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the balance. Hence the natural isochronism of the pendulum is less disturbed by the relatively small inequalities of the maintaining power

3. The elastic force of the spring which impels the circumference of the balance is directly as the tension of the spring : that is, the weights necessary to counterpoise a spiral spring's elastic force, when the balance is wound to different distances from the quiescent point, are in the direct ratio of the arcs through which it is wound.

4. The vibrations of a balance, whether through great or small arcs, are performed in the same time. For the accelerating force is directly as the distance from the point of qui escence: hence, therefore, the motion of the balance is analogous to that of a pendulum vibrating in cycloidal arches (vol. i, art: 276.)

5. The time of the vibration of a balance is the same as if a quantity of matter, whose inertia is equal to that by which the mass contained in the balance opposes the communication of motion to the circumference, described a cycloid whose length is equal to the arc of vibration, described by the circumference, the accelerating force being equal to that of the balance.

6. The times of vibration of different balances are in a ratio compounded of the direct subduplicate ratios of their weights and semidiameters, and the inverse subduplicate ratio of the tenfions of the springs, or of the weights which counterpoise them, when wound through a given angle.

7. The times of vibration of different balances are in a ratio compounded of the direct simple ratio of the radii and direct fubduplicate ratio of their weights, and the inverse subduplicate ratio of the absolute forces of the springs at a given tension.

8. Hence the absolute force of the balance spring, the diameter and weight of the balance being the fame, is inversely as the square of the time of one vibration.

9. The absolute force or strength of the balance fpring, the time of one vibration, and the weight of the balance being the same, is inversely as the square of the diameter.

10. The weight of the balance, the strength of the spring and time of vibration being the same, is inversely as the square of the diameter.

Hence, a large balance, vibrating in the same time with the same spring, will be much lighter than a small one.

!!. If the rim of the balance be always of the fame breadth and thickness, so that the weight shall be as the radius, the strength of the spring must be as the cube of the diameter of the balance, that the time of vibration may continue the fame,

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12. The momentum of the balance is increafed better by in, creasing its diameter than its weight.

13. The longer a detached balance continues its motion the better.

14. The greater the number of vibrations performed by a balance in a given time, the less susceptible is it of external agitations.

15. Slow vibrations are, to a certain extent, preferable to quick vibrations: but there is manifestly a limit; for if the · vibrations be too slow, the watch will be liable to stop.

15. A balance should describe as large arches as possible, as fuppofe 240, 260°, 300', or an entire circle.

First, because the momentum of the balance is thus increafed; and therefore the inequalities in the force of the maintaining power bear a less proportion to it, and of consequence will have lefs influence. 2dly. The balance is less susceptible of external agitations. 3dly. A given variation in the extent of the vi. brations produces a less variation in the going of the machine.

But care must be taken that in these great vibrations, the {pring shall neither touch any obstacle, nor its spires touch each other in contracting.

17The time of the vibration of the balance is increased by heat, and diminished by cold. First, because the length of the spiral spring is increased by heat, and therefore its force diminished; and the contrary by cold. 2dly. The diameter of the balance is increased by heat, and therefore also the time of vibration; and the contrary by cold.

18. That balance is the most perfect which, without the compensation of a thermometer, is most subject to the influ. ence of heat and cold. Because the obstructions from oil and friction act as a compensation to the expansion or contraction of the spring and balance; therefore that balance which is most affected is most free from the influence of oil and friction.

19. The errors in the going of a watch, arising from the change of temperature, may be corrected by varying the length of the balance spring. Nevertheless, as it is extremely difficult to form an isochronal fpiral, any variation in its length is dangerous, because we shall thus probably lose that point which determines its isochronism.

20. The errors in the going of a watch, occasioned by the yariation of temperature, may be corrected by varying the diameter of the balance.

This may be effected by a peculiar contrivance which has obtained the name of the expansion balance, being composed of two different metals which poffefs different degrees of expansbility, as brass and steel, for instance; of which two metals it has been observed, that the increase of dimensions by expanfion, in like clevations of temperature, is nearly as 2 to 1. For, according to Mr. Smeaton's experiments (vol. 48, Phil. Trans.), the corresponding expansions of hard steel and brass wire are as 147 and 232, the expansions being occasioned by a change from a medium temperature to that of 180° of Fahrenheit's thermometer. One of the most approved constructions of an expansion balance, is exhibited in plate VII. and is thus described by Mr. Nicholson: The outer part of the rim is brass, and the inner steel. After this compound rim is brought to its figure by turning, it is cut through in three places, A, B, C, which sets one end of each third part of the periphery at liberty to move outwards, when the temperature is diminished, or in wards when it is increased. D, E, F, are three fimilar and equal masses of metal, fitted upon the circular bars in a proper manner to admit of their being fixed at any required distance from the extremity, where the motion is most confiderable. G, H, I, are three screws, the heads of which may be set nearer to, or further from, the centre, and serve as weights to effect the adjustments for position and rate. The peculiar advantage of this balance may be explained as follows: when an increase of heat diminishes the elastic force of the pendulum spring K, the outer brass rim being lengthened more than the steel, must throw the weights D, E, F, nearer to the axis, and diminish the effect of the inertia of the balance, which consequently is as speedily carried through its vibration as before. And on the contrary, when cold weather adds to the elastic force of the spring, the fame weights are also thrown further out, and prevent the acceleration which would have followed. The exact adjustment of the weights is found by trial of the going of the machine : if it gain by heat, the weights do more than compensate, and must be moved further from the extreme ends of the circular compourd bars; but if the gain be produced by cold, the spring predominates, and the weights will accordingly require to be set further out.

BARK-MỊLL, a mill constructed for the purpose of grinding and preparing bark, till it is fit for the use of a tanner.

Bark-mills, like most other mills, are worked sometimes by means of horses, at others by water, and at others by wind. One of the best mills we have seen described for these purposes, is that invented by Mr. Bagnall, of Worsley in Lancashire : this machine will serve not only to chop bark, to grind, to riddle and pound it; but to beam, or work green hides and skins out of the mastering or drench, and make them ready for the oufe or bark liquor; to beam sheepskins and other skins for the {kinner's use; and to scour and take off the bloom from tanned leather, when in the currying state. The nature and connection of the different parts of this contrivance may be underfood from the three figures on the right-hand side of plate VII. together with the following description.

Fig. 1. is a horizontal plan of the mill. Fig. 2. longitudinal fection of it. Fig. 3. tranfverfe fection of it... .., The water-wheel, by which the whole machinery is worked.

B, The (hafts.

C, The pit-wheel, which is fixed on the water-wheel fhaft B, and turns the upright shaft E, by the wheel F, and works the cutters and hammer by tapets.

D, The fpur and bevil-wheel at the top of upright shafts,
E, The upright shaft.
F, The crown-wheel, which works in the pit-wheel C.
G, The fpur-nut to turn the stones I.

P, The beam, with knives or cutters fixed at the end to chop or cut the bark; which bark is to be put upon the cutters or grating i, on which the beam is to fall.

, The tryal that receives the bark from the cutters i, and conveys it into the hopper H, by which it defcends through the shoe J to the stones ), where it is ground. :, K, The spout, which receives the bark from the stones, and conveys it into the tryal L, which tryal is wired to Gift or drefs the bark, as it defcends from the stones I. . į M. The trough to receive the bark that pafses through the

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tryal L.

:: R, The hammer, to crush or bruise the bark that falls into the dish S, which said dish is on the incline, so that the hammer keeps forcing it out of the lower fide of the faid dish, when bruised. hik, A trough to receive the dust and moss that passes through the tryal Q.

T, I he bevil-wheel, that works in the wheel D, which works the beam-knife by a crank V at the end of the shaft ui

„W; The penetrating rod, which leads from the crank V to the startixo : ;*, The Start, which has several holes in it to lengthen or Thorten the stroke of the beam-knife. .::,

y, The shaft, to which the slide rods b, b, are fixed by the starts 1.950, ...

h, The flide rod, on which the knife.f is fixed.;' which knife is to work the hides, &c., On the knife are two springs a, i to let it have a little play as it makes its stroke backwards and for wards, so that it may not scratch or damage the hides, &c.**

* Is a catch in Aide-rod h, which catches on the arch-head ;

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and the faid arch-head conveys the knife back without touching the hide, and then falls back to receive the catch again.

1, The roller to take up the slide-rod h, while the hides are shifting on the beam b by pulling at the handle m. : b, The b:am to work the hides, &c. on. Each beam has four whecls p, p, working in a trough road g, g, and removed by the levers co's.

When the knife has worked the hide, &c. fufficiently in one part, the beam is then shifted by the lever cas far as is wanted.

d, A press, at the upper end of the beam, to hold the hide fast on the beam while working.

e, An arch head, on which the slide-rod h catches. f, The knife fixed on the side-rod h, to work the hides, &c. i, Cutters or grating to receive the bark for chopring.

The beam P, with knives or cutters, may either be worked by tapers, as described, or by the bevil-wheel T, with a crank, as V, to cut the same is fheurs.

The kuife f is fixed at the bottom of the start, which is fixed on the slide" rod hi the bottom of the start is split open to admit the knife, the width of one foot; the knife should have a gudgeon at each end, to fix in the open part of the start; and the two springs aj a, prevent the knife from giving too much way when working; the knife should be one foot long and four or five inches broad.

The arch-head e will shift nearer to, or further from, the beam k, and will be fixed fo as to carry the knife back as far as is wanted, or it

may

be taken away till wanted. The roller l is taken up by pulling at the handleʻm, which takes up the flide-rod so high as to give head-room under the beam-knife. The handle may be hung upon a hook for that purpose. The slide-rod will keep running upon the roller all the time the hide is shifting ; and when the hide is fixed the knife is put on the beam again by letting it down by the handle m. There may be two or more knives at work on one beam at the same time, by having different slide-rods. There should be two beams, so that the workmen could be shifting one hide, &c. while the other was working. The beam must be flat, and a little on the incline. As to the breadth it does not matter; the broader it is the less shifting of the hides will be wanted, as the lever c will shift them as far as the width of the hide, if required. Mr. Bagnall has formed a kind of press d, to let down, by à lever, to hold the hide fast on each side of the knife if required, so that it will suffer the knife to make its back stroke without pulling the hide up as it comes back. The slide-rod may be weighted, to cause the knife to lay stress on the hide, &c. according to the kind and condition of the goods to be worked.

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