2. Given the argument 10 13° 16' 54", to find the corresponding quantity in table XLVII. Ans. 93° 32 37′′. 10s 13° gives 93° 33′ 40′′.` The difference between 93° 33′ 40′′ and the next following quantity in the table, is 3' 43'. 3. Given the argument 4' 11° 57' 10", to find the corresponding quantity in table XXVII. Ans. 3° 24' 6". 4. Given the argument 3721, to find the corresponding quantity in table. XXXVII. Ans. 4' 52". PROBLEM III. To convert Degrees, Minutes, and Seconds of the Equator into Time. Multiply the quantity by 4, and call the product of the seconds, thirds; of the minutes, seconds; and of the degrees, minutes. EXAM. 1. Convert 72° 17′ 42′′ into time. 72° 17′ 42′′ 4h. 49m. 10sec. 48"". = 4h. 49m. 11sec. nearly. 2. Convert 117° 12' 30" into time. Ans. 7h. 48m. 50sec. 3. Convert 21° 52′ 27′′ into time. Ans. 1h. 27m. 30sec. PROBLEM IV. To convert Time into Degrees, Minutes, and seconds. Reduce the time to minutes, or minutes and seconds; divide by 4, and call the quotient of the minutes, degrees; of the seconds, minutes; and multiply the remainder by 15, for the seconds. * The student can work the proportion, either by common arithmetic, or by logistical logarithms, as he may prefer. EXAM. 1. Convert 5h. 41m. 10sec. into degrees, &c. h. m. sec. 5 41 10 60 4) 341 10 d.° 17′ 30′′ 2. Convert 7h. 48m. 50sec. into degrees &c. Ans. 117° 12′ 30′′. 3. Convert 11h. 17m. 21 sec. into degrees, &c. Ans. 169° 20′ 15′′. PROBLEM V. The Longitude of two Places, and the Time at one of them being given to find the corresponding Time at the other. Express the given time astronomically. Thus, when it is in the morning, add 12 hours, and diminish the number of the day by a unit. When the given time is in the afternoon, it is already in astronomical time. Find the difference of longitude of the two places, by subtracting the less longitude from the greater, when they are both of the same name, that is, both east, or both west; but by adding the two longitudes together when they are of different names. When one of the places is Greenwich, the longitude of the other is the difference of longitude. Then, if the place, at which the time is required, is to the east of the other place, add the difference of longitude, in time, to the given time; but if it is to the west, subtract the difference of longitude from the given time. The sum or remainder is the required time. Note. The longitudes of the places mentioned in the following examples, are given in table VI. EXAM. 1. When it is August 8th, 2h. 12m. 17sec. A. M. at Greenwich, what is the time as reckoned at Philadelphia? 2. When it is April 11th, 3h. 15m. 20sec. P. M. at New York, what is the corresponding time at Greenwich? 3. When it is Sept. 10th, 3h. 20m. 35sec. P. M. at Paris, what is the time as reckoned at new Haven? 4. When it is January 15th, 9h.12m.10sec. P. M. at Washington, what is the corresponding time at Berlin? Ans. Jan. 16, 3h. 13m. 52 sec. A. M. 5. When it is Oct. 5th, 7h. 8m. A. M. at Quebec, what is the time at Richmond? Ans. Oct. 5th. 6h. 43m. 18sec. A. M. 6. When it is noon of the 10th of June at Greenwich, what is the time at Philadelphia? Ans. June 10th, 6h. 59m. 20sec. A. M. PROBLEM VI. For a given mean time, to find the Sun's longitude, Semidiameter, Hourly Motion, the apparent Obliquity of the Ecliptic and the Earth's Radius Vector; also the Sun's Right Ascension and Declination and the Apparent time. For the Longitude. When the given time is not for the meridian at Greenwich, reduce it to that meridian by the last problem. With the mean time at Greenwich, take from Tables XXII, XXIII, and XXIV, the quantities corresponding to the year, month, day, hour, minute and second, and find their sums.* The sum in the column of mean longitudes will be the tabular mean longitude of the sun; the sum in the column of perigee, will be the tabular longitude of the perigee; and the sums in * In adding quantities that are expressed in signs, degrees, &c. reject 12 or 24 signs, when the sum exceeds either of these quantities. In adding any arguments, expressed in 100, or 1000, &c. parts of the circle, when they are expressed by two figures, reject the hundreds from the sun; when by three figures, the thousands; and when by four figures, the ten thousands. the columns I, II, III, and N, will be the arguments for the small equations of the sun's longitude, and for the equation of the equinoxes, which forms one of them. Subtract the longitude of the perigee from the sun's mean longitude, borrowing 12 signs when necessary; the remainder is the sun's mean anomaly. With the mean anomaly, take the equation of the sun's centre from table XXVII, and with the arguments I, II, and III, take the corresponding equations from table XXVIII. The equation of the centre, and the three other equations, added to the mean longitude, gives the true longitude, reckoned from the mean equinox. With the argument N, take the equation of the equinoxes, or which is the same, the nutation in longitude, from table XXX., and apply it, accor ding to its sign, to the true longitude already found, and the result will be the true longitude, from the apparent equinox. For the Hourly Motion and Semi-diameter. With the sun's mean anomaly, take the hourly motion and semi-diameter from tables XXV and XXVI. For the apparent Obliquity of the Ecliptic. To the mean obliquity, taken from table XXIX, apply, according to its sign, the nutation in obliquity, taken from table XXX, with the argument N, and the result will be the Apparent Obliquity. For the Earth's Radius Vector. With the sun's mean anomaly and the arguments I, II, and III, take the corresponding quantities from table XXXI, and the small table on the same page, and the sum of these will be the Radius Vector. For the Right Ascension and Declination. To the cosine of the apparent obliquity of the ecliptic, and the tangent of the sun's true longitude, and reject 10 from the index of the sum; the result will be the tangent of the Right Ascension, which must always be taken in the same quadrant as the longitude. To the sine of the apparent obliquity, add the sine of the longitude and reject 10 from the index of the sum; the result will be the sine of the Declination, which must be taken less than 90°; and it will be north or south according as its sign is affirmative or negative. For the Equation of Time and the Apparent Time. To the sun's tabular mean longitude, increased by 2°, apply according *The terms, Sine Cosine, &c., are to be understood here, and in the subsequent rules as implying the logarithmic Sine, Cosine, &c. * to its sign, the nutation in right ascension, taken from table XXX, with the argument N, and the result will be the sun's mean longitude from the true equinox. Take the difference between this longitude and the sun's right ascension, making it affirmative or negative according as the right ascension is greater or less than the longitude, and the result will be the Equations of Time in arc. This may be converted into time by Prob. III. The equation of time, applied according to its sign to the mean time, gives the apparent time. EXAM. 1. Required the sun's longitude, hourly motion, &c. on the 25th of October, 1836, at 10h. 37m. 10sec. A. M., mean time at Boston. * When one of these quantities is near 0° and the other to 360°, the less must be increased by 360°, and the sun be regarded as its value. |