Problem III.-To determine the length of a line both ends of which are inaccessible.

Let AB, Fig. 5, be the line, the ends A and B of which are inaccessible. Select two points P, Q from which both ends of the line can be seen, and at a distance from each other of about 300

or 400 ft. Measure the line PQ, and the angles K, L, M, and N. Then, from triangle APQ,

EXAMPLE.-If, in Fig. 5, the distance PQ is 400 ft., and the angles, as measured, are K=37° 10′, L=36° 30′, M = 52° 15′, N=32° 55', what is the distance AB?

SOLUTION. In the triangle APQ, R = 180° — (37° 10′+36°30′ +52° 15′) = 54° 05′, and

In the triangle BPQ, S=180°— (36° 30′+52° 15′+32° 55') =58° 20′, M+N=52° 15′+32° 55′ = 85° 10′, and

whence, (X-Y) = 15° 04′, and therefore

(468.30-390.53) cos 18° 35′

AB=

sin 15° 04'

283.58 ft.

LEVELING

SPIRIT LEVELING

Leveling is the process of determining the relative elevations of a series of points. There are three general methods of determining elevations, namely, gravity leveling, commonly called spirit leveling, and also designated as direct leveling; trigonometric leveling, also called indirect leveling; and barometric leveling.

The most highly developed form of the spirit level is the engineers' level. It consists essentially of a telescope, having a very accurate spirit level attached, mounted on a tripod and controlled by leveling screws in such a manner that the line of sight can be made truly horizontal. There are two general classes of engineers' levels, namely, the wye level, also written Y level, in which the telescope rests in Y-shaped supports from which it can be removed, and the dumpy level, in which the telescope is fixed. The wye level is much the more popular with American engineers because of the facility with which it can be adjusted, while the dumpy level is favored in Europe.

THE WYE LEVEL

The telescope

An engineers' wye level is shown in Fig. 1. AB rests in the Y-shaped supports Y, in which it is held firmly by semicircular clasps, commonly called clips; these are hinged at one end, and passing over the telescope are held at the other end by small pins. The lower ends of the wyes pass through the ends of the horizontal bar CD, sometimes called the level bar, and are adjustable vertically by means of the capstan-pattern nuts shown at C and D, which bear against the upper and lower surfaces of the bar. The bar CD is attached rigidly to the center or spindle, which turns in the socket V, permitting the telescope to be revolved in a horizontal plane. The spindle can be clamped by the screw K and the telescope then revolved slowly by means of the tangent screw t, which operates against a short projecting arm having

a spring bearing against its opposite side. The position of the socket V is controlled by the four leveling screws S, which, together with the lower leveling plate M, and the tripod P, are substantially the same as in a transit, except that a level does not commonly have a shifting center.

The telescope is in every respect similar to that of the transit except that it is longer, and having no horizontal axis, it cannot be revolved in a vertical plane.

The spirit level EF is also similar to that attached to the telescope of a transit, but in a leveling instrument, it is usually more accurate and sensitive. It consists of a hermetically sealed glass tube, curved slightly in a manner corresponding to the short upper arc of a large vertical circle, having the upper portion of its inner surface on a longitudinal section ground truly to the arc, and so nearly filled with alcohol, or a mixture of alcohol and ether, as to leave only a small bubble of

air. Alcohol is used extensively for the levels of surveying instruments, but is rather slow acting. Ether, though more sensitive and quick acting, is affected too greatly by changes of temperature to be used in surveying instruments. A mixture of alcohol and ether gives excellent results. Since the air bubble rises to the highest point of the inner surface of the level tube in which it is confined, and since the upper portion of the inner surface of the tube is ground truly to the arc of a circle in the plane of its longitudinal section, it follows that a tangent to this arc at the center of the bubble is a horizontal line. A line tangent to the inner upper surface of the bubble tube at its center is called the axis of the bubble, or axis of the level tube. When the bubble is in the center of the tube, this line will be tangent to the center of the bubble, and consequently, will be a horizontal line. Hence, the axis of the level tube is horizontal when the bubble is in the center of the tube.

Adjustments of the Wye Level.-There are three adjustments of the wye level, as follows:

1. To make the line of sight, or line of collimation, parallel to the axis of the collars, or rings, on the telescope by which it rests in the wyes.

2. To make the axis of the level tube bubble parallel to the axis of the collars, and, consequently, parallel to the line of collimation.

3. To make the axis of the level tube perpendicular to the vertical axis of the instrument, so that when the instru ment is leveled up the bubble will remain centered while the telescope is revolved horizontally.

First Adjustment.-Plant the tripod firmly; choose some distant and clearly defined point, the more distant the better, so long as it is distinctly visible and sharply defined. Remove the pins from the clips, clamp the spindle, and by means of the tangent screw and leveling screws bring the intersection of the cross-hairs to coincide exactly with the point sighted. Revolve or turn the telescope in the wyes through one-half a revolution, that is, until it is bottom side up. If the intersection of the cross-hairs is still on the point of sight, it shows that the line of sight coincides with the axis of the collars. But if, when the

telescope is turned bottom side up, the line of sight defined by the intersection of the cross-hairs is no longer on the point, move the cross-hairs by means of the capstan-headed adjusting screws so as to correct one-half the apparent error, being careful to move them in the opposite direction to which it would appear they should be moved. The apparent error shown by reversing the telescope is double the real error, as is illustrated in Fig. 2.

Suppose that with the instrument at A the line of sight given by the intersection of the cross-hairs is directed to the point B, and that when the telescope has been revolved or turned upside down in the wyes, the line of sight strikes the point C; then the distance BC will be double the real error,

B D

FIG. 2

and the true line of sight will be at D, half way between B and C. Sometimes both the horizontal and the vertical cross-hairs are out of adjustment, in which case they should be moved alternately until their intersection will coincide with the same point throughout a complete revolution of the telescope.

Second Adjustment.-The second adjustment consists of two parts, one lateral and the other vertical.

To adjust the level tube laterally, level up the instrument, remove the pins from the wyes, and open the clips; place the telescope over a pair of leveling screws and clamp the spindle. Bring the bubble exactly to the middle of the tube by means of the leveling screws and revolve the telescope in the wyes, first in one direction and then in the other, through about an eighth of a revolution. If the bubble runs toward one end of the tube when in the first position and toward the other end when in the second, it shows that the longitudinal axis of the bubble tube and the line of collimation, or longitudinal axis, of the telescope do not lie in the same plane. To correct the error, bring the bubble nearly to the center by means of the capstan-headed