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 λ

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.

force at any place; with the daily and other periodical variations in these three elements.

The instrument for observing the declination is called a declinometer or declination magnetometer. Where only the variation of the declination is to be observed, the instrument consists of a horizontal telescope, firmly supported, pointing towards a magnetic needle bearing a mirror so adjusted as to reflect a horizontal scale placed directly under the object glass of the telescope. The least change in the direction of the needle will be indicated by a change in the reading of the scale marked by the middle vertical wire of the telescope.

The best form of instrument for the measurement of absolute declination is a theodolite, or altitude and azimuth instrument, in front of which is suspended a collimator magnet, by fibres of untwisted silk, resting horizontally in a stirrup of gun metal. The collimator magnet is a hollow cylindrical magnet, with a small object glass like a telescope, and a horizontal scale at its focus.

The adjustments of this instrument consist in bringing the collimator magnet into the magnetic meridian without torsion of the thread; in determining the zero division of the scale corresponding to the magnetic axis of the collimator magnet; and in bringing the line of collimation of the theodolite telescope into the magnetic meridian, its vertical wire coinciding with this division. These adjustments are all made at once, by putting in a bar first, equal in weight to the collimating magnet, and adjusting the stirrup approximately; then, after restoring the collimating magnet by repeated trials, making half the necessary correction by moving the theodolite in azimuth, and half by turning the torsion screw at the top of the thread, till the same division of the scale is read, with the collimator in two positions, the second position being produced by turning the collimator over, so that it shall have revolved 180° about its optical axis. There is then no torsion of the thread, the axis of the collimator magnet and of the theodolite are both in the magnetic meridian, and the division read is the zero of the scale.

If in this position the verniers of the azimuth circle of the theodolite be read, and if its telescope then be turned in the direction of some object, whose azimuth is known or can be afterwards determined, the difference of the reading, added to or subtracted from the azimuth of the object, will give the absolute declination. The angular value of one division of the scale is determined by measuring with the theodolite the horizontal angle subtended by a certain number of the divisions, the magnet being temporarily fixed.

H

F

If a denote the angular value of one division of the scale, and

the ratio of the torsion and magnetic forces, the true declination changes

are deduced by multiplying the observed differences of reading by

H

a (1+). The value of is determined by turning the torsion through

F

two large angles, and noting the corresponding differences of reading. If w denotes the angular value of the former, and u that of the latter,

Requires for its determination, 1, experiments of deflexion, 2, of vibration. The former give the ratio of the magnetic moment of the deflecting magnet to the horizontal intensity, the latter the product of the same quantities, and their separate value is obtained by algebraic elimination.

Experiments of deflection consist in placing a magnetic bar, called a deflector, at one side of a freely suspended magnet, in a line drawn horizontally through the centre of the suspended magnet, perpendicularly to the magnetic meridian, its axis coinciding with this line. The deflector should be placed at three different distances from the suspended magnet on this line, in direct and reversed positions, or turned end for end, at each.

Experiments of vibration consist in suspending the same magnet which was used as a deflector, and noting the times at which some central division of the scale passes across the vertical wire of the telescope, at the beginning and end of at least 300 horizontal vibrations of the magnet, the magnet vibrating steadily in a very small arc.

As the time of vibration depends on the form and weight of the suspended mass as well as upon the product of the magnetic moment and horizontal intensity, its moment of inertia must be ascertained by means of a series of vibrations with two cylindrical weights of equal dimensions, whose moment of inertia is known, at opposite ends of the magnet.

If m denote the magnetic moment of the deflecting magnet, x the horizontal intensity, the formulas are

rdist. between centres of deflecting and suspended magnets in feet and decimals.

น=

angle of deflection obtained by multiplying half the mean of each

partial result by the coefficient (see above), a (1 +

H

66

66

=

r-radius of the cylinders in decimal of a foot.
2p their mass in grains."

=

THE INCLINATION

is found directly by the dipping needle, which consists of a magnetic needle, suspended at the centre of a graduated vertical circle. The mean must be taken of results with several needles, reversed on their magnetic axes, and reversed as to their poles by remagnetizing. Observations should be made at different azimuths to test the limb of the instrument, which is often magnetic, in which use the formula

The inclination may also be found by means of the horizontal and vertical components of the intensity, as it would be determined by the direction of their resultant. The vertical component is observed by means of a vertical force magnetometer, which is a needle suspended like the dipping needle, but placed in a plane perpendicular to the magnetic meridian, and made to vibrate in this plane.