EXAMINATION FOR MASTER EXTRA.

THE VERIFICATION OF THE LATITUDE BY DOUBLE ALTITUDES OF THE SUN.

SUMNER'S METHOD.

1. Find the Greenwich date of each observation.

2. From the Nautical Almanac find the Sun's Declination for each date.

3. From the observed find the true altitudes as in the usual method of double altitudes.

4. Select two latitudes, one of which is the degree (without odd minutes) less, and the other the next degree greater than the latitude by dead reckoning. Or, assume two latitudes from 5' to 30′ on each side of the approximate latitude, such that there may be a difference of from 10' to 10 between the two latitudes thus chosen.

5. Find the apparent time from noon in each of the following

cases:

(a.) With the first altitude, the corresponding decli

nation and less latitude.

(b.) With the second altitude, the corresponding declination, and less latitude.

(c.) With the first altitude, the corresponding declination and greater latitude.

(d) With the second altitude, the corresponding declination and greater latitude.

6. Obtain the elapsed time corresponding to each assumed latitude thus-If one observation be A.M. and the other P.M. take the sum of the times (hour angle) found by (a) and (b): if both be A.M. or both P.M take the difference of the results of (a) and (b); call this sum or difference the elapsed time (e) for that assumed latitude. Proceed in the same manner, using the results of (c) and (d) to find the elapsed time (f) for the other assumed latitude.

7. Take the difference of elapsed time (e) for the one assumed latitude and the true apparent elapsed time, calling the remainder too little, if the former is less than the latter, but too much, if the reverse is the case. Find also the difference of the elapsed time (ƒ) for the other assumed latitude and the true apparent time, naming the remainder on the same principle as before.

8.

Whenone elapsed time is too much and the other too little take their sum,-but if both are too much, or both too little, take their difference for the error of elapsed time caused by an error of 10′ 20′ 40′ or 1° of latitude, according to what has been assumed.

Lastly, make this proportion which can be computed by proportional logarithms:

It will be at once seen that when the elapsed time of one assumed latitude is too little and that of the other too much, the true latitude is between the two assumed latitudes; consequently the correction must be added to the less or subtracted from

the greater assumed latitude, according to which is used for determining the correction. But, both the elapsed times of the two assumed latitudes may be too much or both too little (each case is possible), then the correction must be applied to satisfy the following condition: if the elapsed time of the less assumed latitude differs from the true elapsed time by a given quantity, and that of the greater assumed latitude by a less quantity, then the true latitude must be greater than the greater assumed latitude; also if the elapsed time of the greater assumed latitude differs from the true elapsed time by a given quantity, and that of the less assumed latitude by a less quantity, then the true latitude must be less than the less assumed latitude.

Paper II.

To illustrate the foregoing rule the question for double altitude in Paper II. is taken.

RULE FOR CORRECTING THE LATITUDE

FOR THE CHANGE OF DECLINATION IN THE HALF ELAPSED TIME (DOUBLE ALTITUDes.)

To the sine of arc 2,* add the secant of the lat., the log. of the change of declination in the half elapsed time, reduced to seconds, and the co-secant of the half elapsed time, the sum, rejecting tens, is the log. of the correction.

If the second altitude is the less, and the days shortening, the cor. for lat. is+. If either condition is reversed, the cor. is. If both are reversed, the sign remains +.