have the value of es, and that with great exactness, if the stars have been well selected, aro pretty numerous, and have been tolerably observed. To reduce the transits of the other objects observed to apparent R. A. nothing more is required than to add ra + n tan 8 to the observed transit, which, besides being a good deal shorter than the method previously described, only requires a table of natural tangents for computing the corrections.
If the observations are made on or near the pole, where the see 8 varies almost as rapidly as the tangent, a sensible error of collimation would mix itself up in the value of tan 8. If the pivots of the instal. went are exactly equal, two series might be observed in reversed positions of the axis, and as re and a would have the same value in each, while the sign of collimation changes, the determination of the latter would present no difficulty. We have used another plan, which in steady weather, when observations can be made on consecutive nights and in large masses, in perhaps the best for cataloguing. On the first night observe forty or fifty stars, consisting of all the standards which peas and these stars the places of which you wish to determine. These may now lie scattered all over the heavens, so far as the method is concerned. On the following evening reverse the instrument and observe the same stars. The first night, each observed star should be increased by m + a tan 8 + t see 8, while the second night the correc tion is ve + n tan 8—e sec 8, the collimation being supposed (as is found to be the case) to be invariable in a well-made instrument,unless violence is used. Now if wo add tho observed transits of each star on the two nights together, and take a mean, the result requires a correction of In2 2 tan 8; and the collimation is eliminated.
This new correction for the mean of the two nights is exactly of the same form as the original correction, call it at + s tan 8, and sr and are found by comparing the observed places of the standard stars with their known computed places, just as before. If the clock rate is sensible, the values of an preceding and succeeding the mean of the standard stars, must receive a proportional correction, but it is easy to make the rate of a good clock so small that in ordinary circumstances this may be neglected. The rate may be determined near enough by observing the same high star both nights in the same position of the axis, and measuring the inclination by the level. From some trials of this method, we should strongly recommend it in a steady climate and where a large catalogue of stars is to be formed. An error of 0'.I
would, we are convinced, be very rarely found in the n. A. of stars so determined within 40° of the equinoctial; the computations are very abort and can scarcely be wrongly made, and there is only one com putation of a mean place for two complete observations of an apparent place. Writing the separate results under each other, is an excel lent check against those provoking small errors which, when they once get admittance, are so difficult of detection. It is a drawback that the computation does not furnish the absolute time or clock error, if that should be required for other purposes, without further calculation.
In this climate it frequently happens that the star is visible during only a portion of its passage over the wires, or the observer may lose some of the wires ; hence it is necessary to have sonic means of coin pleting the imperfect transits, and ascertaining so far as possible at what time the star would have passed the mean of the wires, if all could have been observed. For this purpose a sufficient number of complete and satisfactory observations is selected (suppose the illumi nated end to be west), and the difference taken between each wire, and the mean of the wiree. Multiplying these numbers respectively by the casino of the corresponding star's declination, we have the differ ences, such as they would have been if the stars observed had been in the equinoctial. A mean of these is taken for each difference, and the proper sign affixed. If a second series be selected of observations made when the illuminated end is cast, and be treated similarly, nearly the same values will be fouiid as before, but in reversed order, and with different signs. A mean is taken when the number is sufficient to give a satisfactory result ; and the logarithms of the intervals be tween each wire and the mean, for an equinoctial star, are set down for future use, discriminating whether the instrument is Illuminated end E. or W. Now suppose a broken set of wires is to be made up : take the logarithms with the proper sign corresponding to each wire, add to each logarithm the log. secant of the star's declination, and take the natural numbers corresponding to the logarithms thus found, and you have the number of seconds and decimals of a second which are to be added (algebraically) to the observation of each wire to reduce it to the mean wire. Of these partial results, a final mean is taken. The number,' for reducing each wire to the mean wire are found in the introductions to all the modern observations.