(2). Let the measured base, instead of being one straight line, consist of two straight lines, a and b, enclosing an angle π-0, where ✪ is very small. Required the correction to find c.

Or if be an angle of n", where n is very small,

[It is seldom necessary, in practice, to apply this cor

rection.]

(3).

From observations made at a point D, which is at a small known distance from a signal C, required to find the angle which A and B subtend at C.

Let CD a, and the observed angle ADB = A,

=

a = ▲ ADC, the angle subtended at D by the left hand object and the signal C.

AC=r, BC = l, the distances of the right and left hand objects from C;—which are approximately known from AB, and the observed angles BAC, ABC.

Let BCA, the required angle, = 0.

Then A+ DBE = exterior angle BEA

=0+ 4CAD;

.. 0-A, the reduction, = DBELCAD

A

{sin (4 + a) _ sin a }

. { sin (4 + a) _ sina},

when expressed in seconds.

r

(4). The altitudes or depressions, (which are small quantities), of two objects having been observed, and also the angle they subtend at a certain point, to find the horizontal angle they subtend. (Encycl. Metrop.)

Let Z be the Observer's zenith, CDRO his horizon, A and B the two objects; BOA = 0; 2 DOC, its reduction to the horizon, 0+x; CA=h, DB=h'.

=

But Cos CZD = cos (0 + x) = cos 0. cos x sin . sin x.

1

{

Ꮎ

4. sin 1" (p' sin 1")". tan- (q' sin 1")". cot

4. sin

2

[If p', q', be the number of seconds in the angles p and q.]

This reduction is not required when a Theodolite is used.

INSTRUMENTS USED IN SURVEYING.

82. THE VERNIER is a contrivance for subdividing equal divisions made on a straight line or a circle.

AB is a portion of a straight line, or a circle, which is divided by straight lines at right angles to it into any number of equal parts. CD, the Vernier, is another scale which slides along AB when AB is a straight line, and revolves round the centre of AB, when it is a circular arc.

If it be required to subdivide each of the divisions of AB into n equal parts, take QM=n-1 of these parts, and divide QM into n equal parts by straight lines at right angles to it, the magnitude of each of these parts will be 1 .a, if a be the length of a division of AB.

n

n

Let R, the 7th division from Q, coincide with a division on AB, then

Ex. If the original divisions were an inch, and nine inches were divided into 10 parts on the Vernier, then if the extremity of a line to be measured came to Q, and the inches marked at P were p, if the third division of the Vernier coincided with a division on the scale, the

length required would be

3

p+

inches.

10

83. THE SPIRIT-LEVEL is a glass tube DABE of circular bore, which is ground into the form of a circular arc of very large radius-sometimes 800 feet. It is then nearly filled with some fluid, and the ends are closed.

B

If the instrument be placed in a vertical plane, and the extremities D and E rest on a horizontal surface, the bubble (AB) of air left in the tube will be at the highest part of it, and if E be gradually raised the bubble will continually keep moving towards E*.

If there be a plane of an instrument, such as a theodolite, which it is necessary to bring into a horizontal position, it is provided with two levels, as nearly equal to one another in every respect as possible, which are placed at right angles to each other, and permanently attached to the plane. The instrument-maker marks the positions of the bubbles when the plane is horizontal, and if therefore on any occasion the bubbles occupy these positions, the plane to which the levels are attached must be horizontal.

If the plane be inclined at any angle to the vertical and the positions of the bubbles are noted, then if at a

* The grinding the bore of a level is done with a plug of metal covered with emery. The elasticity of the glass, assisted probably by that of the metal of which the plug is formed, enables the workman, by means of pressure on the outside, to wear away any particular portion of the interior surface he chooses. If, after the grinding is finished, the bubble be found to be always of the same length, and to move through equal lengths of the tube for equal increments of inclination, the bore of the tube is uniform, and the form of the tube truly circular.

If I be the length through which the bubble moves in consequence of an increase of n" in the inclination, (n being a small quantity,) the radius of the circular arc =

n.sin l'

==> 20625.

n

Ether is the best fluid with which levels can be filled; because in ether the bubble is found to come into a state of rest in the shortest time after a sudden displacement of it.