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Another mode is by calculating the amplitude of the sun at his rising or setting for any day in any latitude, and comparing it with his observed bearing when on the horizon, or rather 34', or about his diameter above it, as his disc is elevated that amount above its true place by refraction.
In the accompanying figure HO is the horizon, P the pole, EQ the equator, PAC the six o'clock hour circle, PEC the meridian, Z the zenith and dd or d'd' the circle of declination of the sun, either north or south of the equator, and supposed to be drawn through his place at the time of sunrise, which is approximately known.
S or S' then, the intersection of this declination circle with the horizon, is the position of the sun at rising, in
the first case before arriving at the 6 o'clock hour circle, and in the second after having past it.
In the triangles ASt or AS't' then,ts or t's' is the sun's declination, and the angle SAt, S'At the co-latitude of the place, from whence we obtain AS, or AS' the amplitude, and also At or Ať, the angular distance before or after 6 o'clock for the time of sunrise. In the same way can be obtained the sun's amplitude at sunset; as also the time, allowing for the change in declination; and if the meridian is to be marked on the ground, it is necessary as before, to observe some object with reference to the magnetic bearing.
A transit instrument placed in the plane of the meridian, of course affords the means of marking out at once a meridian line on the ground; and as a conclusion to these problems a short description, abridged from Dr. Pearson's "Practical Astronomy," is given of the method of adjusting a portable transit approximately in this plane, and of verifying its position when so placed.
1st. The adjustment of the level, and of the axis of the telescope. These two adjustments may be carried on at the
same time, as when the level is made horizontal and parallel to the axis; the axis must be horizontal also. Apply the level to its proper place on the pivots of the axis, and bring it horizontal by the feet screws of the instrument; reverse the level, and mark the difference as shown on the scale attached to it-half this difference must be corrected by the screw of the level, and half by the feet screws, this operation will probably want repeating: if by previous observation, the level has been ascertained to be correct, the feet screws alone must be used in the correction, and if on reversing the instrument in its ys, the level is still correct, the pivots of the axis are of equal size; if not, the instrument should be returned to its maker as imperfect.
2nd. The next object will be to place the spider lines truly vertical, and to determine the equatorial value of their intervals.
Suspend a thick white plumb line on a dark ground, at a distance from the telescope, then the middle wire may be made to coincide with it to insure its verticality, and if a motion in altitude be given to the telescope, and the coincidence continues unaltered by change of elevation, the axis has been truly levelled.
The equatorial value of the intervals between the wires, may be determined by counting the time in seconds and parts occupied by the passage of an equatorial star over all the intervals taken separately and collectively by several repetitions on or near the meridian. If the star observed has any declination, the value of an interval obtained from its passage may be reduced to its equatorial value by multiplying the seconds counted by the cosine of the star's declination; before this method can be used, the telescope must have been placed nearly on the meridian.
3rd. Collimation in azimuth. When the preceding adjustments have been made, the telescope should be directed to a distant object, the middle spider line brought to bisect it, and the axis then turned end for end; if after this reversion, the same point be again besected by the wire, it is a proof that a line passing from the middle spider line to the optical centre of the object glass is at right angles to the axis of the telescope's motion. But if after this reversion of the axis, the visible mark be found on one side of the middle line, half the error thus found must be corrected by the screw which moves the ys in azimuth, and the other half by
the screw for adjusting the wires; several reversions must be made to ensure accuracy. The verification of this adjustment may be proved by the passage of the pole star;-note the time at the preceding and at the middle wire, then reverse the axis, and note the passage over what was the preceding, but is now the following wire; half the difference of the intervals before and after reversion, will show how much the position of the centre wire has been altered by reversion.
4th. Collimation in altitude.—When the telescope is directed to the pole star at the time of its crossing the meridian, or to any well defined distant point by daylight, read the vernier of the altitude circle, while the bubble of the level is at Zero. The axis of the telescope must then be reversed, and the horizontal line again brought to bisect the star, and when the bubble is made to stand at Zero, as before, the reading of the vernier must be again noted; half the sum of these readings will be the true altitude; and half the difference, the error of collimation in altitude. This error may consist of two parts: the spider line may be out of the optical centre of the field of view; and the level (supposing it previously adjusted to reverse properly in position), may not be in its true position as regards the Zero of the circle's divisions; half therefore of the error arising from the half difference of altitudes, must be adjusted by the screws carrying the spider lines, and the other half by the screw that alters the level.
5th. The last and most difficult of all the adjustments, is that by which the instrument is placed in the plane of the meridian of the place of observation. There are many modes of accomplishing this, both by direct and indirect means; but the most convenient and most practised are those in which a circumpolar star is employed; in which two circumpolar stars, differing little in declination, but nearly twelve hours in right ascension, are successively observed; or in which two stars, differing considerably in altitude, and but little in right ascension are observed; but in whatever way the adjustment be made, the clock that gives the times must have its rate previously well determined.
The approximate position of the instrument may be ascertained by calculating the solar time of the pole star's passage over the meridian for any given day, and then the telescope levelled and
pointed at it at the computed time will require but little adjustment. Subsequent observations of circumpolar, or of high and low stars, will gradually rectify the position, provided all the adjustments previously directed continue unaltered for a sufficient length of time, and a meridian mark, capable of adjustment, may be placed at a convenient distance north or south, until their places are definitely fixed by some of the following methods:-At 95.49 yards from the object end of the telescope, one inch will subtend l' or 60", and a scale may be made accordingly, varying of course inversely as the distances; so that when the transit is found to be any number of seconds, say thirty, too much to the east or west, a corresponding distance on the scale shows how much the instrument is to be moved in azimuth, by the proper screws, to effect the correction required.
Method 1st. By a circumpolar star.
azimuthal deviation in seconds at the horizon,
t = the time at upper transit,
t' = at lower passage,
L= the latitude,
and by multiplying by 15, a is converted into space if required.
If the western semicircle is passed through in less time than the eastern, the object end of the telescope points to the west of the true meridian. The clock must be a good one for this method, as it supposes no change of rate for twelve hours.
Method 2nd. By a pair of circumpolar stars.
(t—t'—12h)—(T—T-12h) sin ▲ sin ▲'.
where ▲ and A', the star's polar distances, L = the latitude, t and t the times of the first star's upper and lower passages, T and T' the times of the contrary passages of the second star, following the other at an interval of nearly 12 hours in right ascension or this formula, omitting the 12 hours.
when (t-t-12h) is a greater interval than (r—r—12") the hori
zontal deviation a will be towards the east, and vice versa ; or when (tr) is greater than (t—r) the deviation is also to the east. Method 3rd. By high and low stars.
Where D = (t-t') the difference of the observed times of passage, and D= (Ra—Ra') the difference of the apparent right ascension of the two given stars, & the declination of the higher star, and & that of the lower; the stars for this method ought to be removed from each other at least 40° in declination. When (D-D') is positive, the horizontal deviation is to the east of the south point in northern latitudes, and the contrary when negative. Tables are formed to facilitate the computation of the above formulæ. The times are all supposed to be sidereal; if, therefore, solar time is used in the observations, the acceleration must be added.
The following example is given of the last method, in which, if the difference of the times of the observed passages be exactly equal to the difference of the computed right ascensions of the two stars, the instrument will necessarily be already in the plane of the meridian.
On June 20, 1838, in latitude 51° 23′ 40", the transits of a Corona Borealis, and of Antares, were observed at