or, sin. PZ sin. PSZ (= sin. Pst) :: sin. PS: sin. PZS, sin. PZ Pst: sin. PS: Z; from which proportion, after substituting the above value of Pst, we obtain, the azimuthal deviation required. 97. In a regular observatory, a large circle of brass attached to the east or west face of a wall, or stone pier, is used for obtaining the altitudes of celestial bodies above the horizon, or their distances from the zenith; or again, their distances from the pole of the equator. Such an instrument, called a mural circle, is generally of considerable dimensions (6 feet diameter), and it turns upon a horizontal axis, part of which enters into the supporting wall or pier; either its side or its edge is graduated, and six micrometer microscopes attached to the face of the wall at the circumference of the circle, and at nearly, or exactly, equal distances from each other, are used in reading the subdivisions of the degrees by which the required altitude or distance is expressed. A telescope is made to turn with the circle on the horizontal axis, in making the observation; but it is also capable of being turned independently, on the same axis, and of being made fast to the circle in several different positions, with respect to the zero of the graduations, in order that the angular admeasurement may be read on any part of the circumference at pleasure. 98. In general the horizontality of the axis of motion is verified by means of plumb-lines, or spirit-levels, or by observing both by direct view and by reflexion the transits of a star at the several wires in the eye-piece of the telescope as was mentioned in the account of the transit instrument; and the line of collimation is made perpendicular to the horizontal axis by the aid of meridian marks previously set up towards the north and south of the telescope by means of a transit instrument, which admits of being reversed on its supports. The deviation of the plane of the circle from the meridian may be found by observing the transits of two stars differing considerably in altitude, or of a circumpolar star when at its greatest and least elevation; but great accuracy of adjustment in this respect is evidently of less importance for an instrument which is intended to give altitudes or declinations only than for one by which transits are to be observed. 99. If it be intended to obtain at once, from observation, the polar distances of celestial bodies there must be previously found what is called the place of the pole on the circle, that is, the number of the graduation at which the index stands when the optical axis of the telescope is directed to the pole in the heavens. For this purpose, the altitudes of a circumpolar star at its greatest and least elevations must be obtained by the instrument, and corrected on account of the effects of refraction: then, since the pole is equally distant from the two corrected places of the star, half the sum of the altitudes so corrected will evidently be the required polar point. A mean of the like observations on several circumpolar stars will give the polar point with sufficient accuracy; and this point is to be considered as the zero of the graduations when the polar distance of any celestial body is subsequently to be obtained. 100. If the angular distances of celestial bodies from the zenith are to be immediately obtained from observation with a mural circle, there must be previously found what is called the horizontal point, that is, the number of the graduation at which the index stands when the optical axis of the telescope is in a horizontal position. For this purpose, the altitude of a star on the meridian must be observed by direct view, and also by reflexion in a trough of mercury; both observations being made, and the altitudes read on the circle while the star is in the field of the telescope: the number of the graduation corresponding to the middle point between the places of the index at the two observations will evidently, since both observations are equally affected by refraction, be the required horizontal point, or that which is to be considered as the zero in subsequently determining the altitude of any celestial body; and a mean of many such observations will give it with sufficient accuracy. The following is, however, considered as a more correct process for determining the horizontal point. On any night let the altitudes or zenith distances of two stars be successively observed by direct view, and on the next night let one of them, suppose the first, be observed by direct view, and the other by reflexion from mercury: then, from a table of refractions, find the change which, on account of differences in the density and temperature of the air, may in the interval have taken place in the altitude of the first star; and from the Nautical Almanac find the difference, if any, in its declination. Let the amount of these changes be applied as a correction to the zenith distance of that star on the first night, and let the difference between this corrected zenith B' Бл D S distance and the observed zenith distance of the same star on the second night be considered as a change in the zenith distance of either star on account of a derangement of the instrument. Let BB" represent this change, and let it be applied by addition or subtraction to the zenith distance (suppose ZB) of the second star on the first night, so that ZB" may be considered as the zenith distance of that star on the second night. Now let ZA' be the zenith distance of the same star when observed by reflexion on the second night; then B'A' will represent the double altitude of that star, and the middle point between B" and A' will be the place of the horizontal point on the circle. Several pairs of stars should be observed in like manner, and a mean of the results taken. N 101. In using the mural circle for the purpose of obtaining by observation the declination of a star, if, when the star is bisected by the horizontal wire it is not on the central or meridional wire, but on one of those which are parallel to it, two corrections will be required: one of them depending on the change in the star's polar distance between the time of the observation, and the time at which the star was, or would be on that central wire; and the other on the distance between the parallel of declination apparently described by the star and the great circle passing through it, of which the horizontal wire is considered as a part. The first correction is readily found from the variation of the declination, or polar distance of the celestial body, given in the Nautical Almanac; and the second may be obtained from the formula 4P2. sin. 2PM sin. 1" (art. 71.), in which P is (in seconds of a degree) the equatorial distance from the middle wire to that at which the observation was made, and PM is the star's north polar distance. 102. If the declination or polar distance of the upper or lower limb of the moon is to be obtained from observation when that limb is not entirely enlightened, there will be required a correction which may be thus determined. Let PM be the meridian, s the sun, c the centre of the moon at the time of culminating, and cs an arc of a great circle passing through the moon and sun: the moon's disk having a crescent form, apa'q, let a be the point which is in contact with the horizontal wire of the telescope at the time of the observation. Imagine am to be a horizontal line passing through a, and let b be the nearest extremity of a vertical diameter of the moon; then bm is the correction required: this is manifestly equal to ca versin. acb, in which expression ca is the semidiameter of the moon, the angle acb is equal to the complement of MCs; and this last angle may be found sufficiently near the truth by means of a common celestial globe. In a similar manner may the correction be found when the moon is oval, or gibbous, as ap' a'q: in that case s' being the supposed place of the sun, the required correction b'm' is equal to ca' versin. a'cb'; and the angle a'cb' is the complement of PCS', which may be found as in the other case by means of a celestial globe. 103. The Greenwich mural instrument consists of two circles which, being on the same horizontal axis, act as counterpoises to each other, and neither plumb-lines nor spirit-levels are employed in their adjustment. They are brought as near as possible to the plane of the meridian, and each is provided with six microscopes nearly equally distant from one another at the circumference. Each also is provided with an artificial horizon of mercury, showing as much as possible of the reflected meridian. The meridian circle, which was made by Reichenbach for the observatory at Gottingen, in 1820, serves at once for a transit instrument, and for measuring altitudes: its telescope is five feet long, and it has, at the focus of the object-glass, seven vertical and two horizontal spider threads. The horizontal axis is about three feet long, and it carries, on one side, two concentric circles, three feet in diameter, whose outer surfaces are nearly in one plane: the exterior circle revolves with the telescope, and carries the graduations; and the interior circle would turn on the same axis, were it not that it may be made immovable by means of a clamp attached to the adjacent pier: on this circle are four indices, with verniers, each of which is at forty-five degrees from a vertical line passing through the centre. A suspended level serves to place the axis in a horizontal position; and the zero of the graduations is at the top, or zenith point. The instrument is capable of being reversed on the axis, in order, by making the observations with it in opposite positions, to eliminate the error of collimation: and in observing the circumpolar stars, the zenith distances, both at the superior and inferior culmination, are taken by direct view, and also by reflexion. 104. The most useful instrument in an observatory which may be established for temporary purposes in connection with astronomy and geodesy is the Azimuth and Altitude Circle, since it possesses the properties of a transit telescope and a mural circle together with those of a theodolite. The altitude circle, which carries a telescope, is capable of being fixed in the plane of the meridian; and when necessary, it can be turned so as to allow the observer to obtain, at the same time, the azimuth and altitude of a celestial body or of a terrestrial object. The annexed figures represent two views of the instrument ; in the first is seen the edge, and in the second the face of the vertical, or altitude, circle: the lower part consists of a circular brass frame ABD, which may be from 20 to 24 inches in diameter, and is capable of having its upper surface made parallel to the horizon by means of three screws, two of which are seen at C and c'; these rest on a firm table, or on the top of a stone pedestal, and within the ring AB is the graduated circle which turns on the vertical axis of the whole instrument. The ring is capable of turning in azimuth as much as 4 or 5 degrees, in order that a telescope E, which is attached to the frame, may be directed accurately to a mark made on some distant object in the direction of the meridian, and it carries two micrometer microscopes, F, G, one at each extremity of a diameter, by which the minutes and seconds in the observed azimuthal angle are read. A vertical pillar H of brass, in which the radii of the horizontal circle are fixed, carries a stage K with four pillars, two of which, as MN, on each side, are connected together by a horizontal bar PQ, and on these |