in the almanac, and for the lower marked L. If now the siderial time of transit be observed at any other station, this will be the DR. A. at the instant of observation (see p. 151), and the difference will be her variation in R. A. during the interval between the two transits, viz. that over the meridian of Greenwich and that over the meridian of the station. The meridian of the station (supposing it to be w. of Greenwich), has in this interval of time revolved by the diurnal rotation through an angular space equal to the longitude of the station from Greenwich, plus the distance which the moon has moved in R. A. towards the east. To know this angular space, we have only to compute the time occupied by the moon in changing her right ascension by the difference above mentioned. This may be done by means of the variation of the moon's R. A. for one hour, given at the same page of the Almanac* by proportion. EXAMPLE. Oct. 8th, 1840, sid. time transit D' 1. limb, Error of clock, too slow, True time of transit, R. A. D 1. limb (Nautical Almanac, D culm. stars), 23 192 7.75 23 1 16.95 23 1 23 31 Nautical Almanac gives D var. in R. A. in 1a, 122.59, .. 12259 : 1 :: 6'36 : 3′′ 6*8 which last term is the time occupied by the meridian of the station in revolving to that of Greenwich and 636 further, the last being the angular motion of the D in R. A., since it made its transit at Greenwich, is the longitude of the place of observation. The above method requires an exact knowledge of the siderial time. To obviate this necessity, the Almanac also gives the right ascensions of some stars which make their transit nearly at the same time with the moon, and differ little in declination from her, so as to be conveniently observed * If the distance of the station in long. from Greenwich be great, the variation in R. A. corresponding to the middle interval between the two transits should be used, which may be obtained by interpolation. The numbers in this column of the Almanac include the change of the semidiameter of the limb. in connexion with the moon. If the moon had no motion the difference of her right ascension from that of the star would be constant at all meridians; but in the interval of her transit over two different meridians, her right ascension will have varied, and the difference between the two compared differences will exhibit the amount of this variation,* which added to the difference of the meridians shows the angle through which the westerly meridian must revolve before it comes up with the moon. This angle, as before, will be the time in which the moon is undergoing the observed variation in R. A., which may be computed by means of her hourly variation in R. A. given in the Almanac. The variation of R. a. being subtracted from this result, the remainder will be the difference of longitude required. 12229 : 1 :: 63•27 : 3′′ 4o .. Long. required: 3m 46°•27 2′′ 5773. 17 18 4 6.27 When the meridian to be determined is distant from Greenwich, a very simple and unexceptionable way of proceeding is to assume the longitude which is supposed to be known approximately, and from the culminations of the moon's limb, as given in the Nautical Almanac, to find by interpolation the time of culmination of the limb at the assumed meridian. The difference between this and the observed time of culmination will be the interval of time occupied by the moon's limb in passing from the assumed meridian to the true meridian of the station. The motion of the limb in *For the determination of this variation with great accuracy, observations should be taken simultaneously at the different meridians to be compared. Errors in the computed places of the moon or stars are thereby avoided. The results given in the Almanac may be considered as a very near approximation to what would have been the indication of the Greenwich instruments, had the observations actually been made with them. The traveller has thus the opportunity of rendering his observation immediately available for determining his longitude with considerable accuracy. right ascension during this interval must be computed by first determining the hourly motion in right ascension, by interpolation, for the instant of passing the assumed meridian, and proceeding by proportion, as in the examples above. The result thus obtained being subtracted from the interval, the remainder will be the difference of longitude between the assumed meridian and the meridian of the station. In which A is the element for the noon, midnight, or complete hour preceding the given instant, y is the element required for the given time, m the given number of hours since noon or midnight, or minutes since the even hour (the long. in time of the assumed meridian above). n is 12 hours, 24 hours, or 60 minutes, the interval between the times, for which the element is given in the Nautical Alm. the difference between two consecutive elements in the Naut. Alm. d, the difference between the successive values of ô, For a convenient mode of proceeding, and an example under it where the meridian is distant from Greenwich, see Lee's Tables and Formulas, pp. 69-78, Part III. LONGITUDE BY ECLIPSES OF JUPITER'S SATELLITES. The eclipses of Jupiter's Satellites, especially the first, afford the readiest mode of obtaining the longitude, both from the frequent occurrence of the phenomena, and the simplicity of the calculation. * All that is necessary to be known is the exact time of observation; the difference between this time and the time at Greenwich shows the difference of longitude, and is east or west of Greenwich, according as the time of observation is greater or less than the Greenwich time. EXAMPLE. Suppose the emersion of Jupiter's first satellite to be observed August 8th, 1850, at Paris, and the time of observation there to be 14' 30TM This is given at p. XX. of the Nautical Almanac for each month. At p. 605 of the edition of 1850 is a full description of the page and its use. 17.3 mean time. The emersion takes place at Greenwich (Naut. Alm., p. XX.), at 14 20 558 Greenwich mean time; the difference 9" 215 is the difference of longitude between Greenwich and Paris. And because the time at Paris is greater than that at Greenwich, the former is east of the latter. ASTRONOMICAL DETERMINATION OF AZIMUTHS. In the previous pages the methods of determining difference of azimuths geodetically, or from the triangulation, have been given. But the usefulness of these methods depends on the implied ability to obtain by astronomic observation the azimuths of certain lines from which the others are differentiated. The method of proceeding is to determine by observation the difference of azimuth between the sun or a star, and the line whose azimuth is to be determined, then to find by calculation the azimuth of the sun or star; the sum or difference of these results will be the azimuth required. The difference of azimuth between the sun or star and the line whose azimuth is to be observed is obtained with an altitude and azimuth instrument, or theodolite. The middle vertical wire is made to bisect the star, or to touch the limb of the sun, and the siderial time is observed at the same instant; the reading is then taken on the horizontal limb of the instrument, which is afterwards turned to a signal (bearing a lamp, if at night), which is placed upon one of the sides of the triangulation, or upon any other convenient line, the horizontal angle between which, and the line whose azimuth is required, can be subsequently measured, and the reading of the horizontal limb again taken. The difference of the two readings will be the difference of azimuth between the sun's limb, or star and the signal, at the instant of siderial time above mentioned. With the altitude and azimuth instrument, the transit of both limbs of the sun, or the transit of the star, may be taken over all the wires of the instrument, and the mean of the times taken as the time at which the azimuthal position of the sun's centre, or the star, corresponded to the reading of the horizontal limb. AZIMUTH OF THE SUN OR A STAR. The determination of this requires merely the solution of the triangle zps, in which pz the colatitude of the place of observation, Ps the polar distance of the sun or star, and P the hour angle equal to the difference between the right ascension of the object and the siderial time of observation are given to compute the angle z, which is the azimuth required. Where the object is the sun, of course the value of the semidiameter at the instant must be computed and applied to the reading for the sun's limb to obtain that for his centre. If the altitude of the object is also observed at the same instant, or immediately before P の or after, and reduced to the instant by interpolation, one of the above data, either the hour angle or the latitude, may be replaced by the zenith, distance zs in the triangle. The formula, if the siderial time be unknown, and the altitude The best mode of obtaining the azimuth of a line upon the surface of the earth is by means of the pole star when at its greatest eastern or western elongation. With a telescope as powerful as that of the great theodolite, the necessary observations may be conducted in the day time, |