2. Required the moon's longitude, latitude, equatorial parallax, semi-diameter, and hourly motions in longitude and latitude, on the 27th of April, 1821, at 9h. 43m. 30 sec. P. M. mean time at Baltimore, Ans. Long. 11' 13° 31' 44"; lat. 6′ 55' N. equat. par. 60′ 0′′; semidiam. 16′ 21′′; hor. mot. in long. 36′ 11′′; and hor. mot. in lat. 3' 14", tending north.

3. What will be the moon's longitude, latitude, equatorial parallax, semi-diameter, and hourly motions in longitude and latitude, on the 19th of August, 1822, at 5h. 56m. 14sec. P. M. mean time at Philadelphia? Ans. Long. 6 3° 7' 28"; lat 3° 51' 35" S.; equat. par. 56' 19"; semi-diam. 15' 21'; hor. mot. in long. 32' 7"; and hor. mot. in lat. 2' 1", tending south.

PROBLEM IX.

To find the Moon's Longitude, Latitude, Hourly Motions, Equatorial Parallax, and Semi-diameter, for a given Time, from the Nautical Almanac.

Reduce the given time to Mean time at Greenwich. Then,

For the Longitude.

Take from the Nautical Almanac, the two longitudes, for the noon and midnight, or midnight and noon, next preceding the time at Greenwich, and also the two immediately following these, and set them in succession, one under another. Then, having regard to the signs, subtract each longitude from the next following one, and the three remainders will be the first differences. Call the middle one A. Subtract each first difference from the following, for the second differences. Take the half sum of the second differences, and call it B.

Call the excess of the given time at Greenwich, above the time of the second longitude, T. Then 12 h: T:: A fourth term, which must have the same sign as A.

With the time T at the side, take from table LXVI the quantities corresponding to the minutes, tens of seconds, aud seconds of B, at the top, the sum of these, with a contrary sign to that of B, will be the correction of second differences.

The sum of the second longitude, the fourth term, and the correction of second differences, having regard to the signs, will be the required longitude.

For the Hourly Motion in Longitude.

To the logistical logarithm of of T, add the B, and find the quantity corresponding to the sum. and prefix to it the same sign as that of B.

logistical logarithm of Call this quantity E,

Or E may be found without logarithms; thus 12h.: T:: B: E. Divide the sum of A, B with its sign changed, and E, by 12, and the quotient will be the required hourly motion in longitude.

For the Latitude.

Prefix to north latitudes the affirmative sign, but to south latitudes the negative sign, and then proceed in the same manner as for the longitude. The resulting latitude will be north or south, according as its sign is affirmative or negative.

Note. The Moon's Declination may be found in the same manner.

For the Hourly Motion in Latitude.

With T, and the values that A and B have, in finding the latitude, find the hourly motion in latitude, in the same manner as directed for finding the hourly motion in longitude. When the resulting hourly motion in latitude is affirmative, the moon is tending north, and when it is negative, she is tending south.

For the Semi-diameter and Equatorial Parallax.

The moon's semi-diameter and equatorial, horizontal parallax, may be taken from the Nautical Almanac with sufficient accuracy by simply proportioning for the odd time between noon and midnight, or midnight and

noon.

EXAM. 1. Required the moon's longitude, latitude, equatorial, parallax, semi-diameter, and hourly motions in longitude and latitude from the Nautical Almanac, table LXV, on the 4th of May, 1836, at 4h. 30m. 8sec. P. M. mean time at Philadelphia.

12 9 30 48: 7° 21′ 12".5: 5° 49' 46."8, fourth term.

12

9 30 48 :: -30' 26.3": -24' 7".8, fourth term.

2. Required the moon's longitude, latitude, equatorial parallax, semidiameter, and hourly motions in longitude and latitude, on the 6th of May. 1836, at 1h, 41 m. 40 sec. P. M. mean time at Greenwich.

Ans. Long. 297° 59' 57.1"; lat. 4° 54' 18.6" S; equat. par. 59' 15.0"; semi-diam. 16' 8.7"; hor. mot. in long. 35' 34.9"; hor. mot. in lat. l' 13.7" tending south.

Note 1. When the moon's longitude and latitude are required with great precision, the third and fourth differences should be noticed. To do this, take from the ephemeris, the three longitudes or latitudes, preceding the given time, and the three following it, and find the first, second, third, and fourth differences, as directed in the rule, for the first and second differCall the middle first difference, A, the half sum of the two middle second differences, B, the middle third difference, C, and the half sum of the fourth differences, D. Then taking T, equal the excess of the given time above the time of the third longitude or latitude, find the fourth term and the correction for second differences, as directed in the rule.

ences.

With the time T, and middle third difference, C, take from table LXVII, the correction for third differences, which, when T is less than 6 hours, must have the same sign as C, but a contrary sign, when T is more than 6 hours.

With the time T, and half sum of fourth differences, D, take from table LXVIII, the correction for fourth differences, which must always have the same sign as D.

The sum of the third longitude or latitude, the fourth term and the corrections for second, third, and fourth differences, having regard to the signs of all the quantities, will be the longitude or latitude required.

Note 2. When great precision is required in the moon's parallax and semi-diameter, the corrections for second differences should be applied in the same manner as for the longitude or latitude.

Taking the time, as in the first example, let the moon's longitude be required, as corrected for third and fourth differences.