time of its passage across that field. It is scarcely necessary to remark, that the wires in the eye-piece of the telescope must be adjusted by drawing them towards or from the eye till they appear well defined; and till, at the same time, a star which may be observed near one of them does not change its position on moving the eye towards the right or left hand. The coincidence of the horizontal wire with a diameter passing through the optical axis of the telescope, is of small importance when it is not intended to use the transit instrument for the purpose of obtaining altitudes; but, in the event of this being one of the objects contemplated, the adjustment must be carefully made, or the error accurately determined; the process which is to be employed will be explained in the description of the altitude and azimuth circle (art. 106.). 90. The advantage of having one wire or more parallel to, and on each side of that which is in the plane of the meridian, is, that the unavoidable inaccuracy in estimating the time at which a star, on its transit, appears to be bisected by a wire, may be almost wholly corrected by using a mean of the times at which the bisections take place on all the wires. If the distances between the parallel wires were precisely equal to one another, an arithmetical mean of the times (the sum of all the times divided by the number of wires) might be considered as the correct time of the transit at the middle wire; but, on account of the inequalities of those distances, a mean of the times at which any star appears to be bisected by the several wires is to be taken for the time of the transit at an imaginary wire situated near the central wire; and the differences between the times of the transit at the several wires and at this imaginary wire may then be taken. These differences being multiplied by the cosine of the star's declination (art. 70.), will give the corresponding distance for a star supposed to be in the plane of the equator. 91. In a small transit telescope having five wires, it was found, for a star supposed to be in the plane of the equator, that the differences between the times of the transit at each wire, and at the imaginary mean wire, were as follow First wire Second Third + 51".601 + 25".590 + 0".110 Fourth 25".890 51.410 When, therefore, the transit of a star has not been observed at all the wires of a telescope; if it be required to obtain, from such observations as have been made, the time of transit at the imaginary mean wire, one of the following processes may be used: - From a table formed as above, take the number corresponding to each wire at which the transit has been observed, divide it by the cosine of the star's declination, and add the quotient, according to its sign, to the observed time of transit; the result is the time of transit at the mean wire : then a mean of these separate means will be the required time of transit at the imaginary mean wire. Or, from a table as above, take the number corresponding to each wire at which the observation has been made; then take a mean of these, and divide it by the cosine of the star's declination. Take a mean of the observed times of transit, and add to it, according to its sign, the quotient just found; the result will be the time of transit at the imaginary mean wire. It must be observed, however, that the signs of the numbers in the table formed as above are adapted to the culminations of stars above the pole: those signs, and also the order of the wires, must be reversed when the transits are observed below the pole. 92. The equatorial differences in a table such as that which has been given above, is generally found from the observed transits of stars near the pole, as a Polaris and 8 Ursæ Minoris; and these stars describing very small circles about the pole, their paths are sensibly curved in the passage from the first to the last wire: therefore, though the intervals between the wires were equal, the observed times of passing such equal intervals would not be equal. Thus let abc be part of the arc described by a star in consequence of the diurnal rotation, and a, b, c, places of the star when bisected by three of the wires, of which let the straight line drawn through c be the middle wire: " it is manifest (the intervals am, bn, perpendicular to the middle wire being the distances of the wires at a and b from the mean wire) that the observed intervals between the time of transit at a and c, b and c must be reduced in the ratio of the arcs ac, be to the lines am, 15m.π bn; that is, of 10800 a с n m = 0.00436 m) to sin. 15m, m being the minutes of time in the observed intervals between the transits, and the half circumference of a circle whose radius is unity. 93. In order to obtain the time of transit at the mean wire, when a planet is observed, the number given in the above table, after being divided by the cosine of the planet's declination, must be increased or diminished by the product of that quotient multiplied by the increase or diminution of the planet's right ascension (in time) during one second of time, as a correction on account of the variation in the planet's right ascension while it is passing between the wire at which the transit was observed and the mean wire. When the moon is observed, the number in the table, after being divided by the cosine of her declination, must, in like manner, be increased on account of the increase of her right ascension in such interval of time: but the time in which this celestial body passes between two wires in a telescope is further affected by the small variation of her parallax in that time; and a correction on this account will be presently given (art. 161.). The variations of the right ascensions may be taken from the Nautical Almanac; for the moon, in the pages containing the moon's daily right ascensions and declinations; and for a planet, in the columns of geocentric right ascensions. 94. When it is required to place a transit telescope in the plane of the meridian, an approximate knowledge of the position of a line on the ground in a north and south direction must first be obtained: this may be done in various ways, and one of the most simple depends on observing a star through the telescope of a well-adjusted theodolite at the instants when, in the course of the same night, it has equal altitudes; for the arc apparently described by the star above the horizon being a segment of a circle, if a picket be driven into the ground in the direction of the telescope when it is made to bisect the horizontal angle between its positions at the two times of observation, a line traced on the ground from the picket to a point vertically under the centre of the theodolite will be nearly in the direction of a meridian line. This method, or one similar to it, must be put in practice when the observer is unacquainted with the longitude of his station, and when his watch is not regulated so as to show the time of an observation correctly, or when, though possessing a surveying compass, he may not know the declination of the needle. Little accuracy can be expected in a first operation, but the process may be repeated several times during one night with the same star, and a mean of the bisecting lines will be very nearly in the true direction of the meridian. 95. After a transit telescope has thus been set up very nearly due north and south, when also the horizontality of the axis of motion, the correctness of the line of collimation, and the intervals between the wires at the focus of the object-glass have been ascertained, it is necessary to have the means of making the optical axis of the telescope describe accurately a great circle of the sphere coincident with the plane of the meridian, and also of determining the amount of any accidental deviation of the axis from that plane. Two methods are commonly employed for this purpose, it being supposed that the azimuthal deviation does not exceed a few seconds of a degree: one of them depends on the observed transits of two stars which differ considerably in altitude, and the other upon the observed transits of a circumpolar star above and below the pole. P With respect to the first method, let a hemisphere of the heavens be projected on the plane of the observer's horizon, and let that horizon be represented by the circle AW ME, of which the centre z is the projection of the zenith. Draw the diameter P Z M for the w meridian, in which let p be the pole of the equator, and let AZA' be the projection of the vertical circle in whose plane the optical axis of the telescope moves: let Р S A/ M E also s be the place of a star when it is seen in the telescope, and through it draw the horary circle PS; then (art. 61.) we have in the triangle P Z S, or, since the angle at P and the supplement of the angle at z are very small, we have, employing the number of seconds of a degree in P and z instead of the sines of those angles, and, in order that the angle P may be expressed in seconds of sidereal time, the second member of the equation must be divided by 15. The formula may be represented by P = Zn for the first of two stars s and s', which enter the telescope, and by P'zn' for the second. Now, for the first star, let t be the time of the transit, observed on a sidereal clock, T the true time, or the rightascension of the star, given in the Nautical Almanac; and for the second, let and T' be the corresponding times. Let e represent the error (supposed to be unknown) of the clock by which the times were observed: then and, t-e-T (P, in time) = Z., hence (ť — T′) — (t—T)=z (n'—n), and z —(t'—r')—(t—T) = n'- -n Thus the azimuthal deviation z (=A' Z M) is found. In using the formula, a deviation of the telescope, that is, of the circle AZA', towards the south-west and north-east of the true meridian (as in the figure) is indicated by the value of z being positive and a deviation towards the south-east and northwest by z being negative. This rule holds good whether the upper or the lower star culminate first, and whatever be the positions of the two stars with respect to the zenith and the pole. The value of z thus determined is expressed in seconds of sidereal time, and it must be multiplied by 15 in order to reduce it to seconds of a degree. It may be observed also, that if the star pass the true meridian (as in the figure) before it is seen in the telescope, the value of z n or of z n' (the correction of the time of transit) must be subtracted from the observed time in order to give the time of the transit over the meridian PZ. On the contrary, the correction must be added if the star is seen in the telescope before it comes to the true meridian. The difference between this time of transit over the meridian, and the calculated right ascension of the star, in the Nautical Almanac, is the error of the sidereal clock. Note. In the second equation for P, the lower sign is to be used when the star comes to the meridian below the pole. 96. For the second method, let a hemisphere be projected as before, and let ss' be the places of a star at the times of its observed transits above and below the pole P. Then the angle SPS', expressed in time, measures the least of the two intervals of time between the transits, and, letting fall Pt perpendicularly on the vertical circle zss', the angle SPt will be equal to half that interval, which consequently is known from the observed transits. Now (art. 62. (d')) we have in the right-angled spherical triangle Pst, 7. cos. PS cotan. tPS cotan. Pst. But tps being nearly equal to a right angle, we may, for cotan. tp s, put the S number of seconds in the complement (to 6 hours) of half the least interval of time; let this number, after being multiplied by 15, be represented by D: also the angle Pst being very small, we may write for its cotangent, that is, for COS. PSt sin. Psť the term (Pst being expressed in seconds of a degree), 1 Pst D and then the above equation will become cos. PS = or Psť Again, in the spherical triangle PZS, we |