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kinetic unit, has been explained in Chapter II. (See SS 221–227.) From the measures of force and length we derive at once the measure of work or mechanical effect. That practically employed by engineers is founded on the gravitation measure of force. Neglecting the difference of gravity at London and Paris, we see from the above Tables that the following relations exist between the London and the Parisian reckoning of work:

Foot-pound .'=0:13825 kilogramme-mètre.

Kilogramme-mètre* = 7*2331 foot-pounds." 367. A Clock is primarily an instrument which, by means of a train of wheels, records the number of vibrations executed by a pendulum; a Chronometer or Watch performs the same duty for the oscillations of a flat spiral spring-just as the train of wheel-work in a gas-meter counts the nunber of revolutions of the main shaft caused by the passage of the gas through the machine. As, however, it is impossible to avoid friction, resistance of air, etc., a pendo lum or spring, left to itself, would not long continue its oscillations, and, while its motion continued, would perform each oscillation in less and less time as the arc of vibration diminished : a continuous supply of energy is furnished by the descent of a weight, or the uncoiling of a powerful spring. This is so applied, through the train of wheels, to the pendulum or balance-wheel by means of a mechanical contrivance called an Escapenient, that the oscillations are maintained of nearly uniform extent, and therefore of nearly uniform duration. The construction of escapements, as well as oi trains of clock-wheels, is a matter of Mechanics, with the details of which we are not concerned, although wit may easily be made, the subject of mathematical investigation. The means of avoiding errors introduced by changes of temperature, which have been carried out in Compensation pendulums and balances, will be more properly described in our chapters on Heat. It is to be observed that there is little inconvenience if a clock lose or gain regularly; that can be easily and accurately allowed for : irregular rate is fatal."

368. By means of a recent application of electricity, to be afterwards described, one good clock, carefully regulated from time to time to agree with astronomical observations, may be made (without injury to its own performance) to control any number of other lessperfectly constructed clocks, so as to compel their pendulums to vibrate, beat for beat, with its own.

369. In astronomical observations, time is estimated to tenths of a second by a practised observer, who, while watching the phepomena, counts the beats of the clock. But for the very accurate measurement of short intervals, many instruments have been devised.

Thus if a small orifice be opened in a large and deep vessel full of mercury, and if we know by trial the weight of metal that escapes say in five minutes, a simple proportion gives the interval which elapses during the escape of any given weight. It is easy to con.

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trive an adjustment by which a vessel may be placed under, and withdrawn from, the issuing stream at the time of occurrence of any two successive phenomena.

370. Other contrivances are sometimes employed, called Stopwatches, Chronoscopes, etc., which can be read off at rest, started on the occurrence of any phenomenon, and stopped at the OCcurrence of a second, then again read off; or which allow of the making (by pressing a stud) a slight ink-mark, on a dial revolving at a given rate, at the instant of the occurrence of each phenomenon to be noted. But, of late, these have almost entirely given place to the Electric Chronoscope, an instrument which will be fully described later, when we shall have occasion to refer to experiments in which it has been usefully employed.

371. We now come to the measurement of space, and of angles, and for these purposes the most important instruments are the Vernier and the Screw.

372. Elementary geometry, indeed, gives us the means of dividing any straight line into any assignable number of equal parts; but in

practice this is by no means an accurate or reliable method. It was formerly used

in the so-called Diagonal Scale, of which MITTTTT

the construction is evident from the diagram. The reading is effected by a sliding piece whose edge is perpendicular to the length of the scale. Suppose that it is PQ whose position on the scale is. required. This can evidently cut only one of the transverse lines. Its number gives the number of tenths of an inch (4 in the figure), and the horizontal line next above

the point of intersection gives evidently the number of hundredths in the present case 4). Hence the reading is 7:44. As an idea of the comparative uselessness of this method, we may mention that a quadrant of 3 feet radius, which belonged to Napier of Merchiston, and is divided on the limb by this method, reads to minutes of a degree; no higher accuracy than is now attainable by the pocket sextants made by. Troughton and Simms, the radius of whose arc is virtually little more than an inch. The latter instrument is read by the help of a Vernier.

373. The Vernier is commonly employed for such instruments as the Barometer, Sextant, and Cathetometer, while the Screw is applied) to the more delicate instruments, such as Astronomical Circles,, Micrometers, and the Spherometer.

374. 'The vernier consists of a slip of metal which, slides along a divided scale; the edges of the two being coincident. Hence, when it is applied to a divided. circle, its edge is circular,


and it moves about an axis passing through the centre of the divided limb.

In the sketch let 0, 1, 2, ... 10 denote the divisions on the vernier, 0, 1, 2, etc., any set of consecutive divisions on the limb or scale along whose edge it slides. If, when 0 and o coincide, 10 and 11 coincide also, then 10 divisions of the vernier are equal in length to it on the limb; and therefore each division of the vernier is fiths, or go of a division on the limb. If, then, the ver. l. njer be moved till I coincides with 1, 0 will be oth of a division of the limb beyond o; if 2 coincide with 2, 0 will be sths beyond o; and so on. Hence to read the vernier in any position, note first the division next to o, and behind it on the limb. This is the integral number of divisions to be read. For the fractional part, see which division of the vernier is in a line with one on the limb; if it be the 4th (as in the figure), that indicates an addition to the reading of ths of a division of the limb; and so on. Thus, if the figure represent a barometer scale divided into inches and tenths, the reading is 30-34, the zero line of the vernier being adjusted to the level of the mercury.

375. If the limb of a sextant be divided, as it usually is, to thirdparts of a degree, and the vernier be formed by dividing twenty-one of these into twenty equal parts, the instrument can be read to twentieths of divisions on the limb, that is, to minutes of arc...

If no line on the vernier coincide with one on the limb, then since the divisions of the former are the longer there will be one of the latter included between the two lines of the vernier, and it is usual in practice to take the mean of the readings which would be given by a coincidence of either pair of bounding lines.

376. In the above sketch and description, the numbers on the scale and vernier have been supposed to run opposite ways. This is generally the case with British instruments. In some foreign ones the division; run in the same direction on vernier and limb, and in that case it is easy to see that to read to tenths of a scale division we must have ten divisions of the vernier equal to nine of the scale.

In general to read to the nth part of a scale division, a divisions of the vernier must equal n + I or n - I divisions on the limb, according as these run in opposite or similar directions.

377. The principle of the Screw has been already noticed (8 114). It may be used in either of two ways, i.e. the nut may be fixed, and the screw advance through it, or the screw may be prevented from moving longitudinally by a fixed collar, in which case the nut, if prevented by fixed guides from rotating, will move in the direction 'of the common axis. The advance in either case is evidently proportional to the angle through which the screw has turned about its axis, and this may be measured by means of a divided head fixed perpendicularly to the screw at one end, the divisions being read off by a pointer or vernier attached to the frame of the instrument. The nut carries with it either a tracing point (as in the dividing engine) or a wire, thread, or half the object-glass of a telescope (as in micro meters), the thread or wire, or the play of the tracing-point, being at right angles to the axis of the screw. .

378. Suppose it be required to divide a line into any number of equal parts. The line is placed parallel to the axis of the screw with one end exactly under the tracing-point, or under the fixed wire of a microscope carried by the nut, and the screw-head is read off. By turning the head, the tracing-point or microscope wire is brought to the other extremity of the line; and the number of turns and fractions of a turn required for the whole line is thus ascertained. Dividing this by the number of equal parts required, we find at oộce the number of turns and fractional parts corresponding to one of the required divisions, and by giving that amount of rotation to the screw over and over again, drawing a line after each rotation, the required division is effected.

379. In the Micrometer, the movable wire carried by the nut is parallel to a fixed wire. By bringing them into optical contact the zero reading of the head is known; hence when another reading has been obtained, we have by subtraction the number of turns corresponding to the length of the object to be measured. The absolute value of a turn of the screw is determined by calculation from the number of threads in an inch, or by actually applying the micrometer to an object of known dimensions.

380. For the measurement of the thickness of a plate, or the cur. vature of a lens, the Spherometer is used. It consists of a screw nut rigidly fixed in the middle of a very rigid three-legged table, with its axis perpendicular to the plane of the three feet (or finely rounded ends of the legs,) and an accurately cut screw working in this nut. The lower extremity of the screw is also finely rounded. The number of turns, whole or fractional, of the screw, is read off by a divided head and a pointer fixed to the stem. Suppose it be required to measure the thickness of a plate of glass. The three feet of the instrument are placed upon a nearly enough flat surface of a hard body, and the screw is gradually turned until its point touches and presses the surface. The muscular sense of touch perceives resistance to the turning of the screw when, after touching the hard body, it presses on it with a force somewhat exceeding the weight of the screw. The first effect of the contact is a diminution of resistance to the turning, due to the weight of the screw coming to be borne on its fine pointed end instead of on the thread of the nut. The sudden increase of resistance at the instant when the screw commences to bear part of the weight of the nut finds the sense prepared to perceive it with remarkable delicacy on account of its contrast with the immediately preceding diminution of resistance. The screw-head is now read off,

and the screw turned backwards until room is left for the insertion. beneath its point, of the plate whose thickness is to be measured. The screw is again turned until increase of resistance is again perceived; and the screw-head is again read off. The difference of the readings of the head is equal to the thickness of the plate, reckoned in the proper unit of the screw and the division of its head.

881. If the curvature of a lens is to be measured, the instrument is first placed, as before, on a plane surface, and the reading for the contact is taken. The same operation is repeated on the spherical surface. The difference of the screw readings is evidently the greatest thickness of the glass which would be cut off by a plane passing through the three feet. This is sufficient, with the distance between each pair of feet, to enable us to calculate the radius of the spherical surface.

In fact if a be the distance between each pair of feet, l the length of screw corresponding to the difference of the two readings, R the radius of the spherical surface; we have at once 2R= +1, or, as I is generally very small compared with a, the diameter is, very ap. proximately,

382. The Cathetometer is used for the accurate determination of differences of level--for instance, in measuring the height to which a fuid rises in a capillary tube-above the exterior free surface. It consists of a long divided metallic stem, turning round an axis as nearly as may be parallel to its length, on a fixed tripod stand: and, attached to the stem, a spirit-level. Upon the stem slides a metallic piece bearing a telescope of which the length is approximately enough perpendicular to the axis. The telescope tube is as nearly as may be perpendicular to the length of the stem. By levelling screws in two feet of the tripod the bubble of the spirit-level is brought to one position of its glass when the stem is turned all round its axis. This secures that the axis is vertical. In using the instrument the telescope is directed in succession to the two objects whose difference of level is to be found, and in each case moved (generally by a delicate screw) up or down the stem, until a horizontal wire in the focus of its eye-piece coincides with the image of the object. The difference of readings on the vertical stem (each taken generally by aid of a vernier sliding piece) corresponding to the two positions of the telescope gives the required difference of level. . · 383. The principle of the Balance is generally known. We may note here a few of the precautions adopted in the best balances to guard against the various defects to which the instrument is liable; ånd the chief points to be attended to in its construction to secure delicacy, and rapidity of weighing.

The balance-beam should be very stiff, and as light as possible consistently with the requisite stiffness. For this purpose it is

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