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gramme des Archives' is equal in mass to 15432.349 grains; and theo kilogramme type laiton,' deposited in the Ministère de l'Intérieure in Paris, as standard for French commerce, is 15432:344 grains.

366. The measurement of force, whether in terms of the weight of a stated mass in a stated locality, or in terms of the absolute or kinetic unit, has been explained in Chapter II. (See $$ 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 number 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 pendulum 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 Escapement, that the oscillations are maintained of nearly uniform extent, and therefore of nearly uniform duration. The construction of escapements, as well as of trains of clock-wheels, is a matter of Mechanics, with the details of which we are not concerned, although it 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 peformance) 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 phenomena, 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 contrive 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 P

or reliable method. It was formerly used 7

in the so-called Diagonal Scale, of which 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 Q

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.

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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 0 coincide, 10 and il coincide also, then 10 divisions of the vernier are equal in length to 11 on the limb;

o and therefore each division of the vernier is idths, or 11 of a division on the limb. If, then, the vernier be moved till 1 coincides with 1, 0 will be ioth

30 of a division of the limb beyond 0; if 2 coincide with 2, 0 will be ths beyond 0; and so on. Hence to read the vernier in any position, note first the division next to 0, 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

29 (as in the figure), that indicates an addition to the reading of 4ths 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 divisions 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, n divisions of the vernier must equal n+1 or n-1 divisions on the limb, according as these run in opposite or similar directions.

377. The principle of the Screw has been already noticed ($ 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 micrometers), 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 once 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 curvature of a lens, the Spherometer is used. It consists of a cylindrical stem through the axis of which a good screw works. The stem is supported by three feet, equidistant from each other, and having their extremities in a plane perpendicular to the axis, The lower extremity of the screw, when worked down into this plane, is equidistant from each of the feet—and the extremities of all are delicately pointed. 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 truly flat surface, and the screw is gradually turned until its point just touches the surface. This is determinable with the utmost accuracy, by the whole system commencing to rock, if slightly touched, the instant that the screw-point passes below the plane of the three feet. The reason of this is, of course, that it is geometrically impossible to make a perfectly rigid body stand on four feet, without infinitely perfect fitting. At the instant at which this

rocking (which is exceedingly distinct to the touch, and even to the ear) commences, the point of the screw is just below the plane of the feet of the instrument. 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 now turned until the rocking just recommences, in which case it is evident that if the screw-point were depressed through a space equal to the thickness of the plate, it would be again just below the plane of the feet. From the difference of the readings of the head, we therefore easily calculate the thickness of the plate, the value of one turn of the screw having been, once for all, ascertained.

381. 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 commencement of rocking 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

a2 radius of the spherical surface; we have at once 2R +1, or, as ?

31 is generally very small compared with a, the diameter is, very ap

a2 proximately,

1, 31

382. The Cathetometer is used for the accurate determination of differences of level-for instance, in measuring the height to which a fluid rises in a capillary tube above the exterior free surface. It consists of a divided metallic stem, which can (by means of levellingscrews in its three feet) be placed very nearly vertical. Upon this slides a metallic piece, bearing a telescope whose axis is rendered horizontal by means of a level. This is, of course, perpendicular to the stem; and when the latter is made to revolve in its supports, describes a horizontal plane. The adjustments are somewhat tedious, but present no other difficulty. In using the instrument the telescope is directed first to one of the objects whose difference of level is to be found, then (with its bearing-piece) it is moved 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 vernier attached to the telescope is then read off-and, the process being repeated for the second object, a simple subtraction gives at once the required difference of level.

383. The principle of the Balance is known to everybody. 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 ;

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