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circle "2,'ZN' intersect in Z, it will not come to the latter circle again till after it has passed the meridian at the upper transit.

Hence both the observed transits will be later than the true.

Since however aZ is a smaller arc than a'Z, the one being the sum and the other the difference of the arcs Pa and PZ, the space between the meridian and the circle "Si ZN is less at a than at a, and therefore the observed upper transit occurs nearer the true time than the observed lower transit. Thus the time from the lower to the upper transit is less than it should be, and the result is the same as before.

In general it is advisable to select a star for observation which, as in the first case, makes both its transits on the same side of the zenith, for then the inequality of the times is most perceptible. In fact, the nearer the star is to the pole (its diurnal motion being slower), the greater is the effect on its times of transit of a given deviation from the meridian. On this account the pole-star, commonly called Polaris, which is about a degree-and-a-half from the north pole, is very often observed for the deviation error.

It is obvious that in or near the equator this method is inapplicable, owing to the pole being nearly in the horizon. In this case the same result may be obtained by observing the transits of two stars, one of which passes the meridian near the zenith, and the other near the horizon. It is necessary, however, that the difference of right ascension of the two stars should be known beforehand. Then the time intervening between the true transits of the two stars will be known, being the same part of 24 sidereal hours that their difference in right ascension is of 360 degrees. If therefore the instrument is in adjustment, the interval between the observed transits will be the time so obtained. But if the telescope command the circle TZN' instead of the meridian, the star passing near the zenith will cross this circle much about the same time as it crosses the meridian, while the star near the horizon will have a much larger space to pass through between the two circles. Thus the time of transit of the first star will be much less affected than that of the second, and therefore the interval between the transits will be altered by the error. Thus the error of deviation may be detected when there is no opportunity of observing a circumpolar star.

47. The Astronomical Clock. It is evident from what has preceded, that a clock which can be depended upon is an

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essential part of the furniture of an Observatory. The contrivances by which the necessary accuracy is obtained in clocks used for astronomical observations distinguish them so completely from the coarser time-pieces which are sufficient for ordinary purposes, that the astronomical clock may be looked upon as a separate instrument, and its description may properly occupy a place in an astronomical treatise.

The moving power of the clock is a weight, which is more suitable than a spring, on account of the uniformity of its action. There is no striking apparatus or superfluous machinery of any kind to interfere with the regularity of the working. The pendulum, which vibrates seconds, is made to move with its extremity in a cycloidal arc, and is fitted with a contrivance by which the effects of change of temperature on the rate of going are obviated.

It will be recollected that the period of vibration of a simple pendulum depends on its length. Such a pendulum obviously can only exist in theory, as it is supposed to consist of a heavy particle at the end of a rod without weight. In opposition to this, all pendulums in use are called compound pendulums, and it may be stated, without entering into farther explanations, that the period of vibration of any compound pendulum depends jointly on its form and on the distance of its centre of gravity from the axis about which it revolves. The ordinary pendulum, as is well known, consists of a rod on which slides a piece of metal, technically called a bob, capable of being raised or lowered by means of a screw. The principal weight being in the bob, the centre of gravity of the whole is thus varied in position without much change of form, and the period of vibration is increased or diminished at pleasure. Thus, theoretically, a clock might be regulated to any degree of accuracy, the action of the works on the pendulum being uniform. Since, however, all known metals expand with heat and contract with cold, it is found that the time of vibration of a common pendulum changes with the temperature, the length of the rod being greater in warm weather than in cold. To obviate this, instead of the ordinary bob, a cylindrical glass vessel is used, containing a

V large quantity of mercury. When the rod expands so as to lower the bottom of this vessel, the mercury also expands and rises in the vessel. The centre of gravity of the mercury is therefore farther from the bottom of the vessel than it was before, and this effect compensates that of the lengthening of the rod; so that when the quantity of mercury is properly apportioned to the weight of the pendulum, the position of the centre of gravity of the whole is unchanged, and the time of vibration is unaffected.

By this and other contrivances the astronomical clock has been brought to great accuracy. The grand desideratum is uniformity in the rate of going: for when this is secured we may regulate to any required rate, or by observing the difference between the actual rate and the required rate, we may obtain the means of applying corrections to the observed times. The latter is the method commonly employed. The time lost or gained in a day is determined by observation, and is technically termed the rate. The clock error at any time is the amount to be added to or subtracted from the observed time in order to get the true time. When the clock error and rate are known, the true time at any subsequent observation may be found, by adding to the observed time the error and the amount of the rate multiplied by the number of days and parts of a day intervening since the error was determined. In practice the clock is so regulated as to have a small losing rate, and is set a little too slow. Thus the corrections are additive. The error is never allowed to exceed a small amount.

The error of rate is the difference between the rates of two successive days. If the clock were perfect there would be no such error, and in the present state of mechanical skill it is kept within very small limits. An habitual error of rate amounting to a second would entirely condemn a clock for astronomical purposes.

The astronomical clock is usually set to sidereal time, that is, so as to mark 24 hours between two successive transits of a star. Thus it is independent of the variations in the Sun's position. The sidereal day begins when the vernal equinoctial point, called the first point of Aries (from its being the point where the Sun enters the zodiacal sign Aries), passes the meridian. This occurs about noon when the Sun is in equinox, and earlier every subsequent day by nearly four minutes till it comes round to noon again. It is not, however, to be supposed that any visible point shews by its transit when the sidereal day begins, the first point of Aries being an imaginary point indicated by no appearance in the heavens, and only geometrically determined by its being at the intersection of the equator and the ecliptic. If such a point existed, the sidereal clock might be set at once by observing its transit. The actual mode of setting it is, however, a matter of much greater difficulty, which we are not yet far advanced enough in the subject to explain.

The rate is easily found by observing two successive transits of the same star; and for this purpose it is not necessary that the transit instrument should be in very accurate adjustment, only that it should not be moved between the two observations, for a fixed star occupies exactly 24 sidereal hours in revolving from any given point to the same point again.

A succession of such observations would detect any error of rate.

48. The Equatoreal. This instrument consists of two graduated circles revolving about axes at right angles to one another in the same manner as those of the altitude and azimuth instrument. They are however differently placed, the lower circle having its plane parallel with that of the celestial equator, and consequently inclined to the horizon at an angle equal to the co-latitude. The axis, therefore, which bears the upper circle points to the pole, whereas that of the altitude and azimuth instrument points to the zenith.

The equatoreal, if in exact adjustment, would determine at once the declination of an observed body. For, suppose the telescope to be directed to a star, so that the cross-wires appear to coincide with it; the reading of the upper circle gives the angular distance of the star from the plane of the lower circle, that is, from the celestial equator. On this account the upper circle is called the declination-circle. Supposing also the zero point of graduation of the lower circle to be that marked by the index when the telescope is directed to the meridian, the reading of the lower circle will give the angular distance of the star from the meridian at the time of observation, measured along the celestial equator. This angle will be the same as that between the meridian and the declination-circle of the star. It is commonly called the hour-angle of the star, and the circle on which it is measured is called the hour-circle.

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Let P be the pole, EQ the celestial equator, S a star.

The equatoreal gives the arcs SM and EM, the former of which is the declination, and the latter is equal to the angle EPM, included between PE the meridian, and PS the declination-circle of the star. *

EPM is called the hour-angle, because whatever part it is of 360°, the same part of 24 hours is the time of S from the meridian, that is, the time S will take in getting to the meridian, or the time by which it has passed the meridian, according as it is on the east or the west of the meridian at the time of observation.

In practice the equatoreal is not much used for accurate determinations of declinations and hour-angles, which may be more certainly made by other methods. Its chief use is in giving approximate positions of bodies observed out of the meridian, and in such observations as occupy any considerable time, as those of double stars by micrometers; for by turning the telescope about the polar axis, which is parallel to that of the diurnal motion of the heavens, a star may be kept continually in the field of view, the space commanded by the telescope in its revolution being in fact the star's diurnal path.

In the case of large instruments the telescope is made to move in this manner by clockwork, properly regulated, so that the observer may devote his whole attention to micrometrical measurements, or whatever other object he may have in view.

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