TABLES OF COORDINATE INCREMENTS FOR MINE SURVEYORS

G. F. LYSOV

(Presented by the scientific seminar of the departments of mine surveying and geodesy)

In his practical work, a mine surveyor has to devote a large part of his time to computing the coordinates of points from the results of measurements made.

In accordance with the requirements imposed on the computation of coordinates for underground surveys, in recent years five-place tables of natural values of trigonometric functions and five-place tables of logarithms, compiled by Prof. D. N. Ogloblin, have been published specifically for the needs of mine surveyors.

Meanwhile, a mine surveyor does not always have the opportunity to compute coordinates with the aid of an arithmometer. In this case, the logarithmic method of computing mine-surveying traverses, which is the most labor-intensive, can be replaced by determining coordinates with the aid of tables of coordinate increments.

However, the well-known tables by F. Gauss and Engineer M. L. Rudshtein, published for the needs of geodesists, in some columns of which the increments are given with two decimal places, do not satisfy mine surveyors in terms of accuracy, since in underground surveys coordinate increments are usually expressed in tens of meters and are computed with an accuracy to millimeters. This shortcoming of the tables mentioned has been eliminated in the tables of coordinate increments by I. F. Palenov, published in 1954 by Ugletekhizdat. But I. F. Palenov’s tables also are hardly used in mine-surveying practice. The chief reasons hindering the use of these tables are, first, the difficulty of finding the required arguments because of the very large size of the book (553 pages), and, second, the complexity of interpolation.

In view of the foregoing, we propose to compile and publish special tables of coordinate increments for mine surveyors, of smaller size, the general layout of which, the azimuthal arrangement, and the correction tables are presented in Appendix 1.

The proposed tables should be intended for computing coordinate increments by the formulas:

\[ \Delta x = S \cdot \cos \alpha = \pm S \cdot \cos R, \]

\[ \Delta y = S \cdot \sin \alpha = \pm S \cdot \cos R, \]

where \(S\) is the horizontal projection of the line, and \(\alpha\) and \(R\) are, respectively, the direction angle and the bearing of this line.

Cos 44° Δx (d)

Appendix 1

Sin  44°  \(\Delta y(h)\)

Corrections for tens of seconds

The tables should give products of the cosines and sines of angles from \(0\) to \(90^\circ\), at one-minute intervals, by line measures expressed as whole units from 10 to 99.

Each page of the tables is provided with an azimuthal inscription, owing to which the increment can be assigned for any directional angle of a line, without reducing it to a bearing.

On each left-hand page of the tables the values of \(\Delta x\) are placed for 20 minutes of the angular argument, at one-minute intervals from \(0^\circ\) to \(45^\circ\); on the right-hand page, the values of \(\Delta y\) for the same angle values. For angle values from 45 to \(90^\circ\), the increments \(\Delta x\) and \(\Delta y\) are given, respectively, on the right-hand and left-hand pages.

The required increments are found, for values of the angular argument lying within \(0^\circ\)—\(45^\circ\), by using the row of minutes located at the upper edge of the page (from left to right), and for arguments within \(45\)—\(90^\circ\), by using the row of minutes located at the lower edge of the page, in the direction from right to left.

The increments should be computed with the aid of six-place trigonometric tables and given with three decimal places. In the row “10” the increments should be given with four decimal places, which will make it possible to use this row when the length of the sides is more than 100 meters.

On each page it is also necessary to place two small tables of corrections for tens of seconds (since, in computing the coordinates of underground surveys, it is recommended that directional angles be rounded to tens of seconds), with two entries: (a) tabular differences—in the left vertical column; (b) tens of seconds—in the horizontal row above the corrections. Thus, in the upper small table corrections for 10 and 20 seconds will be given, and in the lower one for 40 and 50 seconds. There are no corrections for 30 seconds in the small tables, since they are easily found mentally (as half the tabular difference).

The proposed arrangement of the tables makes it possible to reduce their volume to 135—137 sheets (270—275 pages), with a sheet size of approximately \(25 \times 33\) centimeters.

Below we give an example of the use of the presented tables:

Example. The length of the line is \(S = 25.172\) meters, and the directional angle is \(\alpha = 44^\circ 43'20''\).

From the part of the tables given in the appendix we find:

\[ \Delta x = 17{,}885_{5}, \qquad \Delta y = 17{,}712_{7} \]

The example of computations given shows that, with the aid of these tables of increments, one can obtain a result accurate to millimeters.

In conclusion, let us note that such tables could be successfully used by mine surveyors for computing coordinates in those cases when it is not possible to use trigonometric tables and an arithmometer, and also when carrying out check computations.