second station, K, and the telescope turned in the direction of the first station, D, the straight-edge is moved by an assistant until both are seen in line from K; their plane then passes through the first course; and if the line AB be prolonged to M and L, the line ML will be directly over the first course, and consequently its bearing will be that of the first course. By measuring the line E, the depth of the shaft may be found.

Let the theodolite, provided with a diagonal eye-piece, be planted over the station D, at the foot of the shaft, and after being levelled, let it be directed on the station K. Then, without changing the plane of vision, let the theodolite be directed to the top of the shaft, and let an assistant plant two flag rods, one at A and the other at B, both in the plane of vision, and let the line AB be prolonged to L and M, as before. The line LM will be in the same vertical plane with the first course, DK. Hence, as before, we may determine the hearing of the first course of the traverse.

Reducing the Traverse.

78. We may now find the azimuths of the various courses. Suppose the bearing of the first course to be S. 23° W.; then will its azimuth be equal to 180°+23°, or, 203°; the azimuths of each of the following courses, in order, may be found by the following

RULE.-Subtract 180° from the horizontal angle, observing that the remainder may be either + or ; add the result to the preceding azimuth; if the sum is negative, add 360° to it; if positive and greater than 360°, subtract 360° from it; the result thus found is the azimuth required.

Thus, in the given example, the horizontal angle between the. first and second courses is 176° 15'; this diminished by 180°, gives - 3° 45', which added to the preceding azimuth, 203°, gives 199° 15' for the azimuth of the second course. In like manner, we find the azimuth of the third course, 251° 30', of the fourth, 300°, of the fifth, 269° 15', and of the sixth, 272° 45'. These are the same as are given in the example, in Article 72.

Having found the data, as given in Article 72, we proceed to reduce the traverse as shown in the following

The form requires but little explanation. From the azimuth, that is, the angle of the course from the north point around by the east, the bearing is easily deduced. The length of the course on the slope, multiplied by the sine of the slope, gives the distance the course rises or falls, in feet, and the length multiplied by the cosine of the slope, gives the reduced length of the course, that is, the length that would have been found had the course been measured on a horizontal line. The bearing and reduced course being found, the latitudes and departures of the courses are found in the usual manner. In the example given, the resultant course descends 60.4 feet, its southing is 550 feet, and its westing 1404.5; hence, its bearing and length on the horizontal, may easily be found by known methods.

If the depth of the shaft is known, this, added to the depression of the resultant, in feet, gives the distance of the last station below the horizontal plane through the mouth of the shaft.

Method of Plotting the Traverse on the Surface.

79. To plot the traverse on the surface of the earth, we lay down the direction of the first course, as already shown: on this, measure off, in the usual manner, the reduced length of the first course, and mark the end of this distance, by a peg; plant the compass over the peg, set it so that the reading of the needle is equal to the bearing of the second course, and in this direction measure off the reduced value of the second course, and so on to the end. Then will the several pegs be exactly over the corresponding stations in the mine.

80. Suppose it were required to sink a shaft, so as to strike the gallery at station 6, in the example given in Article 76. It has been shown how to locate the point exactly over the station, (Art. 75). Let a line of levels be run from the mouth

of the first shaft, to this point, and find the difference of level corresponding to the two points. This with the depth of the station, in the mine, below the mouth of the shaft, will make known the depth of the shaft. This is a problem that frequently occurs in mining.

A similar problem often arises in railroad tunnelling. For example, if a tunnel is to be driven through a hill, it is often desirable to sink shafts, intermediate to the end headings, so as to strike the tunnel. Oftentimes, these shafts are used as starting-points for portions of the tunnel which are intended to meet the parts that are being opened from the headings.

Method of Plotting the Traverse on Paper.

81. To plot the traverse, on paper, we first plot the plan by the usual method of plotting compass-work, using the bearings and the reduced lengths of the courses. This gives the general direction of the horizontal projection of the traverse run; and from the measurements for cross section, the breadths of the gallery, on each side, may be plotted, and thence a complete plan of the mine may be constructed. We next plot the profile of the traverse, using, as in railroad plotting, two scales, one for horizontal distances, and the other and larger one, for vertical distances. The relation between the two scales will depend upon the circumstances of the case. Sometimes, both may be equal. The profile represents the undulation of the traverse, without reference to its horizontal deviations. Let us conceive vertical planes to be passed through all the courses. These will intersect each other in vertical lines. Take the one, through the first course, as the one on which the profile is to be delineated. Then, beginning with the plane through the last course, conceive the other planes to be revolved, in order,

with it, and so on till all are brought into coincidence with the fixed one. The lines of the traverse will then be situated in one plane, and a plot of them, in this position, will be the profile required. The distances from the traverse to the floor and roof of the gallery, at different points, enable us to complete the profile.