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By eclipses, moon's transit.
ascertained, from one place to the other; and if the chronometer did not change its rate during the passage, this method would be perfectly accurate.
Fourth Mcthod. By an eclipse of one of Jupiter's satel
lites. [B. p. 252.]
The signal of the second method cannot be used, when the places are more than 20 or 30 miles apart; and, when the distance is very great, a celestial signal must be used, such as the immersion or emersion of one of Jupiter's satellites. For this purpose, the instant when any such event would happen to an observer at Greenwich is inserted in the Nautical Almanac; and the observer at any other place has only to compare the time of his observation with that of the Almanac to obtain his longitude from Greenwich.
Fifth Method. By an eclipse of the moon. [B. p. 253.]
The beginning or ending of an eclipse of the moon may also be substituted for the signal of the second method to determine the difference of time.
Sixth Method. By a meridian transit of the moon.
[B. p. 431.]
The motion of the moon is so rapid, that the instant of its arrival at a given place in the heavens may be used for the signal. Of the elements of its position its right ascension is changing moșt rapidly, and this element is easily determined at the instant of its passage over the meridian by the difference of time between its passage and that of a known star. The instant of Greenwich time, when the moon's right ascension is equal to the observed right ascension, might be
By a moon's transit.
determined from the right ascension, which is given in the Nautical Almanac for every hour. But this computation involves the observation of the solar time, whereas the observed interval gives at once the sideral time of the observation.
The calculation is then more simple, by means of the table of Moon-Culminating stars given in the Nautical Almanac, in which the right ascensions of the suitable stars and of the moon's bright limb are given at the instant of their upper transits over the meridian of Greenwich, and also the right ascension of the moon's bright limb at the instant of its lower transit. Hence the difference between the right ascensions of the moon's limb, at two successive transits, is the change of its right ascension in passing from the meridian of Greenwich to that which is 12" from Greenwich; so that if the motion in right ascension were perfectly uniform, the right ascension, which corresponded to a given meridian, or the meridian, which corresponded to a given right ascension, might be found by the following simple proportion, 12" : long. of place = diff. of right ascensions for 12 :
diff. of right ascensions for long. of place, (569) in which the longitude of the place may be counted from the meridian 12% from that of Greenwich, provided the change of right ascension for an upper transit is computed from the preceding right ascension, which is that of a lower transit at Greenwich, that is, if the place is in east longitude. Let then T = long., if west,
= 12 - long. (if the long. is east); and let A = diff. of right ascension for the Greenwich
transits, which immediately precede and follow the required or observed transit,
By a moon's transit.
and let 8 A = change of right ascension from the pre
ceding Greenwich transit to the observed
transit, and we have, by (569), 12 : T = A:8 A,
12 A whence 8A= and T=
and if T is reduced to seconds, we have AT
(572) 43200 log. : A = log. A + log. T + (ar. co.) log. 43200 = log. A + log. T + 5.36452
(573) 43200 8 A and
(574) A log. T = 4.63548 + ar. co. log. A + log. : A, (575) and formulas (573) and (575) agree with the parts of the rules in the Navigator, which depend upon A, and are independent of the want of uniformity in the moon's motion.
The corrections which arise from the change of the moon's motion may be calculated, on the supposition that this motion is uniformly increasing or decreasing, so that the mean motion for any interval is equal to the motion which it has at the middle instant of that interval. If we put, then,
B the increase of motion in 12”, (576) A is not the mean daily motion for the interval of longitude T and the instant į T after the meridian transit at Greenwich, but for the interval 12" and the instant 6% after this transit. The mean daily motion for the instant į T is, therefore,
(6— T) B (21600' — 1 T) B
43200 and the correction of : A in (572) is
T(21600—1 T) T(43200-T) &B
B, (579) (43200)
2 (43200) 2 and the value of 8 B is easily calculated and put into tables, like Table XLV of the Navigator.
In correcting the value of T (574), the correction of s A is to be computed from Table XLV by means of the approximate value of T, and the correction of T is then found by the formula to be 43200 8 B
8 T =
It only remains, to show how to find the value of B from the Nautical Almanac. Now if Adenotes the motion in right ascension for the 12" interval of longitude, which precedes that to which A corresponds; and if A" denotes the motion in right ascension for the 12" interval of longitude which follows that of A ; we have
2 B = A". A'
(581) and the calculation agrees entirely with that given in the Navigator.
When the longitude is small, or nearly 12, the correction for the variation of motion may be neglected, provided, in
By a lunar distance.
stead of A, the motion is used which corresponds to the time of the nearest Greenwich transit. Now, in the Nautical Almanac, this motion is given for an hour's interval, of which the middle instant is that of the transit, so that if H= this hourly motion, the motion for the time T may be found by the formula
1: T=H: A, whence
8 A X 1
H log. T = 3.55630 + log. 0 A + (ar. co.) log. H, (583) which agrees with [B. p. 432.]
The formula (583) may be rendered more correct, if the value of H is taken for the instant į T of longitude; and the value can be computed precisely in the same way in which the right ascension was computed for the time T, by noticing the want of uniformity in its increase ; and the formula thus corrected is accurate for small differences of longitude.
The distance of the moon from the sun or a star may be used as the signal; but the true places of these bodies differ from their apparent places, as will be shown in succeeding chapters, so that the observed distance requires to be corrected; and the correction cannot be found without knowing the altitudes of the bodies. It is sufficient, for the present purpose, to know that the difference between the true and apparent places is only a difference of altitude, and not one of azimuth, and that the apparent place of the sun or a star is higher than its true place, while that of the moon is lower.