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By the Nautical Almanac, it appears that the Greenwich mean time answering to this distance, must be between 9 P.M. and midnight-the difference of distance answering to this interval of 3 hours, being

1° 28' 52" Prop. log. 3065* Lunar dist, at 9 P.M. Greenwich 32 3 55 Corrected distance found above 31 16 34

47 21

Prop. log. 5800

Interval of time past 9

2735

1 35 54 9 0 0

.

Greenwich mean time.

10 35 54 Mean time at place of observation 10 38 11.8 Longitude east

0 2 17.8

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The difference between the prop. log. at 9 and midnight being 0, the correction of 2nd differences is nothing.

Mr. Baily's formula for a lunar observation for longitude is as follows:

x the true lunar distance required,

* The interval of time past 9 P.M. might of course have been found by a common proportion, without the aid of prop. logarithms.

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Make B = {(A + H + h)

cos H' cos h'
{(cos B cos B)
a)

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cos H cos h sin a =

cos ž (H' + h) then sin ļ x = cos } (H' + h) cos a.

The following example will also show the method of working out a lunar observation, by Dr. Young's formula, all the terms of which are cosines :

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By Dr. Young's formula, Cos M’S' = {2 cos } (MH + SK + MS) cos, 2 (M HUSK – MS) cos M’H cos SK }

MH cos

cos (M'H + S'K).

MS = 95 50 53
MH = 35 45 4 ar, comp. cos 0·090678
SK =

7 48 1 ar. comp. cos 0:004037

139 23 58

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nat. cos M’S 95° 44' 31"

0.100047 the true lunar distance.

The same example, by Mr. Riddle's first method, which will be found in his “Navigation,” gives 95° 44' 29" for the corrected lunar distance.

By Mrs. Taylor's method, which requires the use of her“ Tables," the true distance is obtained as follows: Table 1 o 1.3873

7533 Table 2

0:5077

- 1'4997

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The apparent altitudes and distance are first obtained from those observed, by correcting them for semidiameter and dip if necessary. Then in Table 1 find the log of the corrections for the altitudes on account of the moon's parallax.

Trom Table 2 take the logs of the effect of the moon's horizontal parallax upon the distance.

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Table 3 gives the minutes and seconds answering to these logarithms.

From Table 4, find the effect of the refractions of both objects on the observed distance.

And from Table 5, if the sun is one of the objects observed, the effect of his parallax.

These corrections, applied, with their proper signs, to the apparent distance, give the true distance as above.

Mr. Airy makes the following remarks upon the effect of errors of observation in taking lunar distances and lunar transits. A certain error of time produces that same error in the deduced longitude; and an error in the measure of one second produces about two seconds of time in the longitude.

An error of one second of time in a lunar transit produces about 30 seconds error in the longitude.

An error of one second of time in a lunar zenith distance will produce at least 30 seconds of time error in longitude-sometimes considerably more. An error of one second in zenith distances produces at least two seconds of time in longitude. An error of one second of time in an occultation produces one second of time in the longitude.

The same with eclipses of Jupiter's satellites.

Instead of measuring the distance between the moon and a star, for a comparison with the time at which the same distance is obtained by calculation for the meridian of Greenwich ; altitudes may be taken simultaneously of the moon and a star, from the latter of which, its right ascension and declination being accurately known, the right ascension of the meridian can be computed. This right ascension applied to the moon's distance from the meridian (the angle P in the astronomical triangle) gives the right ascension of the moon, to be compared with the time at Greenwich at which it is identical, for the difference of longitudes.

Another method, applicable particularly to low latitudes *, is to select, when the moon is on or near the prime vertical, any star whose right ascension and declination are known; it being at the time within 8° or 10° of the zenith.

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* Obtained from Mr. E. K. Horn.

or

Take the distance between this star and the moon; also the moon's altitude, and apply the moon's correction in altitude with a contrary sign as the correction in distance; then, with the corrected distance as a base, and the co-declinations as containing sides, the difference of right ascension, and consequently the moon's right ascension, and Greenwich time, are found.

If a star answering to the above conditions is not available, select any star having the same or nearly the same azimuth as the moon, and not less than 30° or 40° distant; the sum difference of the corrections in altitude would then evidently be the correction in distance. If the star happened to be one of those given in the lunar distance, the Greenwich time is at once found ; if not, with the corrected distance as a base, the problem is worked out as before.

The objection to both these methods is, that the moon's declination is required to be known accurately as an important part of the data, to compute which, it is necessary to know the longitude correctly (the very thing sought), except in cases where the moon's declination on either side of the equinoctial is nearly a maximum, and consequently for some time comparatively stationary. Under these circumstances a good result may be expected from the last method when the moon is on, or nearly on, the prime vertical.

,

.

BY THE METHOD OF MOON CULMINATING STARS.

The proper motion of the moon causing a difference in the interval of time between her transit, and that of any star, over different meridians, affords another method of determining the longitude *. The times of transit (or apparent right ascension) of the moon's enlightened edge, and that of certain stars varying but little from her in declination, are calculated for Greenwich mean time, and given among the last tables in the Nautical Almanac. The transits of the moon's limb, and of one or more of these stars, are observed at the place whose longitude is required, and from the comparison of the differences of the intervals of time, results a most

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* The time of the moon's transit compared with that observed at, or calculated for, another meridian, would be sufficient data for ascertaining differences of longitude ; but by making a fixed star the point of comparison, we obviate any error in the position of the instrument, and also of the clock.

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