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" 30"

* 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.

173 mean time.

The emersion takes place at Greenwich (Naut. Alm., p. XX.), at 14 20 558 Greenwich mean time; the difference 9" 21"5 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 theida. zontal 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

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 observed is

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,

the star being distinctly visible. The mode of proceeding is to commence about 15 minutes before the time of greatest elongation, and place the middle vertical wire alternately upon the star, and upon a signal nearly in the direction of the meridian, a mile or two distant, illuminated if the observation be at night. The readings are taken by the micrometer microscope, on the horizontal limb, both when the middle wire is upon the star and upon the signal, the difference of azimuth of which will be indicated by the difference of the reading, so that when the azimuth of Polaris, at the instant of each observation upon it, is known, the azimuth of the signal becomes known; the mean of all the results is taken as the true azimuth, and thus a line whose azimuth is fixed becomes determined on the ground, from which other azimuths may be differentiated.

The following is the mode of determining, at any instant, the

AZIMUTH OF POLARIS.

If we suppose a spherical triangle having for its three vertices the zenith, the pole, and the star; this triangle, at the time of the star's greatest elongation, will be right angled at the star; for if a cone be conceived having its vertex at the eye of the observer, and for its base the diurnal circle of the star, the tangent plane to this cone, passing through the star, is perpendicular to the declination circle through the star, which is a meridian plane of the cone; the visual or tangent plane through the star at its greatest elongation being a vertical plane, passes through the zenith, and, also passing through the star, determines on the celestial sphere a side zs of the spherical triangle zSP, so that the angle at s is therefore a right angle. In this right angled triangle are known zp the colatitude of the station, and ps the polar distance of the star, to find the hour angle P, and the azimuth z, at the time of greatest elongation. The former, applied to the time of the star's meridian transit or R. A. will give the time of greatest elongation. The formulas are

In which =

For the hour angle cos P = tan cot

For the azimuth, sin z = sin

polar distance, λ = colatitude.

cosec

If the star be observed within 45" of the time of the greatest elongation, the observation may be reduced by the formula

in which c is the correction of the azimuth, t the siderial time from elon

gation, and z the greatest azimuth. The correction is deduced in a manner similar to that on p. 302. Table XXXVI. may be made available as explained at the bottom of that page, or the constant log. 112.5 sin 1" = 6.7367274 may be used with the logs. of t and tan z. This correction being applied subtractively to the azimuth at the time of greatest elongation, computed as above, will give the azimuth at the time of

observation.

If the axis of the telescope be not horizontal, the correction for azimuth is, d being the value of one division of the level scale,

This consists in observing the zenith distances of two stations, and applying the corrections for curvature and refractions, to obtain their difference of level. The theory is simple, and the necessary formulas and table are found at pp. 50 to 54, Part I. of Lee Tables and Formulas.*

The usual mode of observing zenith distances is as follows: the instrument is carefully levelled, i. e., the vertical axis is placed truly vertical; the horizontal wire of the telescope is then pointed at the object, and the vertical circle read off; next the instrument is revolved 180° in azimuth, and the telescope being then moved through the double zenith distance of the object, is pointed again. If we now read off, the difference between the two readings will be 2 z. D.; the operation is, however, repeated (generally six times) if the vertical limb has the repeating motion, before it is read off again. The instrument should be levelled for each set of observations.

MAGNETIC OBSERVATIONS.

These usually accompany the operations of a Geodetic survey. They have for their object to determine, 1. The angle which the magnetic meridian makes with the astronomic meridian, commonly called the variation of the needle, but more properly the Declination. 2. The angle under which a needle suspended by a perfectly flexible thread at its centre of gravity, would be inclined to the horizon, commonly called the dip, but more properly the inclination; and 3. The intensity of the magnetic

* Immediately following (p. 55) are formulæ and tables for the barometric measurement of heights.