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water, by adding one half its thickness to the height of the water column.

Next raise the rod EF, and read the height, first of the top and then of the bottom. The difference will be its length. It is safer to test the result by moving it and repeating. Then bring the rod so that it shall just touch the surface of the mercury, that is, so that the point and its reflection shall coincide, and read the height of D, and of the top of the rod. Their difference added to the length of the rod gives the height of the column. Read the height of the standard barometer placed among the meteorological instruments. Reduce this to millimetres, and subtract from it the other measurement. The difference will be the depression caused by air and the other errors in the barometer D.

13. Hook GAUGE.

Apparatus. A stand, Fig. 12, on which may be placed a vessel of water A, and a micrometer screw B, by which we can raise or lower a rod carrying two points, one turned upwards, the other downwards.

B

A

Experiment. Fill up the vessel until the water just covers the point of the hook. Then turn the screw so that upon looking at the reflection on the surface of some object as a window sash, a slight distortion is produced by the elevation of the water above the hook. Make ten measurements, moving the screw after each, take their mean and compute the probable error of a single observation. When the point is raised it draws the liquid with it. Screw it down until it touches the liquid, and read the micrometer, then raise it until the liquid separates, and take ten readings in each position. fore, the probable error, and reduce to fractions of a millimetre, which is easily done if the pitch of the screw is known. This gives a measure of the comparative accuracy of the hook and simple point. Both are used for determining the exact height of any liquid surface, the hook being employed most frequently in this country, the point abroad. When the surface of a liquid is

Fig. 12.

Compute, as be

rising or falling, and we wish to know the exact time when it reaches a given level, we should use the hook when it descends, otherwise the point; because the former should always be brought up to the surface, the latter down to it.

This instrument is so extremely delicate that it will show the lowering of a surface of water in a few minutes by evaporation. A variety of interesting researches may be conducted with it, by the different students of a class. Thus its comparative accuracy with water, mercury and other liquids, may be measured, their rate of evaporation, and the effect of impurities, such as a drop of oil. The height to which a liquid may be raised by the point, is also a test of its viscosity.

14. SPHEROMETER.

Apparatus. Two lenses, one convex, the other concave, a piece of thick plate glass and a spherometer. The latter consists of a tripod, with a micrometer screw in the centre, whose point may be moved to any desired distance above or below the plane of the three legs on which it rests. The most important qualities are lightness and stiffness, and on this account a very cheap, and quite efficient spherometer may be made with the nut and tripod of wood, using for legs, pieces of knitting needles.

Experiment. Stand the spherometer on the sheet of plate glass and turn the screw until its point is in contact with it. There are three ways of determining the exact position of contact. The first method is dependent on the fact that if the point of the screw is too low the spherometer will stand unsteadily, like a table with one leg too short. The screw is therefore depressed until the instrument rattles, when its top is moved gently from side to side. An exceedingly small motion of this kind is perceptible to the hand. The screw is then turned up and down until the exact point of contact is found. The second, and probably the best method, is to turn the screw slowly, taking care that no greater pressure is exerted on one leg than on the other; as soon as the point touches the glass the pressure is removed from the legs, and the friction of the nut at once makes the whole instrument revolve. Care must be taken not to press on the top of the screw, or the tripod will be bent, and an incorrect reading obtained. The third method of determining contact depends on the sound pro

duced when the instrument slides over the glass, which changes when the screw touches the surface. It should be moved but a short distance and without pressure, for fear of scratching the glass.

Having determined this point with accuracy, read the position of the screw, taking the number of revolutions from the index on one side, and the fraction from the divided circle.

Place the spherometer on each face of the two lenses and measure the position of the point of contact as before. Of course the screw must be raised when the surface is convex, and depressed when it is concave. Subtract each of these readings from that taken on the plate glass, and the difference gives the height of a segment of the sphere to be measured, whose base is a circle passing through the three feet of the spherometer. Call this height h the radius of the circle r, and the radius of the sphere R; then we = h, BD =r, and AC ilar triangles AB : DB

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have, Fig. 13, AB
R. But by sim-
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A

B

E

Fig. 13.

pute in this way the radius of each surface of the lenses, remembering that a negative radius denotes a concave surface. To determiner, measure the distance of each leg of the spherometer from the axis of the screw, and take their mean. Measure also the distances of the three legs from each other and take their mean. They form the three sides of an equilateral triangle; compute by geometry the radius of the circumscribed circle, and see if this value of r agrees with that previously found. Both r and h must be taken in the same unit, as millimetres or inches, and great care should be taken to make no mistake in the position of the decimal point. The reduction of h is effected by multiplying it by the pitch of the screw.

Finally, compute the principal focal distance, F, by the formula

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1)+, in which R and R' are the radii of

R''

the two surfaces, as computed above, and n the index of refraction of the glass. The latter varies in different specimens, but in common lenses is about 1.53.

15.

ESTIMATION OF TENTHS OF A SECOND.

Apparatus. A heavy body carrying a small vertical mirror is suspended by a wire, so that it will swing by torsion, about once in half a minute. A small telescope with cross hairs in its eyepiece, is pointed towards the mirror, and a plate with a pin hole in it is placed in such a position that when the mirror swings, the image of the hole will pass slowly across the field of view of the telescope, like a star. It may be made bright by placing a mirror behind it and reflecting the light of the window. The whole apparatus should be enclosed so as to cut off stray light. A good clock beating seconds is also needed.

Experiment. Twist the mirror slightly, so that it shall turn slowly. On looking through the telescope a point of light or star will be seen to cross the field of view, at equal intervals of about half a minute. Note the hour and minute, and as the star approaches the vertical line take the seconds from the clock and count the ticks of the pendulum. Fix the eye on the star and note its position the second before, and that after, it passes the wire. Subdividing the interval by the eye we may estimate the true time of transit within a tenth of a second. Take twenty or thirty such observations and write them in a column, and in a second column give their first differences. Take their mean and

compute the probable error. It will show how accurately you can estimate these fractions of seconds.

This is called the eye and ear method of taking transits, which form the basis of our knowledge of almost all the motions of the heavenly bodies. It is still much used abroad, although in this country superseded in a great measure by the electric chronograph described on p. 16.

16. RATING CHRONOMETERS.

Apparatus. Two timekeepers giving seconds, one, which may be the laboratory clock, to be taken as a standard, and a second to be compared with it. For the latter a cheap watch may be kept expressly for the purpose, or the student may use his own. If the true time is also to be obtained, a transit or sextant is needed in addition.

Experiment. First, to obtain the true time. As this problem belongs to astronomy rather than physics, a brief description only

will be given. It may be done in two ways, with a transit or a sextant; the former being used in astronomical observations, the latter at sea. A transit is a telescope, mounted so that it will move only in the meridian. With it note by the clock the minute and second when the eastern and western edges of the sun cross its vertical wire, and take their mean. Correct this by the amount that the sun is slow or fast, as given in the Nautical Almanac, and we have the instant of true noon. The interval between this and twelve, as given by the clock, is the error of the latter.

The sextant may be used at any time when the sun is not too near either the meridian or the horizon. A vessel containing mercury is used, called an artificial horizon, and the distance between the sun and its image in this is measured. Since the surface of the mercury is perfectly horizontal, this distance evidently equals exactly twice the sun's altitude. If the observation is made in the morning, when the sun is ascending, the sextant is set at somewhat too great an angle, if after noon at too small an angle, and the precise instant when the two images touch is noted by the clock. The sun's altitude, after allowing for its diameter, is thus obtained. We then have a spherical triangle, formed by the zenith Z, the pole P, and the sun S. In this, PZ is given, being the complement of the latitude; PS, the sun's north polar distance, is obtained from the Nautical Almanac, and ZS is the complement of the altitude just measured. From these data we can compute the angle ZPS, which corrected as before and reduced to hours, minutes and seconds, gives the time before or after noon. The practical directions for doing this will be found given in full in Bowditch's Navigator.

By these methods we obtain the mean solar time, which is that used in every day life. For astronomical purposes sidereal time, or that given by the apparent motion of the stars, is preferable. It is found by similar methods, using a star instead of the sun.

In an astronomical observatory it is found best not to attempt to make the clock keep perfect time, but only to make sure that its rate, or the amount it gains or loses per day, shall be as nearly as possible constant. We can then compute the error E at any given time very easily by the formula E = E' + tr, in which E' was the

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