mile change of latitude. If the azimuths are in the same or opposite quadrants, the difference of these numbers is the total change of longitude for one mile change of latitude. If the azimuths are in adjacent quadrants, the sum of the numbers is the total change of longitude for one mile change of latitude. Hence if the whole difference of longitude between the longitudes found from the observations is divided by the total change of longitude for one mile change of latitude, the result is the correction for latitude. And consequently the correction for longitude will be found by multiplying the number used in getting an azimuth by the correction for latitude. The correction is to be applied to the longitude found from the same observation as the azimuth was found from. In the example just given the numbers are 2.43 and 0.87, and the azimuths are in adjacent quadrants, the total change of longitude for one mile change of latitude is the sum 3.3. The total difference of longitude is 51′·1, and 51'1 divided by 3.3 gives 15' 5, the correction for latitude. The correction for longitude is 0.87 multiplied by 15'5, that is 13'5. Lat. by D.R. 49° 58' N. True lat. 15.5 S. 49 42'5 N. Long. 14° 38′ W. Rule for applying the corrections.—When the azimuths are in adjacent quadrants, add the correction for longitude if the longitude used is the less longitude, but subtract if the longitude used is the greater longitude. When the azimuths are in the same or opposite quadrants, mark the difference of longitude between the observations E. or W., reckoning from the point of less azimuth to that of greater azimuth. Then if the longitude used and difference of longitude have the same name, add the correction for longitude, but subtract if the longitude used and difference of longitude have different names. The correction for latitude takes its name from the line of position drawn through the point; this must be north-easterly and south-westerly or north-westerly and south-easterly. Having determined the direction of the line of position and knowing from the rule for applying the correction for longitude whether it is E. or W., the letter in the line of position connected with the letter of the correction for longitude is the name of the correction for latitude. Thus if the line of position is north-easterly and south-westerly and the correction for longitude is E., then the correction for latitude is N. In the example the azimuths are in adjacent quadrants, the correction for longitude is W. because it is added to the less longitude, which is W. The line of position through the point used is north-easterly and south-westerly, and the correction for longitude is W., therefore the correction for latitude is S. If two observations of the sun have been taken, and the course and distance in the interval noted, then the longitude must be determined from each observation, using the latitude by account at the time of the observation. The longitude found from the first observation must be corrected for the run in the interval, to reduce it to the longitude at the time of taking the second observation. The difference between the reduced longitude and the longitude found from the second observation is the total difference of longitude to be divided by the change of longitude in one mile change of latitude found by the A and B numbers in the A and B Azimuth Tables in Norie's Tables. The A and B numbers will be found combined in Norie's new Tables, Longitude Correction Table. This problem has been made very familiar to navigators by Mr. A. C. Johnson, R.N., through the pamphlet entitled "On Finding the Latitude and Longitude in Cloudy Weather and at other times." If in the case of the sun the second observation is taken near noon within the limits of the Reduction to the Meridian problem, finding the longitude by using the estimated latitude would be altogether useless. The estimated longitude would now be used to find the latitude. Example. December 12th, in lat. by account 30° 12′ N., long. 72° W. The true course and distance in the interval was north 30 miles. The chronometer was estimated to be Im. slow for mean time at Greenwich. Height of the eye 20 feet, I.E.—2′ 10′′. Find the ship's position at the time of taking the second observation. The calculation of the ship's position is now very simple. Since by reference to the Table, p. 379, the hour-angle of the second observation is within the limits of the Reduction to the Meridian, it follows that the latitude found will be the correct latitude, unless the latitude and longitude used in the calculation are both very erroneous. From this it follows that the first observation has been calculated with a latitude 12' in error, and for this observation it has been found that the diff. long. for 1' diff. lat. is 109, the total diff. long. is therefore 1.09 × 12 = 13′′08 and is W. because the true latitude is S. of the lat. by acct. and the position line trends south-westerly. Long. calculated with erroneous lat. 72° Course being N. 72° 2′ W. The position of the ship is therefore lat. 30° 30′ N., long. 72° 15′ W. If the second observation had not been within the limits of the Table on p. 379 and the azimuth had been small, or if the estimated latitude and longitude had been very erroneous, neither the latitude nor the longitude could have been found with any degree of accuracy by the ordinary method COMPUTATION OF ALTITUDES To Compute the Altitude of a Heavenly Body All the necessary elements from the Nautical Almanac must be corrected for the Greenwich date. If the sun is the object, the apparent time from the nearest noon is the sun's hour angle or meridian distance. For the meridian distance, in time, or hour angle of a fixed star, a planet, or the moon.—To the mean time at place add the mean sun's right ascension, and from their sum (which is the right ascension of the meridian subtract the right ascension of the object; the result will be the object's westerly hour angle or meridian distance, which, if greater than 12h., take from 24h. for the easterly (or nearest) hour angle. The method of finding altitudes by computation has been much facilitated by a table of Natural Haversines, which has been included in Norie's Tables and will be found adjacent to the log. haversine of the same arc or angle. The method is mathematically sound, and in consequence of the shortness of the calculation is recommended in preference to other methods, of which there are many. To find the altitude of a celestial object, having given the hour angle. the polar distance, and the latitude For Arc I. add together the log. haversine of the hour angle, the lóg. sine of the co-latitude, and the log. sine of the polar distance; the sum, rejecting tens in the index, is the log. haversine of Arc I. For the zenith distance add together the natural haversine of Arc I. and the natural haversine of the difference of the co-latitude and polar distance; the sum is the natural haversine of the zenith distance, which subtract from 90° for the true altitude. Formula L. hav. Arc I. = L. hav. PL. sin. l' + L. sin. þ 20. polar dist., z = zenith dist. NOTE.-Arc I. need not be taken out, but only the natural haversine of Arc I., which will be found adjacent to the log. haversine of Arc I., as will be seen in the following examples. From the True Altitude to find the Apparent Altitude.-The corrections must be applied in reverse order and with contrary signs to those with which the true is derived from the apparent altitude. For the Sun or for a Planet.-Subtract the parallax in altitude and add the refraction. For a Star.-Add the refraction. For the Moon.-By Computation.-To the log. secant of the moon's true altitude add the proportional log. of moon's reduced horizontal parallax ; the result will be the proportional log. of the approximate parallax in altitude, which take out. Subtract this approximate parallax in altitude from the moon's true altitude, and with the result proceed as before to get the correct parallax in altitude, which subtract from the true altitude; after which add the refraction, to get the moon's apparent central altitude. NOTE. On rare occasions the correction for the parallax in altitude may have to be taken out a third time. The difference between the parallax in altitude and the refraction is the moon's correction of altitude, and can ordinarily be taken from Moon's Correction Table in Norie's Tables. Example 1.-May 27th p.m. 3h. 2m. 49s. mean time at ship. Position by D.R. lat. 33° 54′ S., long. 108° E. Required the moon's true altitude. L. hav. Arc I. Nat. hav. z L. hav. PL. sin. ' + L. sin. p where P = hour angle, l' co-lat., p polar dist., z zenith dist. 20 |