Ex. 5. 1882, February 11th, at 8h 54m 47 P.M., apparent time, long. 11° 4′ W.: find the declination.

297. To find the declination of the sun at the time of its transit over a given meridian.

When the sun is on a meridian in WEST longitude, the Greenwich apparent time is precisely equal to the longitude; that is, the Greenwich apparent time is after the noon of the same date with the ship date by a number of hours, equal to the longitude in time. When the sun is on a meridian in EAST longitude, the Greenwich apparent time is before the noon of the same date as the ship date by a number of hours, equal to the longitude in time. Hence, to obtain the sun's declination for apparent noon at any meridian we have

RULE LXXXI.

Take the declination from the Nautical Almanac (page I of the month) for Greenwich apparent noon of the same date as the ship date, and apply a correction equal to the hourly difference multiplied by the longitude, observing to add or subtract this correction according as the numbers in the Nautical Almanac may indicate for a time before or after noon.

EXAMPLES.

Ex. 1. 1882, September roth, the sun on the meridian, long. 100° 35′ E.: required the sun's declination.

Ex. 2.

1882, June 1st, the sun on the meridian, long. 75° W.: required the sun's decl. Long. 75° W.

As the decl. is increasing, the decl. at 5h after noon will be greater than that for noon.

Ex. 3. In the last question suppose the longitude to be 75° E.

The longitude being 5h E., the Green, A.T. is 5h before the noon of June 1st-the same date as the ship date. The decl. is taken out of the Nautical Almanac, page I of the month; also the var. in 1b, and the work is as follows:

As the decl, is increasing, the decl. at 5h before noon will be less than that for noon.

1*41

298. Interpolation by Second Differences.-The differences between the successive values-given in the Nautical Almanac as functions of time-are called the first differences; the differences between these successive differences are called the second differences; the differences of the second differences are called the third differences, &c. In simple interpolation we assume the function to vary uniformly; that is, we regard the first difference as constant, neglecting the second difference, which is, consequently, assumed to be zero. In interpolation by second differences we take into account the variation in the first difference, but we assume its variations to be constant; that is, we assume the second difference to be constant, and the third difference to be constant.

When the Nautical Almanac is employed we can take the second differences into account in a very simple manner. In this work, since the year 1863, the difference given for a unit of time is always the difference belonging to the instant of Greenwich time against which it stands, and it expresses, therefore, the rate at which the function is changing at that instant. This difference, which we may here call the first difference, varies with the Greenwich time, and (the second difference being constant) it varies uniformly, so that its value for any intermediate time may be found by simple interpolation, using the second differences as first differences. Now, in computing a correction for a given interval of Greenwich time, we should employ the mean, or average value, of the first difference for the interval, and this mean value, when we regard the second differences as constant, is that which belongs to the middle of the interval. Hence, to take into account the second differences, we have only to observe the very simple rule-employ that (interpolated) value of the first difference which corresponds to the middle of the interval for which the correction is to be computed.

299. Degree of Dependence.-The sun's declination changes nearly ' an hour, or 1" in 1, in March and September; hence to insure it to 1" in the extreme case, the Greenwich date must be true to TM.

EXAMPLES FOR PRACTICE.

Required the sun's declination in each of the following examples :

[These are preparatory to working Amplitudes, Azimuths, &c.}

In each of the following examples it is required to find the sun's declination when the sun is on the meridian (at apparent noon):

13. 1882, Jan. 19th, long. 86° 57′ W.

long 156 3 E. long. 72 47 E. long. 45 40 W. long. 110 57 W. long. 129 30 E. long. 1 17 E. long. 172 9 E. long. 68 15 W. long. 4 8 E. long. 100

19. 1882, July 28th,

long. 2° 0' W. long. 156 o E.

300. The Polar Distance of a heavenly body is its angular distance from the elevated pole of the heavens; it is measured by the intercepted arc of the hour circle passing through it, or by the corresponding angle at the centre of the sphere. According as the North or South pole is elevated, we have the North Polar Distance, or the South Polar Distance.

301. To find the Polar distance of a celestial object, proceed according to the following rule:

RULE LXXXII.

When the latitude of the place, and declination of the object, are of the same name subtract the declination from 90°; but when the latitude and declination are of contrary names, add the declination to 90°; the result in either case is the polar distance.

When the latitude is o, the declination, either added to or taken from 90°, is the polar distance.

TO FIND THE EQUATION OF TIME.

302. Apparent Solar Day is the interval between two successive transits of the actual sun's centre over the same meridian; it begins when that point is on the meridian. The apparent solar day is variable in length from two causes; first, the sun does not move uniformly in the ecliptic-its apparent path sometimes describing an arc of 57', and at other times an arc of 61' in a day; second, the ecliptic twice crosses the equinoctial—the great circle whose plane is perpendicular to the axis of rotation-and hence is inclined to it in its different parts; at the point of intersection the inclination is about 23° 27, at two other limiting points they are parallel. A uniform measure of time is obtained by the invention of the Mean Solar Day.

303. Mean Solar Day is the interval between two successive transits of the mean sun over the same meridian; it begins when the mean sun is on the meridian. This fictitious body is conceived to move in the equinoctial with the mean motion of the actual sun in the ecliptic. The length of the mean solar day is the average length of the apparent solar days for the space of a solar year.

304. Equation of Time is the difference between apparent and mean time. It is measured by the angle at the pole of the heavens between two circles passing, the one through the apparent sun's centre, the other through the mean sun. The Equation of Time is so called because it enables us to reduce apparent to mean, or mean to apparent time. In consequence of the motion of the sun in the ecliptic being variable, and the ecliptic not being perpendicular to the axis of the earth's rotation, apparent time is variable, and this fluctuation is considerable, amounting to upwards of half an hour-apparent noon sometimes taking place as much as 16m before mean noon, and at others as much as 141 after. These are the greatest values of the equation of time; it vanishes altogether four times a year-this occurring about April 15th, June 15th, September 1st, and December 24th. It is calculated and inserted in the Nautical Almanac for every day in the year. Cn page I of each month the equation of time given is that to be used in deducing mean from apparent time; that on page II is to be used in deducing apparent from mean time. The difference in the value of the two arises from the one being that at apparent noon, and the other that at mean noon. As these may be separated by an interval of more than a quarter of an hour, the equation of time given in pages I and II may differ by a quarter of the "Var. in 1 hour" given in the adjoining column. The equation of time is itself a portion of mean time.

305. To Reduce Equation of Time to Greenwich date.-The method of correcting the equation of time for the Greenwich date is similar to that for correcting the sun's declination, and the "Variation in 1 hour" may be used for the purpose.

RULE LXXXIII.

1°. Get a Greenwich date, as before.

NOTE.-The time by chronometer when error and rate are applied to it, gives Mean Time at Greenwich.

2o. Take out of Nautical Almanac (page II of the month) the Equation of Time for the noon of Greenwich date, and mark it additive or subtractive, according to the heading of Equation of Time at the top of the column in page I of the month; also note whether it is increasing-when affix the sign +; or decreasing affixing the sign —; at the same time take from the column in page I the "Var. in 1 hour.""

NOTE.-It sometimes happens that the precept for applying the Eq. of Time changes in the course of the month. Thus in April, 1882, a black line is placed between the Eq. T. for the 14th and that for the 15th, indicating that a change of precept occurs between those days. The Equations above the line, page I, have to be added, those below have to be subtracted.

I

3°. Multiply the "Var. in 1 hour" by the hours of Greenwich time, and when great precision is necessary, by the fractional parts of an hour also. The result is the correction to be applied to the equation of time taken from the Nautical Almanac, and is to be added when equation of time is increasing, but subtracted when equation of time is decreasing; the result is the Equation of Time sought. NOTE.-We may, as in reducing the declination (see preceding Rule LXXXIII), take the Eq. T. and "Var. in 1h" from the Nautical Almanac for the nearest noon to the Greenwich time, and multiplying the "Var. in rh" by the time that must elapse before noon; the correction thus obtained must be applied to the Eq. of T. taken out of Nautical Almanac in a contrary way to that directed above, that is to say, when correcting backwards the rule is Eq. T. increasing, subtract-Eq. T. decreasing, add. (See Exs. 3, 4, and 5).

(a) When the correction, being subtractive, exceeds the equation of time itself, subtract the equation of time from the correction; the remainder is the reduced equation of time sought-and it is to be subtracted from apparent time when equation of time at noon is directed to be added, but added to apparent time when equation of time at noon is directed to be subtracted; i.e., the Equation has to be applied to A.T. according to the precept for the day following the given day.

EXAMPLES.

Ex. 1. 1882, January 29th, 6h 53m 498 mean time at Greenwich; find Equation of time to be applied to apparent time in working the chronometer.

In working this example the "Var. for 1 hour" is taken from the Nautical Almanac from the column in page I of the month, and against the given day. The Greenwich date being mean time, take the equation of time from page II of the month, and mark it additive to app. time as directed at the top of the column in page I; also note that the equation is increasing. The Green. time being 6 54 or 6h9; hourly difference is multiplied by 69 giving the product 29187; and since there are three decimal figures in H.D. ('423) and one in Green, time ('9) in all four, four decimal places are marked off from the right hand of the product, the result 29187 or 29 is the correction to be applied to the Eq. of time at noon, and is to be added to it because it is that due to time elapsed since noon while the Eq. T. is increasing.

As the equation of time is not a uniformly varying quantity, it is not quite accurate to compute its correction as above, by multiplying the given hourly difference by the number of hours in the Greenwich time; for that process assumes that this hourly difference in the same for each hour. The variations in the hourly difference are, however, so small that it is only when extreme precision is required that recourse must be had to the more exact method of interpolation for second differences.