Nomograms for Determining the Mean Errors in Measuring Horizontal Angles Due to Incorrect Centering of the Theodolite and Signals

G. F. Lysov

(Presented by the scientific seminar of the Department of Mine Surveying and Geodesy)

When assessing the accuracy of the most critical mine-surveying work, it is often necessary to calculate the squares \(m_i^2\) of the mean errors in measuring horizontal angles due to incorrect centering of the theodolite and signals, which is done by the formula:

\[ m_i^2=\frac{\rho^2 e_c^2}{2a^2}+\frac{\rho^2 e_c^2}{2b^2}+\frac{\rho^2 e_t^2 c^2}{2a^2 b^2}, \tag{1} \]

where \(e_c, e_t\) are the mean linear errors of centering of the signal and the theodolite,
\(a, b\) are the lengths of the sides of the angle being measured,
\(c\) is the distance between the signals,
\(\rho\) is the radian in seconds \((206265'')\).

Calculating the value \(m_i^2\) by formula (1) is too complicated. In this connection, the mine-surveying literature indicates a number of methods that make it possible to find the value \(m_i^2\) with less expenditure of labor and time.

In particular, in the book by Prof. D. N. Ogloblin [2], two graphs are given, with the aid of which the determination of \(m_i^2\) is simplified. But these graphs can be used only under the condition that the linear errors of centering of the signals and the theodolite are equal, which cannot be considered sufficiently justified.

Candidate of Technical Sciences A. N. Belolikov [1], in order to simplify finding the value \(m_i^2\), proposed a graph and special tables. However, for different values of the errors in centering the signals and the theodolite, the use of the graph and tables becomes more complicated, since it becomes necessary to construct auxiliary graphs or to carry out additional calculations.

Prof. F. F. Pavlov and Engineer D. E. Levit [3] created a series of nomograms that considerably facilitate the determination of \(m_i^2\). Without dwelling on the merits of these nomograms, we shall note the following two principal shortcomings of the latter.

1) The difference in the method for determining the quantities \(A_1\), \(A_2\), and \(A_3\) complicates the work of the computer¹.

2) The auxiliary quantity \(A_3\), from the given \(a\), \(b\), \(c\), and \(e_t\), is determined with considerable difficulty.

From the brief review given here of the best-known works on this question, it follows, therefore, that in determining \(m_i^2\) in the general case the methods considered require, in addition to the basic tables or graphs, the performance of additional calculations or graphical constructions, which may be a source of errors.

In the present work, to solve the same problem for any possible values of the variables \(e_c\), \(e_t\), \(a\), \(b\), and \(c\), two combined nomograms are proposed, constructed on the basis of the following considerations.

Let us write formula (1) in the following form:

\[ m_i^2=\frac{A_1+A_2+A_3}{B}, \tag{2} \]

where

\[ A_1=\frac{\rho^2 e_c^2 a^2}{2}, \tag{3} \]

\[ A_2=\frac{\rho^2 e_c^2 b^2}{2}, \tag{4} \]

\[ A_3=\frac{\rho^2 e_t^2 c^2}{2}, \tag{5} \]

\[ B=a^2 b^2. \tag{6} \]

Using formulas (2), (3), (4), (5), (6) and the well-known method for constructing nomograms from aligned points, one can construct two combined nomograms (Fig. 1). The first nomogram is intended for determining the quantities \(A_1\), \(A_2\), and \(A_3\) from given values \(a(b,c)\) within the limits from \(2\ \mathrm{m}\) to \(100\ \mathrm{m}\) and \(e_c(e_t)\) within the limits from \(0.2\ \mathrm{mm}\) to \(10\ \mathrm{mm}\). This nomogram consists of three parallel logarithmic scales “\(a,b,c\),” “\(e\),” and “\(A\),” whose scale and arrangement have been selected in such a way that, when a straightedge is placed on the nomogram at points with the given marks on the scales “\(a,b,c\)” and “\(e\),” the straightedge intersects the scale “\(A\)” at the point with the mark equal to the corresponding value \(A_1(A_2,A_3)\).

The second nomogram serves for determining the quantity \(B\) from given values of \(a\) and \(b\) within the limits from \(2\ \mathrm{m}\) to \(100\ \mathrm{m}\). The scales “\(a\)” and “\(b\)” of this nomogram are combined with the scales “\(a,b,c\)” and “\(e\)” of the first nomogram, and the scale “\(B\)” is placed next to the scale “\(A\).” The scale and arrangement of the scales “\(a\),” “\(b\),” and “\(B\)” have been selected in such a way that the straight line joining the points with the given marks on the scales “\(a\)” and “\(b\)” intersects the scale “\(B\)” at the point with the mark equal to the corresponding value \(B\).

The determination of \(m_i^2\) with the aid of the proposed nomograms is carried out in the following order:

Fig. 1.

Nomogram scales:

• \(a\) (in meters)
• \(a, b, c\) (in meters)
• \(B\)
• \(A\)
• \(b\) (in meters)
• \(e_c, e_t\) (in millimeters)

1) Using the first nomogram, we find the values \(A_1\) (from the given \(a\) in meters and \(e_c\) in mm), \(A_2\) (from the given \(b\) in meters and \(e_c\) in mm), and \(A_3\) (from the given \(c\) in meters and \(e_t\) in mm); we calculate the value of the sum \((A_1 + A_2 + A_3)\). In carrying out this operation it is convenient to use an adding machine, successively adding the values \(A_1\), \(A_2\), and \(A_3\).

2) Using the second nomogram, we find the value \(B\) (from the given \(a\) and \(b\) in meters).

3) With the aid of a slide rule, using formula (2), we calculate the value of \(m_u^2\).

Example. Determine \(m_u^2\) if \(a = 91\) m, \(b = 70\) m, \(c = 140\) m, \(e_c = 1.5\) mm, \(e_t = 2\) mm.

From the first nomogram we find: \(A_1 = 40000\), \(A_2 = 23000\), \(A_3 = 170000\), \((A_1 + A_2 + A_3) = 233000\). From the second nomogram we find \(B = 4000\).

Consequently,

\[ m_u^2 = \frac{233000}{4000} = 58.2 \ \text{sec}^2. \]

In conclusion, let us note that the proposed nomograms make it possible to determine \(m_u^2\) with an error not exceeding one or two units of the second digit, with an average expenditure of 1–2 minutes of time for one determination of \(m_u^2\).

LITERATURE

1. Belolikov A. N. On the question of determining the mean error of measuring angles in a mine, Investigations on Questions of Mine Surveying, collection XXV. Moscow, 1955.

2. Ogloblin D. N. Mine-Surveying Work in the Underground Development of Deposits, Part 1. Moscow, 1950.

3. Pavlov F. F. and Levit D. E. Atlas of Nomograms for Mine-Surveying Calculations. Moscow, 1954.

1. \(A_1=\left(\dfrac{\rho e_c}{a}\right)^2,\quad A_2=\left(\dfrac{\rho e_c}{b}\right)^2,\quad A_3=\left(\dfrac{\rho e_m c}{ab}\right)^2.\) ↩