ANGLE-MEASURING INSTRUMENT

General Theory of Measuring Angular (Spherical) Coordinates and Determining Errors in the Relationships Between Axes

P. A. MASLENIKOV

The four principal axes of an angle-measuring instrument—the axis of the cylindrical level \(U\), the vertical axis of rotation of the alidade of the horizontal circle \(V\), the horizontal axis of rotation of the telescope \(H\), and the line of sight of the telescope \(K\)—as is known, must be in the following relationships to one another:

\[ \begin{aligned} &U \text{ must be perpendicular to } V,\\ &K \quad " \quad " \quad H,\\ &H \quad " \quad " \quad V. \end{aligned} \]

For all three cases noted, one and the same requirement must be fulfilled—mutual perpendicularity for two axes; consequently, a unified solution of the posed problems is possible.

To detect nonperpendicularity of axes (error \(c\)) and to measure the magnitude of the error, counters are needed, the planes of which should preferably be parallel to each pair of axes of the instrument.

In its simplest form, the solution of these problems can be carried out as follows. Suppose it is required to detect and measure the angular magnitude of the nonperpendicularity between the axes \(OA\) and \(OB\) (Fig. 1). Suppose that we can rotate the axis \(OB\) about the axis \(OA\), and assume that at some instant the axis \(OB\) occupies the position \(OB_1\), to which the reading \(R\) on the counter will correspond (the axes \(OA\), \(OB\), and the counter in Fig. 1 are considered to lie in the plane of the drawing; the inscriptions of the divisions on the counter increase clockwise from \(0\) to \(360^\circ\) without a break). From Fig. 1 it follows that, after rotating the axis \(OB\) about the axis \(OA\) by exactly \(180^\circ\), the axis \(OB\) will move from the position \(OB_1\) into the position \(OB_2\), to which the reading \(L\) on the counter will correspond (it is assumed that the counter is stationary).

From the given Fig. 1 it is seen that the rotation by \(180^\circ\) of the axis \(OB\) about the axis \(OA\), from position \(OB_1\) to position \(OB_2\), in a plane perpendicular to the plane of the drawing, is equivalent to rotating the axis \(OB\) (from \(OB_1\) to \(OB_2\)) through an angle \(180^\circ - 2c\) about the point of intersection of the axes in the plane of the drawing (or about the point of intersection of their projections).

If \(OB_0\) is the ideal position of the axis \(OB\), perpendicular to the axis \(OA\), and forming with the actual position \(OB_1\) the angle (error) \(c\), then, as is not difficult to see from Fig. 1, the values of the angle \(c\) and of the readings \(R\) and \(L\) on the counter are related by the following relation:

\[ 2c = R - (L - 180^\circ). \tag{1} \]

Whence, solving equation (1) with respect to \(c\) and denoting by \(L'\) the expression in parentheses, i.e. \(L' = L - 180^\circ\), we find:

\[ c = \frac{R - L'}{2}. \tag{2} \]

Thus, the value of the nonperpendicularity (or error) of the axis \(OB\) to the axis \(OA\) is determined by half the difference of the readings on the counter: the reading \(R\) at the first position of the axis \(OB_1\), and the reading \(L\) at the second position of the axis \(OB_2\), the latter reading being changed by \(180^\circ\), if the division inscriptions on the counter increase from \(0^\circ\) to \(360^\circ\) without interruption.

From Fig. 1 it may be seen that if the axis \(OB\) is rotated about the point \(O\) in the plane of the drawing in the clockwise direction, then the readings on the counter will continually increase. The angle \(c\), under such a rotation of the axis \(OB\), will increase or decrease depending on whether the axis \(OB\) moves away from its ideal position \(OB_0\) or approaches it. We shall call the initial position of the axis \(OB\) that position in which, as the counter readings increase, the angle \(c\) increases. Conversely, we shall call the non-initial position of the axis \(OB\) that position in which, as the counter readings increase, the value of the angle \(c\) decreases.

Fig. 1

It is obvious that the value of the nonperpendicularity of the axis \(OB\) to the axis \(OA\) can also be determined as half the difference of the readings at the initial and non-initial positions of the axis \(OB\), since the readings \(R\) and \(L\) entering formula (2) may be regarded as the readings at the initial and non-initial positions of the axis \(OB\).

Knowing the value of the error \(c\), it will not be difficult to determine on the counter the reading \(MO\) corresponding to the perpendicular position of the axes, namely (see Fig. 1):

\[ MO = R - c = L' + c \tag{3} \]

or, after substituting the value of \(c\) from (2):

\[ MO = \frac{R + L'}{2}, \tag{4} \]

that is, the reading corresponding to the perpendicular position of the axes is equal to the half-sum of the same quantities as in determining the error \(c\).

From formula (2) it is evident that the reading at the initial position of the movable axis differs from the reading at the non-initial position of this axis, corrected by \(180^\circ\), by twice the angle of the nonperpendicularity of the axes. Consequently, if correction of the position of the movable axis is required, then the latter

should be made, by means of the corresponding adjusting screws, by an amount equal to one half of the readings in the initial and non-initial positions.

By the formulas derived, (2) and (4), the above-mentioned quantities are determined—the errors in the relationships between the axes, as well as the angular coordinates, if the fixed axis \(OA\) and the axis \(OB\) in the first and second positions lie in a plane parallel to the plane of the counter.

Let us now suppose that the axis \(OA\) lies in a plane parallel to the plane of the counter, while the axis \(OB\), in its first and second positions, forms with the plane of the counter a certain angle \(\alpha\), not equal to zero, and is projected onto the plane of the counter as the straight lines \(OB'_1\) and \(OB'_2\), which form with the line \(OB_0\) the angle \(\gamma\). Then it is obvious that, instead of the readings \(R\) and \(L\), we shall obtain on the counter other readings, for example, \(R_1\) and \(L'_1\); after substituting these into formula (2) we obtain the value of the angle \(\gamma\), and not the error \(c\), that is:

\[ \gamma=\frac{R_1-L'_1}{2}. \tag{5} \]

The reading \(MO\), corresponding to the perpendicular position of the axes, remains the same, since the quantities \(R_1\) and \(L'_1\) entering into formula (4) have changed—one in the direction of increase, and the other in the direction of decrease—by one and the same amount; and therefore, as before:

\[ MO=\frac{R_1+L'_1}{2}. \tag{6} \]

In this case, to determine the magnitude of the error \(c\), it will be necessary to use another formula, readily obtained from the solution of the triangle \(OB_1B'_1\).

Taking, in the triangle \(OB_1B'_1\), the side \(OB_1\) to be equal to one, for the side \(OB'_1\), knowing its angle of inclination \(\alpha\) to the plane of the counter, we may write the value:

\[ OB'_1=\cos\alpha \]

and, finally, solving the triangle \(OB_1B'_1\) by the two sides \(OB_1\), \(OB'_1\) and the exterior angle \(\gamma\), we find:

\[ \sin c=\sin\gamma\cos\alpha \tag{7'} \]

or, since the error \(c\) is small:

\[ c=\frac{\sin\gamma\cos\alpha}{\sin 1''}. \tag{7} \]

Let us finally suppose that neither of the axes lies in a plane parallel to the plane of the counter, while the axis \(OB\), in its first and second positions, forms with the latter the angle \(\alpha\), and the axis \(OA\) the angle \(\beta\) (see Fig. 2). Let us imagine an auxiliary sphere with center at \(O\), enclosing the counter. Drawing the perpendicular \(OZ\) to the plane of the counting mechanism, we obtain on it, as a result of the intersection of the surface of the sphere by three planes passing pairwise through the axes \(OA\), \(OB_1\), and the perpendicular \(OZ\), an oblique spherical triangle \(AB_1Z\) with sides \(90^\circ-c\), \(90^\circ-\alpha\), and \(90^\circ-\beta\), and angle \(90^\circ-\gamma\).

Solving the oblique spherical triangle \(AB_1Z\), we find:

\[ \sin c=\sin\alpha \sin\beta+\cos\alpha\cos\beta\sin\gamma \tag{8} \]

or, because of the smallness of the error \(c\):

\[ c=\frac{1}{\sin 1''}\left(\sin\alpha \sin\beta+\cos\alpha\cos\beta\sin\gamma\right). \tag{8′} \]

The values of the angle \(\gamma\) and of the reading \(MO\), as is evident from Fig. 2, will in this case also be determined by formulas (5) and (6).

As was already mentioned above, for the four principal axes of an angle-measuring instrument three conditions of their mutual perpendicularity must be satisfied. Let us consider these conditions separately.

First condition.

Perpendicularity of the axis of the level \(U\) to the vertical axis of rotation \(V\).

As is known, the nonperpendicularity of these axes (by the level axis in Fig. 1 is meant the axis \(OB\), and by the vertical axis of rotation—the axis \(OA\)) can be detected and measured with the aid of a scale marked on the tube of the level. For this condition, in formulas (2) and (4), instead of the quantity \(L'\), one should substitute the value of the read reading \(L_U\) (taking signs into account, if the zero of the scale is at the middle of the tube), that is,

Fig. 2

\[ c_U^V=\frac{R_U-L_U}{2} \tag{9} \]

\[ MO_U^V=\frac{R_U+L_U}{2}, \tag{10} \]

where

\[ c_U^V \]
— the angle determining the magnitude of the nonperpendicularity of the level axis \(U\) to the vertical axis of rotation of the instrument \(V\),

\[ MO_U^V \]
— the reading corresponding to the perpendicular position of the axes \(U\) and \(V\),

\[ R_U \text{ and } L_U \]
— readings of the middle of the level bubble.

In this case, simultaneously with the movable axis \(OB\) (the axis of the level \(U\)), which is rotated through \(180^\circ\) about the axis \(OA\), the scale will also rotate through \(180^\circ\), and the initial mark of the scale \(0^\circ\) will move to \(0^\circ_1\), while the index (the middle of the level bubble), opposite which the readings are to be read, in its rotation about \(OA\) will move from point \(B_1\) exactly \(180^\circ\) relative to a certain plumb axis. Since the point \(B_1\) of the level axis is taken as the initial position of the index in Fig. 1, therefore, at this moment

the axis of the level \(OB_1\) is horizontal. It is obvious that the second position of the index will in this case also lie on the horizontal line, and since only one horizontal straight line can be drawn through the same point \(O\), the second position of the index must therefore be considered to be at the point \((B_1)\), diametrically opposite to the point \(B_1\).

After eliminating, by the method indicated above, the nonperpendicularity of the level axis to the vertical axis of rotation of the instrument, the level axis in Fig. 1 will occupy the position \(OB_0\), and the vertical axis of rotation—the position \(OA\). If we now take the straight line \(OB_1\) as the horizontal straight line, then the angle \(c\) will express the angle of inclination of the level axis to the horizon, which, as is not difficult to see, can be computed from formula (9), that is:

\[ c_U^Z=\frac{R_U-L_U}{2}, \tag{11} \]

where \(c_U^Z\) is the angle of inclination of the level axis to the horizon, or the magnitude of the nonperpendicularity of the level axis \(U\) to the plumb direction \(OZ\).

Let now, in Fig. 2, the vertical axis of rotation of the instrument be represented by the straight line \(OZ\), the level axis by the straight line \(OB\), and the plumb direction by the straight line \(OA\). We shall further introduce the notation:

\[ \begin{aligned} c_V^h &= 90^\circ-\beta &&\text{the magnitude of the angle between the vertical axis of rotation and the plumb direction}\\ &&&\text{(or the magnitude of the nonperpendicularity of the vertical axis of rotation to the horizontal plane);}\\ c_U^V &= \alpha &&\text{the magnitude of the nonperpendicularity of the level axis to the vertical axis of rotation;}\\ c_U^Z &= c &&\text{the angle of inclination of the level axis to the horizon (the nonperpendicularity of the level axis}\\ &&&\text{to the plumb direction).} \end{aligned} \]

Finally, let us replace the angle \(\gamma\) by the equal quantity \(\gamma=R_h-MO_{(h)U}^{VZ}\), where \(R_h\) is the reading on the horizontal circle corresponding to the initial position of the level axis, and \(MO_{(h)U}^{VZ}\) is the reading on the horizontal circle corresponding to the perpendicular position of the level axis to the vertical axis and to the plumb direction (the vertical axis does not coincide with the plumb direction). Then formula (8′) will be rewritten in the form:

\[ \sin c_U^Z=\sin c_U^V\cos c_V^h+\cos c_U^V\sin c_V^h\sin\left(R_h-MO_{(h)U}^{VZ}\right). \tag{12} \]

If the level axis \(U\) has been brought into a position perpendicular to the vertical axis of rotation \(V\), then it is obvious that \(c_U^V=0\), and formula (12) in this case takes the form:

\[ \sin c_U^Z=\sin c_V^h\sin\left(R_h-MO_{(h)U}^{VZ}\right). \tag{13} \]

Having measured the angles of inclination of the level axis to the horizon \((c_{U_1}^Z\) and \(c_{U_2}^Z)\) in two mutually perpendicular positions of it and substituting the values of these angles, computed by formula (11), into formula (13) in order to determine the magnitude of the angle between the vertical axis and the plumb direction, we obtain two such equations:

\[ \sin c_{U_1}^Z=\sin c_V^h\sin\left(R_h-MO_{(h)U}^{VZ}\right) \tag{14} \]

and

\[ \sin c_{U_2}^Z=\sin c_V^h\sin\left(90^\circ+R_h-MO_{(h)U}^{VZ}\right), \tag{15} \]

the sum of whose squares gives:

\[ \sin^2 c_V^h=\sin^2 c_{U_1}^Z+\sin^2 c_{U_2}^Z . \tag{16} \]

Hence, owing to the smallness of the angles entering formula (16), for the angle \(c_V^h\) between the vertical axis and the plumb direction we find:

\[ c_V^h=\sqrt{(c_{U_1}^Z)^2+(c_{U_2}^Z)^2}. \tag{17} \]

or, after substituting the values of \(c_{U_1}^Z\) and \(c_{U_2}^Z\), determined by formula (11), we obtain:

\[ c_V^h=\sqrt{\left(\frac{R_{U_1}-L_{U_1}}{2}\right)^2+ \left(\frac{R_{U_2}-L_{U_2}}{2}\right)^2}, \tag{18} \]

In exactly the same way, for the reading on the horizontal circle corresponding to the perpendicular position of the level axis relative to the vertical axis and the plumb direction, proceeding from (14) and (15) and taking into account that \(c_{U_1}^Z\) and \(c_{U_2}^Z\) are small quantities, we obtain the expression:

\[ \operatorname{tg}(R_h-MO_{h(U)})= \frac{c_{U_1}^Z}{c_{U_2}^Z} = \frac{R_{U_1}-L_{U_1}}{R_{U_2}-L_{U_2}} . \tag{19} \]

The derivation given is valid for all levels of an angle-measuring instrument. Therefore, measurement of the angle between the vertical axis and the plumb direction and determination of the reading on the horizontal circle defining the direction of inclination of the vertical axis may be performed with any sufficiently accurate level of the instrument.

Fig. 3

The formulas (9), (10) given above were derived under the assumption that the initial position of the level axis was horizontal. Let us now consider the general case, when the level axis in its initial and non-initial positions will not be horizontal.

Suppose that the circle shown in Fig. 3 represents the scale of the level; the straight line \(OZ\) is the plumb line; the straight line \(OV\) is the vertical axis of rotation of the instrument, making with the plumb direction the angle \(90^\circ-c_V^Z\), where \(c_V^Z\) is the angle of inclination of the vertical axis to the horizon. Let, in the initial position, the level axis (the tangent at the highest טאָ

point of the inner curve of the longitudinal section of the cylindrical level by a vertical plane) occupies position \(a_1 b_1\), and in the second position, after rotation by \(180^\circ\) about the axis \(OV\), position \(a_2 b_2\).

Let us introduce the notation:

\(\varepsilon\) — the reading fixing the position of the level axis (zero—the point of the level);

\(c_U^V\) — the magnitude of the nonperpendicularity of the level axis to the vertical axis;

\(c_{U1}^Z\) and \(c_{U2}^Z\) — the magnitude of the nonperpendicularity of the level axis to the plumb direction in the first and second positions, or the angles of inclination of the level axis to the horizon;

\(R_U\) and \(L_U\) — readings of the middle of the level bubble in the first and second positions of the level axis (the middle of the level bubble is considered in both cases to be located on the plumb line at point \(2\));

\(O_1^0\) and \(O_2^0\) — the initial mark of the level counter in the first and second positions;

\[ MO_U^V = c_U^V + \varepsilon \]

— the reading corresponding to the perpendicular position of the level axis and the vertical axis of rotation.

Proceeding from Fig. 3, for the reading corresponding to the perpendicular position of the level axis and the vertical axis of rotation, one may write the same expression

\[ MO_U^V=\frac{R_U+L_U}{2}, \tag{20} \]

as in the case when in the initial position the level axis occupies a horizontal position.

For the magnitude of the nonperpendicularity of the level axis to the vertical axis of rotation we obtain the expression

\[ c_U^V=MO_U^V-\varepsilon=\frac{R_U+L_U}{2}-\varepsilon, \tag{21} \]

For the angles of inclination of the level axis to the horizon, or the magnitude of the nonperpendicularity of the level axis to the plumb direction, we find the values:

\[ c_{U1}^Z=R_U-\varepsilon, \tag{22} \]

\[ c_{U2}^Z=L_U-\varepsilon, \tag{23} \]

And, finally, for the angle between the vertical axis of rotation and the plumb direction, or the magnitude of the nonperpendicularity of the vertical axis to the horizon, we write the expression:

\[ 90^\circ-c_V^Z=c_V^h=\frac{R_U-L_U}{2}=R_U-MO_U^V=MO_U^V-L_U, \tag{24} \]

where \(c_V^h\) is the magnitude of the nonperpendicularity of the vertical axis of rotation of the instrument to the horizon.

Bringing the level axis into a position perpendicular to the vertical axis, as is evident from the derivations, can theoretically be performed without setting, as is customary, the level axis in the initial position into a horizontal position. In order that the level axis be perpendicular to the vertical axis, it must be (see Fig. 3):

\[ \varepsilon=MO_U^V=\frac{R_U+L_U}{2}. \tag{25} \]

This can be achieved if, in the initial position of the axis of the level, by acting on it with the adjusting screws, the middle of the level bubble is set to a reading equal to

\[ R_{U_0}=R_U-C_U^V=\frac{R_U-L_U}{2}+\varepsilon . \tag{26} \]

The derivation of the last formulas (20–26) was carried out under the assumption that the axis of the level and the vertical axis of rotation lie in one plumb plane. However, from Fig. 4 it follows that this conclusion will be valid for formulas (20), (21), (25), and (26) also in the case when the axis of the level and the vertical axis of rotation of the instrument do not lie in one vertical plane. In this latter case, the right-hand sides of the equalities in formulas (22)–(24) will express the projections \(\gamma_{U_1}^Z\) and \(\gamma_{U_2}^Z\) of the angles of inclination of the axis of the level to the horizon \((c_{U_1}^Z\) and \(c_{U_2}^Z\), Fig. 3), while \(90^\circ-\gamma_V^Z\) is the projection of the angle between the vertical axis and the plumb direction, that is,

\[ \gamma_{U_1}^Z=R_U-\varepsilon , \tag{22'} \]

\[ \gamma_{U_2}^Z=L_U-\varepsilon , \tag{23'} \]

\[ 90^\circ-\gamma_V^Z=\frac{R_U-L_U}{2}. \tag{24'} \]

In Fig. 4 the position of the axis of the level in the initial position is marked by one point of tangency of the axis with the circle passing through the vertical axis and the scale of the level tube—point \(U_1\); in the non-initial position, by point \(U_2\); and the index (the middle of the level bubble), by point \(Z_1\).

Fig. 4

The latter is the highest point of the circle of the level scale, obtained as a result of the intersection of this circle with the plumb plane perpendicular to it and passing through the plumb direction \(OZ\).

Starting from the spherical triangle \(ZZ_1V\), right-angled at the point \(Z_1\), with acute angle \(A\), leg \(90^\circ-\gamma_V^Z\), and hypotenuse \(90^\circ-c_V^Z=c_V^h\),

for the angle \(c_V^h\) between the vertical axis and the plumb direction one may write the expression

\[ \cos A=\operatorname{ctg} c_V^h \operatorname{tg} (90^\circ-\gamma_V^Z) =\operatorname{ctg} c_V^h \operatorname{tg}\frac{R_U-L_U}{2} \]

or, since the angles \(c_V^h\) and \(\dfrac{R_U-L_U}{2}\) are small:

\[ c_V^h\cos A=\frac{R_U-L_U}{2}. \tag{27} \]

Taking readings for the middle of the level bubble in the initial and non-initial positions of its axis for two mutually perpendicular positions of the level counter, and substituting the readings obtained into formula (27), we obtain two equations:

\[ c_V^h\cos A=\frac{R_{U_1}-L_{U_1}}{2}, \tag{28′} \]

\[ c_V^h\sin A=\frac{R_{U_2}-L_{U_2}}{2}, \tag{28″} \]

The square root of the sum of the squares of (28′) and (28″) will give us the value of the angle between the vertical axis and the plumb direction, or the magnitude of the nonperpendicularity of the vertical axis to the horizon:

\[ c_V^h=\sqrt{\left(\frac{R_{U_1}-L_{U_1}}{2}\right)^2+ \left(\frac{R_{U_2}-L_{U_2}}{2}\right)^2} \tag{29} \]

Let \(O_h\) be the initial mark on the horizontal circle, \(R_h\) the reading on the horizontal circle corresponding to the first position of the level counter, and \(MO_{(h)U}^{VZ}\) the reading on the horizontal circle corresponding to the position of the level axis perpendicular (if this axis is horizontal) to the vertical axis of rotation and to the plumb direction. Then for the angle \(A\) one may write the following equality:

\[ A=90^\circ+R_h-OM_{(h)U}^{VZ}, \tag{30} \]

after substituting which into equations (28′) and (28″) and dividing these equations by one another, we find the value \(MO_{(h)U}^{VZ}\) from the expression:

\[ \operatorname{tg}(R_h-MO_{(h)U}^{VZ})= \frac{R_{U_1}-L_{U_1}}{R_{U_2}-L_{U_2}}. \tag{31} \]

It follows from formula (29) that the angle between the vertical axis of rotation of the instrument and the plumb direction can be measured with a level whose axis has not previously been set in a position perpendicular to the vertical axis of rotation. From formula (31), however, we find that bringing the level axis into a position perpendicular to the vertical axis (carrying out the adjustment of the level) can be reduced to setting the level to the reading \(MO_{(h)U}^{VZ}\) on the horizontal circle and subsequently bringing its axis into the horizontal position by acting with the adjusting screws of the level.

Second condition

Perpendicularity of the line of sight \(K\) to the horizontal axis of rotation \(H\).

The nonperpendicularity of these axes (by the line of sight in Figs. 1 and 2 is meant the axis \(OB\), and by the horizontal axis of rotation—the axis \(OA\)) can be detected and measured with the aid of the reading device of the horizontal circle of an angle-measuring instrument. (For a Kipregel this reading device is the plane table, on which, along the beveled edge of the Kipregel ruler, straight lines are drawn corresponding to the readings on the horizontal circle in a theodolite.)

In Fig. 1, when sighting with the circle on the right at some point \(T\) in space, the line of sight occupied the position \(OB_2\), to which the reading \(R\) on the horizontal circle corresponded. When the telescope is transited through the zenith and the line of sight is set parallel to the plane of the reading device of the horizontal circle, the latter will occupy the position \(OB_1\), to which the previous reading \(R\) will correspond on the horizontal circle, since when the telescope is transited through the zenith (when \(OB_2\) is rotated about \(OA\)) the index against which the reading on the horizontal circle was made remains fixed in this case. In order, with the circle on the left, to sight the same point of space as with the circle on the right, the line of sight must be turned from the position \(OB_1\) through \(180^\circ - 2c\) about the point \(O\) into the position \(OB_2\). Obviously, in this case the horizontal axis of rotation \(OA\) (having occupied the position \(OA'\)) and the index (having occupied the position \(B_2'\)), against which we shall now see not the reading \(R\), but a reading greater by \(180^\circ - 2c\), will turn through the same angle and in the same direction, that is, a reading equal to:

\[ L = R + 180^\circ - 2c, \]

whence

\[ c = \frac{R - L'}{2}. \]

It follows from this that the formulas given above are also valid for the case when, for the movable axis \(OB\), two rotations have to be made: the first about the axis \(OA\) in a plane perpendicular to the plane of the reading device, and the second about the point \(O\) of intersection of these axes through \(180^\circ - 2c\) in a plane parallel to the plane of the reading device.

In the present case we have a fixed reading device with a rotating index; the first, initial position of the index is taken at the point \(B'\), and the second, noninitial one, at the point \(B_2'\). The straight line \(OB_2\) should be taken as the initial position of the line of sight, since with an increase of the angle \(c\) in this case the readings on the horizontal circle increase (an increase of the angle \(c\), which can be produced by the tangent screws at the reticle when sighting at the point \(T\) of space (in Fig. 1), will force the line of sight to turn, and with it also the index \(B'\), in the direction of increasing readings).

Depending on the relative arrangement of the axes and the reading device, different formulas should be used for computing the nonperpendicularity (error \(c\)) and the reading corresponding to the perpendicular position of the axes.

If both axes are parallel to the plane of the reading device, then, proceeding from formulas (2) and (4), one should use the formulas:

\[ c_K^H = \frac{R_h - L'_h}{2} \tag{32} \]

\[ MO_{(h)K}^{H}=\frac{R_h+L'_h}{2}, \tag{33} \]

where \(c_K^{H}\) is the magnitude of the nonperpendicularity of the sighting axis to the horizontal axis of rotation (collimation error); \(MO_{(h)K}^{H}\) is the reading on the horizontal circle corresponding to the perpendicular position of the sighting and horizontal axes; \(R_h\) and \(L_h\) are the readings on the horizontal circle obtained when sighting the same point in space at two positions of the vertical circle.

If, however, the sighting axis \(K\) (axis \(OB\)) is not parallel, while the horizontal axis of rotation \(H\) (axis \(OA\)) is parallel to the plane of the counter, then, on the basis of (5), (6), and (7), we apply the formulas:

\[ \gamma_{(h)K}^{H}=\frac{R_h-L'_h}{2} \tag{34} \]

\[ MO_{(h)K}^{H}=\frac{R_h+L'_h}{2} \tag{35} \]

\[ c_K^{H}=\frac{\sin \gamma_{(h)K}^{H}\cdot \cos \alpha}{\sin 1''} \tag{36} \]

or, because of the smallness of the angle \(\gamma_{(h)K}^{H}\):

\[ c_K^{H}=\gamma_{(h)K}^{H}\cos \alpha, \tag{37} \]

where \(\gamma_{(h)K}^{H}\) is the projection onto the plane of the horizontal circle of the angle \(c_K^{H}\), and \(\alpha\) is the angle between the sighting axis and the plane of the horizontal circle.

If the plane of the horizontal circle is horizontal, then the angle \(\alpha\) may be regarded as the angle of inclination of the sighting axis to the horizon, or as the angular magnitude of the nonperpendicularity of the sighting axis to the plumb direction \((c_K^{Z})\). Then formula (37) is rewritten as:

\[ c_K^{H}=\gamma_{(h)K}^{H}\cdot \cos c_K^{Z}. \tag{38} \]

And, finally, in the case when both axes (the general case) are not parallel to the plane of the counter and make different angles with it, the computations should be carried out on the basis of (5), (6), and (8) by the formulas:

\[ \gamma_{(h)K}^{H\,V}=\frac{R_h-L'_h}{2} \tag{39} \]

\[ MO_{(h)K}^{H}=\frac{R_h+L'_h}{2} \tag{40} \]

\[ c_K^{H}=\frac{1}{\sin 1''}\left\{\sin \alpha \sin \beta+\cos \alpha \cos \beta \sin \gamma_{(h)K}^{H\,V}\right\} \tag{41} \]

or, because of the smallness of the angles \(\beta\) and \(\gamma_{(h)K}^{H\,V}\):

\[ c_K^{H}=\beta \sin \alpha+\gamma_{(h)K}^{H\,V}\cdot \cos \alpha, \tag{42} \]

where \(\gamma_{(h)K}^{H\,V}\) is the error of reading on the horizontal circle due to the combined influence of: the nonperpendicularity of the sighting axis to the horizontal-

…of this axis of rotation and the angle of inclination of the horizontal axis to the plane of the horizontal circle; \(\alpha\) and \(\beta\) are the angles made by the sighting and horizontal axes with the plane of the horizontal circle.

If the plane of the horizontal circle is horizontal, then the angle \(\alpha\) is the angle of inclination of the sighting axis to the horizon, or the magnitude of the nonperpendicularity (which we shall denote by \(c_K^Z\)) of the sighting axis to the plumb direction; while the angle \(\beta\) is the angle of inclination of the horizontal axis to the horizon, or the magnitude of the nonperpendicularity (which we shall denote by \(c_H^Z\)) of the horizontal axis to the plumb direction. In this case formula (42) is rewritten as follows:

\[ c_K^H = c_H^Z \sin c_K^Z + \gamma_{(h)K}^{\gamma}\cdot c_K^Z \tag{43} \]

Third condition

Perpendicularity of the horizontal axis of rotation \(H\) to the vertical axis of rotation \(V\).

The nonperpendicularity of these axes is usually detected and measured by means of the vial of a striding level, or else by means of special observations that graphically fix, on a plumb plane—for example, the wall of a building—the position of the axis of rotation.

Let us imagine in Fig. 3 the ideal position of the horizontal axis, perpendicular to the vertical axis of rotation, in the form of the straight line \(OH\), and its actual position as \(OH_1\); let us denote the angle between these straight lines by \(c_H^V\) (the magnitude of the nonperpendicularity of the horizontal axis of rotation to the vertical). Obviously, if we further imagine the straight line \(OV_1\), perpendicular to \(OH_1\) and making with the plumb direction \(OZ\) the angle \(c_{V_1}^{h}\), then the magnitude of the nonperpendicularity of the horizontal axis of rotation to the vertical, \(c_H^V\), will be measured by the angle between the straight lines \(OV\) and \(OV_1\), which may be represented in the form:

\[ c_H^V = c_{V_1}^{h} - c_V^{h}. \tag{44} \]

The values of the quantities \(c_V^h\) and \(c_{V_1}^h\) entering into the last formula are computed by formula (24), into which we substitute the readings of the center of the bubble of the striding level obtained when measuring \(c_V^h\) and \(c_{V_1}^h\).

To determine the quantities \(c_V^h\) and \(c_{V_1}^h\), we place the striding level in its initial position on the horizontal axis of rotation and take the reading \(R_U\) on it. We rotate the instrument together with the level through \(180^\circ\) about \(OV\) and, with the level not reversed, take the reading \(L_U\). We return the instrument with the level to its initial position and take the level reading \(R_{U_1}=R_U\). We rotate the striding level through \(180^\circ\) about the axis \(OV_1\) (we reverse the level on the horizontal axis) and take the reading \(L_{U_1}\). Then, using formula (24), we obtain:

\[ c_V^h = \frac{R_U - L_U}{2}, \tag{45} \]

\[ c_{V_1}^h = \frac{R_{U_1} - L_{U_1}}{2}. \tag{46} \]

Substitution of the last expressions into formula (44) gives:

\[ c_H^V = \frac{R_{U_1} - L_{U_1}}{2} - \frac{R_U - L_U}{2} \tag{47} \]

or, since \(R_U=R_{U_1}\):

\[ c_H^V=\frac{L_U-L_{U_1}}{2}. \tag{48} \]

Obviously, observations for determining \(c_V^h\) and \(c_{V_1}^h\) can be organized in such a way that \(L_U=L_{U_1}\) and \(R_U\ne R_{U_1}\); then

\[ c_H^V=\frac{R_{U_1}-R_U}{2}. \]

From consideration of Fig. 4 it follows that, if the horizontal axis of rotation does not lie in one plumb plane passing through the plumb direction \(OZ\) and the vertical axis of rotation \(OV\), then the formulas (48) and (49) given above will also be valid in this case. In this case, with the attached level, instead of the quantities \(c_V^h\) and \(c_{V_1}^h\), the quantities \(90^\circ-\gamma_V^Z\) and \(90^\circ-\gamma_{V_1}^Z\), determined by formula \((24')\), will be measured. Comparing (24) and \((24')\), it is not difficult to see that:

\[ c_H^V=c_{V_1}^h-c_V^h=(90^\circ-\gamma_{V_1}^Z)-(90^\circ-\gamma_V^Z). \tag{50} \]

The formulas (48) and (49) given above will be valid only in the absence of conicity of the pivots of the horizontal axis of rotation, since the angle \(c_{V_1}^h\), in the presence of conicity, will be the angle of inclination to the horizon for the upper generatrix of the cone, and not for its axis, which represents the horizontal axis of rotation of the instrument.

In order to determine the magnitude of the conicity \(\omega\), and with it the actual magnitude of the nonperpendicularity of the horizontal axis to the vertical, one must measure the angle of inclination to the horizon \(c_{V_2}^h\) for the lower generatrix of the cone. To do this, having taken the level reading \(R_{U_2}=R_{U_1}=R_U\) in the initial position, we rotate it through \(180^\circ\) about some axis \(OV_2\), perpendicular to the lower generatrix, which is equivalent to reversing the horizontal axis, and take the reading \(L_{U_2}\) in the noninitial position of the level. After this the value of the conicity is computed from the formulas

\[ \omega=c_{V_2}^h-c_{V_1}^h =\frac{R_{U_2}-L_{U_2}}{2}-\frac{R_{U_1}-L_{U_1}}{2} =\frac{L_{U_1}-L_{U_2}}{2}. \tag{51} \]

If, however, \(R_{U_2}=R_{U_1}\), \(R_{U_2}\ne R_{U_1}\), and \(L_{U_2}=L_{U_1}\):

\[ \omega=\frac{R_{U_2}-R_{U_1}}{2}. \tag{52} \]

The magnitude of the nonperpendicularity of the horizontal axis to the vertical will be found from the formulas:

\[ c_H^V=c_{V_1}^h-c_V^h+\frac{1}{2}\omega =\frac{1}{4}(-L_{U_2}-L_{U_1}+2L_U) \tag{53} \]

or, if \(L_{U_2}=L_{U_1}=L_U\),

\[ c_H^V=\frac{1}{4}(-2R_U+R_{U_1}+R_{U_2}) \tag{54} \]

To determine the angle of inclination of the horizontal axis to the horizon, or, what is the same thing, the magnitude of the nonperpendicularity of the horizontal axis to

plumb direction \(c_H^Z\), in the absence of conicity of the pivots, one may use the formula for the inclination of the upper generatrix (46), that is

\[ c_H^Z = c_{V_1}^h=\frac{RU_1-LU_1}{2}. \tag{55a} \]

In the case where conicity is present, however, one must take the mean of the inclinations of both generatrices, that is

\[ c_H^Z=\frac{1}{2}\left(c_{V_2}^h+c_{V_2}^h\right) =\frac{1}{4}\left(RU_1-LU_1+RU_2-LU_2\right) \tag{56} \]

In order to set the horizontal axis of rotation in a position perpendicular to the vertical axis, it is necessary, by means of the adjusting screws at the supports of the horizontal axis, to bring the bubble of the striding level to the reading \(MO_{(U)H}^{V}\), corresponding to the perpendicular position of the horizontal and vertical axes, determined, in the absence of conicity, by the formula:

\[ MO_{(U)H}^{V}=R_U-c_H^V =R_U-\frac{RU_1-RU}{2} =\frac{3RU-RU_1}{2}, \tag{57} \]

and, when conicity is present, by the formula:

\[ MO_{(U)H}^{V}=\frac{1}{4}\left(6RU-RU_1-RU_2\right) \tag{58} \]

Thus, theoretically, in order to bring the horizontal axis into a position perpendicular to the vertical axis, and also to measure the conicity and inclination of the horizontal axis, there is no need to have a previously adjusted level or to bring the upper generatrix of the axis into a horizontal position.

If the axis of the level is first brought into a position perpendicular to the vertical axis of rotation, and the latter into the plumb position, then, obviously, at that moment \(c_U^V=0\) and \(c_V^h=0\), and formulas (47), (53), (57), and (58) take the form:

\[ c_H^Z=c_H^V=\frac{RU_1-LU_1}{2}, \tag{47′} \]

\[ c_H^Z=c_H^V=\frac{1}{4}\left(RU_1-LU_1+RU_2-LU_2\right), \tag{53′} \]

\[ MO_{(U)H}^{V}=\frac{RU_1+LU_1}{2}, \tag{57′} \]

\[ MO_{(U)H}^{V}=RU_1-\frac{1}{4}\left(RU_1-LU_1+RU_2-LU_2\right). \tag{58′} \]

In the case where the axis of the striding level does not lie in the same plumb plane passing through the vertical and horizontal axes of rotation, but forms with it some angle \(A\), then in the formulas given above it would be necessary everywhere to compute the quantities \(c_V^h\), \(c_{V_1}^h\), and \(c_{V_2}^h\), proceeding from (26), by the formula

\[ c_V^h=\frac{RU-LU}{2\cos A}. \tag{59} \]

If, however, the values of these quantities are computed by formula (24), then in this case they will be in error by the amount \(\Delta c\), determined as the difference between (59) and (24), that is:

\[ \Delta c=\frac{R_v-L_v}{2}\left(\frac{1}{\cos A}-1\right) = c_v^h\left(\frac{1}{\cos A}-1\right), \tag{60} \]

whence the angle \(A\) may be computed from the expression

\[ \cos A=\frac{c_v^h}{c_v^h+\Delta c}. \]

Let us consider, by means of a concrete example, what the value of the angle \(A\) may be for the maximum value of the quantity \(c_v^h\) that can be measured with a striding level. Suppose that we have a level with a scale interval \(\tau=5''\), by means of which quantities \(c_v^h\) up to \(30'\) can be measured. Then, if we require that the value \(\Delta c\) not exceed the accuracy of reading the level, we must put \(\Delta c=0.2\tau=1''\). In this case for \(\cos A\) we obtain the value

\[ \cos A=\frac{30'}{30'+1''}=0.9994447, \]

which corresponds to \(A=1^\circ54'30''\). In order to eliminate the angle \(A\), that is, to bring the axis of the level into one plane with the horizontal axis, it is necessary, for the given angle, by acting with the horizontal adjusting screws, to displace the level, for a level of length \(20\ \mathrm{cm}\), by \(3.3\ \mathrm{mm}\).

The analysis carried out shows that, when working with a striding level, there is no need to bring the level axis into a position perpendicular to the vertical axis and to bring the level axis exactly into one plumb plane with the horizontal axis of rotation.

If the second condition is satisfied in the instrument, the vertical axis of rotation is brought into a plumb position and the horizontal axis is set parallel to some plumb plane, for example, the wall of a building, then, when the line of sight is rotated about the horizontal axis, the line of sight describes a plane making with the horizon the angle \(90^\circ-c_H^v=90^\circ-c_H^z\). Marking on the wall the traces of the sighting plane for the two positions of the horizontal axis (with the circle on the right and on the left), we thereby graphically represent on the wall two perpendiculars to the two positions of the horizontal axis, forming an angle equal to \(2c_H^v=2c_H^z\). Without dwelling on the question of correcting the nonperpendicularity of the axes, which is examined in detail for this method in courses on geodesy, let us turn to the question of the joint verification of the second and third conditions.

Suppose that, with the vertical axis in a plumb position, at the moment of sighting some point in space, the line of sight \(K\) makes with the horizon an angle \(\alpha=c_K^z\), and the horizontal axis of rotation an angle \(\beta=c_H^z\) (by the line of sight in Fig. 2 is meant the axis \(OB\), and by the horizontal axis of rotation, the axis \(OA\)). Then the nonperpendicularity of the line of sight to the horizontal axis of rotation (the quantity \(c_K^H\)) and the angle of inclination of the horizontal axis to the horizon (the angle \(\beta=c_H^z\)) can be determined by formulas (5) and (8). For this, it is obviously necessary, at two positions of the vertical circle, to sight two points in space located at different heights and to take the corresponding readings on the horizontal and vertical circles (for the first point—\(R_{h_1}, L_{h_1}\) and \(R_{v_1}, L_{v_1}\), and for the second—\(R_{h_2}, L_{h_2}\) and \(R_{v_2}, L_{v_2}\)). Then, from the readings on the vertical

on the circle we determine the angles of inclination \(\alpha_1 = c^Z_{K_1}\) and \(\alpha_2 = c^Z_{K_2}\) for two sighting lines, and from the readings on the horizontal circle we compute, by formula (5), the values \(\gamma_1\) and \(\gamma_2\):

\[ \gamma_1=\frac{R_{h_1}-L'_{h_1}}{2} \]

and

\[ \gamma_2=\frac{R_{h_2}-L'_{h_2}}{2}. \]

Having determined the values \(\alpha_1=c^Z_{K_1}\), \(\gamma_1\), \(\alpha_2=c^Z_{K_2}\), and \(\gamma_2\), and substituting them into formula (8), we obtain two equations:

\[ c^H_K=\frac{1}{\sin 1''} \left( \sin c^Z_{K_1}\cdot \sin c^Z_H+ \cos c^Z_{K_1}\cdot \cos c^Z_H\cdot \sin \gamma_1 \right), \]

\[ c^H_K=\frac{1}{\sin 1''} \left( \sin c^Z_{K_2}\cdot \sin c^Z_H+ \cos c^Z_{K_2}\cdot \cos c^Z_H\cdot \sin \gamma_2 \right), \]

which, owing to the smallness of the angles \(c^Z_H\), \(\gamma_1\), and \(\gamma_2\), may be represented in the following form:

\[ c^H_K=c^Z_H\cdot \sin c^Z_{K_1}+\gamma_1\cdot \cos c^Z_{K_1}, \]

\[ c^H_K=c^Z_H\cdot \sin c^Z_{K_2}+\gamma_2\cdot \cos c^Z_{K_2}. \]

Solving the last two equations simultaneously, for the quantities \(c^H_K\) and \(c^Z_H\) we find the values:

\[ c^H_K= \frac{ \gamma_1\cos c^Z_{K_1}\cdot \sin c^Z_{K_2} - \gamma_2\cdot \sin c^Z_{K_1}\cdot \cos c^Z_{K_2} }{ \sin c^Z_{K_2}-\sin c^Z_{K_1} } \tag{61} \]

and

\[ c^Z_H= \frac{ \gamma_1\cdot \cos c^Z_{K_1} - \gamma_2\cdot \cos c^Z_{K_2} }{ \sin c^Z_{K_2}-\sin c^Z_{K_1} }. \tag{62} \]

Below is given a numerical example of the simultaneous determination of the magnitude of the nonperpendicularity \(c^H_K\) of the sighting axis to the horizontal axis, and of the magnitude of the nonperpendicularity \(c^Z_H\) of the horizontal axis of rotation to the plumb direction (or of the angle of inclination of the horizontal axis to the horizon). The determination was carried out under laboratory conditions with a theodolite of the “Geodesia” factory, No. 4106, with a reading accuracy on both circles equal to one minute.

Table 1

Observation for correcting the nonperpendicularity of the axes

Table 2

Observations after correcting the nonperpendicularity of \(K\) to \(H\)

After processing the observations obtained by formulas (61) and (62), the following values were found for \(c_K^H\) and \(c_H^Z\):

Table 3

Comparing the values of the nonperpendicularity of the axes \(c_K^H\) and \(c_H^Z\), computed by formulas (61) and (62), with the values obtained from direct measurements of these quantities, one may note a satisfactory agreement, of the order of the accuracy of reading on the instrument.

Measurement of Vertical Angles

The theory of determining the inclinations of the observed rays (vertical angles) can easily be obtained on the basis of the theory set forth above for the errors of the angle-measuring instrument in the relationships among its axes.

Let us suppose that (Fig. 1) the axis \(OA\) represents a plumb line; \(OB_0\), a horizontal line perpendicular to \(OA\); \(OB_1\), the sighting axis of the telescope; and the counter, the vertical circle.

As is not difficult to see, the vertical angle is the magnitude of the nonperpendicularity of the sighting axis \(OB_1(K)\) to the plumb line \(OA(Z)\), that is, the angle \(c = c_K^Z\) is determined, proceeding from (2), by the formula:

\[ c_K^Z=\frac{R_v-L'_v}{2}, \tag{63} \]

where \(c_K^Z\) is the magnitude of the nonperpendicularity of the sighting axis to the plumb line, or the vertical angle, and \(R_v\) and \(L_v\) are readings on the vertical circle.

The reading corresponding to the position of the axis \(OB\) perpendicular to the line \(OA\), determined from (4), is given by the formula

\[ MO_{(v)K}^Z=\frac{R_v+L'_v}{2}. \tag{64} \]

It is nothing other than the place of zero, that is, the reading on the vertical circle corresponding to a horizontal sighting ray.

In deriving formulas (63) and (64), in Fig. 1 the counter—the vertical circle—was assumed to be fixed; in reality, however, it is usually only the index that is fixed (on the alidade of the vertical circle), against which the readings are taken. In this case, in order to derive the formulas determining the values of the vertical angle and the zero point, let us assume that the zero of the scale on the vertical circle is located at the point \(0^\circ_K\), which is displaced from the former \(0^\circ\) by \(180^\circ - 2c\), reckoned clockwise, and that the index of the alidade of the vertical circle is situated at the point \(B_2\).

Let us further assume that, at the moment of sighting with the circle on the right at a certain point in space, the line of sight occupied the position \(OB_2\), and opposite the index \(B_2\) on the vertical circle the reading \(R_v\) was obtained. Transferring the line of sight through the zenith into the position \(OB_1\), that is, rotating \(OB_2\) about the point \(O\) through \(180^\circ - 2c\), we shall rotate through the same angle also the zero of the scale on the vertical circle (that is, the point \(0^\circ_K\) will occupy the position \(0^\circ\)), and at the index \(B_2\) we shall read the reading \(L = L_v\).

If \(OB_1\) is rotated about \(OA\) through \(180^\circ\), the line of sight will occupy the position \(OB_2\), and we shall thereby carry out sighting on the same point in space, but now with the circle on the left. It is obvious that, when the line of sight together with the vertical circle is rotated about \(OA\), the readings on the vertical circle do not change, and we have the right to write the equality

\[ L_v - R_v = 180^\circ - 2c = 180^\circ - 2c^Z_K, \]

from which, remembering that \(L_v - 180^\circ = L'_v\) and \(MO_{(v)K}^Z = R_v - c^Z_K\), we obtain formulas (63) and (64).

The formulas (63) and (64) given above are, of course, valid if the line of sight \(OB_1\) and the vertical circle are in parallel vertical planes. If, however, the line of sight, when the vertical circle is plumb, makes with the latter some small angle \(a = c^H_K\), then, as is clear from the preceding, the vertical angle \(c^Z_K\) should be computed, on the basis of (7′), by the formula:

\[ \sin c^Z_K = \sin \gamma \cdot \cos c^H_K, \tag{65} \]

having first determined the value of the projection of the vertical angle, that is, the angle \(\gamma\), on the basis of (5), by the formula:

\[ \gamma = \frac{R_v - L'_v}{2}, \]

and the zero point, on the basis of (6), by the formula:

\[ MO_{(v)K}^Z = \frac{R_v + L'_v}{2}. \tag{66} \]

But since usually the angle \(c^H_K\), which in the present case expresses the magnitude of the nonperpendicularity of the line of sight to the horizontal axis of rotation, is very small, its cosine in formula (65) may therefore be taken as equal to unity and, without committing a large error, the value of the vertical angle may be expressed as

\[ c^Z_K = \gamma = \frac{R_v - L'_v}{2}. \tag{67} \]

In the case where the plumb line and the line of sight make with the plane of the vertical circle the angles \(\beta\) and \(\alpha\), respectively (in Fig. 2 \(OA\) is the plumb line, \(OZ\) is the horizontal axis of rotation of the telescope, \(OB_1\) is the sighting ...

axis), the position of zero is still determined, on the basis of (6), by formula (66). The magnitude of the vertical angle \(c_K^Z\), on the basis of \((8')\), will be determined by the formula:

\[ \sin c_K^Z=\sin\alpha \sin\beta+\cos\alpha \cos\beta \sin\gamma, \tag{68} \]

which, because of the smallness of the angles \(\alpha\) and \(\beta\) (in the present case \(\alpha=c_K^H\)—the magnitude of the nonperpendicularity of the sighting axis to the horizontal axis, and \(\beta=c_H^Z\)—the magnitude of the nonperpendicularity of the horizontal axis to the plumb direction, or the inclination of the horizontal axis to the horizon), may be represented, analogously to the preceding case, in the form:

\[ c_K^Z=\gamma=\frac{R_v-L'_v}{2}. \tag{69} \]

Measurement of Horizontal Directions

The question of determining horizontal directions is resolved much more simply, since in this case it is not necessary to find the magnitudes of the nonperpendicularity of the axes.

Understanding by a horizontal direction the reading on the horizontal circle corresponding to the horizontal projection of the line of sight to a point in space, with the vertical axis of rotation in a plumb position and with the three conditions of mutual perpendicularity of the axes of the angle-measuring instrument noted above fulfilled, for determining the horizontal direction, on the basis of (4) and (6), one may write a single value:

\[ MO_{(h)}=\frac{R_h+L'_h}{2}, \tag{70} \]

where \(MO_{(h)}\) is the required horizontal direction to a point in space, and \(R_h\) and \(L'_h\) are readings on the horizontal circle for the two positions of the vertical circle.

If the values of the magnitudes of nonperpendicularity of the axes of the angle-measuring instrument are known, then the value of the horizontal direction may be determined using only one reading on the horizontal circle (for example \(R_h\)), guided by the formula following from the features given:

\[ MO_{(h)}=R_h-\gamma_{(h)}, \tag{71} \]

where \(\gamma_{(h)}\) is a certain correction depending on the quantities expressing the nonperpendicularity of the instrument axes.

In the presence of nonperpendicularity of the sighting axis to the horizontal axis (the quantity \(c_K^H\)), the value of the correction \(\gamma_{(h)K}^{H}\) is obtained from (7), taking into account that \(c_K^H\) and \(\gamma_{(h)K}^{H}\) are small quantities:

\[ \gamma_{(h)K}^{H}=\frac{c_K^H}{\cos\alpha}=\frac{c_K^H}{\cos c_K^Z}. \tag{72} \]

In the presence of nonperpendicularity of the horizontal axis of rotation to the vertical axis (the quantity \(c_H^V\)), the value of the correction is obtained from \((8')\) and taking into account that \(c=c_K^H=0\), while \(\gamma=\gamma_{(h)H}^{V}\) and \(\beta=c_H^V\) are small quantities:

\[ \gamma_{(h)H}^{V}=-c_H^V\cdot \operatorname{tg}\alpha=-c_H^V\cdot \operatorname{tg} c_K^Z. \tag{73} \]

And, finally, when there is simultaneously nonperpendicularity of the sighting axis to the horizontal axis and of the horizontal axis to the vertical axis, proceeding from this same formula (8′), for the correction \(\gamma_{(h)}\) we find the value:

\[ \gamma_{(h)}=\frac{c_{K}^{H}}{\cos\alpha}-c_{H}^{V}\cdot \operatorname{tg}\alpha =\frac{c_{K}^{H}}{\cos c_{K}^{Z}}-c_{H}^{V}\cdot \operatorname{tg}c_{K}^{Z}. \tag{74} \]

Conclusion

The theory of determining the errors of an angle-measuring instrument in the relationships between its axes, and the theory of determining angular coordinates—the vertical angle and the horizontal direction—in existing courses of geodesy are usually given piecemeal, in different parts of the course, in the form of special explanations (with the attachment of special drawings and the introduction of special designations for each case) for each question separately, without establishing the internal connection between them; this makes the exposition cumbersome and the noted questions difficult to assimilate.

The uniformity of the requirements in the relationships between the geometrical elements of an angle-measuring instrument—the requirement of mutual perpendicularity for each pair of axes of the instrument—has enabled the author to substantiate very simply the known general rules for checking an angle-measuring instrument and to derive the general formulas:

\[ c=\frac{R-L'}{2} \tag{75} \]

and

\[ MO=\frac{R+L'}{2}, \tag{76} \]

which determine: 1) the magnitude of the error in the relationships between the axes, 2) the value of the reading corresponding to the proper relationship of the axes, and 3) the correct values of the angular coordinates.

The generalization carried out further made it possible to prove the possibility of measuring angles with a level without its preliminary check, to establish new methods for checking a level, a method for measuring the inclination of the vertical axis, a method for the joint determination of the nonperpendicularity of the sighting axis to the horizontal axis and of the horizontal axis to the vertical axis, and a method for measuring the inclination of the horizontal axis to the horizon in the absence of a striding level.

In concluding here the solution of the problems posed above, it is necessary to note that the numerous designations existing in the literature for the same quantities—often not symbolizing these quantities—compelled the author, in writing this work, to use special designations for angular quantities, the meaning of which is explained by indices characterizing the names of the geometrical elements of the angle-measuring instrument.