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To find the Mean Solar Time corresponding to a given Sidereal Time.
Find, as in the last problem, the sidereal time at mean noon, subtract it from the given sidereal time (adding 24h. if necessary), and the remainder will be the interval of sidereal time from noon. Add together the mean solar equivalents for the hours, minutes and seconds of this interval, taken from table XI., and the sum will be the mean time required.
EXAMPLE. Required the mean solar time at Philadelphia on the 27th of May, 1836, at 1h. 21m. 47.5sec., sidereal time.
Sid. time at M. N., Green.
h. m. sec.
4 20 16.68
4 21 6.07
25 21 47.5
21 0 41.43
20 56 33.579
20 57 14.896
To find, from the Tables, the Moon's Longitude, Latitude, Equatorial Parallax, Semidiameter, and Hourly Motions, in Longitude and Latitude, for a given time.
When the given time is not for the meridian of Greenwich, reduce it to that meridian; and when it is apparent time, reduce it to mean time.
With the mean time at Greenwich, take out from tables XXXII. to XXXVI., the arguments numbered 1, 2, 3, &c., to 20, and find their sums, rejecting the ten thousands in the first nine, and the thousands in the others. The resulting quantities will be the arguments for the first twenty equations of Longitude.
With the same time, and from the same tables, take out the remaining arguments and quantities, entitled Evection, Anomaly, Variation, Longitude, Supplement of the Node, II., V., VI., VII., VIII., IX., and X.; and add the quantities in the column for the Supplement of the Node.
For the Longitude.
With the first twenty arguments of longitude, take, from tables XXXVII. to XLII., the corresponding equations, and place their sum in the column of Evection. Then, the sum of the quantities in this column will be the corrected argument of Evection.
With the corrected argument of Evection, take the Evection from table XLIII., and add it to the sum of the preceding equations. Place the resulting sum in the column of Anomaly. Then, the sum of the quantities in this column will be the corrected Anomaly.
With the corrected Anomaly, take the Equation of the Centre from table XLIV., and add it to the sum of all the preceding equations. Place the resulting sum in the column of variation. Then, the sum of the quantities in this column will be the corrected argument of variation.
With the corrected argument of Variation, take the variation from table XLV., and add it to the sum of all the preceding equations; the result will be the sum of the first twenty-three equations of the Longitude. Place this sum in the column of Longitude. Then, the sum of the quantities in this column will be the Orbit Longitude of the Moon, reckoned from the mean equinox.
Add the Orbit Longitude to the Supplement of the Node. The result will be the argument of the Reduction. It will also be the first argument of Latitude.
With the argument of Reduction, take the reduction from table XLVI., and add it to the Orbit Longitude. Also, with the 19th argument, which is the same as argument N, for the Sun's Longitude, take the Nutation in Longitude, from table XXX., and apply it, according to its signs, to the last sum. The result will be the Moon's true Longitude from the Apparent equinox.
For the Latitude.
Place the sum of the first twenty-three equations of Longitude, taken to the nearest minute, in the column of Arg. II. Then the sum of the quantities in this column will be Arg. II. of Latitude, corrected. The Moon's true Longitude is the 3d argument of Latitude. The 20th argument of Longitude is the 4th argument of Latitude. Convert the degrees and minutes, in the sum of the first twenty-three equations of Longitude, into thousandth parts of the circle, by taking from table L. the number corresponding to them. Place this number in the columns V., VI., VII., VIII., and IX.; but not in column X. Then the sums of the quan
tities in columns V., VI., VII., VIII., IX., and X., rejecting the thousands, will be the 5th, 6th, 7th, 8th, 9th, and 10th arguments of Latitude.
With the sum of the Supplement of the Node, and the Moon's Orbit Longitude, which is Arg. 1. of Latitude, take the Moon's distance from the North Pole of the Ecliptic, from table XLVII., and with the remaining nine arguments, take the corresponding equations from tables XLVIII., XLIX., and LI. The sum of these ten quantities will be the Moon's true distance from the North pole of the Ecliptic. The difference between this distance and 90°, will be the Moon's true latitude; which will be north or south according as the distance is less or greater than 90°.
For the Equatorial Parallax.
With the corrected arguments Evection, Anomaly, and Variation, take the corresponding quantities from tables LII., LIII., and LIV. Their sum will be the Equatorial Parallax.
For the Semidiameter.
With the Equatorial Parallax take the Moon's Semidiameter from table LV.
For the Hourly Motion in Longitude.
With the arguments 2, 3, 4, and 5, of Longitude, rejecting the two right hand figures in each, take the corresponding equations from table LVI. Also, with the correct argument of Evection, take the equation from table LVII.
With the sum of the preceding equations at top, and the correct anomaly at the side, take the equation from table LVIII. Also, with the correct anomaly, take the equation from table LIX.
With the sum of all the preceding equations at the top, and the correct argument of Variation at the side, take the equation from table LX. With the correct argument of Variation, take the equation from table LXI. And, with the argument of Reduction, take the equation from table LXII. These three equations added to the sum of all the preceding ones, will give the Moon's Hourly Motion in Longitude.
For the Hourly Motion in Latitude.
With the 1st and 2nd arguments of Latitude, take the corresponding quantities from table LXIII. and LXIV., and find their sum, attending to the signs. Then 32′ 56′′ the moon's true hourly motion in Longitude :: this sum: the moon's true hourly motion in Latitude. When the sign
is affirmative, the moon is tending north; and, when it is negative, she is tending south.
EXAM. 1. Required the moon's longitude, latitude, equatorial parallax, semidiameter, and hourly motions in longitude and latitude, on the 6th of August, 1821, at 8h. 46m. 33sec. A. M. mean time at Philadelphia.
Mean time at Philadelphia, August,
Mean time at Greenwich, August,
d. h. m. sec.
6 1 47 13
1821 August 6 d.
1 h. 47 m. 13 sec.
Sum of Equa.
10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 0027 83655389 1368 697077146319 70247800 620 917 842 142979|067923|331|134 036 036 5804 7776 518 838 1134 8874 22332742 5148 873 599 938 245 22471 759917503 31| 97 137 3249 5201 1435 1678 1860 289 1952 121 351 156 352 271 43 12 1 14 16 1 21 34 9 11 12
171 496 153 183 210 130 1 2
3 1 4 1 2
2 16 1 13
2 1. 3
5970 9438 1185|3662|9807|8476 8845 1747|3071|849 674 137 560 504|692 867460 768 68 135|
1s 19° 43′ 47" 5 59
1 26 34 57
11 16 51 306 27 6 27
9° 54' 17" 10s 22° 4′ 2′′
9 46 42
1 47 41
7 1 31
46 32 3 4 23 35
7 18 58 30
Nutat. in long.
Moon's true Longitude 7 18 58 38
0 24 7 18
33 11 55 15
8 13 28 26
Arg. 1 of Latitude.