In fixed observatories the meridian error is obtained when possible from consecutive transits of ' Polaris above and below pole. The observations are corrected for collimation and inclination, and for the rate of the clock. If the times of passage thus corrected differ twelve hours, there Is no meridian error ; but if the difference is greater or less than twelve hours, the deviation may be computed thus : Let a and s' be observed times of upper and lower culmination, corrected for collimation, inclination, and rate: let a and a' be the apparent right ascension of Polaris, taken from the Nautical Almanac ; e the deviation as before ; –p and p' the values of for Polaris cos a above and below pole, which have different signs. Then, exactly as before, a– p x and a'+p'x will be the corrected times of transit, and (s'+p' x) – (a p x) s• a' – a, (a' – a) – (s' – a) ors= + r • When three consecutive transits are observed, there is no need of taking any account of the rate of the clock (which is supposed to go uniformly), or the change in right ascension of Polaris in the interval ; only a mean is taken of the first and third transits, which is compared with the second. The difference between these, divided by p' +p, gives the value of as The sign may be made out by seeing whether tho passage from upper to lower culmination is too small, and from lower to upper culmination too large, when the deviation is to the west of north, and therefore to the cast of south, and the correction is to be added, or rend vice, when it is to be eubtracted.t In well-regulated observa tories Polaris is always observed when it is convenient, and often when it is not ; and the right ascension of the star, as well as the position of the instrument, is deduced from these double transits. When the right ascension is thus perfectly known, single transits of Polaris may be safely used in combination with other stars, to determine the azimuthal error for those times of the year when the star at one of its culminations passes altogether unseasonably. It is assumed, in this method, that the position of the instrument is the same for twenty. four hours; or that it changes uniformly.
If the reader has fully understood what precedes, he will have no difficulty in comprehending the mode of observing and reducing the observations which is followed at Greenwich and our principal observatories; but the private observer cannot always command a site which is wholly to be relied upon, and very seldom can afford the time for such an uninterrupted aeries of observations as is required to give full effect to the system just described. The precautions to be taken will depend mainly upon the objects he has in view, but generally speaking, the private transit observer will do wisely to take the places of the principal fixed stars from the lists furnished by Greenwich, though he may subsequently modify the values enter se by his own observations. Having done this, it will be easy for him to fix any other object with perfect accuracy, or to determine his time most scrupulously, although the steadiness of his transit is not beyond suspicion, and his avocations or inclination should only allow him to observe by fits and starts. We shall proceed to show how he may pro ceed under different hypotheses.
If he has a distant mark ors collimating telescope [etc COLLIMATOR] with a micrometer to his transit, he proceeds exactly as we have described above to measure the quantity and direction of the collima tion, the quantity and direction of the inclination, and then observes away, only taking care to note as many standard stars as will give him a correct clock-error, and at least some so far apart as will enable him to detect his azimuthal error. If he observe one or two stars near the pole, so much the better. Now calling e the correction for the clock at the mean of the time of the observations of standard stars ; a, a' a", &c., as before the observed times of transit, corrected for collimation and level as aforesaid; p, p', p", &c., with their proper signs, the correcting factors of the unknown azimuth x; and a, a', a", &c., the apparent right ascensions of the standard stars, he has the following series of simple equations :— Group together the equations in which the coefficient of x is nearly of the same magnitude and with the same sign,* and dividing each group by the number of its component parts, so as to have e with unity as a coefficient, form at least two equations in which the coefficients of x differ considerably, and subtracting one from the other, a value of x will be found with its proper sign. Substituting this value of x in each of the equations e = a — a—px, you will have as many values of t as you have equations ; and taking the mean of those which are derived from the quick-moving stars, you have a good clock correction. Now if any other objects have been determined, the right ascension of which is required, add (speaking algebraically) the sum of all the corrections for collimation, inclination, declination, and clock-error to the observed transit, and you will Iowa the apparent right ascension.
But it most frequently happens that the observer has no collimating mark and no micrometer to his instrument ; nay, he may only have a view out of a window t which commands no distant or distinct object, and not be able to see even the zenith, much less the polo. This last is the greatest objection; for the accuracy of the meridian adjustment depends chiefly upon getting stars near the pole. To detect the error of collimation the observer must proceed thus :—After having carefully determined the inclination of the axis, he observes as many well-known stars as he can, especially getting them as high and as low as possible for ascertaining azimuthal error. He must then reverse the axis, repeating the measurement of the inclination, but by no means touching the elevating screw of the instrument, and make a similar act of observations. A series of equations must be formed for each position of the instrument, which will be of the following form : supposing s, a' a", to be the observed transits of the stars corrected for inclination only ; q, q' q", to be the values of the secant of declination for each star respectively ; c, the unknown value of the collimation error in the first position (which becomes—c when the instrument is reversed); x, the error of deviation; and e the clock correction,— re vcrseu.