2. The observed altitude of a star, when the barometer stood at 30.5 inches, and the thermometer, at 62°, was 15° 6′ 30".

Required the cor

From the observed altitude of a star, or the under or upper limb of the sun, to obtain the true altitude.

For a star. The observed altitude, corrected for refraction by the last problem, gives the true altitude of the star.

For the sun. Correct the observed altitude for refraction, by the last problem. Find the sun's semidiameter by prob. VI, or take it from the Nautical Almanac* or other ephemeris in which it is given. Then, if the lower limb was observed, add the semidiameter to the corrected altitude; but if the observation was on the upper limb, subtract the semidiameter; and the result will be the altitude of the centre, corrected for refraction. To this, add the parallax in altitude, taken from table VIII, and the sum will be the true altitude.

EXAMPLES.

1. Suppose the observed altitude of the sun's lower limb at a certain time was 18° 48' 5"; the barometer standing at 29.7 inches, the thermometer at 70°, and the sun's semidiameter being 15′ 47′′.4. Required the true altitude.

The United States Almanac, just published in Philadelphia, and designed to be continued annually, gives the sun's semidiameter for each 6th day of the year.

the

2. The observed altitude of the sun's upper limb being 21° 7' 12", barometer 30.3 inches, the thermometer 40°, and the sun's semidiameter 16' 17.2; required the true altitude.

Ans. 20° 48' 28.9".

PROBLEM XXVI.

To find the apparent right ascension and declination of any of the stars in the small catalogue, tab. IX, for a given day.

1. To find the Variations in mean right ascension and declination.

Reduce the months and days of the given time to the decimal of a year, by means of the small table at the foot of the second page of table IX,* and annexing it to the years, find the interval between this time and the date of the table, marking the interval negative when the given time is prior to that date. Take from the table the annual variations of the given star, and multiplying each by the interval, the products will be the variations of the mean right ascension and declination, respectively.

2. To find the Aberrations.

Find L', the sun's longitude, for the given day, by prob. VI, or take it from an ephemeris, and take from tab. IX, the values of 4, log. m, 0, and log. n, for the given star. Then,

log. (aber. in right ascen.) = log. m + sin (I+Q),
log. (aber. in decl.)

=

log. n + sin (L' + 0).

3. To find the Nutations.

Find N, the mean longitude of the moon's ascending node, for the given day, by taking the supplement of the node, obtained as in prob. VIII, from 12 0° 7', or take it from an ephemeris.t Take from tab. IX, the values of p', log. m', ', and log. n', for the given star. Then,

log. (nut. in right ascen.) = log. m' + sin (N + p'),

log. (nut. in decl.)

=log. n' + sin (N +0').

4. Attending to the signs, add to the mean right ascension of the star, given in the table, the variation, aberration and nutation in right ascension, and the sum will be the apparent right ascension. In like manner, find the apparent declination, observing that the declination is to be regarded as negative when it is south.

The U. S. Almanac gives the decimal of a year for each day.

The mean longitude of the moon's ascending node is given in the U. S. Almanac, for each 10th day, and may be easily obtained for any intermediate day, by proportion

EXAMPLES.

1. Required the apparent right ascension and declination of a Bootis, (Arcturus), the 1st of May, 1837; the sun's longitude, at that time, being 40° 52′, and the mean longitude of the node 31° 14'.

2. Required the apparent right ascension and declination of a Leonis, (Regulus), the 19th of August, 1842; the sun's longitude being 146° 2′, and the mean longitude of the moon's ascending node 288° 43'.

To find the Latitude of a place, having given the corrected altitude of a star, its apparent right ascension and declination, and the mean time of observation.

Find the sidereal time corresponding to the given mean time, by Prob.

*The altitude may be taken with a sextant and artificial horizon. For the method of adjusting the instrument and making the observation, the student is referred to Simms' small work on instruments mentioned in a note on page 27.

VII, or obtain it from an ephemeris. Take the difference between this time and the star's apparent right ascension; and if the difference exceeds 12 hours, subtract it from 24 hours. The result, converted into arc, will be the distance of the star from the meridian. Call this distance H, the star's apparent declination, regarded affirmative whether north or south, D, and the corrected altitude A. Then, find two arcs B and C, neither of them exceeding 90°, from the formulæ.

tang B = cot D + cos H, or sin (H-90°) when H exceeds 90°. sin C = Ar. Co. sin D + sin A + cos B.

When H, the star's distance from the meridian, exceeds 90°, the sum of B and C, is the latitude of the place. When the star's declination is of the same name with the latitude of the place and less than it, and its position at the time of observation is on the opposite side of the prime vertical, from the elevated pole, the supplement of the sum of B and C, is the latitude. In all other cases the latitude is equal to the difference between B and C.

Note. The observation should not be made so near the prime vertical as to make the side on which the star is situated, doubtful. It is always best, when convenient, to make it near the meridian; as then, a small error in the clock or in the longitude of the place, required in finding the sidereal time, produces but very slight influence on the computed latitude.

Several observations of the altitude and corresponding time should be taken, and the latitude be deduced from each. The mean of these, that is, their sum divided by their number, may be regarded as more accurate than the latitude obtained from a single observation. The probable accuracy of the determination, will be still further increased, if, near the same time, the latitude be deduced in like manner from observations on a star* on the opposite side of the zenith, and the half sum of the two latitudes thus obtained, be taken for the latitude.

EXAMPLES.

1. Given the corrected altitude of a Ursae Minoris 41° 33' 21.4", obtained from an observation at a place, long. 5 h. 1 m. 15 sec. W. and lat. about 40° N., on the 25th of November, 1839, at 8 h. 34 m. 17 sec. P. M., mean time, to find the latitude; the apparent right ascension of the star being 1 h. 2 m. 22.63 sec., and declination 88° 27′ 39.3" N.

The sidereal time corresponding to the given mean time is found to be 0 h. 51 m. 29.96 sec.

* A star whose altitude is within a few degrees of the former, is to be preferred.

2. Given for the same place and same night as in the last example, the corrected altitude of ẞ Orionis, 41° 29′ 36′′.1, at 12 h. 35 m. 19 sec. P. M., mean time, to find the latitude; the apparent right ascension of the star being 5 h. 6m. 52.40 sec., and declination 8° 23′ 17′′.1 S.

PROBLEM XXVIII.

Ans. 40° 0′ 54′′.9.

Given the true altitude of the sun, obtained from observation at a given place, and the time of observation as indicated by a clock, to find the time, and the error of the clock.

Find the sun's declination for the given time, and subtracting from 90° when it is of the same name with the latitude of the place, but adding when of a contrary name, we have his polar distance. Call the polar distance D, the altitude A, the latitude of the place L, and the hour angle or distance of the sun from the meridian H. Add the values of A, D and L together, and call the sum S.

Add together Ar. Co. sin D, Ar. Co. cos L, cos S, and sin (S — A) without rejecting any 10 from the index, and taking half the sum, it will be sin H. When the observation is made in the afternoon the hour angle H, converted into time, is the apparent time; but when it is made in the forenoon, the difference between this interval and 12 hours, is the apparent time. To the apparent time, apply the equation of time and we obtain the mean time. The difference between this and the time shown by the clock, is the error of the clock.

Note 1. The observations for finding the time should be made when the the sun is several hours from the meridian; the nearer the prime vertical, the better, provided the altitude is not less than 12 or 15o.

2. The time and error of the clock may be obtained in nearly a similar manner, from the corrected altitude of a star. Having found the star's apparent right ascension and declination, and computed the hour angle H, using the star's polar distance and altitude, add it to the right ascension when the observation is made to the west of the meridian, but subtract it from the right ascension when the observation is made to the east. The result will be the sidereal time of the observation. From this, subtract the sidereal time found for mean noon of the given day, and converting the remainder into mean time by means of tab. XI, it will be the mean time of the observation.