
chosen half-way between those planes of obedience, in 
opposite quadrants, then the secondary plane was found 
to lie half way in the alternate quadrants, at right angles 
to the primary. 
How to explain this phenomenon was a puzzle, till my 
father hinted that its law might lie in a difference of 
periods of oscillation in those two planes of obedience, 
caused-probably by the curved shape of the twig or per- 
haps by its elliptic section, at any rate caused by some 
condition which made the twig vibrate as a short spring 
with short period in one direction, and as a long spring 
with long period in another direction at right angles to 
the first. 
This hint gave the key to the puzzle, and it was easy to 
demonstrate that all the phenomena would necessarily 
follow on such an assumption. Laying down two systems 
of rectangular co-ordinates to represent the spaces de- 
scribed in so many units of time (the motion of the twig 
being resolved in those two directions at right angles), and 
making 7 such spaces in one direction and 7 - 1 in the 
other, we had a diagram on which we could trace the 
twig’s path, beginning at one corner and drawing the 
diagonals in the successive rectangular spaces. If there 
Were 7 such spaces in both directions (which would repre- 
sent equal periods of oscillation), our course of diagonals | 
would only carry us into the opposite corner, with no 
alternative but to retrace the same line to and fro without 
deviation ; but since in one direction there remains one 
sp.ce over when we reach the border of our diagram, our 
course of diagonals carries us across the corner, and our 
path returns wich the width of one space between it and 
i's former self; in like manner, on reaching the border of 
the diagram near the starting-corner, the course of the 
diagonals carries us across to the other side of our first 
track, and we make a second journey only to wander still 
farther from our first path in the return. The error 
increases at every turn, till at last the path of our imagi- 
nary twig finds itself wholly forgetful of the corners with 
which its shuttle-play began, and giving all its allegiance 
to the alternate pair. At last our diagonals are all 
described, and we find that they end in one corner or the 
other according as 7 is even or odd, and the twig must 
then be supposed to retrace its maze. If we make our 
spaces all equal, the track of our twig looks very angular, 
like the path of a cracker ; but if we endeavour to imitate 
the truth by greatly diminishing the marginal spaces, our 
diagonal track becomes bent into a series of quasi-elliptic 
curves, which represent with tolerable accuracy the path 
of our twig, if we suppose it to vibrate without frictional 
retardation (see Fig. 13). 

We shall get the due diminution of the marginal spaces 
by drawing our two sets of parallel lines through two sets 
of points in the circumference of a circle, equidistant for 
each set, but allowing only 7 equal spaces in the semi- 
circumference for the 7 period, and 7 + 1 for the other. 
Introducing friction, we have a gradual diminution of 
the orbit, which brings our twig finally to rest in the 
centre of the diagram. But this friction has greater effect 
in the direction of shorter period, because our twig has to 
make 2 + 1 journeys in that direction to 2 in the other, 
consequently the range of the orbit in the former di- 
rection will undergo more rapid contraction than in the 
latter, and the twig will sooner come to rest in the one 
plane than in the other ; so that if there is large dispro- 
portion between 7 and z + 1, there will remain a residue 
of surplus vibration in the direction corresponding to the 
long period after all motion in the cross-plane has been 
arrested. This is easily seen by experiment ona twig 
that vibrates much more rapidly in one direction than in 
the other. 
Having a desire to get a permanent record of the fleet- 
ing footsteps of my acacia-twig, I forced the butt-end of a 
small dance-pencil into the soft pith in the centre of the 
top-section, and set the twig vibrating with one hand, while 
with the other I held a sheet of note-paper in contact 
with the pencil-point. As might be supposed, the result 
was not satisfactory, but very suggestive. The twig was 
not strong enough to overcome the resistance of friction 
between pencil and paper, and the hand-suspension for 
the latter was very inefficient. I soon found an upright 
hazel-stem nearly an inch in diameter, possessing all the 
vibratile properties of my slender acacia-twig with much 

Fic. 13.—Diagram showing approximately the theoretical path of a spring 
vibrating without friction, with periods of vibration in cross-planes in the 
proportion of xto’” + 1. (z = 10.) A and B are the beginning and end ofthe 
cycle, perpetually retraced, and are analogous to the two cusps of Fig. 9 or 
Fig. 11. 
greater strength, and transferred my pencil to its new 
abode. For suspension of paper I erected a wigwam of 
four poles round the hazel, and stretched a quarto leaf by 
india-rubber bands from the four poles to the four corners 
close above the pencil. Then pulling the hazel aside, I 
adjusted the paper-suspension till I was sure of good con- 
tact with the pencil, and then let go:—buzz—a momen- 
tary rustle under the paper, and the thing was done ; and, 
on loosing the elastic bands, I found the path of my 
pencil-point faithfully traced in delicate lines, which the 
eye could follow from the starting-point till lost in the 
mazy confusion of the centre where the manifold crossings 
and recrossings were inextricably entangled. By starting 
the hazel again and again, leaving the paper undisturbed, 
