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and the approximate longitude, and for that time take, with the greatest exactness, from the Nautical Almanac the sun's right ascension, and the moon's polar distance, semidiameter, and parallax, applying all corrections.
To the apparent time, add the sun's right ascension, and the difference between this sum, and the star's right ascension, will be the meridian distance of the latter. Call this distance P; the star's polar distance p; its right ascension R; the reduced co-latitude 7; the moon's polar distance m; her reduced horizontal parallax H; and her semidiameter 8.
I+P, cos~P, and cot
Then add together sec 2 2 2' sum, rejecting twenty, will be the tangent of arc a, of the same 1 + P. affection as 2
Add together cosec
1 + P. sin 2
and the sum,
rejecting twenty, will be the tan of arc b (always acute). When I is greater than p, a + b = arc c; and when l is less than p, a barc c.
sin~, and cot
Add together tan c, cosec l, cosec P, and prop. log H, and the sum, rejecting the tens, is prop. log of arc d. When arc c is obtuse, pd arce; and when c is acute, p darce.
Add together cosec l, cosec P, prop. log H; and with the sum S, and p, take the correction from the subjoined table, and applying it with its proper sign to e, call the sum or the remainder é. The difference of m and e' is arc f.
To S add sin e', and the sum, rejecting the tens, is the prop. log of arc g.
To the prop. logs of s+f, and sf, add twice the sine of arc e, and half the sum, rejecting the tens, is the prop. log. of arc h.
Then the moon's right ascension = R±g±h, where g is additive west of the meridian, and subtractive east; and h is additive at an emersion, and subtractive at an immersion.
Having found the moon's right ascension, the corresponding Greenwich time is to be found from the Nautical Almanac, the comparison of which with the local time gives the longitude of the place of observation.
In the spherical triangle Z PS, already alluded to as the astronomical triangle; and in which the co-latitude Z P, and the time represented by the angle P, were ascertained by the method of absolute altitudes in pages 191 and 195; the azimuth of any celestial body S is measured by the angle Z, which is found from knowing either the time, or
Star's Polar Distance p.
TO DETERMINE THE DIRECTION OF A MERIDIAN LINE* AND
THE VARIATION OF THE COMPASS.
* The method of ascertaining the direction of the meridian with an altitude and azimuth instrument, or a large theodolite, has been already described at page 155.
the latitude, in addition to the observed altitude. This calculated azimuth compared with the magnetic bearing of the object observed at the same instant, and determined with reference to some well-defined terrestrial mark, affords the means of laying down a meridian line, and gives the variation of the compass.
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 when he is 34 minutes, or about his own 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, d 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 passed 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 At, 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.—If the meridian is to be marked on the ground, it is necessary, as before stated, 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; the following short description, abridged from Dr. Pearson's "Practical Astronomy," explains 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 footscrews 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 footscrews, which operation will probably want repeating—if by previous observation, the level has been ascertained to be correct, the foot-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
* In an Observatory, the principal uses to which a transit is applied, are the obtaining true time, and the determination of right ascensions-very excellent directions for using and adjusting a portable transit for the determination of longitudes, &c., drawn up by Mr. Airy, will be found in the Narrative of the North American Boundary, by Major Robinson, from which one example is given at the end of this chapter, to show the form there adopted for recording transit observations.
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 bisected 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.