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its rod, so that the rate or "period" of swing may be adjusted or altered. The nearer the weight is brought to the point of suspension, the oftener will the pendulum swing to and fro in a given time—usually taken as one minute. From this it is obvious that the rates of swing of the two pendulums can be adjusted relatively to one another. If they are exactly equal, they are said to be in unison, and under these conditions the instrument would trace figures varying in outline between the extremes of a straight line on the one hand and a circle on the other. A straight line would result if both pendulums were released at the same time, a circle,[1] if one were released when the other had half finished a swing, and the intermediate ellipses would be produced by various alterations of "phase," or time of the commencement of the swing of one pendulum relatively to the commencement of the swing of the other.

[Footnote 1: It should be pointed out here that the presence of friction reduces the "amplitude," or distance through which a pendulum moves, at every swing; so that a true circle cannot be produced by free swinging pendulums, but only a spiral with coils very close together.]

But the interest of the harmonograph centres round the fact that the periods of the pendulums can be tuned to one another. Thus, if A be set to swing twice while B swings three times, an entirely new series of figures results; and the variety is further increased by altering the respective amplitudes of swing and phase of the pendulums.

We have now gone far enough to be able to point out why the harmonograph is so called. In the case just mentioned the period rates of A and B are as 2: 3. Now, if the note C on the piano be struck the strings give a certain note, because they vibrate a certain number of times per second. Strike the G next above the C, and you get a note resulting from strings vibrating half as many times again per second as did the C strings—that is, the relative rates of vibration of notes C and G are the same as those of pendulums A and B—namely, as 2 is to 3. Hence the "harmony" of the pendulums when so adjusted is known as a "major fifth," the musical chord produced by striking C and G simultaneously.

In like manner if A swings four times to B's five times, you get a "major third;" if five times to B's six times, a "minor third;" and if once to B's three times, a "perfect twelfth;" if thrice to B's five times, a "major sixth;" if once to B's twice, an "octave;" and so on.

So far we have considered the figures obtained by two pendulums swinging in straight lines only. They are beautiful and of infinite variety, and one advantage attaching to this form of harmonograph is, that the same figure can be reproduced exactly an indefinite number of times by releasing the pendulums from the same points.

[Illustration: FIG. 169.—Goold's Twin Elliptic Pendulum Hamonograph.]

But a fresh field is opened if for the one-direction suspension of pendulum B we substitute a gimbal, or universal joint, permitting movement in all directions, so that the pendulum is able to describe a more or less circular path. The figures obtained by this simple modification are the results of compounded rectilinear and circular movements.

[Illustration: FIG. 170.—Benham's miniature Twin Elliptic Pendulum
Harmonograph.]

The reader will probably now see even fresh possibilities if both pendulums are given universal movement. This can be effected with the independent pendulums; but a more convenient method of obtaining equivalent results is presented in the Twin Elliptic Pendulum invented by Mr. Joseph Goold, and shown in Fig. 169. It consists of—(1) a long pendulum, free to swing in all directions, suspended from the ceiling or some other suitable point. The card on which the figure is to be traced, and the weights, are placed on a platform at the bottom of this pendulum. (2) A second and shorter free pendulum, known as the "deflector," hung from the bottom of the first.

This form of harmonograph gives figures of infinite variety and of extreme beauty and complexity. Its chief drawback is its length and weight, which render it more or less of a fixture.

Fortunately, Mr. C. E. Benham of Colchester has devised a Miniature Twin Elliptic Pendulum which possesses the advantages of the Goold, but can be transported easily and set up anywhere. This apparatus is sketched in Fig. 170. The main or platform pendulum resembles in this case that of the Rectilinear Harmonograph, the card platform being above the point of suspension.

Value of the Harmonograph.—A small portable harmonograph will be found to be a good means of entertaining friends at home or elsewhere. The gradual growth of the figure, as the card moves to and fro under the pen, will arouse the interest of the least scientifically inclined person; in fact, the trouble is rather to persuade spectators that they have had enough than to attract their attention. The cards on which designs have been drawn are in great request, so that the pleasure of the entertainment does not end with the mere exhibition. An album filled with picked designs, showing different harmonies and executed in inks of various colours, is a formidable rival to the choicest results of the amateur photographer's skill.

Practical Instructions for making Harmonographs.

Pendulums.—For the Rectilinear type of harmonograph wooden rods 5/8 to 3/4 inch in diameter will be found very suitable. They cost about 2d. each. Be careful to select straight specimens. The upper pendulum of the Miniature Twin Elliptic type should be of stouter stuff, say a broomstick; that of the Goold apparatus stouter still.

All pendulums on which weights are slid up and down should be graduated in inches and fractions, reckoning from the point of suspension as zero. The graduation makes it easy to re-establish any harmony after the weights have been shifted.

Suspensions.—For a harmonograph to give satisfaction it is necessary that very little friction should be set up at the point of suspension, so that the pendulums may lose amplitude of swing very slowly.

One-way suspensions are easily made. Two types, the point and knife-edge respectively, are shown in Fig. 168 and the top part of Fig. 172. The point suspension is most suitable for small rods and moderate weights; the knife-edge for large rods and heavy weights which would tend to crush a fine point.

[Illustration: FIG. 171.—Gimbal giving universal movement: point suspension.]

Points should rest in cup-shaped depressions in a metal plate; knife-edges in V-shaped grooves in a metal ring.

[Illustration: FIG. 172.—Knife-edge universal-motion gimbal.]

Screws turned or filed to a sharp end make convenient points, as they can be quickly adjusted so that a line joining the points lies exactly at right angles to the pendulum. The cups to take the points should not be drilled until the points have been thus adjusted. Make a punch mark on the bedplate, and using this as centre for one of the points, describe an arc of a circle with the other. This will give the exact centre for the other cup. It is evident that if points and cup centres do not coincide exactly there must be a certain amount of jamming and consequent friction.

In making a knife-edge, such as that shown in Fig. 172, put the finishing touches on with a flat file drawn lengthwise to ensure the edge being rectilinear. For the same reason the V slots in the ring support should be worked out together. If they are formed separately, the chances are against their being in line with one another.

Gimbals, or universal joints, giving motion in all directions, require the employment of a ring which supports one pair of edges or points (Fig. 172), and is itself supported on another pair of edges or points set at right angles to the first. The cups or nicks in the ring should come halfway through, so that all four points of suspension shall be in the same plane. If they are not, the pendulum will not have the same swing-period in all directions. If a gimbal does not work with equal freedom in all ways, there will be a tendency for the pendulum to lose motion in the direction in which most friction occurs.

By wedging up the ring of a gimbal the motion of the pendulum is changed from universal to rectilinear. If you are making a harmonograph of the type shown in Fig. 168, use a gimbal for the platform pendulum, and design it so that the upper suspension gives a motion at right angles to the pen pendulum. The use of two little wedges will then convert the apparatus in a moment from semirectilinear to purely rectilinear.

Weights.—The provision of weights which can be slipped up and down a rod may present some difficulty. Of iron and lead, lead is the more convenient material, as occupying less space, weight for weight, and being more easily cast or shaped. I have found thin sheet roofing lead, running 2 lbs. to the square foot, very suitable for making weights, by rolling a carefully squared strip of the material round the rod on which it will have to move, or round a piece of brass tubing which fits the rod. When the weight has been rolled, drill four holes in it, on opposite sides near the ends, to take nails, shortened so that they just penetrate all the laps but do not enter the central circular space. These will prevent the laps sliding over one another endways. A few turns of wire round the weight over the heads makes everything snug.

Just one caution here. The outside lap of lead should finish at the point on the circumference where the first lap began, for the weight to be approximately symmetrical about the centre.

An alternative method is to melt up scrap lead and cast weights in tins or flowerpots sunk in sand, using an accurately centred stick as the core. This stick should be very slightly larger than the pendulum rod, to allow for the charring away of the outside by the molten metal. (Caution.—The mould must be quite dry.)

Failing lead, tin canisters filled with metal scrap may be made to serve. It will in this case be necessary to bore the lid and bottom centrally and solder in a tube fitting the rod, and to make an opening through which the weighting material can be inserted.

Adjustment of Weights.—As lead is too soft a metal to give a satisfactory purchase to a screw—a thread cut in it soon wears out—it is better to support a leaden weight from underneath by means of a brass collar and screw. A collar is easily made out of a bit of tubing thickened at the point where the screw will pass by soldering on a suitably shaped piece of metal. Drill through the reinforcement and tubing and tap to suit the screw used, which may well be a camera tail screw, with a large flat head.

I experienced some trouble from the crushing of wooden rods by a screw, but got over it as follows. The tubing selected for the collar was large enough to allow a piece of slightly smaller tubing to be introduced between it and the rod. This inner piece was slit from one end almost to the other, on opposite sides, and soldered at one end to the outer tube, a line joining the slots being at right angles to the axis of the screw. The pressure of the screw point was thus distributed over a sufficient area of the wood to prevent indentation. (See Fig. 173.)

[Illustration: FIG. 173.]

[Illustration: FIG. 174.—Pivot for pen lever.]

Pen Levers.—The pen lever, of whatever kind it be, must work on its pivots with very little friction, and be capable of fine adjustment as regards balance. For the Rectilinear Harmonograph the form of lever pivot shown in Fig. 174 is very suitable. The spindle is a wire nail or piece of knitting needle sharpened at both ends; the bearings, two screws filed flat at the ends and notched with a drill.

The brass standard should be drilled and tapped to fit the screws fairly tight, so that when once adjusted they may not slacken off. If the lever is made of wood, the tail may be provided with a number of metal pegs on which to place the weights; if of wire, the tail should be threaded so that a brass weight and lock screw may be moved along it to any desired position. It is very important that the pressure of the pen