Scholium. The corrections of the times of contact on account of parallax, obtained as above, may be regarded as very nearly true. But the times of contact obtained for the earth's centre, and consequently those for a given place, cannot be depended on as equally correct; as an error of three or four seconds in the longitude of the sun or Mercury, may produce an error of a minute in time.

PROBLEM XXVI.

To correct the observed altitude of a heavenly body on account of

Refraction.

With the given altitude, take the corresponding mean refraction* from table VII., and subtract it from the altitude. The remainder will be the corrected altitude, very nearly.

If greater accuracy is desired and the states of the barometer and Fahrenheit's thermometer have been observed, take from the table, the numbers corresponding to the given altitude, that are in the two columns following that of the mean refraction. Multiply the first of these by the number of inches in the height of the barometer, less 30, and the second by 50, less the number of degrees in the height of the thermometer. The products will be the corrections of the refraction in seconds, depending on the states of the barometer and thermometer respectively. Add these, attending to their signs, to the mean refraction, and the result will be the true refraction; which being subtracted from the observed altitude, gives the correct altitude.

EXAMPLES.

1. The observed altitude of a body being 35° 25′ 35′′, what is its altitude, corrected for mean refraction?

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 corrected altitude.

* The mean refraction is that which corresponds to a height of 30 inches of the barometer and 50° of Fahrenheit's thermometer.

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

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 XXVIII.

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 p, log. m, 0, and log. n, for the given star. Then

log. (aber. in right ascen.) = log. m + log. sin (L' + ¢),
log. (aber. in decl.)
log. nlog. 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. X., from 12s 0° 7′, or take it from an ephemeris. 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' + log. sin (N+),
log. (nut. in decl.)

log. n + log. 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 regarded as negative when it is south, and positive when it is north.

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'.

3. Required the apparent right ascension and declination of ẞ Libræ,

PROBLEM XXIX.

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. 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.

meridian, exceeds 90°, the sum When the star's declination is

log. tang B log. cot D + log. cos H, or log. sin (H— 90°) when H exceeds 90°. log. sin C Ar. Co. log. sin D + log. sin A + log. cos B. When H, the star's distance from the of B and C, is the latitude of the place. 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 start on the opposite side of the zenith, and the half sum of the two latitudes thus obtained, be taken for the latitude. (See Art. 183, 1st method.)

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