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 returned to the plane in which it started, by a complete recantation of its former errors, though the gradually failing 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 the 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 perhaps 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 described in so many units of time (the motion of the twig being resolved in those two directions at right angles), and making such spaces in one direction and n+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 n such spaces in both directions (which would represent 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 space over when we reach the border of our diagram, our course of diagonals carries us across the corner, and our path returns with 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 imaginary 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 n 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 n equal spaces in the semicircumference for the n period, and n + 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 n + 1 journeys in that direction to n in the other, consequently the range of the orbit in the former direction 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 disproportion between " and " + 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 on a twig that vibrates much more rapidly in one direction than in the other.

Having a desire to get a permanent record of the fleeting 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

FIG. 13.-Diagram showing approximately the theoretical path of a spring vibrating without friction, with periods of vibration in cross-planes in the proportion of n to n + 1. (n = 10.) A and B are the beginning and end of the 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 contact with the pencil, and then let go:-buzz-a momentary 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,

FIG. 14.-Specimen (obtained from nature) of the path of a stiff vibrating hazel shoot. Much friction. A the starting-point. E, the end of the first cycle x x', the path of the twig set vibrating exactly in the plane of slow vibration. Y y', nearly in the plane of quick vibration.

planes of obedience (xx', YY'.) A is the starting point, and B the point where the first retrograde step begins in the secondary plane. It will be seen on examination that from A to B the twig has accomplished exactly six quasielliptic journeys resolved parallel to the plane xx', and six and a half resolved parallel to vy'. 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 oscillation 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 acaciatwig. 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 intermediary and the rod. In any intermediate plane the rod could only swing by both 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 acaciatwig.

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 permanent 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 on the paper. By suction I drew up a small quantity of ink into the tube through the microscopic pore at the point, and then fastened my pen in a groove at the end of the pendulum-rod by elastic bands, so that it could be raised or lowered within short 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 accommodate the nearly spherical "locus" of the pen-point, 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 imposed by its two-fold suspension. The very first backstroke 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 oscillated slowly from one side to the other of the plane of slow vibration,

FIG. 15.-Curve traced by double-jointed pendulum, with proportion near unity, about 50: 51.

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

with a little care it was not difficult to adjust the level of the paper, so that very little displacement was called for to meet the requirements of the pen's descent towards the centre of the figure, and that little was granted at once by the facility with which the india-rubber yielded to the demand. When the suspension was ill-adjusted, so that the pen-point pressed too heavily on the paper, there was a slight lateral displacement; but this danger was reduced almost to nil by using at each corner two elastic bands at right angles, instead of only one, ensuring resistance to any rotatory jerk in either direction.

A little more practice in the manufacture of the glass pen enabled me to improve the delicacy and uniformity of the lines. The chief difficulty lay in breaking the capillary tube square to the axis. The tube delighted in oblique fracture, which gave an oblique pore when the edges were smoothed in the flame, and the oblique pore was apt to write unequally in different directions, often refusing to write at all on one side of the figure, when the pore was not facing its work. Only now and then was the first fracture fortunately square; generally I had to pick at it with the finger-nail to reduce its obliquity. Latterly, I tried to ensure success by coating the point with a thin layer of bees' wax, tracing a fine groove in the wax round the slender tube, and allowing a drop of solution of fluoric acid to adhere round the groove until the acid had eaten into the glass and made it ready to break at the ring of corrosion. Then the remainder of the wax was washed off with turpentine, and the point was ready for its "baptism of fire." In this way I succeeded well with one or two pens, but the process was rather troublesome. That "baptism of fire" was another dangerous crisis in the early life of the pen, for the risk was great that it might be exposed to the flame just a fraction of a second too long, sealing the liquid lips for ever. A good way of testing the size of the pore-it was much too small to be examined by the naked eye-was to blow through it and notice the size of the current of air disturbing the pale blue flame of the Bunsen's burner. Alternately dipping the point for the tenth of a second in the outskirts of the furnace, and quickly withdrawing it, and trying its calibre by the breath, it was seen that the air-current grew smaller and smaller after every dip in the flame, till I dared not dip again, and then I had recourse to a powerful pocket-lens to examine the size of the pore and the smoothness of its lip. The diameter of pore of the best pen I have succeeded in making is 1-500th of an inch.

“

I grew discontented with common black ink for my pendulum-curves; it was apt to coagulate and choke the pore, frequently requiring the solvent power of sulphuric acid to restore free passage. Besides, I wished to have several figures superposed on the same paper, yet so that each should remain distinct. So I procured a set of coloured inks at the stationer's, price id. per bottle, and with these I was able to give additional interest to the sheets that were rapidly accumulating from all these trials of new ink, new pens, and new pendulums. For I soon grew discontented also with my first pendulum; its tripod was not strong enough, and its cord-hinges were very weak, and were fast fraying under the strain of 50 lbs. of lead in habitual oscillation below, and I feared a snap and a crash. I kept it in my bedroom, and at midnight I heard it creak, and could not rest until I had insinuated a rush-bottomed chair between the legs of the tripod, immediately below the lead, to break the fall which I fully expected. However, nothing happened, and in the morning I changed the frayed string for a trustworthy cord, and slept securely next night. I also made a new tripod with the aid of three surveying-poles, and improved the attachment of the pen by making it slide in a hole bored in the end of the rod, with a lateral screw to fix it at any required height.

(To be continued.)

HUBERT AIRY

THE BRITISH ASSOCIATION MEETING AT EDINBURGH

EDINBURGH, Friday morning THE work and play- of the Edinburgh meeting of the British Association is now over; the visitors have all left, except such as have remained to do honour to the memory of the great Magician of the North; indeed, for the last two days the Southerners have dividea their allegiance between the British Association and the Scott Centenary. Instead of Spontaneous Generation and the Germ Theory of Disease, the Solar Eclipse and the great Dredging Expedition, we have heard quite as much of Abbotsford and Dryburgh, Jock o' Hazeldean and the Laird of Dumbiedikes.

As announced in our letter of last week, the concluding meeting of the General Committee was held in the University on Wednesday at one o'clock, Sir William Thomson in the chair. Dr. Hirst read certain regulations which were proposed by the Committee on Recommendations to be adopted relative to the proceedings of the sections. They had reference to the organisation and constitution of the Sectional Committees, but were merely in regard to matters of detail. In an appended circular, authors of papers were reminded that, under an arrangement dating from 1871, the acceptance of memoirs, and the days on which they were to be read, were now as far as possible determined by organising committees for the several sections before the beginning of the meeting. It had therefore become necessary that an author should prepare an abstract of his memoir, of a length suitable for insertion in the published transactions of the Association, and that he should send it, together with the original memoir, to the general secretaries in London a certain time before the meeting. If it should be inconvenient to the author that his paper should be read on any particular day, he was requested to send information thereof to the secretaries in a separate note. These resolutions, after some discussion, were adopted. The next subject referred to the General Committee on Recommendations had reference to Dr. King's proposal that there should be a subsection of Anthropology. Before the consideration of that suggestion was concluded, another came from Section D of a more definite nature; and, on considering both propositions together, the Committee on Recommendations decided that they could not recommend the adoption of Dr. King's motion, but that they could strongly recommend the adoption of the other. They therefore recommended-"That in future the division of the Section of Biology into the three departments of Anatomy and Physiology, Anthropology, and Zoology and Botany should be recognised in the programme of the Association meetings; and that the president, two vice-presidents, and at least three secretaries shall be appointed; and that the vicepresidents and secretaries, who shall take charge of the organisation of the several departments, should be designated respectively before the publication of each programme." That would virtually amount to the direct recognition of the three departments of Section D. Logically, it would be impossible to take any of these departments from Biology to make a separate section of it; but they were recognised distinctly, and the gentlemen who would preside over these departments would be stated by name. The recommendation was agreed to.

The following recommendations were then read and unanimously adopted:

"That the President and Council of the British Association be authorised to co-operate with the President and Council of the Royal Society, in whatever manner may seem to them to be best, for the promotion of the circumnavigation expedition specially fitted out for carrying the physical and biological Exploration of the Deep-sea into all the great oceanic centres.

"That the President and general officers, with power to