star by 8, multiplying the means of the proportion and dividing by the last term, we have may be found as at p. 150, by supposing s to be on the meridian, without sensible error, by means of the latitude of the station and declination of the star. Represent the fractional part of the above formula by n. It becomes Suppose now another star crossing the meridian nearly at the same time. For this we have Let now t represent the observed time of transit of the later star, a its right ascension, t' and a' the same for the other star, and let e denote the error of the clock which may be supposed unknown. For the former star (since a is the time of its passing the meridian), we shall have for the value of the hour angle when it makes the transit of the middle wire, By subtraction of (5) from (4) the error of the clock e is eliminated, and there results Substituting for P and P' in (6) their values given by (2) and (3), (6) becomes The value of the azimuth error or deviation from the meridian z is thus found in time. To convert it into space this value must be multi In the later catalogues of stars, their north polar distances (N.P.D.) are given instead of their declinations. In the above formula sin (N.P.D.) would of course be in place of cos d. When the declination of the star is south, we have still sin PS= cos ¿, for sin (90° + d) = sin (90° — ¿) = cos d. (See App. Art. 13.) plied by 15. The instrument may then be adjusted to the meridian by turning the screw of the y, which admits of horizontal motion, and which has usually upon it a graduated arc, by means of which the movement of the instrument in azimuth is indicated. To compute the effect of the azimuth error upon the time of transit take the value of z as given by (7) in time, and substitute it before converting it into space, in either (2) or (3), which will give the value of P or p' the hour angle in time, n or n' being an abstract number expressing the ratio of two trigonometrical lines. The value of P is the correction to be applied to the observed time of transit of the later star, to obtain the time of its meridian transit. That of p' the same for the other star. If the deviation from the meridian be southwest and northeast, as in the diagram* where we have supposed the zenith to be south of the pole, the correction for a star south of the zenith will be subtractive, for one north additive, unless the latter make an inferior transit or sub polo, in which case the motion being in the opposite direction, the correction is subtractive. This is evident from an inspection of the diagram. The only remaining correction is for error of collimation. The line of collimation when this error exists describes a cone about the supporting axis as an axis, and the point in which it pierces the surface of the celestial sphere, describes a small circle of that sphere parallel to the meridian, and at a very short distance from it. The distances between these circles measured on parallels of declination may be considered every where the same without sensible error, and the time of traversing this distance by any star will be inversely as the cosine of the star's declination. The equatorial interval between the circles in question may be found by moving the instrument in azimuth, after reversing, by means of the screw in that y which gives horizontal motion, till the intersection of the wires is brought back to the terrestrial point on which it was placed before reversing. The degrees and fractions of a degree passed over on the graduated arc on the y will indicate double the error of collimation, which, divided * Which will evidently be indicated by the value of z being positive. If the deviation be s.E. and N.w. then z is negative. This may be seen by trying various cases by the diagram, such as one star passing, 1st, N. of the zenith, 2nd, sub polo, &c., first writing the numerator in the value of z in (7) under the form (t- a) — (t' — a'). The only difference in the above diagram for a star north of the zenith would be that the angle of deviation itself instead of its supplement would be the angle of the triangle, but the proportion would be the same. by 15, will give the equatorial value of it in time. This multiplied by the cosine of any star's declination will give the effect of the error of collimation on the time of the star's transit. The error of collimation is best measured by means of a movable vertical wire, to which motion is given by a micrometer screw, as described in another place. Should no distant terrestrial object be visible from an observatory, owing to intervening objects near at hand, a small telescope in the building having its object glass turned towards that of the transit instrument may serve as a collimator. The rays of light proceeding from the wires at the focus of the object glass of the small telescope strike this object glass, are refracted by it, and emerge in parallel lines; they then strike the object glass of the transit instrument, and are conveyed to the focus of parallel rays, which is the astronomical focus; so that in looking through the eye end of the transit instrument the wires of the small telescope will be distinctly seen. Care should be taken to throw the light of a window or lamp in at the eye end of the small telescope. A similar contrivance may be employed for a meridian mark. But the transit instrument may be made its own collimator, by placing a vessel of mercury underneath, and turning the object end of the telescope downwards. If the axis be horizontal, and the instrument truly collimated, the wires being illuminated by an orifice in the side of the eye piece, the rays of light will pass from them to theobject glass, emerge in parallel lines, strike the surface of the mercury vertically, be reflected back in the same lines, and converge to the focus of the object glass at the same points which they left, so that the reflected image of the wires will be seen coinciding with the direct image. If not, there is either error of collimation or of level, or both. If the axis had previously been made horizontal by the striding level, it is the latter, and the diaphragm containing the wires must be moved till there is coincidence between their direct and reflected images; or a movable wire may serve to measure the interval between them. This interval is double the collimation error, because the angle of incidence is equal to the angle of reflection, the former being on one side the vertical, the latter on the other. If, therefore, the direct image of the wire be brought to the vertical by the screws of the diaphragm by a movement over half the distance between the direct and reflected image, the reflected image. will be brought there too. The striding level need not be used at all, if the instrument be reversed in the y3, in using the collimating eye piece with a basin of mercury; for in one position of the instrument the angle obtained by taking half the distance between the direct and reflected image of the wires is the sum, and in the reverse position is the difference of level error and error of collimation. The well-known algebraic formula, "to half the sum add half the difference for the greater of the two quantities, and from half the sum subtract half the difference for the less," will serve to determine those two errors separately. To know which is the greater, the level or collimation error, we have this rule:-If the reflected image in both positions appears on the same side of the direct, then the level error is the greater of the two, but if on different sides, the collimation error is the greater. All this will appear evident if the student make a diagram with a line to represent the supporting axis with level error exaggerated, a line perpendicular to this at the middle, to represent true line of collimation, another line from the same middle point oblique to represent the erroneous line of collimation, a horizontal line below to represent the surface of the mercury, and from the point where the erroneous line of collimation meets it, a vertical and also a line making the same angle with it as does the erroneous line of collimation. In reversing the instrument the only change will be in the erroneous line of collimation, which will now make the same angle on the other side of the true. The following example will serve to illustrate all the foregoing rules for applying the corrections to an observation with the transit instrument. |