Chapter XIV: Part 14

A steel wire, about one tenth of an inch in diameter, is fixed in the usual manner to the spring by which the pendulum is suspended. This wire passes to the bottom of an interior brass tube, in the centre of which it is firmly screwed. The top of this tube is closed, the steel rod passing freely through a hole in the centre. Into the top of this interior tube two steel wires, of one tenth of an inch in diameter, are screwed into holes made in that diameter, which is at right angles to the motion of the pendulum. These wires pass down the tube without touching either it or the central rod, through holes made in the piece which closes the bottom of the interior tube. The lower extremities of these wires, which project a little beyond the inner tube, are securely fixed in a piece which closes the bottom of an exterior brass tube, which is of such a diameter as just to allow the interior tube to pass freely through it, and of a sufficient length to extend a little above it. The top of the exterior tube is closed like that of the interior, having also a hole in its centre, to allow the first steel rod to pass freely through it. Into the top of the exterior tube, in that diameter which coincides with the motion of the pendulum, a second pair of steel wires of the same diameter as the former are screwed, their distance from the central rod being equal to the distance of each from the first pair. They consequently pass down within the interior tube, and through holes made in the pieces closing the lower ends of both the interior and exterior tubes. The lower ends of these wires are fastened to a short cylindrical piece of brass of the same diameter as the exterior tube, to which the bob is suspended by its centre.

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_Fig. 209._ is a full sized section of the rod; the three concentric circles represent the two tubes, and the rectangular position of the two pair of wires round the middle one is shown by the five small circles.

_Fig. 210._ is the part which closes the upper end of the interior tube. The two small circles are the two wires which proceed from it, and the three large circles show the holes through which the middle wire and the other pair of wires pass.

_Fig. 211._ is the bottom of the interior tube. The small circle in the centre is where the central rod is fastened to it, the others the holes for the other four wires to pass through.

_Fig. 212._ is the part which closes the top of the external tube. In the large circle in the centre a small brass tube is fixed, which serves as a covering for the upper part of the middle wire, and the two small circles are to receive the wires of the last expansion.

_Fig. 213._ represents the bottom of the exterior tube, in which the small circles show the places where the wires of the second expansion are fastened, and the larger ones the holes for the other pair of wires to pass through.

_Fig. 214._ is a cylindrical piece of brass, showing the manner in which the lower ends of the wires of the last expansion are fastened to it, and the hole in the middle is that by which it is pinned to the centre of the bob. The upper ends of the two pair of wires are, as we have observed, fastened by screwing them into the pieces which stop up the ends of the tubes, but at the lower ends they are all fixed as represented in _fig. 214._ The pieces represented by _figs. 213._ and _214._ have each a jointed motion, by means of which the fellow wires of each pair would be equally stretched, although they were not exactly of the same length.

The action of this pendulum is evidently the same as that of the gridiron pendulum, as we have three lengths of steel expanding downwards, and two of brass expanding upwards. The weight of the pendulum has a tendency to straighten the steel rods, and the tubular form of the brass compensation effectually precludes the fear of its bending; an advantage not possessed by the gridiron pendulum, in which brass rods are employed.

Mr. Troughton, to the account he has given of this pendulum in Nicholson’s Journal, for December, 1804, has added the lengths of the different parts of which it was composed, and the expansions of brass and steel from which these lengths were computed. The length of the interior tube was 31·9 inches, and that of the exterior one 32·8 inches, to which must be added 0·4, the quantity by which in this pendulum the centre of oscillation is higher than the centre of the bob. These are all of brass. The parts which are of steel are,--the middle wire, which, including 0·6, the length of the suspension spring, is 39·3 inches. The first pair of wires 32·5 inches; and the second pair, 33·2 inches. The expansions used were, for brass ·00001666, and for steel ·00000661, in parts of their length for one degree of temperature.

_Benzenberg’s Pendulum._

This pendulum is mentioned in Nicholson’s Journal for April, 1804, and is taken from Voigt’s Magazin für den Neuesten Zustande der Naturkunde, vol. iv. p. 787. The compensation appears to have been effected by a single rod of lead in the centre, of about half an inch thick; the descending rods were made of the best thick iron wire.

As this pendulum deserves attention from the ease with which it may be made, and as others which have since been produced resemble it in principle, we have given a representation of it at _fig. 215._, where A B C D are two rods of iron wire riveted into the cross pieces A C B D. E F is a rod of lead pinned to the middle of the piece B D, and also at its upper extremity to the cross piece G H, into which the second pair of iron wires are fixed, which pass downwards freely through holes made in the cross piece B D. The lower extremities of these last iron wires are fastened into the piece K L, which carries the bob of the pendulum.

To determine the length of lead necessary for the compensation, we must recollect, as before, that the distance from the point of suspension to the centre of the bob (speaking always of a pendulum intended to vibrate seconds) must be 39 inches. Let us suppose the total length of the iron wire to be 60 inches; then, from the table which we have given, we have ·4308 for the length of a rod of lead, the expansion of which is equivalent to that of an iron rod whose length is unity. Multiplying 60 inches by ·4308, we have 25·84 inches of lead, which would compensate 60 inches of iron; but this, taken from 60 inches, leaves only 34·16 instead of 39 inches. Trying again, in like manner, 68·5 inches of iron, we find 29·5 inches of lead for the length, affording an equivalent compensation, and which, taken from 68·5 inches, leaves 39 inches.

The length of the rod of lead then required as a compensation in this pendulum is about 29-1/2 inches.

The writer of this article would suggest another form for this pendulum, which has the advantage of greater simplicity of construction.

S A, _fig. 216._, is a rod of iron wire, to which the pendulum spring is attached. Upon this passes a cylindrical tube of lead, 29-1/2 inches long, which is either pinned at its lower extremity to the end of the iron rod S A, or rests upon a nut firmly screwed upon the extremity of this rod.

A tube of sheet iron passes over the tube of lead, and is furnished at top with a flanche, by which it is supported upon the leaden tube; or it may be fastened to the top of this tube in any manner that may be thought convenient.

The bob of the pendulum may be either passed upon the iron tube (continued to a sufficient length) and secured by a pin passing through the centre of the bob, or the iron tube may be terminated by an iron wire serving the same purpose.

Here we have evidently the same expansions upwards and downwards as in the gridiron form, given to this pendulum by Mr. Benzenberg, joined to the compactness of Troughton’s tubular pendulum.

_Ward’s Compensation Pendulum._

In the year 1806, Mr. Henry Ward, of Blandford in Dorsetshire, received the silver medal of the Society of Arts for the compensation pendulum which we are about to describe.

_Fig. 217._ is a side view of the pendulum rod when together. H H and I I are two flat rods of iron about an eighth of an inch thick. K K is a bar of zinc placed between them, and is nearly a quarter of an inch thick. The corners of the iron bars are bevelled off, which gives them a much lighter appearance. These bars are kept together by means of three screws, O O O, which pass through oblong holes in the bars H H and K K, and screw into the rod I I. The bar H H is fastened to the bar of zinc K K, by the screw _m_, which is called the adjusting screw. This screw is tapped into H H, and passes just through K K; but that part of the screw which passes K K has its threads turned off. The iron bar I I has a shoulder at its upper end, and rests on the top of the zinc bar K K and is wholly supported by it. There are several holes for the screw _m_, in order to adjust the compensation.

The action of this pendulum is similar to that last described, the zinc expanding upwards as much as the iron rods expand downwards, and consequently the instance from the point of suspension to the centre of oscillation remains the same.

_London, Pubd. by Longman & Co._]

Mr. Ward states that the expansion of the zinc he used (hammered zinc) was greater than that given in the tables. He found that the true length of the zinc bar should be about 23 inches; our computation would make it nearly 26.

_The Compensation Tube of Julien le Roy._

We mention this merely to state that it is similar in principal to the apparatus represented at _fig. 204._, with merely this difference, that, instead of the steel rod being fixed to a cross piece proceeding from the brass bar B R, it is attached to a cap fixed upon a brass tube (through which it passes) of the same length as that of the brass rod B R. Cassini spoke well of this pendulum, and it was used in the observatory of Cluny about the year 1748.

_Deparcieux’s Compensation._

This was contrived in the same year as that invented by Julien le Roy. It is represented at _fig. 218._, where A B D F is a steel bar, the ends of which are to be fixed to the lower sides of pieces forming a part of the cock of the pendulum. G E I H is of brass, and stands with its extremities resting on the horizontal part B D of the steel frame. The upper part E I of the brass frame passes above the cock of the pendulum, and admits the tapped wire K, to which the pendulum spring is fixed through a squared hole in the middle. A nut upon this tapped wire gives the adjustment for time. The spring passes through the slit in the cock in the usual manner.

It may be easily perceived that this pendulum is in principle the same as that of Le Roy; the expansion of the total length of steel A B S C downwards being compensated by the equivalent expansion of the brass bar G E upwards. It is, however, preferable to Le Roy’s, because the compensation is contained in the clock case.

Deparcieux had previously published, in the year 1739, an improvement of an imperfectly compensating pendulum, proposed in the year 1733 by Regnauld, a clockmaker of Chalons. In this pendulum Deparcieux employed a lever with unequal arms to increase the effect of the expansion of the brass rod, which was too short.

We may here remark, that all fixed compensations are liable to the same objection, namely, that of not moving with the pendulum, and therefore not taking precisely the same temperature.

_Captain Kater’s Compensation Pendulum._

In Nicholson’s Journal, for July, 1808, is the description of a compensation pendulum by the writer of this article. In this pendulum the rod is of white deal, three quarters of an inch wide, and a quarter of an inch thick. It was placed in an oven, and suffered to remain there for a long time until it became a little charred. The ends were then soaked in melted sealing-wax; and the rod, being cleaned, was coated several times with copal varnish. To the lower extremity of the rod a cap of brass was firmly fixed, from which a strong steel screw proceeded for the purpose of regulating the pendulum for time in the usual manner.

A square tube of zinc was cast, seven inches long and three quarters of an inch square; the internal dimensions being four tenths of an inch. The lower part of the pendulum rod was cut away on the two sides, so as to slide with perfect freedom within the tube of zinc. To the bottom of this zinc tube a piece of brass a quarter of an inch thick was soldered, in which a circular hole was made nearly four tenths of an inch in diameter, having a screw on the inside. A cylinder of zinc, furnished with a corresponding screw on its surface, fitted into this aperture, and a thin plate of brass screwed upon the cylinder, served as a clamp to prevent any shake after the length of zinc necessary for compensation should have been determined. A hole was made through the axis of the cylinder, through which passed the steel screw terminating the pendulum rod.

An opening was made through the bob of the pendulum, extending to its centre, to admit the square tube of zinc which was fixed at its upper extremity to the centre of the bob. The pendulum rod passed through the bob in the usual manner, and the whole was supported by a nut on the steel screw at the extremity.

In this form the compensation acts immediately upon the centre of the bob, elevating it along the rod as much as the rod elongates downwards: the method of calculating the length of the required compensation is precisely the same as that we have before given.

Assuming the length of the deal rod to be 43 inches, and multiplying this by ·1313 from Table II., we have 5·64 inches for the length of the zinc necessary to counteract the expansion of the deal. The length of the steel screw between the termination of the pendulum rod and the nut was two inches, and that of the suspension spring one inch. Now, 3 inches of steel multiplied by ·3682 would give 1·10 inches for the length of zinc which would compensate the steel, and, adding this to 5·64 inches, we have 6·74 inches for the whole length of zinc required.

In this pendulum, the length of the compensating part may be varied by means of the zinc cylinder furnished with a screw for that purpose. The bob of this pendulum and its compensation are represented at _fig. 219._

It has been objected to the use of wooden pendulum rods, that it is difficult, if not impossible, to secure them from the action of moisture, which would at once be fatal to their correct performance. The pendulum now before us has, however, been going with but little intermission since it was first constructed: it is attached to a sidereal clock, not of a superior description, and exposed to very considerable variations of moisture and dryness; yet the change in its rate has been so very trifling as to authorize the belief that moisture has little or no effect upon a wooden rod prepared in the manner we have described. Its rate, under different temperatures, shows that it is over-compensated; the length of the zinc remaining, as stated in Nicholson’s Journal 7·42 inches, instead of which it appears, by our present compensation, that it should be 6·78 inches.

_Reid’s Compensation Pendulum._

Mr. Adam Reid of Woolwich presented to the Society of Arts, in 1809, a compensation pendulum, for which he was rewarded with fifteen guineas. This pendulum is the same in principle with that last described; the rod, however, is of steel instead of wood, and the compensation possesses no means of adjustment. This pendulum is represented at _fig. 220._, where S B is the steel rod, a little thicker where it enters the bob C, and of a lozenge shape to prevent the bob turning, but above and below it is cylindrical.

A tube of zinc D passes to the centre of the bob from below, and the bob is supported upon it by a piece which crosses its centre, and which meets the upper end of the tube.

The rod being passed through the bob and zinc tube, a nut is applied upon a screw at the lower extremity of the rod in the usual manner. If the compensation should be too much, the zinc tube is to be shortened until it is correct.

The length of the zinc tube will be the same in this pendulum as in that of Mr. Ward--about 23 inches, if his experiments are to be relied upon.

The objection to this pendulum appears to be its great length, which amounts to 62 inches. We conceive it would be preferable to place the zinc above the bob, as in the modification which we have suggested of Benzenberg’s pendulum.

_Ellicott’s Pendulum._

It appears that the idea of combining the expansions of different metals with a lever, so as to form a compensation pendulum, originated with Mr. Graham; for Mr. Short, in the Philosophical Transactions for 1752, states that he was informed by Mr. Shelton, that Mr. Graham, in the year 1737, made a pendulum, consisting of three bars, one of steel between two of brass; and that the steel bar acted upon a lever so as to raise the pendulum when lengthened by heat, and to let it down when shortened by cold.

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This pendulum, however, was found upon trial to move by jerks, and was therefore laid aside by the inventor to make way for the mercurial pendulum.

Mr. Short also says that Mr. Fotheringham, a quaker of Lincolnshire, caused a pendulum to be made, in the year 1738 or 1739, consisting of two bars, one of brass and the other of steel, fastened together by screws with levers to raise or let down the bob, and that these levers were placed above the bob.

Mr. John Ellicott of London had made very, accurate experiments on the relative expansions of seven different metals, which, however, will be found to differ more or less from the results of the experiments of others. It is not, however, from this to be concluded that Ellicott’s determinations were erroneous; for the expansion of a metal will suffer considerable change even by the processes to which it is necessarily subjected in the construction of a pendulum. It is therefore desirable, whenever a compensation pendulum is to be made, that the expansions of the materials employed should be determined after the processes of drilling, filing, and hammering have been gone through.

It had been objected to Harrison’s gridiron pendulum, that the adjustments of the rods was inconvenient, and that the expansion of the bob supported at its lower edge would, unless taken into the account, vitiate the compensation. These considerations, it is supposed, gave rise to Ellicott’s pendulum, which is nearly similar to those we have just mentioned.

Ellicott’s pendulum is thus constructed:--A bar of brass and a bar of iron are firmly fixed together at their upper ends, the bar of brass lying upon the bar of iron, which is the rod of the pendulum. These bars are held near each other by screws passing through oblong holes in the brass, and tapped into the iron, and thus the brass is allowed to expand or contract freely upon the iron with any change of temperature. The brass bar passes to the centre of the bob of the pendulum, a little above and below which the iron is left broader for the purpose of attaching the levers to it, and the iron is made of a sufficient length to pass quite through the bob of the pendulum.

The pivots of two strong steel levers turn in two holes drilled in the broad part of the iron bar. The short arms of these levers are in contact with the lower extremity of the brass bar, and their longer arms support the bob of the pendulum by meeting the heads of two screws which pass horizontally from each side of the bob towards its centre. By advancing these screws towards the centre of the bob, the longer arms of the lever are shortened, and thus the compensation may be readily adjusted. At the lower end of the iron rod, under the bob, a strong double spring is fixed, to support the greater part of the weight of the bob by its pressure upwards against two points at equal distances from the pendulum rod. Mr. Ellicott gave a description of this pendulum to the Royal Society in 1752, but he says the thought was executed in 1738. As this pendulum is very seldom met with, we think it unnecessary to give a representation of it.

_Compensation by means of a Compound Bar of Steel and Brass._

Several compensations for pendulums have been proposed, by means of a compound bar formed of steel and brass soldered together. In a bar of this description, the brass expanding more than the steel, the bar becomes curved by a change of temperature, the brass side becoming convex and the steel concave with heat. Now, if a bar of this description have its ends resting on supports on each side the cock of the pendulum, the bar passing above the cock with the brass uppermost, if the pendulum spring be attached to the middle of the bar, and it pass in the usual manner through the slit of the cock, it is evident that, by an increase of temperature, the bar will become curved upwards, and the pendulum spring be drawn upwards through the slit, and thus the elongation of the pendulum downwards will be compensated. The compensation may be adjusted by varying the distance of the points of support from the middle of the bar.

Such was one of the modes of compensation proposed by Nicholson. Others of the same description (that is, with compound bars) have been brought before the public by Mr. Thomas Doughty and Mr. David Ritchie; but as they are supposed to be liable to many practical objections, we do not think it requisite to describe them more particularly.

There is, however, a mode of compensation by means of a compound bar, described by M. Biot in the first volume of his Traité de Physique, which appears to possess considerable merit, of which he mentions having first witnessed the successful employment by the inventor, a clockmaker named Martin. At _fig. 221._, S C, is the rod of the pendulum, made, in the usual manner, of iron or steel; this rod passes through the middle of a compound bar of brass and steel (the brass being undermost), which should be furnished with a short tube and screws, by means of which, or by passing a pin through the tube and rod, it may be securely fixed at any part of the pendulum rod.

Two small equal weights W W slide along the compound bar, and, when their proper position has been determined, may be securely clamped.

The manner in which this compensation acts is thus:--Suppose the temperature to increase, the brass expanding more than the steel, the bar becomes curved, and its extremities carrying the weights W and W are elevated, and thus the place of the centre of oscillation is made to approach the point of suspension as much, when the compensation is properly adjusted, as it had receded from it by the elongation of the pendulum rod.

There are three methods of adjusting this compensation: the first, by increasing or diminishing the weights W and W; the second, by varying the distance of the weights W and W from the middle of the bar; and the third, by varying the distance of the bar from the bob of the pendulum, taking care not to pass the middle of the rod. The effect of the compensation is greater as the weights W and W are greater or more distant from the centre of the bar, and also as the bar is nearer to the bob of the pendulum.

M. Biot says that he and M. Matthieu employed a pendulum of this kind for a long time in making astronomical observations in which they were desirous of attaining an extreme degree of precision, and that they found its rate to be always perfectly regular.

In all the pendulums which we have described, the bob is supposed to be fixed to the rod by a pin passing through its centre, and the adjustment for time is to be made by means of a small weight sliding upon the rod.

_Of the Mercurial Pendulum._

We have been guided, in our arrangement of the pendulums which we have described, by the similarity in the mode of compensation employed; and we have now to treat of that method of compensation which is effected by the expansion of the material of which the bob itself of the pendulum is composed.

On this subject, as we have before observed, an admirable paper, from the pen of Mr. Francis Baily, may be found in the Memoirs of the Astronomical Society of London, which leaves nothing to be desired by the mathematical reader. But as our object is to simplify, and to render our subjects as popular as may be, we must endeavour to substitute for the perfect accuracy which Mr. Baily’s paper presents, such rules as may be found not only readily intelligible, but practically applicable, within the limits of those inevitable errors which arise from a want of knowledge of the exact expansion of the materials employed.

At _fig. 222._, let S B represent the rod of a pendulum, and F C B a metallic tube or cylinder, supported by a nut at the extremity of the pendulum rod, in the usual manner, and having a greater expansibility than that of the rod. Now C, the centre of gravity, supposing the rod to be without weight, will be in the middle of the cylinder; and if C B, or half the cylinder, be of such a length as to expand upwards as much as the pendulum rod S B expands downwards, it is evident that the centre of gravity C will remain, under any change of temperature, at the same distance from the point of suspension S. M. Biot imagined that, in effecting this, a compensation sufficiently accurate would be obtained; but Mr. Baily has shown that this is by no means the fact.

Let us suppose the place of the centre of oscillation to be at O, about three or four tenths of an inch, in a pendulum of the usual construction, below the centre of gravity. Now, the object of the compensation is to preserve the distance from S to O invariable, and not the distance from S to C.

The distance of the centre of oscillation varies with the length of the cylinder F B, and hence suffers an alteration in its distance from the point of suspension by the elongation of the cylinder, although the distance of the centre of gravity C from the point of suspension remains unaltered.

We shall endeavour to render this perfectly familiar. Suppose a metallic cylinder, 6 inches long, to be suspended by a thread 36 inches long, thus forming a pendulum in which the distance of the centre of gravity from the point of suspension is 39 inches: the centre of oscillation in such a pendulum will be nearly one tenth of an inch below the centre of gravity. Now let us imagine cylindrical portions of equal lengths to be added to each end of the cylinder, until it reaches the point of suspension; we shall then have a cylinder of 78 inches in length, the centre of gravity of which will still be at the distance of 39 inches from the point of suspension. But it is well known that the centre of oscillation of such a cylinder is at the distance of about two thirds of its length from the point of suspension. The centre of oscillation, therefore, has been removed, by the elongation of the cylinder, about 13 inches below the centre of gravity, whilst the centre of gravity has remained stationary.

Now the same thing as that which we have just described takes place, though in a very minor degree, with our former cylinder, employed as a compensating bob to a pendulum. The rod expands downwards, the centre of gravity remains at the same distance from the point of suspension, and the cylinder elongates both above and below this point; the consequence of which is, that though the centre of gravity has remained stationary, the distance of the centre of oscillation from the point of suspension has increased. It is, therefore, evident that the length of the compensation must be such as to carry the centre of gravity a little nearer to the point of suspension than it was before the expansion took place; by which means the centre of oscillation will be restored to its former distance from the point of suspension.

Let us suppose the expansions to have taken place, and that the centre of gravity, remaining at the same distance from the point of suspension, the centre of oscillation is removed to a greater distance, as we have before explained. It is well known that the product obtained by multiplying the distance from the point of suspension to the centre of gravity, by the distance from the centre of gravity to the centre of oscillation, is a constant quantity; if, therefore, the distance from the centre of gravity to the point of suspension be lessened, the distance from the centre of gravity to the centre of oscillation will be proportionally, though not equally, increased, and the centre of oscillation will, therefore, be elevated. We see, then, if we elevate the centre of gravity precisely the requisite quantity, by employing a sufficient length of the compensating material, that although the distance from the centre of gravity to the point of suspension is lessened, yet the distance from the point of suspension to the centre of oscillation will suffer no change.

The following rule for finding the length of the compensating material in a pendulum of the kind we have been considering will be found sufficiently accurate for all practical purposes:--

_Find in the manner before directed the length of the compensating material, the expansion of which will be equal to that of the rod of the pendulum. Double this length, and increase the product by its one-tenth part, which will give the total length required._ We shall give examples of this as we proceed.

_Graham’s Mercurial Pendulum._

It was in the year 1721 that Graham first put up a pendulum of this description, and subjected it to the test of experiment; but it appears to have been afterwards set aside to make way for Harrison’s gridiron pendulum, or for others of a similar description. For some years past, however, its merits have been more generally known, and it is not surprising that it should be considered as preferable to others, both from the simplicity of its construction, and the perfect ease with which the compensation may be adjusted.

We have already alluded to Mr. Baily’s very able paper on this pendulum, and we shall take the liberty of extracting from it the following description:--

At _fig. 223._ is a drawing of the mercurial pendulum, as constructed in the manner proposed by Mr. Baily.

“The rod S F is made of steel, and perfectly straight; its form may be either cylindrical, of about a quarter of an inch in diameter, or a flat bar, three eighths of an inch wide, and one eighth of an inch thick: its length from S to F, that is, from the bottom of the spring to the bottom of the rod at F, should be 34 inches. The lower part of this rod, which passes through the top of the stirrup, and about half an inch above and below the same, must be formed into a _coarse_ and _deep_ screw, about two tenths of an inch in diameter, and having about thirty turns in an inch. A steel nut with a milled head must be placed at the end of the rod, in order to support the stirrup; and a similar nut must also be placed on the rod _above_ the head of the stirrup, in order to screw firmly down on the same, and thus secure it in its position, after it has been adjusted _nearly_ to the required rate. These nuts are represented at B and C. A small slit is cut in the rod, where it passes through the head of the stirrup, through which a steel pin E is screwed, in order to keep the stirrup from turning round on the rod. The stirrup itself is also made of steel, and the side pieces should be of the same form as the rod, in order that they may readily acquire the same temperature. The top of the stirrup consists of a flat piece of steel, shaped as in the drawing, somewhat more than three eighths of an inch thick. Through the middle of the top (which at this part is about one inch deep) a hole must be drilled sufficiently large to enable the screw of the rod to pass _freely_, but without _shaking_. The inside height of the stirrup from A to D may be 8-1/2 inches, and the inside width between the bars about three inches. The bottom piece should be about three eighths of an inch thick, and hollowed out nearly a quarter of an inch deep, so as to admit the glass cylinder freely. This glass cylinder should have a brass or iron cover G, which should fit the mouth of it freely, with a shoulder projecting on each side, by means of which it should be screwed to the side bars of the stirrup, and thus be secured always in the same position. This cap should not _press_ on the glass cylinder, so as to prevent its expansion. The measures above given may require a slight modification, according to the weight of the mercury employed, and the magnitude of the cylinder: the final adjustment, however, may be safely left to the artist. Some persons have recommended that a circular piece of thick plate glass should float on the mercury, in order to preserve its surface uniformly level.[7] The part at the bottom marked H is a piece of brass fastened with screws to the front of the bottom of the stirrup, through a small hole, in which a steel wire or common needle is passed, in order to indicate (on a scale affixed to the case of the clock) the arc of vibration. This wire should merely rest in the hole, whereby it may be easily removed when it is required to detach the pendulum from the clock, in order that the stirrup might then stand securely on its base. One of the screw holes should be rather larger than the body of the screw, in order to admit of a small adjustment, in case the steel wire should not stand exactly perpendicular to the axis of motion. The scale should be divided into _degrees_, and not _inches_, observing that with a radius of 44 inches (the estimated distance from the bend of the spring to the end of the steel wire) the length of each degree on the scale must be 0·768 inch.”

[7] The variation produced in the height of the column of mercury (supposed to be 6-1/2 inches high) by an alteration of ± 16° in the temperature will be only ± 1/100 of an inch, or in other words, 1/100 of an inch will be the total variation from its _mean_ state, by an alteration of 32° in the temperature. It is therefore probable that, in most cases of moderate alteration in the temperature, the _centre_ only of the column of mercury is subject to elevation and depression, whilst the exterior parts remain attached to the sides of the glass vessel. It was with a view to obviate this inconvenience that Henry Browne, Esq. of Portland Place (I believe) first suggested the piece of floating glass.

In order to determine the length of the mercurial column necessary to form the compensation for this pendulum, we must proceed in the following manner:--

Let us suppose the length of the steel rod and stirrup together to be 42 inches. The absolute expansion of the mercury is ·00010010; but it is not the absolute expansion, but the vertical expansion in a glass cylinder, which is required, and this will evidently be influenced by the expansion of the base of this cylinder. It is easily demonstrable that, if we multiply the linear expansion of any substance (always supposed to be a very small part of its length) by 3, we may in all cases take the result for the cubical or absolute expansion of such substance. In like manner, if we multiply the linear expansion by 2, we shall have the superficial expansion.

If we want the apparent expansion of mercury, the absolute or cubical expansion of the glass vessel must be deducted from the absolute expansion of the mercury, which will leave its excess or apparent expansion. In like manner, deducting the superficial expansion of glass from the absolute expansion of mercury, we shall have its relative vertical expansion. Now, taking the rate of expansion of glass to be ·00000479, and multiplying it by 2, the relative vertical expansion of the mercury in the glass cylinder will be ·00010010 - ·00000958 = ·00009052.

The expansion of a steel rod, according to our table, is ·0000063596; which, divided by ·00009052, gives ·0703 for the length of a column of mercury, the expansion of which is equal to that of a steel rod whose length is unity.

We have now to multiply 42 inches by ·0703, which gives 2·95 inches; and this, deducted from 42, leaves 39·1 inches; so that the length of rod we have chosen is sufficiently near the truth. Now, double 2·95 inches, and add one tenth of its product, and we shall have 6·49 inches for the length of the mercurial column forming the requisite compensation. Mr. Baily’s more accurate calculation gives 6·31 inches.

A mercurial compensation pendulum may be formed, having a cylinder of steel or iron, with its top constructed in the same manner as the top of the stirrup, so as to receive the screw of the rod. To find the length of the mercurial column necessary in a pendulum of this description (that is, with a cylinder made of steel), we must double the linear expansion of steel, and take it from the absolute expansion of mercury to obtain the relative vertical expansion of the mercury. This will be ·00010010 - ·00001272 = ·00008738; and, proceeding as before, we have ·0000063596/·00008738 = ·07279.

Let the length of the steel rod be, as before, 42 inches. Multiplying this by ·07279, we have 3·057, which being doubled, and one tenth of the product added, we obtain 6·72 inches for the length of the compensating mercurial column; which Mr. Baily states to be 6·59.

A mercurial compensation pendulum having a rod of glass has been employed by the writer of this article, who has had reason to think well of its performance. Its cheapness and simplicity much recommend it. It is merely a cylinder of glass of about 7 inches in depth, and 2-1/2 inches diameter, terminated by a long neck, which forms the rod of the pendulum, the whole blown in one piece. A cap of brass is clamped by means of screws to the top of the rod, and to this the pendulum spring is pinned.

We have unquestionable authority for saying, that the mercurial pendulum of the usual construction, that is, with a steel rod and glass cylinder, is not affected by a change of temperature simultaneously in all its parts. Now, the pendulum of which we are treating being formed throughout of the same material in a single piece, and in every part of the same thickness, it is presumed it cannot expand in a linear direction, until the temperature has penetrated to the whole interior surface of the glass, when it is rapidly diffused through the mass of mercury. M. Biot mentions that a pendulum of this kind was formerly used in France, and expresses his surprise that it was no longer employed, as he had heard it very highly spoken of. The writer of this article has also used a pendulum with a glass rod, which differs from that we have just mentioned, in having the lower end of the rod firmly fixed in a socket attached to the centre of a circular iron plate, on the circumference of which a screw is cut, which fits into a collar of iron, supporting the cylinder (to which it is cemented) by means of a circular lip.

This arrangement, though perhaps less perfect than that we have just described, the pendulum not being in one piece, has the advantage of allowing a circular plate of glass to be placed upon the surface of the mercury, as practised by Mr. Browne. To determine the length of a column of mercury for a glass pendulum, let us suppose the glass, including the cylinder, to be 41 inches in length. Multiplying this by ·0529, the number taken from Table II. for a glass rod and mercury in a glass cylinder, we have 2·17 inches for the uncorrected length of mercury, which compensates 41 inches of glass. Suppose the steel spring to be one inch and a half long: multiplying this by ·0703, the appropriate decimal taken from Table II., we have 0·1, the length of mercury due to the steel, making with the former 2·27 inches, which, being doubled, and the product increased by its one-tenth part, we obtain five inches for the length of the required column of mercury.

_Compensation Pendulum of Wood and Lead, on the Principle of the Mercurial Pendulum._

If by any contrivance wood could be rendered impervious to moisture, it would afford one of the most convenient substances known for a compensation pendulum. It does not appear that sufficient experiments have been made upon this subject to decide the question. Mr. Browne of Portland Place, who has devoted much of his time and attention to the most delicate enquiries of this kind, has, we believe, found that if a teak rod is well gilded, it will not afterwards be affected by moisture. At all events, it makes a far superior pendulum, when thus prepared, to what it does when such preparation is omitted.

Mr. Baily, in the paper we have before alluded to, proposes an economical pendulum to be constructed by means of a leaden cylinder and a deal rod. He prefers lead to zinc, on account of its inferior price, and the ease with which it may be formed into the required shape; and as there is no considerable difference in their rates of expansion, it is equally applicable to the purpose.

Let the length of the deal rod be taken at 46 inches. Then, to find the length of the cylinder of lead to compensate this, we have, in Table II., ·1427 for such a pendulum; which, being multiplied by 46, the product doubled, and one tenth of the result added to it, gives 14·44 inches for the length of the leaden cylinder. Mr. Baily’s compensation gives 14·3 inches.

_London, Pubd. by Longman & Co._]

The rod is recommended to be made of about three eighths of an inch in diameter: the leaden cylinder is to be cast with a hole through its centre, which will admit with perfect freedom the cylindrical end of the rod. The cylinder is supported upon a nut, which screws on the end of the rod in the usual manner. This pendulum is represented at _fig. 224._

Mr. Baily proposes that the pendulum should be adjusted nearly to the given rate by means of the screw at the bottom, and that the final adjustment be made by means of a slider moving along the rod. Indeed, this is a means of adjustment which we would recommend to be employed in every pendulum.

_Smeaton’s Pendulum._

We shall conclude our account of compensation pendulums with a description of that invented by Mr. Smeaton. The compensation for temperature in this pendulum is effected by combining the two modes, which have been so fully described in the preceding part of this article.

The pendulum rod is of solid glass, and is furnished with a steel screw and nut at the bottom in the usual manner. Upon the glass rod a hollow cylinder of zinc, about the eighth of an inch thick, and about 12 inches long, passes freely, and rests upon the nut at the bottom of the pendulum rod.

Over the zinc cylinder passes a tube made of sheet-iron. The edge of this tube at the top is turned inwards, and is notched so as to allow of this being effected. A flanche is thus formed, by which the iron tube is supported, upon the zinc cylinder. The lower edge of the iron tube is turned outwards, so as to form a base destined to support a leaden cylinder, which we are about to describe.

A cylinder of lead, rather more than 12 inches long, is cast with a hole through its axis, of such a diameter as to allow of its sliding freely, but without shake, upon the iron tube over which it passes, and by the lower extremity of which it is supported.

Now the zinc, resting upon the nut and expanding upwards, will raise the whole of the remaining part of the compensation. This expansion upwards will be slightly counteracted by the lesser expansion downwards of the iron tube, which carries with it the leaden cylinder. The cylinder of lead now acts upon the principle of the mercurial pendulum, and, expanding upwards, contributes that which was wanting to restore the centre of oscillation to its proper distance from the point of suspension.

This pendulum, we have been informed, does well in practice, and we are not aware that any description of it has been before published.

The method of calculating the length of the tubes required to form the compensation is very simple; nothing more is necessary than to find the length of zinc, the expansion of which is equal to that of the pendulum rod.

Let the pendulum rod be composed of 43 inches of glass, the spring being an inch and a half long, and the screw between the end of the glass rod and the nut half an inch, making in the whole two inches of steel and 43 inches of glass.

Now to find the length of zinc that will compensate the glass, we have, from Table II., for glass and zinc ·2773, which, multiplied by 43, gives 11·92 inches. In like manner we obtain as a compensation for two inches of steel 0·74 of zinc, which, added to 11·92, gives 12·66 inches for the total length of the zinc cylinder.

Now if the iron tube and the lead cylinder be each made of the same length as the zinc, and arranged as we have described, the compensation will be perfect.

To prove this, find, by means of the expansions given in Table I., the actual expansion of each of the substances employed in the pendulum, and we shall have the following results:--

The expansion of 12·66 inches of zinc expanding
upwards is ·0002186

Deduct that of 12·66 inches of iron expanding
downwards ·0000869
--------
Remaining effect of expansion upwards, referred
to the lower extremity of the iron tube ·0001317

Now, for the lead.--On the principle of the
mercurial compensation, subtract one tenth part
of the length of the cylinder, and take half
the remainder, and we shall have six inches of
lead, the expansion of which upwards is ·0000955
--------
Total expansion of the compensation upwards ·0002272
--------
To find the expansion of the rod, we have
the expansion of 43 inches of glass ·0002059

Of two inches of steel ·0000127
--------
Total expansion of the pendulum rod ·0002186

Agreeing near enough with that of the compensation before found.

As we conceive we have been sufficiently explicit in our description of this pendulum, in the construction of which no difficulty presents itself, we think an engraved representation of it would be superfluous.

We have hitherto treated only of compensations for temperature; but there is another kind of error, which has been sometimes insisted upon, arising from a variation in the density of the atmosphere. If the density of the atmosphere be increased, the pendulum will experience a greater resistance, the arc of vibration will in consequence be diminished, and the pendulum will vibrate faster. This, however, is in some measure counteracted by the increased buoyancy of the atmosphere, which, acting in opposition to gravity, occasions the pendulum to vibrate slower. If the one effect exactly equalled the other, it is evident no error would arise; and in a paper by Mr. Davies Gilbert, President of the Royal Society of London, published in the Quarterly Journal for 1826, he has proved that, by a happy chance, the arc in which pendulums of clocks are usually made to vibrate is the arc at which this compensation of error takes place. This arc, for a pendulum having a brass bob, is 1° 56′ 30″ on each side of the perpendicular; and for a mercurial pendulum, 1° 31′ 44″, or about one degree and a half.

It is well known that, if a pendulum vibrates in a circular arc, the times of vibration will vary nearly as the squares of the arcs; but if the pendulum could be made to vibrate in a cycloid, the time of its vibration in arcs of different extent would then remain the same. Huygens and others, therefore, endeavoured to effect this by placing the spring of the pendulum between cheeks of a cycloidal form.

When escapements are employed which do not insure an unvarying impulse to the pendulum, the force may be unequally transmitted through the train of the clock in consequence of unavoidable imperfections of workmanship, and the arc of vibration may suffer some increase or diminution from this cause. To discover a remedy for this is certainly desirable.

The writer of this article some years ago imagined a mode, which he believes has also been suggested by others, by which he conceived a pendulum might be made to describe an arc approaching in form to that of a cycloid. The pendulum spring was of a triangular form, and the point or vertex was pinned into the top of the pendulum rod, the base of the triangle forming the axis of suspension. Now it is evident that when the pendulum is in motion, the spring will resist bending at the axis of suspension, with a force in some sort proportionate to the base of the triangle.

Suppose the pendulum to have arrived at the extent of its vibrations; the spring will present a curved appearance; and if the distance from the point of suspension to the centre of oscillation be then measured, it will evidently, in consequence of the curvature of the spring, be shorter than the distance from the point of suspension to the centre of oscillation, measured when the pendulum is in a perpendicular position, and consequently when the spring is perfectly straight.

The base of the triangle may be diminished, or the spring be made thinner; either of which will lessen its effect. We cannot say how this plan might answer upon further trial, as sufficient experiments were not made at the time to authorize a decisive conclusion.

We have thus completed our account of compensation pendulums; but before we conclude, it may not be unacceptable if we offer a few remarks on some points which may be found of practical utility.