P Let x be the place of a heavenly body near the meridian. Draw the circle of declination P X and circle of altitude z x through x, then in the spherical triangle P ZX are given the hour angle P, the polar distance PX, and the zenith distance zx, to find the colatitude P Z. This may be done by dropping a perpendicular from x upon PQ, in the manner pointed out in Problem 131 of the Astronomical Problems: but the direct method of solving it being long and tedious an analytical formula is obtained for this purpose (see astronomical problems, p. 201), from which the following rule is deduced. Rule XXXIV. To find the latitude from an altitude of the sun near the meridian. 1. Find the Greenwich date in mean time. 2. Take out the declination and equation of time for this date. 3. To find the sun's hour angle. To the Greenwich mean time found as accurately as possible apply the longitude in time; subtracting if west, and adding if east; the result will be ship mean time: to this apply the equation of time with its proper sign to reduce mean time into apparent time; the result will be the sun's hour angle. 4. Add together the following logarithms,— Constant log., 6·301030 Log. haversine hour angle.* reject 30 in the index, and look for the result as a logarithm, and take out its natural number. * Or, instead of log. haversine, twice the log. sine of half the hour angle (rejecting in this case 40 from the index). 5. Correct the observed altitude for index correction, dip, semidiameter, correction in altitude, and thus get a zenith distance. 6. From the versine of zenith distance subtract the natural number found as above. The remainder will be the versine of a meridian zenith distance, which find from the tables. 7. Under the meridian zenith distance put the declination, and proceed to find the latitude by one of the preceding rules for finding the latitude by a meridian altitude. NOTE.-If the latitude thus found differ much from the estimated latitude used in the question, the work should be corrected by using the last latitude found, in place of the former one. EXAMPLES. August 22, 1853, A.M., in latitude by account 50° 48' N., and long. 1° 6' W., a chronometer showed 11h 50m 22s, error on Greenwich mean time being 40.2s fast, when the observed altitude of the sun's lower limb (in artificial horizon) was 101° 14′ 10′′ (Z. N.), index correction + 30", required the latitude. As this latitude differs from the estimated latitude, one part of the above operation should be repeated, using lat. 50° 47′ 49′′ instead of 50° 48', thus Constant log. Log. hav. H. A. The same natural number as before, which shows that the erroneous latitude used in the first operation produced no practical error in the resulting latitude. The above example worked by formula, p. 134. (135.) May 10, 1853, A.M., in latitude by account 50° 50 N., and long. 2° 10' W., a chronometer showed 11h 51m 58s, error on Greenwich mean time being 11m 31s fast, when the observed altitude of the sun's lower limb was 56° 19′ 30′′ (Z. N.), index correction 3' 20", and height of eye 18 feet, required the latitude. Ans., lat. 50° 51′ 34′′ N. (136.) Nov. 14, 1853, P.M., in lat. by account 87° 41' S. and long. 1° 0′ W., a chronometer showed Oh 25m 27s, error on Greenwich mean time being fast 5m 56.7, when the observed altitude of the sun's lower limb was 20° 26′ 20′′ (Z. S.), index correction 2' 20", and height of eye 10 feet, required the latitude. Ans., lat. 87° 42′ 15′′ S. (137.) June 30, 1853, A.M., in lat. by account 63° 20′ N. and long. 23° 30′ W., a chronometer showed 11h 30m 15s, error on Greenwich mean time being 7m 32s fast, when the observed altitude of the sun's upper limb was 44° 20′ 22′′ (Z. N.), index correction + 2′ 20′′, and height of eye 14 feet, required the latitude. Ans., lat. 63° 21′ N. (138.) July 10, 1853, A.M. in lat. by account 57° 24' N. and long. 3° 40′ W., a chronometer showed 11h 20m 153, error on Greenwich mean time being 30m 30s slow, when the observed altitude of the sun's lower limb was 54° 17′ 19′′ (Z. N.), index correction2' 40", and height of eye 20 feet, required the latitude. Ans., lat. 57° 25′ 25′′ N. (139.) May 20, 1853, A.M., in lat. by account 79° 48' N., and long. 44° 30′ E., a chronometer showed 11h 30m 0s, error on Greenwich mean time being 15m 20s slow, when the observed altitude of the sun's lower limb (in artificial horizon) was 54° 30' 20" (Z. N.), index correction - 4' 30", required the latitude. Ans., lat. 79° 48' 30" N. (140.) June 16, 1853, P.M., in lat. by account 52° 25′ N., and long. 1° 6′ W., a chronometer showed 1h 2m 9s error on Greenwich mean time being 40m 30s fast, when the observed altitude of the sun's lower limb was 60° 37′ 50′′ (Z. N.), index correction - 2' 10", and height of eye 17 feet, required Ans., lat. 52° 24′ 15′′ N. the latitude. To find the latitude by Inman's rule for double altitude. The most general rule for finding the latitude by a double altitude of a heavenly body is the one selected by Dr. Inman: the labour of reducing the observations is somewhat greater than in the one known as Ivory's Rule, which follows: but the great advantage of the method adopted by Inman is that it may be applied to the same or different heavenly bodies, observed at the same instant or at different times: we will W N Q S X E give examples of its application to all the cases that usually occur, referring the student for more complete information on the subject to the Appendix to "Inman's Navigation." Let P be the pole, z the zenith, x and y the same heavenly Ly body observed at different times; or different heavenly bodies ob served at the same instant, or different heavenly bodies |
