W. long. Decl. increasing, add correction. E. long. Decl. increasing, subtract correction. NOTE. In March the sun's declination changes from S. to N., and in September it changes from N. to S.; if the correction is subtractive and greater than the declination, subtract the declination from the correction, and give the remainder the contrary name to the declination; these cases must be specially noted. Example. Find the sun's declination from the Nautical Almanac for apparent noon at place on the following dates, the longtitude being 100° 30′ W. 6h. 42m. = Var. in 1h. = 57":56 = 6.7h. 6.7 40292 34536 5° 1′ 19′′-2 N. 5 7 449 N. Example. Find the sun's declination from the Nautical Almanac for apparent noon at place on the following dates, the longitude being 154° 30′ E.=10h. 18m. = 10.3h. N.B. If the declination at Greenwich noon is o° o' o"; in east longitude the correction will be the declination of the same name as that of the day before; in west longitude the correction will be the declination of the same name as that of the day after. To find the Declination and Right Ascension of the Moon, at any given Greenwich date The declination and right ascension of the moon are given for every hour of Greenwich date, together with the "Var. for 10 minutes." Consequently by removing the decimal point of the variation one place to the left, you have the " Var. for 1 minute"; then RULE.-Multiply the "Var. for Im." by the minutes and decimal of a minute of the Greenwich time; this gives the correction required, which is to be added to, or subtracted from the quantity, according to whether it is increasing or decreasing. Example.-—Given the Greenwich date May 5d. 9h. 8m. 30s., required the moon's right ascension and declination. For the right ascension the variation in 10m. is 23.503s., hence for im. it is 2.3503s.; also 8m. 30s. of Greenwich time = 8.5m. For the declination the variation in 10m. is 105" 08, hence for Im. it is 10"-508. Here, also, as in the case of the sun, if the Greenwich time is nearer to a subsequent than to a preceding Almanac date, it will not only be more convenient, but more accurate, to interpolate back from the subsequent date Example.-Given the Greenwich date May 5d. 9h. 47m. 42s., required the moon's right ascension and declination. The variation for the right ascension in 10m. at 10 hours is 23.563s.; for Im. = 2.3563s. The variation for the declination in 10m. at 10 hours is 104"-11; for. IM.=10".411. Also 9h. 47m. 42s. is 12m. 18s. (or 12.3m.) from 10 hours; hence Var. in Im. = 2.3563 Moon's R.A. May 5d. at Ioh. 15 48 9:54 12.3 Cor. for 12 3m. 28-98 Interpolation by Second Differences.-This method is wholly unnecessary for sea computations, which assume the first differences variation in 1h. oi variation in 10m., to be constant; but for great accuracy interpolation is required by second differences. NOTE. The differences between successive first differences are called the second differences. To Correct the Sun's Declination by Second Differences.-The Nautical Almanac says the " Var. in 1 hour" (formerly called the hourly difference) is intended to facilitate the reduction of the quantities from noon to any other time; but it is the variation at noon, and requires to be reduced to midway between noon and the time at which the declination is required. The simplest and surest way to make this correction is to take the difference between the "Var. in 1 hour" for the given Greenwich day and that for the following day; divide this difference by 4, and the quotient by 12 (since 4 times 12=48), then multiply the last quotient by the Greenwich time and apply the product to the "Var. in 1 hour" for the given day, adding it if the "Var. in 1 hour" is increasing, otherwise subtracting it. Suppose the sun's declination and variation in 1 hour to be as follows Let it be required to find the sun's declination for Greenwich date mean time, December 19d. 14h. 24m. The difference between 2"-78 and 1" 60 is 1"-18, which divided by 4 gives 295, and 295 divided by 12 gives 0246; then 0246 multiplied by 14.4 gives 35424; and 35 (using only two decimals) subtracted from 2" 78 (since variation in 1 hour is decreasing) leaves 2" 43 for the correct "Var. in 1 hour." Finally, in the usual way, multiply 2" 43 by 14.4 (the Greenwich time), and 34" 99 is the correction of declination, to be added because declination is increasing; hence, declination for December 19d. 14h. 24m.=23° 26′ 31′′·2 S. Remember that twice in a year, viz., once in June, and once in December, the sum of the "Var. in 1 hour" for the two days will be their difference, = because in the interval the sun has attained its greatest declination and has begun to decrease. In the present case this occurs between the 21st and 22nd of December, when 43 +75 I-18 is the difference of the two variations. Then, proceeding as already directed, if the correction of variation is less than the "Var. in 1 hour" the correction of declination will be additive to the declination; but if the correction of variation is greater than the "Var. in 1 hour," subtract the variation from its correction, and the correction of declination when obtained must be subtracted from the declination, for the sun has passed its maximum declination. When the declination is about to change from S. to N. or from N. to S. the "Var. in 1 hour" is large and the difference of two successive variations very small; but when the declination is near a maximum the "Var. in 1 hour is small, and the difference of two successive variations large; at the maximum the sum of two variations will probably be their difference. To Correct the Equation of Time by Second Differences.-The remarks already made in respect to the sun's declination equally apply to the correction of the equation of time. Suppose, for two consecutive days, the equation of time and variation in 1 hour to be— and the equation be required for 15d. 14h. 24m. or 14·4h. Then, 023 is the difference of variation, which divided by 4 gives 00575, and this divided by 12 gives 00048; multiply 00048 by 14.4 and you get 006912; or, say 007 to be subtracted from 585, which leaves 578 for the corrected" Var. in 1 hour"; finally 578 multiplied by 14.4 gives correction of equation 8.32s. to be added to 13m. 46.78s.: hence, corrected equation of time will be 13m. 55·1s. Remember that four times in the year, viz., once in February, once in May, once in July, and once in November, the sum of the "Var. in I hour" for the two days will be their difference. When the equation of time is about to change from sub. to add. or from add. to sub. the "Var. in 1 hour" is large and the difference of two successive variations very small; but when the equation is near a maximum, the “ Var. in 1 hour" is small, and the difference of two successive variations large; at the maximum the sum of the two variations will probably be their difference, as already explained in connection with the sun's declination. To Correct the Moon's Declination and Right Ascension by Second Differences.-The moon's declination and right ascension are given for every hour of the day, Greenwich date, with the variation in 10 minutes; but this "Var. in 10m." must be reduced to midway between the time for which the declination or right ascension is required and the preceding hour. RULE. Take the difference between the "Var. in 10m." for the given Greenwich hour and that for the next hour; multiply this difference by the minutes (and decimal of a minute) of Greenwich time, and divide the product by 120 (twice 60); apply the result to the "Var. in 10m." for the given hour, adding it if the variation is increasing, otherwise subtracting it. Suppose the moon's right ascension and declination to be as follows Suppose it is required to find the declination and right ascension for Greenwich time, 1h. 40m. 42s., or 1h. 40.7m. For the declination the difference between 39" 17 and 37"-67 is 1"-5, which multiplied by 40·7 gives 61-05; then 61.05 divided by 120 gives ·509, or say 51 to be subtracted from 39′′-17, leaving 38" 66 as the proper variation in rom. for the declination. Proceeding in a similar manner for the variation in rom. for the right ascension you get 25-660s. as the proper variation in 10m. Having obtained the variation in 10m. you can understand that by removing the decimal point one figure to the left you get the “Var. in 1 minute.' Thus 3" 866 will be the Var. in 1 min. for the Decl. ; For the Correction of the Declination or Right Ascension. Multiply the variation in Im. by the given minutes (and decimal of a minute) of Greenwich time, and apply this correction in the usual way. Thus, for the declination, 3"-866 multiplied by 40.7 gives 157′′ 3462, or 2' 37" 3 to be added to the declination for 1 hour, since the declination is increasing; and the corrected declination for 1h. 40m. 42s. is 22° 56′ 10′′·9 S. When the correction is subtractive and exceeds the declination, take the declination from the correction, and change the name of the declination. Similarly, for the right ascension, 2.566s. multiplied by 407 gives 104.4362s. or Im. 44:44s. to be added to right ascension for 1 hour, since the right ascension is increasing; and the corrected right ascension for 1h. 40m. 42s. is 16h. 6m. 16·61s. The correction of right ascension is always additive, and when the application of the correction causes the right ascension to exceed 24 hours, reject the 24 hours, leaving only the minutes and seconds. When the moon's declination is about to change from N. to S., or from S. to N., the " Var. in rom." is large and the difference between two successive variations small; but when the declination is near a maximum (N. or S.) the "Var. in rom." is small but the difference between two successive variations is large. At the maximum the sum of two variations will be their difference; for example, suppose the " Var. in rom." to be as follows— |