A General Method for Calculating and Estimating the Accuracy of Orientation Through Two Vertical Mine Shafts

V. I. Akulov

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

The methods for calculating and estimating the accuracy of orientation through two vertical mine shafts recommended in the mine-surveying literature are applicable to one particular case of its solution, namely: to the case when the orientation is calculated without using the side of the orientation traverse between the plumb lines.

Orientation through two vertical mine shafts may be calculated without any side—including the side between the plumb lines—and without any angle of the orientation traverse; that is, from an orientation performed once for each side of the underground connecting traverse, consisting of \(n\) vertices, not counting the plumb lines \(A\) and \(B\). In the coordinate system adopted on the surface, \(2(n+1)\) values of the direction angle can be calculated, and for each of its vertices \(2(n+1)\) coordinate values.

When calculating the orientation by the method recommended in all mine-surveying literature, for each side of the underground connecting traverse only one value of the direction angle can be calculated from its \(2(n+1)\) values, and for each of its vertices likewise only one coordinate value from their \(2(n+1)\) values; the remaining \(2n+1\) values of the direction angle for each side and the \(2n+1\) coordinate values for each vertex cannot be determined from the apparatus of formulas recommended in the mine-surveying literature. Therefore, when calculating the orientation by the method recommended in the mine-surveying literature, it is impossible to determine, for each side of the underground connecting traverse—including the first—the value of the direction angle in the coordinate system adopted on the surface which, for the given accuracy of measurements of the angles and lengths of the sides both of the surface theodolite traverses to plumb lines \(A\) and \(B\) and of the underground connecting traverse, and for their specific form, has the minimum error.

The proposed general method for calculating orientation through two vertical shafts makes it possible, for each side of the underground connecting traverse, to determine the value of the direction angle in the coordinate system adopted on the surface that has the minimum error.

In this article, owing to the complicated structure of the formulas for estimating the accuracy of orientation calculated without the angle of the underground connecting—

…of the polygon; only the computation of the orientation without the side of the orienting polygon is considered.

General method for computing orientation through two vertical mine shafts without the side of the orienting polygon

When orienting underground surveys through two communicating vertical mine shafts, as is known, one plumb line is lowered from the surface to the horizon being oriented in each shaft, and between the indicated plumb lines on the horizon being oriented a theodolite traverse \(A—1—2\ldots—n—B\) is run through the mine workings (Fig. 1), in which all angles and sides are measured.

The coordinates of the plumb lines \(A\) and \(B\) in the system adopted on the surface are computed, respectively, from the surface theodolite traverses \(R—1'—2'—A\) and \(S—1''—2''—B\).

In what follows, the theodolite traverse \(A—1—2\ldots—n—B\) (Fig. 1), run on the horizon being oriented between the plumb lines \(A\) and \(B\), will be called the underground connecting polygon; and the closed polygon \(A—1—2\ldots—n—B—A\), formed by the underground connecting polygon and the side \(AB\), computed from the surface survey, will be called the orienting polygon.

Fig. 1

Introduce the following notation:

\(\beta\) — the measured angle of the underground connecting polygon;

\(l\) — the measured length of a side, reduced to the mean level surface and to the plane in the Gauss projection;

\(l_k\) — the side of the orienting polygon without which the orientation is computed;

\(\alpha_{ik}\) — the direction angle, in the coordinate system adopted on the surface, of the \(i\)-th side of the underground connecting polygon, computed without the side numbered “\(k\)”;

\(\alpha_k\) — the direction angle of the side numbered “\(k\)”;

\(\alpha'_i\) — the conditional direction angle of the \(i\)-th side of the underground connecting traverse relative to the conditional abscissa axis, coinciding with the direction of its first side;

\((AB)'_k\) — the conditional direction angle of the alignment line of plumbs \(AB\), computed without the side with number “\(K\)”;

\(X'_{k-1}, Y'_{k-1}\) — the conditional coordinates of the vertex with number “\(K-1\)” relative to the conditional origin of coordinates, coinciding with plumb \(A\);

\(X''_B, Y''_B\) — the conditional coordinates of plumb \(B\) relative to the second conditional origin of coordinates, coinciding with vertex “\(K\)”;

\((AB)\) — the direction angle of the alignment line of the plumbs from the surface survey;

\(X_A, Y_A;\ X_B, Y_B\) — the coordinates of plumbs \(A\) and \(B\) from the surface survey;

\(X_{k-1}, Y_{k-1};\ X_k, Y_k\) — the coordinates of the vertices bounding the side without which the orientation was computed;

\(n\) — the number of vertices in the underground connecting traverse, not counting plumbs \(A\) and \(B\);

\(N=n+1\) — the number of sides in the underground connecting traverse;

\(C_p\) — the distance between the plumbs from the surface survey;

\(C_u\) — the distance between the plumbs from the underground connecting traverse;

\(\rho = 206265''\).

To compute the underground connecting traverse in the coordinate system adopted on the surface, regardless of whether all the sides in this traverse have been measured or one fewer, it is necessary first, in the indicated coordinate system, to determine the direction angle of its first side.

When computing the orientation without the side “\(K\),” the direction angle, in the coordinate system adopted on the surface, of the first side of the underground connecting traverse is determined from the equation:

\[ \operatorname{tg}(\alpha_{1k}+\alpha'_k)= \frac{Y_k-Y_{k-1}}{X_k-X_{k-1}} . \tag{1} \]

After transformation, equation (1) takes the form:

\[ \operatorname{tg}(\alpha_{1k}+\alpha'_k)= \frac{(Y_B-Y_A)-(X''_B+X'_{k-1})\sin\alpha_{1k}-(Y''_B+Y'_{k-1})\cos\alpha_{1k}} {(X_B-X_A)-(X''_B+X'_{k-1})\cos\alpha_{1k}+(Y''_B+Y'_{k-1})\sin\alpha_{1k}} . \tag{2} \]

The conditional coordinates \(X'_{k-1}, Y'_{k-1}\) and \(X''_B, Y''_B\) are computed by the formulas:

\[ X'_{k-1}=\sum_{1}^{k-1} l_i\cos\alpha'_i, \tag{3} \]

\[ Y'_{k-1}=\sum_{1}^{k-1} l_i\sin\alpha'_i, \tag{4} \]

\[ X''_{\mathrm{B}}=\sum_{k+1}^{n+1} l_i \cos \alpha_i', \tag{5} \]

\[ Y''_{\mathrm{B}}=\sum_{k+1}^{n+1} l_i \sin \alpha_i', \tag{6} \]

\[ \alpha_i'=\sum_{1}^{i-1}\beta_i \pm (i-1)180^\circ . \tag{7} \]

Solving equation (2), we obtain:

\[ \sin \varphi_k = \frac{(X''_{\mathrm{B}}+X'_{k-1})\sin \alpha_k' - (Y''_{\mathrm{B}}+Y'_{k-1})\cos \alpha_k'}{C_n}, \tag{8} \]

\[ \varphi_k=\alpha_{1k}+\alpha_k'-(AB). \tag{9} \]

From (9) we have

\[ \alpha_{1k}=(AB)-(\alpha_k'-\varphi_k). \tag{10} \]

Since

\[ \alpha_k=\alpha_{1k}+\alpha_k', \]

expression (9) will take the form:

\[ \varphi_k=\alpha_k-(AB). \tag{11} \]

It follows from formula (11) that the angle \(\varphi_k\) is the angle between the line of the plumb bobs and the side without which the orientation has been computed.

The difference \(\alpha_k' - \varphi_k\) corresponds to the conditional direction angle of the line of the plumb bobs, computed without side “\(K\),” i.e.,

\[ (AB)_k'=\alpha_k'-\varphi_k. \tag{12} \]

Therefore formula (10) will take the form:

\[ \alpha_{1k}=(AB)-(AB)_k'. \tag{13} \]

The direction angles of the sides of the underground connecting traverse in the coordinate system adopted on the surface are computed by the formula:

\[ \alpha_{ik}=\alpha_{1k}+\alpha_i'. \tag{14} \]

The coordinates of the vertices of the underground connecting traverse in the system adopted on the surface, from the first through “\(K-1\)” inclusive, are computed from the coordinates of plumb bob A, and the remaining vertices—from the coordinates of plumb bob B.

Thus, proceeding from formulas (8), (12), and (13), one can compute the orientation through two vertical shafts without any side of the underground connecting traverse.

If all angles and sides are measured in the underground connecting traverse, then formula (8) takes the form:

\[ \sin \varphi_k=\frac{X'_B \sin \alpha'_k-Y'_B \cos \alpha'_k}{C_n}; \tag{8a} \]

\[ X'_B=\sum_{1}^{n+1} l_i \cos \alpha'_i; \tag{15} \]

\[ Y'_B=\sum_{1}^{n+1} l_i \sin \alpha'_i, \tag{16} \]

where \(X'_B, Y'_B\) are the conventional coordinates of plumb line \(B\) relative to a conventional origin of coordinates coinciding with plumb line \(A\).

The angle \(\varphi_k\), computed by formula (8a), does not depend on the side without which the orientation was computed, i.e., on side “\(K\).”

Let us consider, on the basis of formulas (8a), (12), and (13), the computation of the orientation without the side of the orientation traverse between the plumb lines, i.e., without side \(AB\) (Fig. 1), computed from the surface survey.

For the case when the orientation is computed without the side of the orientation traverse between the plumb lines, formula (8a) takes the form:

\[ X'_B \sin (AB)'_{AB}-Y'_B \cos (AB)'_{AB}=0. \]

Hence

\[ \operatorname{tg}(AB)'_{AB}=\frac{Y'_B}{X'_B}. \tag{17} \]

From formulas (12) and (13) we obtain:

\[ \alpha_{1AB}=(AB)-(AB)'_{AB}. \tag{18} \]

The method we have considered for computing the orientation without the side of the orientation traverse between the plumb lines is recommended by all mine-surveying literature as the sole method of computing it.

Thus, the method recommended by all mine-surveying literature for computing the orientation through two vertical mine shafts is a particular case of the general method of computing the orientation without any side of the orientation traverse.

For the case when all angles and sides are measured in the underground connecting traverse, formula (13) may be written in the following form:

\[ \alpha_{1K}=\alpha_{1AB}+\frac{\rho \Delta C \operatorname{tg}\varphi_k}{C_n}; \tag{13a} \]

\[ \Delta C=C_u-C_n; \tag{19} \]

\[ C_u=\frac{Y'_B}{\sin (AB)'_{AB}}=\frac{X'_B}{\cos (AB)'_{AB}}. \tag{20} \]

\[ C_n=\frac{Y_B-Y_A}{\sin(AB)}=\frac{X_B-X_A}{\cos(AB)} . \tag{21} \]

The correctness of the computation of the orientation is checked by the equality of the direction angles of the side without which the orientation was computed, found from the formulas:

\[ \alpha_k=\alpha_{1k}+\alpha'_k; \tag{22} \]

\[ \tg \alpha_k=\frac{Y_k-Y_{k-1}}{X_k-X_{k-1}} . \tag{23} \]

If the coordinates of all vertices of the underground connecting polygon, including plumb line \(B\), are computed from the coordinates of plumb line \(A\) and all its measured angles and side lengths, including the side without which the orientation was computed, then the correctness of the computation of the orientation is checked by the equalities:

\[ f_{x_k}=\frac{\Delta C\cos \alpha_k}{\cos \varphi_k}, \]

\[ f_{y_k}=\frac{\Delta C\sin \alpha_k}{\cos \varphi_k}. \tag{24} \]

The left-hand sides of equalities (24), i.e., the coordinate misclosures of plumb line \(B\) from the underground connecting polygon and the surface survey, are equal to

\[ f_{x_k}=\sum_{1}^{n+1} l_i\cos \alpha_{ik}-(X_B-X_A), \tag{25} \]

\[ f_{y_k}=\sum_{1}^{n+1} l_i\sin \alpha_{ik}-(Y_B-Y_A). \tag{26} \]

The linear misclosure of the underground connecting polygon, on the basis of formulas (24), is equal to:

\[ f_{s_k}=\frac{\Delta C}{\cos \varphi_k}. \tag{27} \]

The relative linear misclosure of the underground connecting polygon for the case when the orientation is computed without side “\(K\)” of the orientation polygon is equal to:

\[ \frac{f_{s_k}}{P}=\frac{\Delta C}{P\cos \varphi_k}, \tag{28} \]

where \(P\) is the perimeter of the underground connecting polygon.

The angle \(\varphi_k\) in formulas (13a), (24), (27), and (28) is calculated by the approximate formula

\[ \varphi_k=\alpha_k'-(AB)'_{AB}. \tag{29} \]

Check (24) verifies the correctness of the computation of the underground connecting traverse, taking into account its redundantly measured side, both in the conditional coordinate system and in the coordinate system adopted on the surface.

Check (24) is carried out before computing the coordinates of the vertices of the underground connecting traverse.

When computing the orientation without the side of the orienting traverse between the plumb lines, i.e., by the method recommended in mine-surveying literature, check (24) and formulas (27) and (28) take the form:

\[ f_{x_{AB}}=\Delta C\cos(AB); \]

\[ f_{y_{AB}}=\Delta C\sin(AB); \tag{30} \]

\[ f_{s_{AB}}=\Delta C; \tag{31} \]

\[ \frac{f_{s_{AB}}}{P}=\frac{\Delta C}{P}. \tag{32} \]

The correctness of the measurements of the elements (angles and side lengths) of the underground connecting traverse is checked by the inequality:

\[ \Delta l_k < \Delta l_{k_{\mathrm{dop.}}}, \tag{33} \]

where \(\Delta l_k\) is the actual difference between the computed and measured length of the side without which the orientation was computed;
\(\Delta l_{k_{\mathrm{dop.}}}\) is the permissible difference.

The actual difference “\(\Delta l_k\)” is determined by the formula:

\[ \Delta l_k=\sqrt{(X_k-X_{k-1})^2+(Y_k-Y_{k-1})^2}-l_k. \tag{34} \]

Let us take as \(\Delta l_{k_{\mathrm{dop.}}}\) twice the mean-square error of the difference \(\Delta l_k\), determined by formula (34); then

\[ \Delta l_k < \pm 2M_{\Delta l_k}, \tag{35} \]

where \(M_{\Delta l_k}\) is the mean-square error of the difference \(\Delta l_k\), depending on the errors of measuring the angles and side lengths of the underground connecting traverse.

Proceeding from (34), we obtain:

\[ M_{\Delta l_k}=\pm \frac{1}{\cos\varphi_k} \sqrt{ \sum_1^n R_i''^{\,2}\frac{m_{\gamma_i}^2}{\rho^2} + \sum_1^{n+1}\cos^2\delta_i\,m_{l_i}^2 }; \tag{36} \]

\[ R_i''=R_i\sin\left[(AB)'-\gamma_i'\right]; \tag{37} \]

\[ \delta_i=(AB)'-\alpha_i', \tag{38} \]

where \(R_i\) is the shortest distance from the \(i\)-th vertex of the underground connecting traverse to plumb line \(B\);

\(R_i''\) is the projection of the ray \(R_i\) onto the direction perpendicular to the line of alignment of the shafts;

\(t_i'\) is the conditional direction angle of the ray \(R_i\);

\(\delta_i\) is the angle between the “\(i\)-th” side of the traverse and the line of alignment of the shafts;

\(m_\beta\) is the root-mean-square error of angle measurement;

\(m_l\) is the root-mean-square error of measuring the length of a side.

Substituting \(M_{\Delta l_k}\) from (36) into inequality (35), we obtain:

\[ \Delta l_k < \pm \frac{2}{\cos \varphi_k} \sqrt{ \sum_{1}^{n} R_i''^{\,2}\frac{m_{\beta_i}^{2}}{\rho^{2}} + \sum_{1}^{n+1}\cos^{2}\delta_i\, m_{l_i}^{2} }. \tag{39} \]

The relation between \(\Delta l_k\) and \(\Delta C\) is expressed by the formula:

\[ \Delta C = -\Delta l_k \cos \varphi_k . \tag{40}^{1)} \]

Therefore inequality (39) will take the form:

\[ \Delta C < \Delta C_{\mathrm{dop.}}; \tag{41} \]

\[ \Delta C_{\mathrm{dop.}} = \pm 2 \sqrt{ \sum_{1}^{n} R_i''^{\,2}\frac{m_{\beta_i}^{2}}{\rho^{2}} + \sum_{1}^{n+1}\cos^{2}\delta_i\, m_{l_i}^{2} }. \tag{42} \]

Thus, independently of the side of the orienting traverse without which the orientation has been computed, the correctness of the measurements of the elements of the underground connecting traverse is checked by inequality (41).

It is necessary to note that fulfillment of inequality (41) may also occur in the presence of a gross error in measuring a side of the underground connecting traverse perpendicular to the line of alignment of the shafts, or an angle—in the case of an elongated form of the traverse; therefore it is necessary to choose such a method of field measurements as would completely exclude gross errors.

A check of the correctness of measurements when orienting through two vertical shafts, expressed by inequality (41), is recommended in all mine-surveying literature.

In all cases when inequality (41) is satisfied, the actual coordinate misclosures of shaft \(B\) from the underground connecting traverse, as follows from check (24), must be less than the permissible ones computed from \(\Delta C_{\mathrm{dop.}}\). Therefore, if one proceeds only from the accuracy of orienting underground surveys, there is no need, in addition to checking the correctness of measurements by (41), to establish standards for the relative linear misclosure of the underground connecting traverse, as is recommended in all mine-surveying literature. On the other hand, if the relative linear misclosure of the underground connecting traverse is not limited, then, with permissible accuracy of orienting underground surveys, insufficient accuracy of their centering may occur. Therefore, proceeding from the accuracy of centering underground surveys, we believe that,

\(^{1)}\) Equality (40) is valid for sides not perpendicular to the line of alignment of the shafts.

that the relative linear misclosure of the underground connecting traverse must not exceed \(1/5000\), i.e.

\[ \frac{\sqrt{f_{x_k}^{\,2}+f_{y_k}^{\,2}}}{P}<\frac{1}{5000}. \tag{43} \]

Taking into account the check (24), inequality (43) will take the form:

\[ \frac{\Delta C}{P\cos\varphi_k}<\frac{1}{5000}. \]

Hence

\[ \Delta C<\frac{P}{5000\sec\varphi_k}. \tag{44} \]

From inequalities (41) and (44) it follows that \(\Delta C_{\text{adm.}}\), computed by formula (42), must satisfy the inequality:

\[ \Delta C_{\text{adm.}}<\frac{P}{5000\sec\varphi_k}. \tag{45} \]

In all cases when inequalities (41) and (45) are simultaneously satisfied, the actual relative linear misclosure of the underground connecting traverse will be less than \(\frac{1}{5000}\) of the traverse perimeter, i.e. inequality (43) will hold.

When computing the orientation without the side of the orientation traverse between the plumb lines, i.e. by the method recommended in mine-surveying literature, inequality (45) will take the form:

\[ \Delta C_{\text{adm.}}<\frac{P}{5000}. \tag{46} \]

General method for estimating the accuracy of orientation through two vertical mine shafts

Orientation through two vertical mine shafts, as we have established, can be computed without any side of the orientation traverse; the set of formulas recommended throughout the mine-surveying literature for estimating the accuracy of orientation is applicable to only one particular solution, namely: to the case when the orientation is computed without the sides of the orientation traverse between the plumb lines.

Let us establish, proceeding from the general method of computing orientation, a general method for estimating the accuracy of orienting underground surveys through two vertical mine shafts.

The direction angle of the side of the underground connecting traverse with number “\(q\)” (Fig. 1), in the coordinate system adopted on the surface, for the case when the orientation is computed without the side “\(K\)” of the orientation traverse, proceeding from formulas (12), (13), and (14), is equal to

\[ \alpha_{q_k}=(AB)+\varphi_k+\alpha'_q-\alpha'_k. \tag{47} \]

From formula (47) it follows that \(\alpha_{q_k}\) is a function of the initial data and of all the measured angles and side lengths both of the surface theodolite traverses to plumb lines A and B, and of the underground connecting traverse, with the sole exception of the side without which it is computed.

In what follows, we shall investigate the influence of errors in the measurements of the angles and side lengths of the underground connecting traverse on the accuracy of deter-

determination of the directional angles of its sides in the coordinate system adopted on the surface.

The root-mean-square error of the directional angle of side “\(q\),” depending on errors in the measurements of the angles and lengths of the sides of the underground connecting traverse, on the basis of formula (47), is equal to:

\[ M_{\alpha q_k}=\pm \]

\[ \pm \rho \sqrt{ \sum_{1}^{n} \left\{ \frac{\partial \alpha'_q}{\partial \beta_i} - \frac{R'_{i_k}}{C\cos \varphi_k} \right\}^{2} \frac{m_{\beta_i}^{2}}{\rho^{2}} + \frac{1}{C^{2}\cos^{2}\varphi_k} \sum_{1}^{n+1} \sin^{2}\delta_{i_k}\, m_{l_i}^{2} }; \tag{48} \]

\[ R'_{i_k}=R_i\cos(\alpha'_k-\gamma'_i), \tag{49} \]

\[ \delta_{i_k}=\alpha'_k-\alpha'_i, \tag{50} \]

where \(C\) is the distance between the plumb lines;

\(R'_{i_k}\) is the projection of the ray \(R_i\) onto the direction parallel to the side without which the orientation has been computed;

\(\delta_{i_k}\) is the angle between the \(i\)-th side of the underground connecting traverse and the side of the orienting traverse without which the orientation has been computed.

Formula (48) expresses, for the case when the orientation is computed without a side of the orienting traverse, the general law of the influence of errors in the measurements of the angles and lengths of the sides of the underground connecting traverse on the accuracy of orienting underground surveys through two vertical mine shafts.

Formula (48) may be represented in the form:

\[ M_{\alpha q_k}=\pm \frac{\rho}{C\cos\varphi_k}\sqrt{m_{A_k}^{\prime\,2}+m_{B_k}^{\prime\,2}}; \tag{48a} \]

\[ m'_{A_k}=\pm \sqrt{ \sum_{1}^{q-1} R_{A_i k}^{\prime\,2}\frac{m_{\beta_i}^{2}}{\rho^{2}} + \sum_{1}^{q-1}\sin^{2}\delta_{i_k}\,m_{l_i}^{2} }; \tag{51} \]

\[ m'_{B_k}=\pm \sqrt{ \sum_{q}^{n} R_{i_k}^{\prime\,2}\frac{m_{\beta_i}^{2}}{\rho^{2}} + \sum_{q}^{n+1}\sin^{2}\delta_{i_k}\,m_{l_i}^{2} }; \tag{52} \]

\[ R'_{A_i k}=R_{A_i}\cos(\alpha'_k-\gamma'_{A_i}), \tag{53} \]

where \(m'_{A_k}\) is the error of the position of plumb line \(A\) in the direction perpendicular to the side without which the orientation has been computed, from the hanging traverse from vertex “\(q-1\)” to plumb line \(A\), considering in this traverse the directional angle of the side numbered “\(q\)” to be fixed;

\(m'_{B_k}\) — the error in the position of plumb line B in the direction perpendicular to the side without which the orientation is computed, from the hanging traverse from vertex “\(q-1\)” to plumb line B, taking in this traverse the direction angle of the side numbered “\(q\)” as fixed;

\(R_{A_i}\) — the shortest distance from the “\(i\)-th” vertex to plumb line A;

\(\gamma'_{A_i}\) — the conditional direction angle of the ray \(R_{A_i}\);

\(R'_{A_{ik}}\) — the projection of the ray \(R_{A_i}\) onto the direction parallel to the side without which the orientation is computed.

The error of the direction angle of the first side of the underground connecting traverse, on the basis of formulas (48) and (48a), is equal to:

\[ M_{\alpha_{1_k}} = \pm \frac{\rho}{C\cos\varphi_k} \sqrt{ \sum_{1}^{n} R_{i_k}^{\prime 2}\frac{m_{\beta_i}^{2}}{\rho^{2}} + \sum_{1}^{n+1}\sin^{2}\delta_{ik}\,m_{l_i}^{2} }. \tag{54} \]

\[ M_{\alpha_{1_k}} = \pm \frac{\rho\,m_{B_k}}{C\cos\varphi_k}; \tag{54a} \]

\[ m_{B_k} = \pm \sqrt{ \sum_{1}^{n} R_{i_k}^{\prime 2}\frac{m_{\beta_i}^{2}}{\rho^{2}} + \sum_{1}^{n+1}\sin^{2}\delta_{ik}\,m_{l_i}^{2} }, \tag{55} \]

where \(m_{B_k}\) is the error in the position of plumb line B in the direction perpendicular to the side without which the orientation is computed from the underground connecting traverse, considering it as a hanging traverse with a fixed point (plumb line) A and a fixed direction angle of the first side.

When computing the orientation without the side of the orienting traverse between the plumb lines, i.e., by the method recommended in mine-surveying literature, formulas (48) and (54) take the form:

\[ M_{\alpha_{q\,AB}} = \pm \rho \sqrt{ \sum_{1}^{n} \left\{ \frac{\partial \alpha'_c}{\partial \beta_i} - \frac{R'_{iAB}}{C} \right\}^{2} \frac{m_{\beta_i}^{2}}{\rho^{2}} + \frac{1}{C^{2}} \sum_{1}^{n+1}\sin^{2}\delta_{iAB}\,m_{l_i}^{2} }. \tag{56} \]

\[ M_{\alpha_{1\,AB}} = \pm \frac{\rho''}{C} \sqrt{ \sum_{1}^{n} R_{iAB}^{\prime 2}\frac{m_{\beta_i}^{2}}{\rho^{2}} + \sum_{1}^{n+1}\sin^{2}\delta_{iAB}\,m_{l_i}^{2} }; \tag{57} \]

\[ R'_{iAB}=R_i\cos\left[(AB)'-\gamma'_i\right]; \tag{58} \]

\[ \delta_{iAB}=(AB)'-\alpha'_i. \tag{59} \]

Formulas (56) and (57), obtained from the general formulas (48) and (54) for the particular solution of the problem of orienting underground surveys through two vertical mine shafts without the side of the orienting-

of the traverse between the plumb lines were first established by Professor I. M. Bakhurin¹ and at the present time are recommended throughout mine-surveying literature as the only formulas for estimating the accuracy of orientation through two vertical mine shafts.

Formulas (48) and (48a) for underground connecting traverses of arbitrary form cannot be simplified; for elongated traverses with approximately equal sides, however, the indicated formulas can be transformed.

Formula (48) for an elongated equilateral underground connecting traverse (Fig. 2), independently of the side of the orientation traverse without which the orientation is computed, will take the form:

\[ M_{\alpha_q}=\pm m_\beta \sqrt{ \frac{(N-1)(2N-1)}{6N} - \frac{(N-q)(q-1)}{N} }, \tag{60} \]

where \(N\) is the number of sides in the underground connecting traverse.

Fig. 2.

It follows from formula (60) that, for an elongated form of the underground connecting traverse, errors in measuring the lengths of its sides do not affect the accuracy of orientation of underground surveys.

Let us transform formula (48a) for an elongated equilateral underground connecting traverse.

Formula (48a) for an elongated underground connecting traverse, independently of the side of it without which the orientation is computed, will take the form:

\[ M_{\alpha_q} = \pm \frac{\rho}{C} \sqrt{ m_{A_{AB}}^{\prime 2}+m_{B_{AB}}^{\prime 2} }. \tag{61} \]

The error of plumb line \(A\) in the direction perpendicular to the line of alignment of the plumb lines, from the hanging traverse from vertex “\(q-1\)” to plumb line \(A\) (Fig. 2), is equal to:

\[ m_{A_{AB}}^{\prime 2} = \frac{q(q-1)(2q-1)l^2 m_\beta^2}{6\rho^2}. \tag{62} \]

The error of plumb line \(B\) in the direction perpendicular to the line of alignment of the plumb lines, from the hanging theodolite traverse from vertex “\(q-1\)” to plumb line \(B\) (Fig. 2), is equal to:

\[ m_{B_{AB}}^{\prime 2} = \frac{(N-q)(N-q+1)(2N-2q+1)l^2 m_\beta^2}{6\rho^2}. \tag{63} \]

¹ Bakhurin I. M. Course in Mine-Surveying Art. Special Part, ONTI, 1932.

Substituting \(m_{\lambda_{AB}}\) and \(m'_{\beta_{AB}}\) from (62) and (63) into formula (61), we obtain:

\[ M_{\alpha_q} = \pm \frac{m_\xi}{N} \sqrt{ \frac{q(q-1)(2q-1)}{6} + \frac{(N-q)(N-q+1)(2N-2q+1)}{6} }. \tag{64} \]

Formula (64) is known in mine-surveying literature as the formula of Engineer M. S. Cheremisin.

Formula (64), by its construction, is rather complex. If the expression \(3N^2 m_\xi^2\) is added to and subtracted from the expression under the radical, then formula (64) takes the form of formula (60).