DETERMINATION OF WEIGHTS AND ERRORS IN MEASURING HORIZONTAL ANGLES, PRECALCULATION OF THE ORIENTATION ERROR THROUGH TWO VERTICAL MINE SHAFTS AND OF THE ERROR OF BREAKTHROUGH OF MINE-WORKING FACES (IN PLAN) BY A GRAPHICAL METHOD

V. I. Akulov

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

The form and length of the sides of underground theodolite traverses, when orienting through two vertical mine shafts and driving mine workings by opposing faces, are determined by the arrangement of the mine workings. Therefore, the lengths of the sides of underground theodolite traverses vary within very broad limits (from 5 to 100 meters and more).

The unequal length of the sides causes unequal-accuracy measurement of horizontal angles in underground theodolite traverses. In this connection, in the precalculation of expected errors in major mine-surveying operations—for example, orientation through two vertical mine shafts and the holing-through of a mine working—the horizontal angles must be regarded as measured with unequal accuracy.

The error¹ of each horizontal angle of an underground theodolite traverse must be determined by the well-known formula

\[ m_{\beta} = \pm \sqrt{ m_i^2 + \frac{e_c^2 \rho^2}{2a^2 b^2}(a^2+b^2) + \frac{e_T^2 \rho^2}{2a^2 b^2}(a^2+b^2-2ab\cos\beta) }, \tag{1} \]

where \(m_i\) is the error of the measurement of the angle itself;
\(a, b\) are the sides of the angle;
\(\beta\) is the angle being measured;
\(e_c, e_T\) are the linear errors of centering the signals and the theodolite;
\(\rho = 206265''\).

The error of measuring the angle itself (the instrumental error) is determined by the method of measurement and by the accuracy of the instrument, and is calculated by the formula

\[ m_i = \pm \sqrt{\frac{m_b^2}{2n^2}+\frac{m_v^2}{n}}, \tag{2} \]

or

\[ m_i = \pm \sqrt{\frac{m_b^2}{n}+\frac{m_v^2}{n}}, \tag{2a} \]

where \(m_o\) is the reading error;
\(m_v\) is the sighting error;
\(n\) is the number of repetitions, or sets.

By formula (2) the error of an angle measured by the method of repetitions is determined, and by formula (2a)—by the method of sets.

Calculating the error of angle measurement analytically by formula (1) is rather complicated. Therefore in practice, and also in the Technical Instructions for Mine-Surveying Work, 1959 edition, in preliminary calculations of expected errors in capital mine-surveying work, horizontal angles are assumed to have been measured with equal accuracy. In this connection the expected errors are established with insufficient precision.

The analytical method of precalculating expected errors in capital mine-surveying work, besides the complexity of the calculations, does not guarantee against possible gross mistakes.

In the literature, no checking methods have been developed for precalculating expected errors in capital mine-surveying work that would provide reliable control of the analytical method.

This paper proposes a graphical method for precalculating expected errors in capital mine-surveying work, which reliably checks the analytical method and is performed simply and easily with the aid of a ruler and compass.

1. Transformation of the general formula for the error of measuring a horizontal angle

We shall write formula (1) as follows:

\[ m_\beta = \pm \sqrt{m_i^2 + m_{\mathrm{ц}}^2}; \tag{1a} \]

\[ m_{\mathrm{ц}} = \pm \frac{\rho}{\sqrt{2ab}} \sqrt{e_c^2(a^2 + b^2) + e_t^2(a^2 + b^2 - 2ab\cos\beta)}, \tag{3} \]

where \(m_{\mathrm{ц}}\) is the error of angle measurement depending on the centering errors of the theodolite and signals.

Assuming

\[ p_\beta m_\beta^2 = p_i m_i^2 = p_{\mathrm{ц}} m_{\mathrm{ц}}^2 = m^2, \]

we obtain

\[ m_\beta = \pm m \sqrt{\frac{1}{p_\beta}}; \tag{1б} \]

\[ m_i = \pm m \sqrt{\frac{1}{p_i}}; \tag{2б} \]

\[ m_{\mathrm{ц}} = \pm m \sqrt{\frac{1}{p_{\mathrm{ц}}}}, \tag{3a} \]

where \(m\) is the mean square error of measuring an angle whose weight is equal to unity;
\(p_\beta\) is the weight of the measured angle;
\(p_i\) is the weight of the error of the angle measurement proper;
\(p_{\mathrm{ц}}\) is the weight of the angle error depending on the centering errors of the theodolite and signals.

From (1a, 1б, 2б, and 3a), we obtain

\[ \sqrt{\frac{1}{p_\beta}} = \sqrt{ \left(\sqrt{\frac{1}{p_i}}\right)^2 + \left(\sqrt{\frac{1}{p_{\mathrm{ц}}}}\right)^2 }. \tag{4} \]

Transform formula (3) as applied to the graphical method. Since

\[ c^2=a^2+b^2-2ab\cos\beta, \]

then

\[ m_{\mu}=\pm \frac{\rho}{\sqrt{2ab}}\sqrt{e_c^2(a^2+b^2)+e_T^2c^2}, \tag{3б} \]

where \(c\) is the distance between the end vertices of the angle being measured (Fig. 1).

Assuming

\[ \lambda=\frac{e_c}{e_T}\quad(\text{for } e_c \ll e_T) \tag{5} \]

and

\[ d=\sqrt{a^2+b^2}, \tag{6} \]

we obtain

\[ m_{\mu}=\pm \frac{\rho e_T}{\sqrt{2ab}}\sqrt{\lambda^2d^2+c^2}. \]

Let us denote

\[ D=\sqrt{\lambda^2d^2+c^2}, \tag{7} \]

then

\[ m_{\mu}=\pm \frac{\rho e_T D}{\sqrt{2ab}}. \tag{3г} \]

Introduce the notation

\[ x=\frac{ab}{D}. \tag{8} \]

Formula (3г), taking (8) into account, will take the form

\[ m_{\mu}=\pm \frac{\rho e_T}{\sqrt{2x}}. \tag{3д} \]

From (3а and 3д) we obtain

\[ \sqrt{\frac{1}{P_{\mu}}}=\frac{\rho e_T}{\sqrt{2mx}}. \tag{9} \]

Let us denote

\[ B=\frac{\rho e_T}{\sqrt{2m}}, \tag{10} \]

then

\[ \sqrt{\frac{1}{P_{\mu}}}=\frac{B}{x}. \tag{3е} \]

2. Determination of \(\sqrt{\frac{1}{P_{\beta}}}\) by the graphical method

In the graphical method for determining \(\sqrt{\frac{1}{P_{\beta}}}\), a diagram of the theodolite traverse is drawn at a scale, for example, \(1:500\), \(1:1000\), or \(1:2000\) (Fig. 1).

On the basis of the adopted method of measuring the horizontal angles of the traverse, the error of the actual angle measurement \((m_i)\) and the linear errors of centering the theodolite \((e_T)\) and the signals \((e_c)\) are established.

At point \(O\) (Fig. 2) we construct a rectangular coordinate system \((XOY)\). From the origin we draw the straight line \(OU\) at an angle \(\varphi\) to the \(x\)-axis, determined from the equation

\[ \cos \varphi = \lambda \;(\lambda \leqslant 1). \]

Fig. 1. Diagram of a theodolite traverse when orienting through two vertical shafts.

For example, with \(e_{\mathrm{T}}=\sqrt{2}\,e_c\), we have

\[ \lambda=\frac{e_c}{e_{\mathrm{T}}}=\frac{1}{\sqrt{2}}; \]

\[ \cos \varphi=\frac{1}{\sqrt{2}}; \]

\[ \varphi=45^\circ. \]

The determination of \(\sqrt{\dfrac{1}{P_{\beta_5}}}\), for example for angle \(\beta_5\) of the theodolite traverse (Fig. 1), is performed graphically in the following sequence.

Fig. 2. Determination of the value \(X\) for an angle by graphical means.

Along the coordinate axes (Fig. 2) we lay off the sides of angle \(\beta_5\), taken by the measurer from the traverse diagram (Fig. 1), namely: side \(a_5\) (segment \(4\!-\!5\), Fig. 1) is laid off from the origin along the \(x\)-axis, and side \(b_5\) (segment \(5\!-\!O_2\), Fig. 1) along the \(Y\)-axis; we obtain points 1 and 2 (Fig. 2). Segment \(1\!-\!2\) is laid off by the measurer from the origin along the line \(OU\),

we obtain point 3. We project point 3 onto the abscissa axis and obtain point 4. Side \(c_5\) (segment \(4—O_2\), Fig. 1), taken with a scale from the traverse diagram, is laid off from the coordinate origin along the ordinate axis, and point 5 is obtained. Segment \(4—5\) is laid off along the ordinate axis from point \(O\), and point 6 is obtained. We connect points 1 and 6 by a straight line. From point 6 we lay off, along the ordinate axis, side \(b_5\) (segment \(5—O_2\), Fig. 1, or segment \(0—2\), Fig. 2), and obtain point 7. From point 7 we draw the straight line \(7—8\) parallel to the abscissa axis until it intersects line \(1—6\), and obtain point 8.

It follows from the constructions that segment \(7—8\) is equal to \(x_5\), computed by formula (8).

To determine \(\sqrt{\dfrac{1}{P_{\mathrm{ц}}}}\), through point \(O\) (Fig. 3) we draw two mutually perpendicular straight lines \(OO_2\) and \(OA\). On line \(OA\) we lay off segment \(OA\), equal to a decimeter, and obtain point \(A\). Through point \(A\) we draw the straight line \(AC\), perpendicular to \(OA\).

By formula (10) we determine the value \(B\) and lay it off from point \(O\) along line \(OO_2\) (Fig. 3), at the scale at which the diagram of the theodolite traverse has been drawn, obtaining point \(O_1\). For example, for \(m_\beta = \pm 20''\) and \(\varepsilon_{\mathrm{т}} = 0.001\ \mathrm{m}\), the value \(B\), proceeding from formula (10), will be

\[ B=\frac{206265\cdot 0.001}{\sqrt{2\cdot 20}}=7.3\ \mathrm{m}. \]

If the diagram of the theodolite traverse has been drawn at a scale of \(1:500\), then segment \(OO_1\) (Fig. 3) is equal to \(14.6\ \mathrm{mm}\); if at a scale of \(1:1000\), then segment \(OO_1\) is equal to \(7.3\ \mathrm{mm}\).

Through point \(O_1\) we draw the straight line \(O_1B\) parallel to \(OA\).

Segment \(7—8\) (Fig. 2), i.e., \(x_5\), is laid off with a scale from point \(A\) on the straight line \(AC\), and point \(a_5\) is obtained (Fig. 3). We connect points \(O\) and \(a_5\) by a straight line. We establish point \(b_5\)—the point of intersection of the straight lines \(Oa_5\) and \(O_1B\).

Segment \(O_1b_5\), expressed in decimeters, as follows from the constructions and formula (3e), is the value

\[ \sqrt{\frac{1}{P_{\mathrm{ц}}}} \]

for the given angle.

Fig. 3. Determination of \(\sqrt{\dfrac{1}{P_{\mathrm{з}}}}\) by graphical means.

The angular error dependent on the centering error of the theodolite and signals is determined by formula (3a).

To determine \(\sqrt{\dfrac{1}{P_{\beta}}}\), we lay off from point \(O_1\) (Fig. 3), along the line \(O_1O_2\), \(\sqrt{\dfrac{1}{P_i}}\) in fractions of a decimeter, obtaining point \(O_2\).

The value \(\sqrt{\dfrac{1}{P_i}}\), proceeding from (2b), is determined by the formula

\[ \sqrt{\frac{1}{P_i}}=\frac{m_i}{m}. \tag{11} \]

Thus, for example, with \(m_i=\pm 13''\) and \(m=\pm 20''\),

\[ \sqrt{\frac{1}{P_i}}=\frac{13}{20}=0.65. \]

Therefore the segment \(O_1O_2\) (Fig. 3) will be equal to \(0.65\ \mathrm{dm}\), or \(65\ \mathrm{mm}\). We lay off the segment \(O_2b_i\) (Fig. 3) from point \(O\) on the line \(OA\), and obtain point 5. From the constructions and formula (4) it follows that the segment \(0\text{–}5\) (Fig. 3), expressed in decimeters, is \(\sqrt{\dfrac{1}{P_{\beta}}}\) for the given angle.

The total error in measuring the angle is computed by formula (1b).

3. Precalculation of the orientation error through two vertical mine shafts by a graphical method

The orientation error through two vertical mine shafts, as is known, is understood as the error of the direction angle of the initial side of the underground survey.

The initial side of the underground survey is taken to be the side of the underground connecting traverse from which the theodolite survey of the horizon is developed.

The error of the direction angle of the initial side of the underground connecting traverse (without taking into account the errors of connection to the surface and projection) is determined by the formulas:

\[ M_{\alpha}=\pm \sqrt{M_{q_{\beta}}^{2}+M_{q_l}^{2}}; \tag{12} \]

\[ M_{q_{\beta}}=\frac{1}{C}\sqrt{\sum_{1}^{q-1} S_y^{2}m_{\beta}^{2}+\sum_{q}^{n-1} R_y^{2}m_{\beta}^{2}}; \tag{13} \]

\[ M_{q_l}=\pm \frac{\rho}{C}\sqrt{\sum_{1}^{n} m_l^{2}\sin^{2}\varphi}, \tag{14} \]

where \(M_{q_{\beta}}\) and \(M_{q_l}\) are the errors of the direction angle of the initial side as a function of the errors in measuring angles and side lengths, respectively;

\(m_{\beta}\) is the error in measuring a horizontal angle;

\(m_l\) is the error in measuring the length of a side as a function of random errors;

\(R_y\) — the projection, onto the direction of the plumb-line alignment \(O_1O_2\), of the distance from the vertex of the underground connecting traverse to plumb line \(O_2\);

\(S_y\) — the projection, onto the direction of the plumb-line alignment \(O_1O_2\), of the distance from the vertex of the underground connecting traverse to plumb line \(O_1\);

\(C\) — the distance between the plumb lines;

\(q\) — the ordinal number of the initial side;

\(n\) — the number of sides in the underground connecting traverse;

\(\rho = 206265''\).

To precompute the error of the direction angle of the initial side of the underground survey, we draw up, to scale—for example, \(1:500\), \(1:1000\), or \(1:2000\)—a scheme of orientation through two vertical mine shafts and determine, on the basis of the position of the opening mine workings, the initial side of the underground survey. For example, for the scheme (Fig. 1), side 3–4 \((q=4)\) is taken as the initial side.

A. Determination, by a graphical method, of the error of the direction angle of the initial side as a function of the error of angle measurement

Formula (13), taking (16) into account, is written as follows:

\[ M_{q_i}=\pm \frac{m}{C} \sqrt{ \sum_{1}^{q-1}\left(S_y\sqrt{\frac{1}{P_\zeta}}\right)^2 + \sum_{q}^{n-1}\left(R_y\sqrt{\frac{1}{P_\zeta}}\right)^2 }. \tag{13a} \]

Since

\[ \begin{aligned} S_y &= s_y \cdot N;\\ R_y &= r_y \cdot N;\\ C &= c \cdot N, \end{aligned} \tag{15} \]

then

\[ M_{q_i}=\frac{m}{c} \sqrt{ \sum_{1}^{q-1}\left(s_y\sqrt{\frac{1}{P_\zeta}}\right)^2 + \sum_{q}^{n-1}\left(r_y\sqrt{\frac{1}{P_\zeta}}\right)^2 }, \tag{13б} \]

where \(r\) is the magnitude of the projection \(R\) on the scheme;

\(s\) is the magnitude of the projection \(S\) on the scheme;

\(C\) is the distance between the plumb lines on the scheme;

\(N\) is the denominator of the scale of the orientation scheme.

Let us denote

\[ s' = s_y\sqrt{\frac{1}{P_\zeta}}; \]

\[ r' = r_y\sqrt{\frac{1}{P_\zeta}}. \tag{16} \]

In the adopted notation, formula (13б) will have the form

\[ M_{q_i}=\pm \frac{m}{c} \sqrt{ \sum_{1}^{q-1}(s')^2 + \sum_{q}^{n-1}(r')^2 }. \tag{13в} \]

For convenience in determining the error \(M_{g_\beta}\) by a graphical method, let us put

\[ s' = k s''; \]

\[ r' = k r'', \tag{17} \]

where \(k\) is a constant coefficient.

From (13в and 17) we shall have

\[ M_{g_\beta} = \pm \frac{m \cdot k}{c} R_\beta, \tag{13г} \]

\[ R_\beta = \pm \sqrt{\sum_1^{q-1}(s'')^2 + \sum_q^{n-1}(r'')^2}. \tag{18} \]

The error of the direction angle of the initial side, for example \(3—4\) (Fig. 1), as a function of the error in measuring the angles, is determined graphically in the following way.

For each angle of the underground connecting traverse, by the graphical method considered in § 2, we determine \(\sqrt{1/P_\beta}\) and lay them off in fractions of a decimeter on the line \(OA\) (Fig. 3); we obtain points 1, 2, 3, 4, and 5.

On the line \(O_2O\) (Fig. 3) we lay off the segment \(OO_3\), equal to \(k\) decimeters, for example, 1, 2, 3. We project the vertices of the underground connecting traverse onto the line of alignment of the plumb lines \(O_1O_2\) (Fig. 1); we obtain the points \(1'\), \(2'\), \(3'\), \(4'\), and \(5'\).

As the initial side of the underground survey we have taken the side \(3—4\); therefore the projections onto the line of alignment of the plumb lines must be taken toward the plumb line \(O_1\) for vertices 1, 2, and 3, and toward the plumb line \(O_2\) for vertices 4 and 5, namely: \(s_1 = O_1 1'\), \(s_2 = O_1 2'\), \(s_3 = O_1 3'\), \(r_4 = O_2 4'\), and \(r_5 = O_2 5'\).

The projections \(s_1\), \(s_2\), \(s_3\), \(r_4\), and \(r_5\), taken with a measuring device from the orientation scheme (Fig. 1), are laid off from point \(O_3\) on the line \(O_3O\) (Fig. 3); we obtain the points \(1'\), \(2'\), \(3'\), \(4'\), and \(5'\). We connect point \(O_3\) by straight lines with points 1, 2, 3, 4, and 5. From the points \(1'\), \(2'\), \(3'\), \(4'\), and \(5'\) we draw, perpendicular to the line \(O_3O\), the straight lines \(1'1_0\), \(2'2_0\), \(3'3_0\), \(4'4_0\), and \(5'5_0\) until they intersect, respectively, the lines \(O_3 1\), \(O_3 2\), \(O_3 3\), \(O_3 4\), and \(O_3 5\); we obtain the points \(1_0\), \(2_0\), \(3_0\), \(4_0\), and \(5_0\).

From the segments \(1'1_0\), \(2'2_0\), \(3'3_0\), \(4'4_0\), and \(5'5_0\) we construct a rectangular polygon (Fig. 4). The rectangular polygon is constructed in the following way.

Through point \(O\) (Fig. 4) we draw the straight line \(O1\), on which we lay off with the measuring device the segment \(1'1_0\) (Fig. 3); we obtain point 1. We draw, perpendicular to the line \(O1\), the straight line \(1—2\) and lay off on it the segment \(2'2_0\) (Fig. 3); we obtain point 2, which we connect with point \(O\). We draw, perpendicular to the line \(O2\), the straight line \(2—3\), on which we lay off the segment \(3'3_0\) (Fig. 3); we obtain point 3. We connect point 3 by a straight line with point \(O\). The further construction of the rectangular polygon is performed analogously, as for point 3.

Fig. 4. Rectangular polygon for determining \(R_\beta\)

In the rectangular polygon (Fig. 4), the sides \(O—1,\ 1—2,\ 2—3,\ 3—4\), and \(4—5\) are respectively equal to the segments \(1'1_0,\ 2'2_0,\ 3'3_0,\ 4'4_0,\ 5'5_0\) (Fig. 3), and the angles \(O—1—2,\ O—2—3,\ O—3—4\), and \(O—4—5\) are right angles. From the constructions and formula (18) it follows that the closing side of the rectangular polygon \(O—5\) (Fig. 4) is equal to \(R_\beta\) (formula 18).

To determine the error \(M_{q\beta}\), through point \(O\) (Fig. 5) we draw the straight line \(OB\). On \(OB\) we set off the distance between plumb lines, taken by measurement from the orientation diagram (Fig. 1), and obtain point \(A\) (segment \(OA\) is equal to segment \(O_1O_2\), Fig. 1); setting off the closing side of the rectangular polygon \(R_\beta\) (\(R_\beta\) is equal to segment \(O—5\), Fig. 4), we obtain point \(B\). Through point \(A\) we draw the line \(AD\) perpendicular to \(OB\). On the line \(AD\) we set off, to scale—for example, \(1\ \text{sec}—1\ \text{mm}\)—the value \(\kappa m\), where \(\kappa\) is the length of the segment \(OO_3\) (Fig. 3) in decimeters, and \(m\) is the error of measurement of an angle whose weight is equal to unity; we obtain point \(D\). For convenience in the graphical constructions, the error \(m\), taking into account that it enters into formula (10), must be taken equal to \(20''\).

Through point \(B\) (Fig. 5) we draw the straight line \(BE\) parallel to \(AD\), up to its intersection with the line \(OD\), and obtain point \(E\). From the constructions and formula (13г) it follows that the segment \(BE\) corresponds to the error \(M_{q\beta}\) on the scale in which the segment \(AD\), i.e., the value \(\kappa m\), was constructed.

Fig. 5. Determination of the error \(M_{q\beta}\) by the graphical method.

When precomputing the error of the direction angle of the first side of the underground connecting polygon, the graphical constructions are carried out analogously to those used in determining the error of the direction angle of the initial side with number \(q\), except that for all vertices of the polygon the projections \(R_y\) are taken to the plumb line \(O_2\).

В. Determination by the graphical method of the error of the direction angle of the initial side as a function of the error in measuring side lengths

The error of the direction angle of the sides of the underground connecting polygon as a function of the errors in measuring the lengths of its sides is the same for all sides of the polygon and is computed by formula (14).

The error in measuring the side lengths is determined by the formula

\[ m_l=\pm a\sqrt{L}, \tag{19} \]

where \(L\) is the length of the side in meters,

\(a\) is the coefficient of random influence \(\left(m^{\frac{1}{2}}\right)\).

For the graphical determination of the error \(m_l\), we write formula (19) as follows:

\[ m_l=\pm a\sqrt{10^3Nl}\ (\text{mm}), \tag{19a} \]

where \(l\) is the length of the side, taken from the orientation diagram, in mm;

\(N\) is the denominator of the scale of the orientation diagram.

The error \(m_l\) (formula 19a) is determined graphically as follows. From point \(O\) (Fig. 6) we draw two mutually perpendicular straight lines \(OA\) and \(OE\). On the straight line \(OE\) we set off from point \(O\) 10, 20, 30, 40, 50, etc. millimeters, obtaining points \(a,\ b,\ c,\ d,\ e\), etc.

Through the indicated points we draw the straight lines \(aa_1, bb_1, cc_1, dd_1, ee_1\), etc., perpendicular to the line \(OE\). On these straight lines we lay off the segments \(aa_1, bb_1, cc_1, dd_1, ee_1\), etc., respectively equal to \(\sqrt{10}, \sqrt{20}, \sqrt{30}, \sqrt{40}, \sqrt{50}\), etc., millimeters, obtaining the points \(a_1, b_1, c_1, d_1, e_1\), etc. By joining the indicated points and the point \(O\) with a smooth curve, we obtain the curve \(OF\).

On the straight line \(OA\) we lay off the segment \(OA\), equal to \(a\sqrt{10^3 N}\) decimeters. For example, for \(a = 0.0005\) and \(N = 2000\), the segment \(OA\) will be

\[ OA = 0.0005\sqrt{10^3 \cdot 2000} = 0.70 \text{ dm}. \]

In view of the small magnitude of the error \(m_l\), in what follows, for convenience in graphical constructions, we shall determine the error of the length of a side increased \(Q\) times \((Q = 1, 2, 3, 4, 5)\).

Fig. 6. Determination of the error \(m_l\) by a graphical method.

On the line \(OA\) (Fig. 6) we lay off the segment \(OB\), equal to \(\dfrac{1}{Q}\) decimeters. For example, for \(Q = 5\), \(OB = \dfrac{1}{5} = 0.2\) dm. Through the points \(A\) and \(B\) we draw the straight lines \(AD\) and \(BC\) parallel to \(OE\).

To determine the error of the length of a side, for example 4–5, we take from the orientation scheme (Fig. 1), with a ruler, the length of side 4–5 and lay it off on the line \(OE\) from the point \(O\) (Fig. 6); we obtain point 1. Through point 1 we draw the straight line \(1—1_0\), parallel to \(OA\), to its intersection with the curve \(OF\); we obtain point \(1_0\). We measure the segment \(1—1_0\) and lay it off on the line \(BC\) from the point \(B\); we obtain point 2. We draw the straight line \(O—2\) to its intersection with the line \(AD\); we obtain point 3.

The segment \(A3\), as follows from the constructions and formula (19a), is equal to \(Qm_l\), which we denote by \(M_l\), i.e.

\[ M_l = Qm_l. \tag{20} \]

Hence

\[ m_l = \frac{M_l}{Q}. \tag{21} \]

From (14 and 21) we obtain

\[ M_{q_l}=\pm \frac{\rho}{CQ}\sqrt{\sum M_l^2 \sin^2 \varphi}, \tag{14a} \]

or

\[ M_{q_l}=\pm \frac{\rho}{cNQ}\,R_l; \tag{14b} \]

\[ R_l=\sqrt{\sum M_l^2 \sin^2 \varphi}, \tag{22} \]

where \(c\) is the distance between the plumb-line axes from the orientation diagram, in mm;
\(N\) is the denominator of the scale of the orientation diagram.

The error \(M_{q_l}\) is determined graphically in the following manner. Through point \(O\) (Fig. 7) we draw the straight lines \(O1, O2, O3, O4, O5\), and \(O6\), respectively parallel to the sides \(O_1—1, 1—2, 2—3, 3—4, 4—5\), and \(5—O_2\) of the underground connecting traverse (Fig. 1). On the indicated straight lines we lay off the errors of measuring the lengths of the sides, taken with a compass from Fig. 6; for example, the segment \(O5\) (Fig. 7) is equal to the segment \(A3\) (Fig. 6). We obtain the segments \(O1, O2, O3, O4, O5\), and \(O6\). Through point \(O\) we draw the straight line \(AB\) parallel to the line of the plumb-line alignment \(O_1O_2\). We project points 1, 2, 3, 4, 5, and 6 (Fig. 7) onto the line \(AB\), obtaining the points \(1', 2', 3', 4', 5'\), and \(6'\).

Fig. 7. Determination of the projections of the errors in measuring side lengths onto the direction perpendicular to the plumb-line alignment.

Fig. 8. Rectangular polygon for determining \(R_l\).

From the segments \(11', 22', 33', 44', 55'\), and \(66'\) (Fig. 7), i.e., the projections of the errors in measuring the side lengths onto the direction perpendicular to the plumb-line alignment, we construct a rectangular polygon (Fig. 8), in which \(Oa=11'\), \(ab=22'\), \(bc=33'\), \(cd=44'\), \(de=55'\), and \(ef=66'\).

The closing side of the rectangular polygon \(Of\) (Fig. 8), as follows from the construction and formula (22), is \(R_l\) (formula 22).

On the straight line \(OB\), from point \(O\) (Fig. 9), we lay off the distance between the plumb-line axes, taken with a compass from the orientation diagram (Fig. 1), obtaining point \(A\), and from the closing side \((R_l=Of)\) of the rectangular polygon (Fig. 8), we obtain point \(B\). Through points \(A\) and \(B\) we draw the straight lines \(AC\) and \(BD\) perpendicular to \(OB\). On the straight line \(AC\) we lay off, to scale, for example, \(1\ \text{sec}=1\ \text{mm}\), the value

\[ \frac{\rho}{NQ}, \]

obtaining point \(C\). For example, when \(N=2000\) and \(Q=5\), we have

\[ \frac{\rho}{NQ}=\frac{206265}{2000\cdot 5}=20.6\ \text{sec}. \]

On the scale of 1 sec—1 mm, the segment \(AC\) (Fig. 9) is equal to 20.6 mm. We determine point \(D\) (Fig. 9)—the point of intersection of the straight lines \(OC\) and \(BD\).

The segment \(BD\), as follows from the construction and formula (14б), corresponds to the error \(M_{q_l}\) on the scale on which the segment \(AC\) was constructed.

Fig. 9. Determination of the error \(M_{q_l}\) by the graphical method.

We construct a right triangle with legs \(M_{q_\beta}\) and \(M_{q_l}\). The hypotenuse of the triangle corresponds to the error \(M_q\), computed by formula (12).

In accordance with § 269 of the Technical Instructions for Mine-Surveying Work, 1959 edition, the error of the direction angle of the initial side (without taking into account the errors of design and of transfer to the surface) must not exceed \(\pm 60''\).

4. Precalculation of the Error of Meeting of Mine Working Faces in Plan by the Graphical Method

The error of meeting of the faces of a mine working along the responsible direction in plan is computed by the formula:

\[ M_x = \pm \sqrt{M_{x_\beta}^{2} + M_{x_l}^{2}}, \tag{23} \]

where \(M_{x_\beta}\), \(M_{x_l}\) are the errors of meeting of the faces along the responsible direction in plan due to errors in measuring the angles and the lengths of the sides, respectively.

Fig. 10. Diagram of a theodolite traverse for the breakthrough of a mine working.

In the case of a closed theodolite traverse (Fig. 10), the errors \(M_{x_\beta}\) and \(M_{x_l}\) are computed by the formulas:

\[ M_{x_\beta} = \pm \sqrt{\sum \frac{m_\beta^{2}}{\rho^{2}} R_y^{2}}; \tag{24} \]

\[ M_{x_l} = \pm \sqrt{\sum m_l^{2} \cos^{2} \gamma}, \tag{25} \]

where \(R_y\) is the projection of the distance from the vertex of the theodolite traverse to the assumed meeting point of the faces onto the direction perpendicular to the responsible one;

\(\alpha\) is the bearing angle of the side relative to the abscissa axis, coinciding with the responsible direction.

For the graphical determination of the error \(M_{x_z}\), we transform formula (24).

Taking (16) into account, we write formula (24) as follows:

\[ M_{x_z}=\pm \frac{m}{\rho}\sqrt{\sum\left(R_y\cdot \sqrt{\frac{1}{P_i}}\right)^2}. \tag{24a} \]

But

\[ R_y=r_y\cdot N, \]

therefore

\[ M_{x_z}=\pm \frac{mN}{\rho}\sqrt{\sum\left(r_y\cdot \sqrt{\frac{1}{P_i}}\right)^2}, \tag{24б} \]

where \(r_y\) is the magnitude of the projection \(R_y\) on the breakthrough plan of the mine working;

\(N\) is the denominator of the scale of the breakthrough plan.

Denote

\[ r'_y=r_y\sqrt{\frac{1}{P_i}}. \tag{26} \]

Then

\[ M_{x_z}=\pm \frac{mN}{\rho}\sqrt{\sum (r'_y)^2}. \tag{24в} \]

For convenience of graphical constructions, set

\[ r'_x=\kappa r''_y, \tag{27} \]

where \(\kappa\) is a constant coefficient equal to the length of the segment \(OO_3\) (Fig. 3) in decimeters.

From (24б and 27) we obtain

\[ M_{x_z}=\pm \frac{\kappa mN}{\rho}\,R_\beta; \tag{24г} \]

\[ R_\beta=\sqrt{\sum (r''_y)^2}. \tag{28} \]

The quantity \(R_\beta\) is determined graphically in the same sequence as the quantity \(R_z\) entering formula (13г), namely: for each angle of the theodolite traverse the value \(x\) is determined analogously to that for the angle \(\beta_z\) (Fig. 2), a graph is constructed (Fig. 3), and for each angle of the theodolite traverse, as for the angle \(\beta_z\) (Fig. 3), the values \(\sqrt{\frac{1}{P_i}}\) and \(\sqrt{\frac{1}{P_z}}\) are determined; the quantities \(\sqrt{\frac{1}{P_i}}\) are laid off for all traverse angles on the straight line \(OA\) (Fig. 3). We obtain points 1, 2, 3, etc., corresponding to the numbers of the angles of the theodolite traverse (Fig. 10), which we connect by straight lines with point \(O_3\) (Fig. 3).

Through point \(k\) (Fig. 10), the meeting point of the faces, we draw conditional coordinate axes \(X\) and \(Y\) in such a way that the conditional abscissa axis coincides with the responsible direction in plan, i.e., with the direction perpendicular to the axis of the mine working being driven for breakthrough \(1\!-\!8\) (Fig. 10).

We project all vertices of the designed theodolite traverse \(k—1—2—3—4—5—6—7—8—k\) (Fig. 10) onto the conditional ordinate axis, obtaining the points \(k, 1', 2', 3', 4', 5', 6', 7'\), and \(8\).

The projections \(r_y\) of the traverse vertices (Fig. 10) are expressed by the segments \(r_{1y}=k-1'\), \(r_{2y}=k-2'\), \(r_{3y}=k-3'\), \(r_{4y}=k-4'\), \(r_{5y}=k-5'\), \(r_{6y}=k-6'\), \(r_{7y}=k-7'\), and \(r_{8y}=k-8\).

We take with a measuring ruler from the breakthrough scheme the projections \(r_y\) for each vertex of the traverse and lay them off from point \(O_3\) along the line \(O_3O\) (Fig. 3); we obtain points \(1'\), \(2'\), \(3'\), etc., from which we draw the straight lines \(1'-1_0\), \(2'-2_0\), \(3'-3_0\), etc., to their intersections respectively with the lines \(O_3 1\), \(O_3 2\), \(O_3 3\), etc.; we obtain the points \(1_0\), \(2_0\), \(3_0\), etc. (Fig. 3).

From the segments \(1'1_0\), \(2'2_0\), \(3'3_0\), etc., we construct a rectangular polygon, analogously to the orientation through two vertical mine shafts (Fig. 4). The closing side of the rectangular polygon is the magnitude \(R_\beta\).

To determine the error \(M_{x_\beta}\), through point \(O\) (Fig. 11) we draw the straight line \(OA\), on which we lay off the segment \(OA=1\) dm and

\[ OB=\frac{k m N}{\rho}\ \text{dm}. \]

For example, for \(k=2\) (the segment \(OO_3\), Fig. 3, equals 2 dm), \(m=\pm 20''\) and \(N=1000\), the segment \(OB\) will be

\[ OB=\frac{2\cdot 20\cdot 1000}{206265}=0.2\ \text{dm}. \]

Fig. 11. Determination of the error of meeting of the faces of a mine working as a function of the error of measuring angles by a graphical method.

On the straight line \(AC\), drawn perpendicular to \(OA\), we lay off from point \(A\), with a measuring ruler, the closing side \(R_\beta\) of the rectangular polygon (Fig. 4), obtaining point \(C\). From point \(B\) we draw the straight line \(BD\) parallel to \(AC\) to its intersection with the straight line \(OC\), obtaining point \(D\). The segment \(BD\), as follows from the constructions and formula (24г), corresponds to the error \(M_{x_\beta}\), expressed in millimeters.

The errors of the side lengths entering into formula (25) are computed by formula (19).

Formula (25), taking (21) into account, is written as follows:

\[ M_{x_l}=\pm \frac{R_l}{Q}; \tag{25a} \]

\[ R_l=\pm \sqrt{\sum M_l^2 \cos^2 \alpha}. \tag{29} \]

The error \(M_l\), i.e., the error \(m_l\) increased \(Q\) times, is determined graphically for each side of the traverse in the same way as for the sides of the underground connecting polygon during orientation through two vertical mine shafts, § 2.

To determine \(M_l\cos\alpha\), i.e., the projection of the error \(M_l\) onto the corresponding direction of the breakthrough, we proceed as follows. From point \(O\) (Fig. 12) we draw lines \(O1\), \(O2\), \(O3\), etc., parallel to the sides of the theodolite traverse (Fig. 10). The errors \(M_l\), taken with a measuring ruler from Fig. 6 for each side of the traverse, are laid off from point \(O\) (Fig. 12) in the corresponding directions; we obtain points 1, 2, 3, etc. We project points 1, 2, 3, etc., onto the corresponding direction of the breakthrough \((x)\); we obtain points \(1_0\), \(2_0\), \(3_0\), etc. From the segments \(O1_0\), \(O2_0\), \(O3_0\), etc., i.e.

of the projections of the errors \(M_l\) onto the responsible direction of the breakthrough, we construct a rectangular polygon, analogous to the rectangular polygon (Fig. 8). The closing line of the rectangular polygon corresponds to \(R_l\) (formula 25a). Dividing \(R_l\) by \(Q\), we obtain the error \(M_{xl}\) in millimeters.

The error of closure of opposing faces of a mine working (in plan) along the responsible direction (the error \(M_x\)) corresponds to the hypotenuse of a right triangle with legs \(M_{x_\beta}\) and \(M_{xl}\).

The proposed graphical method for precalculating the error of the direction angle of the side of the connecting polygon when orienting

Fig. 12. Determination of the projection of errors in measuring side lengths onto the responsible direction of the breakthrough.

through two vertical mine shafts and the error of closure of the faces of a mine working in plan has been used by students of the mine-surveying specialty of the Tomsk Polytechnic Institute since 1955 in carrying out laboratory work, course projects, and diploma projects. In all cases of its application, the results of the precalculation corresponded to the analytical method.

With the graphical method of precalculation, as a rule, gross errors did not occur; with the analytical method, however, the indicated errors were encountered.

The students easily and quickly mastered the technique of the graphical method of precalculation and successfully used it in their work.

The graphical method of precalculating the error of orientation through two vertical mine shafts and of the breakthrough of a mine working, having undergone seven years of testing within the institute, deserves broad introduction into the practice of mine-surveying organizations.

1. By “error” throughout is meant the root-mean-square error. ↩