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easy and accurate determination of the difference of meridians *; of which the following example is sufficiently explanatory.
At Chatham, March 9, 1838, the transit of a Leonis was observed by chronometer at 10h 52m 463, and of the moon's bright limb, at 10h 20m 7; the gaining rate of chronometer being 1*5.
Difference of sidereal time between the intervals Due to change in time of moon's semidiameter passing the meridian
Difference in D's right ascension
0 27 30.58
0 27 25.96
+0 0 ⚫01
0 0 4.63
The variation of D's right ascension in 1 hour of terrestrial longitude is, by the Nautical Almanac, 112.77 seconds. Therefore as 112-77s. 1h:: 4.63: 147·80,= 2′ 27′′-8, the difference of longitude.
But when the difference of longitude is considerable, instead of using the figures given in the list of moon-culminating stars for the
* For a more rigid method of computing the difference of meridians by lunar transits, see Baily's Formulæ and Problems, pp. 239 to 247.
variation of the moon's right ascension in one hour of longitude, the right ascension of her centre at the time of observation should be found, by adding to, or subtracting from the right ascension of her bright limb at the time of Greenwich transit, the observed change of interval, and the sidereal time in which her semidiameter passes the meridian. The Greenwich mean time corresponding to such right ascension being then taken from the Nautical Almanac, and converted into sidereal time, will give, by its difference from the observed right ascension, the difference of longitude required. For instance, in the above example :—
▷ Right ascension at Greenwich transit Sidereal time of semidiameter passing meridian of place
▷ Right ascension at Greenwich transit Observed difference
▷ Right ascension at the time, and sidereal time at the place, of observation
Greenwich mean time correspond- H.
ing to the above right ascension. 11 17 0.5
Or sidereal time at Greenwich 10 25 46.5
Difference of longitude
0 2 27.9
BY OCCULTATIONS OF FIXED STARS BY THE MOON.
The rigidly-accurate mode of finding the longitude from the occultation of a fixed star by the moon, involves a long and intricate calculation, an example of which will be found in the 37th chapter of Woodhouse's "Astronomy:" and the different methods of calculating occultations, are analyzed at length by Dr. Pearson in his "Practical Astronomy," commencing at page 600, v. ii.
The following rule, however, taken from Riddle's "Navigation," will give the longitude very nearly, without entering into so long a computation :
Find the Greenwich mean time from knowing the local time
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 1; the moon's polar distance m; her reduced horizontal parallax H; and her semidiameter s.
Then add together sec
1 + P 1~P, and cot 2?
sum, rejecting twenty, will be the tangent of arc a, of the same
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, ab
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 e'. 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±gh, 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.
TO DETERMINE THE DIRECTION OF A MERIDIAN LINE* AND
THE VARIATION OF THE COMPASS.
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
* 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.
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 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.-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.