In the accompanying figure Z represents the zenith, P the pole, M the observed place of the moon, and S that of the sun or star. The data given are M S, the measured angular distance; and and ZM and ZS the two zenith distances (or co-altitudes) from whence the angle MZS is found, the value of which is evi dently not affected by refraction or parallax, which, acting in vertical lines, cause the true place of the moon to be elevated above its apparent place (the parallax, from her vicinity to the earth, being a greater quantity than the correction for refraction), and that of the sun or star, to be depressed below its apparent place. Let M' and S' represent the corrected places of these bodies, and we have then Z M' and Z S-the zenith distances corrected for refraction and parallax-and the angle Z' before found, to find the true lunar distance M'S' in the triangle Z M'S'. The apparent time represented by the angle ZPS may be found in the triangle ZP S, having SS, P S, and ZP the co-latitude, if the exact error of the chronometer at the moment is not already known; and this time, compared with the Greenwich time at which the lunar distance is found from the Nautical Almanac to be the same, gives the difference of longitude east or west of the meridian of that place. The example below will show all the steps of the opera tion. On May 4, 1838, at 10h 41m 45s-8 by chronometer, the following observations were taken in latitude 51° 23'40 north, to find the longitude; the chronometer having been previously ascertained the same evening to be 3m 34s too fast. Double altitude-D 74° 42′ 35", taken with a sextant; index error- -22′′. Altitude Spica Virginis 28° 15′ 50′′-with alt. and az. inst.; index error-28". Distance 31° 25′ 55′′-with repeating circle. Barometer standing at 29′′-9, and thermometer at 61o. 1st-Then in the triangle Z M S we have the three sides to find the angle M Z S. S M M P Then to correct the zenith distances for refraction and parallax: ZS' = 61 46 23 to find M'S' the corrected lunar distance. By the Nautical Almanac, it appears that the Greenwich mean time answering to this distance, must be between 9 P.M. and midnight-the difference of distance answering to this interval of 3 hours, being Lunar dist. at 9 P.M. Greenwich 1°28′52′′ Prop. log. 3065* The difference between the prop. log. at 9 and midnight being 0, the correction of 2nd differences is nothing. Mr. Baily's formula for a lunar observation for longitude is as follows: x the true lunar distance required, *The interval of time past 9 P.M. might of course have been found by a common proportion, without the aid of prop. logarithms. The following example will also show the method of working out a lunar observation, by Dr. Young's formula, all the terms of which are cosines: Cos M'S' 2 cos(MH+ SK + MS) cos 1⁄2 (M H + SK — MS) cos M'H cos S'K cos MH cos SK -cos (M'H + S'K). MS 95 50 53 MH = 35 45 4 ar. comp. cos 0.090678 139 23 58 |