2. Required the eclipses that may be expected in 1823, and the times, nearly, at which they will take place, expressed in mean time at Greenwich. Ans. One of the moon on the 26th of January, at 5h. 24m. P. M.; one of the sun on the 11th of February, at 3h. 12m. A. M.; one of the sun on the 8th of July, at 6h. 50m. A. M.; and ore of the moon on the 23d of July, at 3h. 33m. A. M. PROBLEM XIV. To calculate an Eclipse of the Moon. Find the approximate time of full moon, by prob. XII., and, for this time, compute the sun's longitude, semidiameter, and hourly motion, and the moon's longitude, latitude, equatorial parallax, semidiameter, and hourly motions in longitude and latitude. Subtract the sun's longitude from the moon's, and call the remainder R. Also, subtract the hourly motion of the sun from that of the moon. Then, as the difference of the hourly motions: the difference between R and VI. signs :: 60 minutes: a correction. The correction, added to the approximate time of full moon, when R is less than VI. signs, but subtracted when it is greater, will give the true time of full moon for the meridian at Greenwich. Reduce this time to the time at the place for which the calculation is to be made, and call the reduced time T. For the Semidiameter of the Earth's Shadow. To the moon's equatorial parallax, add the sun's, which may be taken 9", and from the sum subtract the semidiameter of the sun. Increase the result by part, and it will be the semidiameter of the earth's shadow, which call S. 50 For the Inclination of the Moon's Relative Orbit. To the arithmetical complement of the logarithm of the difference between the hourly motions in longitude of the moon and sun, add the logarithm of the moon's hourly motion in latitude, and the result will be the log. tangent of the inclination, which call I. Add together the constant logarithm 3.55630, the log. cosine of I., and the arithmetical complement of the log. difference between the hourly motions of the moon and sun, in longitude, rejecting the tens in the index, and call the resulting logarithm R. For the Time of the Middle of the Eclipse. Add together the logarithm R, the logarithm of the moon's latitude at the true time of full moon, and the sine of I., rejecting the tens in the index, and the result will be the logarithm of an interval t, in seconds of time, which, added to T, when the latitude is decreasing, but subtracted when its increasing, will give the time of the middle of the eclipse. For the times of Beginning and End. To the logarithm of the moon's latitude at the true time of full moon, add the log. cosine of I., rejecting the tens in the index, and the result will be the logarithm of an arc, which call c. Call the moon's semidiameter d. To, and from, the sum of S and d, add and subtract c. Then add together the logarithms of the results, S + d + c and S + d C, divide the sum by 2, and to the quotient add the logarithm R, and the result will be the logarithm of an interval x, in seconds of time, which, subtracted from, and added to, the time of the middle, will give the times of the beginning and end. Note. If c is equal to, or greater than, the sum of S and d, there cannot be an eclipse. For the Times of Beginning and End of the Total Eclipse. To and from the difference of S and d, add and subtract c. Then add together the logarithms of the results, Sd+c and Sdc, divide the sum by 2, and to the quotient add the logarithm R, and the result will be the logarithm of an interval a', in seconds of time, which, subtracted from, and added to, the time of the middle, will give the times of the beginning and end of the total eclipse. Note. When c is greater than the difference of S and d, the eclipse cannot be total. For the Quantity, of the Eclipse. Add together the constant logarithm 0.77815, the logarithm of (S+ d —c), and the arithmetical complement of the logarithm of d, rejecting the tens in the index, and the result will be the logarithm of the quantity of the eclipse, in digits. Note 1. In partial eclipses of the moon, the southern part of the moon is eclipsed when the latitude is north, and the northern part when the latitude is south. 2. When the eclipse commences before sunset, the moon rises about the same time the sun sets. To obtain the quantity of the eclipse nearly, at the time the moon rises, take the difference between the time of sunset and the middle of the eclipse. Then, as 1 hour: this difference: : difference between the hourly motion of the moon and sun, in longitude : a fourth term. Add together the squares of this fourth term and of the arc c, both in seconds, and extract the square root of the sum. Use this root 1 instead of c, in the above rule, and it will give the quantity of the eclipse at the time of the moon's rising very nearly. When the eclipse ends after sunrise in the morning, the quantity at the time of the moon's setting may be found in the same manner, only using sunrise instead of sunset. 3. The relative positions of the earth's shadow and moon, at the time of the eclipse, may be easily represented. Let AB, Fig. 42, be a part of the ecliptic, and C the position of the centre of the earth's shadow at the time of full moon. Draw LCK perpendicular to AB, and make CM equal to the moon's latitude at the time of full moon, taken from a scale of equal parts, above AB if the latitude is north, but below if it is south. Draw Ma parallel to AB, and make it equal to the difference between the hourly motions of the moon and sun in longitude; and draw ac parallel to LK, above Ma, when the latitude is tending north, but below, when it is tending south. Then PQ, drawn through M and c, will represent the moon's relative orbit. Draw CN perpendicular to PQ, meeting it in H. Then will I be the place of the moon's centre at the middle of the eclipse. With the centre C and a radius equal to S, the semidiameter of the earth's shadow, describe the circle LNK, to represent the shadow. With the same centre and a radius equal to (S + d), describe arcs, cutting PQ in D and E, which will be the positions of the moon's centre at the beginning and end of the eclipse. With the centres D, H, and E, and a radius equal to d, the moon's semidiameter, describe circles to represent the moon's disc, at the beginning, middle, and end of the eclipse. When the eclipse is total, describe, with the centre C and a radius equal to (Sd), arcs, cutting PQ in F and G, which will be the positions of the moon's centre at the beginning and end of the total eclipse. EXAM. 1. Required to calculate, for the meridian of Philadelphia, the eclipse of the moon in July, 1823, The approximate time of full moon, is July 22, at 15h. 33m. |