PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Volume 131　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1965

GENERAL METHODS FOR SOLVING BASIC COMPUTATIONAL PROBLEMS ON THE TERRESTRIAL SPHEROID

B. F. KRUTOI

(Presented by the mine-surveying and geodetic section
of the Jubilee Conference of TPI in February 1961)

The main purpose of the present article is to set forth general, chiefly new methods for solving, in geodetic coordinates, the basic computational problems on the terrestrial spheroid: the computation of point positions, shortest distances, directions along the shortest path, direct resection by rays, areas of coordinate trapezia, and certain other problems.

Part 1. Statement of the Problem

1. We first introduce new, more precise names for certain concepts that arise in the study of surfaces.

We shall call a surface thread a continuous one-dimensional set of points of a surface.

Among surface threads, by virtue of their special properties, we shall distinguish those threads \(\Gamma\) at every point of which the principal normal coincides with the normal to the surface. Such threads \(\Gamma\) are customarily called geodesic threads, since they are widely used in solving various problems of geodesy. However, such a designation of these threads is purely accidental, for it does not reflect a number of their remarkable intrinsic properties:

a) If an arc \(\Delta \Gamma_{ij}\) of a geodesic thread \(\Gamma_{ij}\) on a surface \(S\), drawn between some two points \(i, j\) of this surface, contains no vertices of the thread \(O_l\), then this arc \(\Delta \Gamma_{ij}\) is the shortest on the surface \(S\) between the indicated points \(i, j\);

b) If, on a smooth surface \(S\), one end \(i\) of a flexible material thread is fixed, and the other end is passed through a small ring at a second point \(j\) of the surface \(S\), then, under the action of a tension applied to its free end, the flexible material thread between the indicated points \(i, j\) of the surface \(S\) will lie along a geodesic curve and at the same time will be the most straightened thread between these points;

c) If at every point of a geodesic thread \(\Gamma_{ij}\) on a surface \(S\) a tangent plane to the surface \(S\) is drawn, and then an enveloping surface \(D\) is constructed for these planes, then the geodesic thread \(\Gamma_{ij}\) of the surface \(S\) will also be geodesic for the surface \(D\). Therefore, when the surface \(D\) is developed onto a plane, the geodesic thread of the surface \(S(D)\) will pass into a geodesic thread of the plane, i.e., will turn into a straight line.

The considerations presented above speak quite convincingly to the fact that surface threads with coinciding principal and surface normals will more justifiably be called not geodesic, but leveled threads of the surface. We shall adhere to this designation for surface threads of this kind in what follows.

1. Let us also note that the geodesic coordinates \(B, L\) of surface points of the terrestrial spheroid are a special case of surface coordinates \(u, v\), for which here two angular quantities have been taken: latitude \(B\) and longitude \(L\). In the indicated reference frame the positions of surface points \(i\) of the terrestrial spheroid are determined by the intersection of two families of coordinate threads: meridians \(L=L_i\) and parallels \(B=B_j\). It is easy to establish that the threads of the first family \(L=L_i\) are leveled on the spheroid, whereas the threads of the second family \(B=B_j\) will not be leveled; moreover, any two threads \(L=L_i\) and \(B=B_j\) of these families intersect at a right angle. Therefore the geodesic reference frame \(B, L\) on the spheroid may be assigned to the class of rectangular semi-leveled reference frames.

Let us note further that two pairs of coordinate threads \(L=L_i, L=L_h\) and \(B=B_j, B=B_k\) of the spheroid, taken in pairs from each family, form at their intersection a spheroidal coordinate trapezoid \(ihkj\) with four right angles. The determination of the area \(S_{ij}^{hk}\) of such a coordinate trapezoid is one of the basic computational problems on the spheroid.

1. We now introduce, for the surface of the terrestrial spheroid, a number of notations:

\(B_i, L_i\) — the geodesic latitude and longitude of point \(i\) of the terrestrial spheroid;

\(A_{ij}\) — the geodesic azimuth at point \(i\) of the leveled thread \(\Gamma_{ij}\), drawn on the spheroid through point \(i\) and the neighboring point \(j\);

\(s_{ij}\) — the length of the arc \(\Delta \Gamma_{ij}\) of the leveled thread \(\Gamma_{ij}\) between points \(i, j\) of the spheroid;

\(\Delta L_{ij}\) — the difference of longitudes \(L_i, L_j\) of points \(i, j\) of the spheroid;

\(x_{ij}\) — the length of the arc \(\Delta X_{ij}\) of the meridian \(X_{ij}\) between points \(i, j\) of the spheroid with latitudes \(B_i, B_j\);

\(\pi_{ij}\) — the length of the arc \(\Delta \Pi_{ij}\) of the parallel \(\Pi_{ij}\) between points \(i, j\) of the spheroid with latitudes \(B_i, B_j\) and longitude difference \(\Delta L_{ij}\);

\(S_{ij}^{hk}\) — the area of a spheroidal trapezoid bounded by two meridians \(X_{ih}, X_{jk}\) with longitude difference \(\Delta L_{ij}=\Delta L_{hk}\) and two parallels \(\Pi_{ij}, \Pi_{hk}\) with latitudes \(B_i=B_j, B_h=B_k\).

The notations introduced here and the concepts associated with them require certain clarifications and additions, caused chiefly by the fact that the leveled curves \(\Gamma_{ij}\) of the spheroid are not, in general, closed and are tangent by their successive vertices to two bounding parallels: the northern \(\Pi^{(ij)}\) and the southern \(\overline{\Pi}^{(ij)}\), equidistant from the equatorial plane. We shall now dwell on these clarifications and additions.

a) Since between two points \(i, j\) of the spheroid one can draw two leveled arcs: the shortest \(\Delta \Gamma_{ij}\) and the longer \(\Delta \check{\Gamma}_{ij}\), we shall denote the distance along the arc \(\Delta \Gamma_{ij}\) by \(s_{ij}\), and along the arc \(\Delta \check{\Gamma}_{ij}\) by \(\check{s}_{ij}\), and \(s_{ij}\) and \(\check{s}_{ij}\) will be considered positive. In accordance with this, the geodesic azimuths at point \(i\) of the arcs \(\Delta \Gamma_{ij}, \Delta \check{\Gamma}_{ij}\) will be denoted by \(A_{ij}\) and \(\check{A}_{ij}\); as a rule, \(A_{ij}\ne \check{A}_{ij}\pm 180^\circ\), which will be established below [Del. 7].

b) The geodetic latitudes of points \(i\) of the spheroid shall be reckoned in both directions from the plane of the equator, from \(0\) to \(\pm \dfrac{\pi}{2}\). For the northern half of the spheroid we shall denote the latitudes of points \(i\) by \(B_i\) and take them as positive; for the southern half we shall denote the latitudes of points \(i\) by \(\overline{B}_i\) and take them as negative. Thus the latitudes \(B_i, \overline{B}_i\) vary within the following limits:

\[ 1)\quad 0 \leq B_i \leq +\frac{\pi}{2}; \qquad 2)\quad 0 \geq \overline{B}_i \geq -\frac{\pi}{2}. \]

c) The geodetic longitudes of points \(i\) of the spheroid shall be reckoned in two ways: eastward and westward from Greenwich, from \(0\) to \(\pm 2\pi\), and we shall denote them respectively by \(L_i\) and \(\mathring{L}_i\), assuming here that \(L_i > 0\), while \(\mathring{L}_i < 0\). Hence, for point \(i\), we shall have two longitude differences:

\[ 1)\quad \Delta L_{ij}=L_j-L_i, \qquad 2)\quad \mathring{\Delta L}_{ij}=\mathring{L}_j-L_i=\mathring{L}_{ij}. \]

Such a double method of reckoning longitudes and their differences is convenient in solving direct intersections on the spheroid, and also in certain other cases.

d) The northern and southern vertices of the adjusted curve \(\Gamma_{ij}\), i.e., the points on this curve with the smallest absolute value of latitude, shall be denoted: east of the initial point \(i\)—by \(O_{ij}^{(s)}, \overline{O}_{ij}^{(s)}\), and west of the initial point—by \(\mathring{O}_{ij}^{(s)}, \mathring{\overline{O}}_{ij}^{(s)}\), where \(s=1,2,\ldots\) is the order of remoteness of the vertex of the given kind relative to the initial point \(i\). Correspondingly, the latitude of the northern vertices \(O_{ij}^{(s)}, \mathring{O}_{ij}^{(s)}\) shall be denoted by \(B_0^{(ij)}\), and the latitude of the southern vertices \(\overline{O}_{ij}^{(s)}, \mathring{\overline{O}}_{ij}^{(s)}\)—by \(\overline{B}_0^{(ij)}\), where evidently \(B_0^{(ij)}=-\overline{B}_0^{(ij)}\). The longitudes \(L\) of the points \(O_{ij}^{(s)}\) and \(\overline{O}_{ij}^{(s)}\) shall be denoted by \(L_0^{(ij.s)}\) and \(\overline{L}_0^{(ij.s)}\); the longitudes \(\mathring{L}\) of the points \(\mathring{O}_{ij}^{(s)}, \mathring{\overline{O}}_{ij}^{(s)}\) shall be denoted by \(\mathring{L}_0^{(ij.s)}, \mathring{\overline{L}}_0^{(ij.s)}\).

e) The points of intersection of the adjusted curve \(\Gamma_{ij}\) with the equator shall be denoted: east of the initial point \(i\)—by \(\mathcal{E}_{ij}^{(s)}\), and west of \(i\)—by \(\mathring{\mathcal{E}}_{ij}^{(s)}\), where \(s=1,2,\ldots\) is the order of remoteness of the intersection point relative to the initial point \(i\). Correspondingly, the longitudes \(L\) of the eastern points \(\mathcal{E}_{ij}^{(s)}\) shall be denoted by \(L_{\mathcal{E}}^{(ij.s)}\), and the longitudes \(\mathring{L}\) of the western points \(\mathring{\mathcal{E}}_{ij}^{(s)}\)—by \(\mathring{L}_{\mathcal{E}}^{(ij.s)}\). The constant azimuth of the adjusted curve \(\Gamma_{ij}\) at the points \(\mathcal{E}_{ij}^{(s)}, \mathring{\mathcal{E}}_{ij}^{(s)}\) shall be denoted by \(A_{\mathcal{E}}^{(ij)}\).

f) Sometimes, for points \(i,j\) of the northern and southern parts of the spheroid, we shall introduce special notation: points of the northern part shall be denoted by \(i,j\), and points of the southern part—by \(\overline{i}, \overline{j}\). These and all the above notations for the various points on the adjusted curve \(\Gamma_{ij}\) are shown in Fig. 1.

Using the notation given above, we shall now briefly write down the conditions of six computational problems, which may be regarded as basic for the surface of the terrestrial spheroid:

1. Direct problem for the arc \(\Delta\Gamma_{ij}\) of the adjusted thread \(\Gamma_{ij}\).
   Given \(B_1, L_1, A_{1.2}, s_{1.2}\); find \(B_2, L_2, s_{1.2}\).

2. Inverse problem for the arc \(\Delta\Gamma_{ij}\) of the adjusted thread \(\Gamma_{ij}\).
   Given \(B_1,L_1\) and \(B_2,L_2\); find \(A_{1.2}, A_{2.1}\) and \(s_{1.2}\).

3. Direct adjusted-ray intersection \((i=1,2;\ j=3)\).
   Given \(B_1,L_1,A_{1.3}\) and \(B_2,L_2,A_{2.3}\); find \(B_3,L_3\), and also \(A_{3.1}, s_{1.3}\) and \(A_{3.2}, s_{2.3}\).

4. Direct problem for the arc \(\Delta\Pi_{ij}\) of the parallel \(\Pi_{ij}\) \((i=1,\ j=2)\). Given \(B_1=B_2\) and \(\Delta L_{1.2}\); find \(p_{1.2}\).

b) We shall reckon the geodetic latitudes of points \(i\) of the spheroid in both directions from the plane of the equator, from 0 to \(\pm \dfrac{\pi}{2}\). For the northern half of the spheroid we shall denote the latitudes of points \(i\) by \(B_i\) and take them to be positive; for the southern half we shall denote the latitudes of points \(i\) by \(\overline{B}_i\) and take them to be negative. Thus the latitudes \(B_i,\overline{B}_i\) vary within the following limits:

\[ 1)\quad 0 \leqslant B_i \leqslant +\frac{\pi}{2}; \qquad 2)\quad 0 \geqslant \overline{B}_i \geqslant -\frac{\pi}{2}. \]

c) We shall reckon the geodetic longitudes of points \(i\) of the spheroid in two ways: eastward and westward from Greenwich, from 0 to \(\pm 2\pi\), and denote them respectively by \(L_i\) and \(\overset{\circ}{L}_i\), assuming here that \(L_i>0\), while \(\overset{\circ}{L}_i<0\). Hence, for point \(i\), we shall have two longitude differences:

\[ 1)\quad \Delta L_{ij}=L_j-L_i,\qquad 2)\quad \overset{\circ}{\Delta L}_{ij}=\overset{\circ}{L}_j-L_i=\overset{\circ}{L}_{ij}. \]

Such a dual method of counting longitudes and their differences is convenient in solving direct intersections on the spheroid, and also in some other cases.

d) We shall denote the northern and southern vertices of the adjusted curve \(\Gamma_{ij}\), i.e. the points on this curve with the smallest absolute value of latitude, as follows: eastward from the origin \(i\)—by \(O_{ij}^{(s)}, \overline{O}_{ij}^{(s)}\), westward from the origin—by \(\overset{\circ}{O}_{ij}^{(s)}, \overset{\circ}{\overline{O}}_{ij}^{(s)}\), where \(s=1,2,\ldots\) is the order of remoteness of a vertex of the given kind relative to the initial point \(i\). Accordingly, we shall denote the latitude of the northern vertices \(O_{ij}^{(s)}, \overset{\circ}{O}_{ij}^{(s)}\) by \(B_0^{(ij)}\), and the latitude of the southern vertices \(\overline{O}_{ij}^{(s)}, \overset{\circ}{\overline{O}}_{ij}^{(s)}\) by \(\overline{B}_0^{(ij)}\), where obviously \(B_0^{(ij)}=-\overline{B}_0^{(ij)}\). The longitudes \(L\) of the points \(O_{ij}^{(s)}\) and \(\overline{O}_{ij}^{(s)}\) will be denoted by \(L_0^{(ij.s)}\) and \(\overline{L}_0^{(ij.s)}\); the longitudes \(\overset{\circ}{L}\) of the points \(\overset{\circ}{O}_{ij}^{(s)}, \overset{\circ}{\overline{O}}_{ij}^{(s)}\) will be denoted by \(\overset{\circ}{L}_0^{(ij.s)}, \overset{\circ}{\overline{L}}_0^{(ij.s)}\).

e) We shall denote the points of intersection of the adjusted curve \(\Gamma_{ij}\) with the equator as follows: eastward from the origin \(i\)—by \(\mathcal{E}_{ij}^{(s)}\), and westward from \(i\)—by \(\overset{\circ}{\mathcal{E}}_{ij}^{(s)}\), where \(s=1,2,\ldots\) is the order of remoteness of the point of intersection relative to the initial point \(i\). Accordingly, the longitudes \(L\) of the eastern points \(\mathcal{E}_{ij}^{(s)}\) will be denoted by \(L_{\mathcal{E}}^{(ij.s)}\), and the longitudes \(\overset{\circ}{L}\) of the western points \(\overset{\circ}{\mathcal{E}}_{ij}^{(s)}\) by \(L_{\mathcal{E}}^{(ij.s)}\). The constant azimuth of the adjusted curve \(\Gamma_{ij}\) at the points \(\mathcal{E}_{ij}^{(s)}, \overset{\circ}{\mathcal{E}}_{ij}^{(s)}\) will be denoted by \(A_{\mathcal{E}}^{(ij)}\).

f) Sometimes, for points \(i,j\) of the northern and southern parts of the spheroid, we shall introduce special notation: points of the northern part will be denoted by \(i,j\), points of the southern part—by \(\overline{i},\overline{j}\). These and all the above notations for the various points on the adjusted curve \(\Gamma_{ij}\) are shown in Fig. 1.

Using the notation introduced above, we shall now write briefly the conditions of the six computational problems that may be regarded as basic for the surface of the terrestrial spheroid:

1. The direct problem for an arc \(\Delta\Gamma_{ij}\) of an adjusted thread \(\Gamma_{ij}\).
   Given \(B_1,L_1,A_{1.2},s_{1.2}\); find \(B_2,L_2,s_{1.2}\).

2. The inverse problem for an arc \(\Delta\Gamma_{ij}\) of an adjusted thread \(\Gamma_{ij}\).
   Given \(B_1,L_1\) and \(B_2,L_2\); find \(A_{1.2}, A_{2.1}\) and \(s_{1.2}\).

3. Direct adjusted-ray intersection \((i=1,2;\ j=3)\).
   Given \(B_1,L_1,A_{1.3}\) and \(B_2,L_2,A_{2.3}\); find \(B_3,L_3\), as well as \(A_{3.1},s_{1.3}\) and \(A_{3.2},s_{2.3}\).

4. The direct problem for an arc \(\Delta\Pi_{ij}\) of the parallel \(\Pi_{ij}\) \((i=1,\ j=2)\). Given \(B_1=B_2\) and \(\Delta L_{1.2}\); find \(p_{1.2}\).

a) finite

\[ S_{ij}^{hk}=b^2\cdot \Delta L_{ij}\int_{B_i}^{B_h} \frac{\cos B\,dB}{(1-e^2\sin^2 B)^2}= \]

\[ =\frac{b^2}{2}\cdot \Delta L_{ij} \left[ \frac{\sin B}{1-e^2\sin^2 B} +\frac{1}{2e}\ln\frac{1+e\sin B}{1-e\sin B} \right]_{B_i}^{B_h}. \tag{3} \]

b) in the form of a convergent series

\[ S_{ij}^{hk}=b^2\cdot \Delta L_{ij}\int_{B_i}^{B_h} \sum_{\lambda=0}^{n}\binom{-2}{\lambda} (-e^2\sin^2 B)^\lambda \cos B\,dB= \]

\[ =b^2\cdot \Delta L_{ij} \sum_{\lambda=0}^{n} (-1)^\lambda \frac{\binom{-2}{\lambda}}{\lambda+1} e^{2\lambda}\sin^{2\lambda+1}B \Bigg|_{B_i}^{B_h}, \tag{4} \]

where \(a\) and \(b\) are the major and minor semiaxes of the terrestrial spheroid, and

\[ 1)\ e^2=\frac{a^2-b^2}{a^2}, \qquad 2)\ \binom{m}{\lambda}= \frac{m(m-1)\ldots [m-(\lambda-1)]}{1\cdot 2\ldots \lambda}, \]

\[ \binom{m}{0}=\binom{m}{m}=1. \tag{5} \]

It is clear from this that principal attention must be devoted to the first three problems (1)—(3), over the improvement and generalization of whose solution many geodesists in all countries are working.

Div. 3. Derivation of general expressions underlying the solution of the first three problems

1. For the solution of the first two problems (1), (2), more than a dozen special methods had been proposed, suitable for distances \(s_{1.2}\) not exceeding 1000–3000 km, and one general method, due to Bessel [1], for any distances \(s_{1.2}\). As for the third problem (3), sufficiently simple and at the same time completely general methods for its solution had not yet been found.

Taking the above into account, I have discovered and developed yet another general method for solving the first two problems (1), (2), in which the question of the possible relations between the initial and the quantities to be determined is considered with the utmost completeness. On the basis of the working expressions derived in solving these problems, two independent methods have been found for solving the last problem—the direct spheroidal intersection. The essence of the proposed methods is considered below; at the beginning we shall obtain a compilation of the initial closed expressions, from which we shall then derive the corresponding working expressions for solving the three basic problems (1)—(3).

1. The new methods are based on a compilation of three ordinary first-order differential equations with variable coefficients, which defines the level line \(\Gamma\) of the terrestrial spheroid drawn through the point \(C(B,L)\) under the azimuth \(A\):

\[ \begin{cases} M\,dB+0\cdot dL-\cos A\,ds+0\cdot dA=0,\\ 0\cdot dB+r\,dL-\sin A\,ds+0\cdot dA=0,\\ 0\cdot dB+\sin A\,dL-0\cdot ds-1\cdot dA=0. \end{cases} \tag{6} \]

In addition, the well-known Clairaut equation that follows from this set is used:

\[ r\sin A=h=r_0=\text{const.} \tag{7} \]

Here \(M\) and \(r\) are the radii of curvature of the meridian and the parallel at the variable point \(C\) of the given straightened thread \(\Gamma\) of the spheroid, emerging from \(C\) at azimuth \(A\); the constant \(h\) is, obviously, the radius of the parallel \(r_0\) at the vertex \(O\) of the straightened curve \(\Gamma\), i.e., at the point where the azimuth \(A=A_0=90^\circ\) or \(270^\circ\).

1. Using the indicated collection of equalities (6), (7), we first find the values of \(ds\), \(dL\), and \(dA\) as functions of the latitude \(B\). First of all, we have:

\[ \begin{gathered} 1)\ \sin A=\frac{h}{r}; \qquad 2)\ \cos A=\frac{\cos A}{|\cos A|}\,\frac{\sqrt{r^2-h^2}}{r};\\[6pt] 3)\ \beta=\frac{\cos A}{|\cos A|}=\frac{dB}{|dB|}=\pm1;\\[6pt] 4)\ \beta\,ds=Mr\,\frac{dB}{\sqrt{r^2-h^2}}; \qquad 5)\ \beta\,dL=h\,\frac{M}{r}\,\frac{dB}{\sqrt{r^2-h^2}};\\[6pt] 6)\ dA=h\,\frac{M}{r}\,\frac{\sin B\,dB}{\sqrt{r^2-h^2}}. \end{gathered} \tag{8} \]

The multiplier \(\beta=\pm1\) is introduced here because \(ds\) is always taken \(>0\), \(\sqrt{r^2-h^2}\) is here regarded as \(>0\), and, consequently, under these conditions we have:

Table 1

\(0\le A<\dfrac{\pi}{2}\)	\(\sin A\ge0\)	\(\cos A\ge0\)	\(dB\ge0\)	\(dL\ge0\)	\(dA\ge0\)
\(\dfrac{\pi}{2}\le A<\pi\)	\(\sin A\ge0\)	\(\cos A\le0\)	\(dB\le0\)	\(dL\ge0\)	\(dA\ge0\)
\(\pi\le A<\dfrac{3}{2}\pi\)	\(\sin A\le0\)	\(\cos A\le0\)	\(dB\le0\)	\(dL\le0\)	\(dA\le0\)
\(\dfrac{3}{2}\pi\le A<2\pi\)	\(\sin A\le0\)	\(\cos A\ge0\)	\(dB\ge0\)	\(dL\le0\)	\(dA\le0\)

Taking into account now that

\[ \begin{gathered} 1)\ M=\frac{a(1-e^2)}{W^3} =\frac{a(1-e^2)}{(1-e^2\sin^2 B)^{3/2}}; \qquad 3)\ e^2=\frac{a^2-b^2}{a^2};\\[6pt] 2)\ r=\frac{a\cos B}{W} =\frac{a\cos B}{(1-e^2\sin^2 B)^{1/2}}; \qquad 4)\ e'^2=\frac{a^2-b^2}{b^2}, \end{gathered} \tag{9} \]

we shall compute separately the quantities

\[ Mr,\qquad \frac{M}{r},\qquad \sqrt{r^2-h^2}, \]

included in (8). After simple transformations we obtain

\[ \begin{gathered} 1)\quad Mr=a(1-e^{2})\frac{\cos B}{W^{4}};\qquad 2)\quad \frac{M}{r}=\frac{1-e^{2}}{W^{2}\cos B};\\[4pt] 3)\quad \sqrt{r^{2}-h^{2}}=\frac{a}{W}\sqrt{\cos^{2}B-\frac{h^{2}}{a^{2}}W^{2}} . \end{gathered} \tag{10} \]

In (10.3) let us introduce the notation:

\[ \frac{r^{2}}{a^{2}}\sin^{2}A=\frac{h^{2}}{a^{2}}=\sin^{2}A_{\mathrm e}=\nu^{2}\leqslant 1, \tag{11} \]

where \(A_{\mathrm e}\) is, evidently, the azimuth of the rectified curve \(\Gamma\) at the point \(\mathrm E\) of its intersection with the equator. Then, instead of (10.3), we obtain:

\[ \sqrt{r^{2}-h^{2}} = \frac{a\sqrt{1-\nu^{2}}}{W} \sqrt{1-\frac{1-e^{2}\nu^{2}}{1-\nu^{2}}\sin^{2}B}. \]

If we then introduce the new notation

\[ \frac{1-e^{2}\nu^{2}}{1-\nu^{2}}=\tau^{2}\geqslant 1, \tag{12} \]

then for \(\sqrt{r^{2}-h^{2}}\) we shall finally have:

\[ \sqrt{r^{2}-h^{2}} = \frac{a\sqrt{1-\nu^{2}}}{W}\sqrt{1-\tau^{2}\sin^{2}B}. \tag{13} \]

We now substitute the expressions found for \(Mr\), \(\dfrac{M}{r}\), and \(\sqrt{r^{2}-h^{2}}\), namely (10.1), (10.2), and (13), into the initial equalities (8). Then, taking into account the notations (11), (12), we obtain:

\[ \left\{ \begin{aligned} 1)\quad \beta\,ds &= \frac{a(1-e^{2})}{\sqrt{1-\nu^{2}}}\, \frac{\cos B\,dB}{(1-e^{2}\sin^{2}B)^{3/2}\sqrt{1-\tau^{2}\sin^{2}B}}, \\[6pt] 2)\quad \beta\,dL &= \frac{\nu(1-e^{2})}{\sqrt{1-\nu^{2}}}\, \frac{dB}{\cos B\sqrt{(1-e^{2}\sin^{2}B)(1-\tau^{2}\sin^{2}B)}}, \\[6pt] 3)\quad dA &= \frac{\nu(1-e^{2})}{1-\nu^{2}}\, \frac{\tg B\,dB}{\sqrt{(1-e^{2}\sin^{2}B)(1-\tau^{2}\sin^{2}B)}} . \end{aligned} \right. \tag{14} \]

The equalities (14) are the desired expanded representation of the differentials \(ds\), \(dL\), and \(dA\) as functions of the latitude \(B\) of the current point \(C\) on the rectified thread \(\Gamma\) of the spheroid. The quantities \(\nu^{2}\) and \(\tau^{2}\) entering into these equalities are determined according to (7), (11), and (12).

1. Let us also express \(ds\) and \(dL\) as functions of the azimuth \(A\) of the rectified thread \(\Gamma\) at the same current point \(C\) of the spheroid. From (6.1) and (6.2), first of all we find:

\[ 1)\quad ds=\frac{r\,dL}{\sin A} = a\,\frac{r}{a}\,\sin A\,\frac{dL}{\sin^{2}A} = a\nu\,\cosec^{2}A\,dL; \]

\[ 2)\quad dL=\frac{dA}{\sin B}. \tag{15} \]

But from the same equality (11) we obtain, in more expanded form:

\[ \nu^{2}=\frac{r^{2}}{a^{2}}\sin^{2}A = \frac{\sin^{2}A\cos^{2}B}{\sqrt{1-e^{2}\sin^{2}B}}, \]

whence, conversely,

\[ \sin B=\sqrt{\frac{\sin^2 A-\nu^2}{\sin^2 A-e^2\nu^2}}. \tag{16} \]

Thus, finally,

\[ \left\{ \begin{aligned} 1)\quad ds&=a\nu\sqrt{\frac{\sin^2 A-e^2\nu^2}{\sin^2 A-\nu^2}}\operatorname{cosec}^2 A\,dA,\\[6pt] 2)\quad dL&=\sqrt{\frac{\sin^2 A-e^2\nu^2}{\sin^2 A-\nu^2}}\,dA. \end{aligned} \right. \tag{17} \]

Equations (17) are the required ones.

1. Equations (14), expressing \(ds\), \(dL\), and \(dA\) as functions of the latitude \(B\), have a complicated form and therefore, before integration, must be simplified by introducing new variables.

Let us first simplify equations (14.1) and (14.2) for \(ds\) and \(dL\). Introducing, instead of the latitude \(B\), a new variable \(\varphi\) by means of the substitution

\[ \tau\sin B=\sin\varphi, \tag{18} \]

we obtain from this, taking into account the structure of equations (14.1) and (14.2):

\[ \begin{aligned} 1)\quad &\cos B\,dB=\frac{1}{\tau}\cos\varphi\,d\varphi;\\[4pt] 2)\quad &\sqrt{1-\tau^2\sin^2 B}=\sqrt{1-\sin^2\varphi}=\cos\varphi;\\[4pt] 3)\quad &1-e^2\sin^2 B=1-\frac{e^2}{\tau^2}\sin^2\varphi. \end{aligned} \qquad \begin{aligned} 4)\quad &\frac{dB}{\cos B}=\frac{\cos\varphi\,d\varphi}{\tau^2\cos^2 B};\\[6pt] 5)\quad &\cos^2 B=1-\sin^2 B=\\ &\hspace{2.7em}=1-\frac{1}{\tau^2}\sin^2\varphi. \end{aligned} \tag{19} \]

Substituting now expressions (19) into equations (14.1), (14.2), and taking (12) into account, we find:

\[ 1)\quad \beta\,ds= \frac{a(1-e^2)}{\sqrt{1-e^2\nu^2}}\cdot \frac{d\varphi}{\left(1-\dfrac{e^2}{\tau^2}\sin^2\varphi\right)^{3/2}}; \tag{20} \]

\[ 2)\quad \beta\,dL= \frac{\nu(1-e^2)}{\sqrt{1-e^2\nu^2}}\cdot \frac{d\varphi}{ \left(1-\dfrac{1}{\tau^2}\sin^2\varphi\right) \sqrt{1-\dfrac{e^2}{\tau^2}\sin^2\varphi} }. \]

Introduce in (20) the notations:

\[ 1)\quad \frac{1-e^2}{\sqrt{1-e^2\nu^2}}=\mu;\qquad 2)\quad \frac{e^2}{\tau^2}=\kappa^2\le e^2;\qquad 3)\quad \frac{1}{\tau^2}=m^2\le 1. \tag{21} \]

Then, instead of (20), we obtain the following final expressions for \(ds\) and \(dL\) as functions of the new variable \(\varphi\), determined by relation (18):

\[ \left\{ \begin{aligned} 1)\quad \beta\,ds&=a\mu\,\frac{d\varphi}{(1-\kappa^2\sin^2\varphi)^{3/2}};\\[6pt] 2)\quad \beta\,dL&=\nu\mu\, \frac{d\varphi}{(1-m^2\sin^2\varphi)\sqrt{1-\kappa^2\sin^2\varphi}}. \end{aligned} \right. \tag{22} \]

6. We shall now simplify the expression found above, (14.3), for the differential \(dA\) in terms of the latitude \(B\). For this purpose we introduce the substitution

\[ \sec^2 B=y. \tag{23} \]

Then, after simple transformations of equality (14.3), we obtain the following final expression for the differential \(dA\) as a function of the new variable \(y\):

\[ dA=\frac{\nu(1-e^2)}{2\sqrt{1-\nu^2}}\, \frac{dy}{\sqrt{\lambda_1 y^2+2\lambda_2 y+\lambda_3}}, \tag{24} \]

where the following notations have been introduced:

\[ 1)\quad \lambda_1=(1-e^2)(1-\tau^2)= -\frac{\nu^2(1-\nu^2)^2}{1-\nu^2}\leqslant 0, \]

\[ 2)\quad 2\lambda_2=(1-e^2)\tau^2+(1-\tau^2)e^2 = \frac{(1-e^2)(1-2e^2\nu^2)}{1-\nu^2}, \tag{25} \]

\[ 3)\quad \lambda_3=e^2\tau^2= \frac{e^2(1-e^2\nu^2)}{1-\nu^2}. \]

7). Having found the final expressions (14.1), (14.2) for \(ds\) and \(dL\), and also having obtained expression (24), with account taken of (23), (25), for \(dA\), we pass from them, finally, to the corresponding integral relations. The same integral transition is carried out with equalities (17) for \(ds\) and \(dL\). Then, carrying out along the way the solution of the simple algebraic integral that follows from equality (24), and also using Clairaut’s equation (7), we obtain the following set of closed expressions for the arc \(\Delta\Gamma_{1.2}\) of the straightened string \(\Gamma_{1.2}\) of the terrestrial spheroid:

\[ \left. \begin{aligned} 1)\quad &\beta\cdot s_{1.2} = a\nu\int_{\varphi_1}^{\varphi_2} \frac{d\varphi}{(1-\kappa^2\sin^2\varphi)^{3/2}}; \\[1.2em] 2)\quad &s_{1.2} = a\nu\int_{A_{1.2}^{\prime}}^{A_{2.1}^{\prime}} \sqrt{\frac{\sin^2 A-e^2\nu^2}{\sin^2 A-\nu^2}}\, \cosec A\,dA; \\[1.2em] 3)\quad &\beta\cdot\Delta L_{1.2} = \gamma\int_{\varphi_1}^{\varphi_2} \frac{d\varphi}{(1-m^2\sin^2\varphi)\sqrt{1-\kappa^2\sin^2\varphi}}; \\[1.2em] 4)\quad &\Delta L_{1.2} = \int_{A_{1.2}^{\prime}}^{A_{2.1}^{\prime}} \sqrt{\frac{\sin^2 A-e^2\nu^2}{\sin^2 A-\nu^2}}\,dA; \\[1.2em] 5)\quad &2\cdot\Delta A_{1.2} = \left.\arc\sin\left[2\nu^2(1-\nu^2)\sec^2 B-(1-2e^2\nu^2)\right]\right|_{B_1}^{B_2}; \\[0.8em] 5a)\quad &2\cdot\Delta A_{1.2} = \arc\sin(1-2\sin^2 A_{1.2}) - \arc\sin(1-2g^2\sin^2 A_{1.2}); \\[0.8em] 6)\quad &\operatorname{ctg} A_{1.2} = \frac{V_2\cos B_1-V_1\cos B_2\cos\Delta A_{1.2}} {V_2\cos B_1\sin\Delta A_{1.2}} = \frac{g-\cos\Delta A_{1.2}}{\sin\Delta A_{1.2}}; \end{aligned} \right\} \tag{26} \]

\[ \left. \begin{aligned} 7)\quad -\operatorname{ctg} A'_{2.1} &= \frac{V_1\cos B_2 - V_2\cos B_1\cos \Delta A_{1.2}} {V_1\cos B_2\sin \Delta A_{1.2}} \\ &= \frac{(1:g)-\cos \Delta A_{1.2}} {\sin \Delta A_{1.2}}; \\[1.2em] 8)\quad \sin A_{1.2} &= \frac{a\nu}{r_1} = \sqrt{1-e^2}\,\frac{\nu V_1}{\cos B_1}; \\[1.2em] 9)\quad \sin A'_{2.1} &= \frac{a\nu}{r_2} = \sqrt{1-e^2}\,\frac{\nu V_2}{\cos B_2}; \\[1.2em] 10)\quad L_2 &= L_1+\Delta L_{1.2}; \qquad 11)\quad A_{2.1}=A'_{2.1}\pm 180^\circ =(A_{1.2}+\Delta A_{1.2})\pm 180^\circ . \end{aligned} \right\} \tag{26} \]

In equalities (26) the following notation has been introduced:

\[ \left\{ \begin{aligned} 1)\quad &V=\sqrt{1+e'^2\cos^2 B}; \\[0.8em] 2)\quad &\nu=\frac{r_1}{a}\sin A_{1.2} \\ &=\sqrt{1+e'^2}\,\frac{\cos B_1\sin A_{1.2}}{V_1} =\nu_{1.2}; \\[0.8em] 3)\quad &\mu=\frac{1-e^2}{\sqrt{1-e^2\nu^2}}=\mu_{1.2}; \\[0.8em] 4)\quad &\tau^2=\frac{1-e^2\nu^2}{1-\nu^2}=\tau^2_{1.2}; \\[0.8em] 5)\quad &\kappa^2=e^2:\tau^2=\kappa^2_{1.2}; \\[0.8em] 6)\quad &m^2=1:\tau^2=m^2_{1.2}; \\[0.8em] 7)\quad &\sin\varphi=\tau\sin B; \\[0.8em] 8)\quad &g=\frac{V_2\cos B_1}{V_1\cos B_2}; \\[0.8em] 9)\quad &e^2=\frac{a^2-b^2}{a^2}; \\[0.8em] 10)\quad &e'^2=\frac{a^2-b^2}{b^2}; \\[0.8em] 11)\quad &\beta= \frac{\cos A_{1.2}}{|\cos A_{1.2}|} = \frac{B_2-B_1}{|B_2-B_1|} = \frac{\varphi_2-\varphi_1}{|\varphi_2-\varphi_1|} =\beta_{1.2}=\pm 1 . \end{aligned} \right. \tag{27} \]

\(a\) and \(b\) are the major and minor semiaxes of the terrestrial spheroid.

The system (26) has two noteworthy features:

a) equality (26.1) has the same form as the well-known expression for the length \(x_{1.2}\) of a meridian arc \(\Delta X_{1.2}\):

\[ \beta x_{1.2} = a(1-e^2) \int_{B_1}^{B_2} \frac{dB}{(1-e^2\sin^2 B)^{3/2}} . \tag{28} \]

It is easy to see that (28) is obtained from (26.1), if in (27.2)—(27.7) one introduces the azimuth \(A_{1.2}\) of the rectified arc \(\Delta X_{1.2}\), equal to \(0\);

b) the quantities \(s_{1.2}\) and \(\Delta L_{1.2}\) are expressed not only through the latitude \(B\) at the current point \(C\) of the arc \(\Delta \Gamma_{1.2}\) of the rectified thread \(T_{1.2}\) of the spheroid, but also as functions of the azimuth \(A\) of the arc \(\Delta \Gamma_{1.2}\) at the same current point \(C\). This makes it possible to check the sought quantities in several ways.

1. The system (26) is a unified basis for the new general methods I have proposed for solving the first three computational problems (1)—(3) on the terrestrial spheroid. For lack of space, I cannot give a detailed solution of the elliptic integra-

of (26.1)—(26.4), in what follows I shall confine myself only to applying, in the problems (1)—(3) indicated above, the corresponding working expressions for the integrals (26.1)—(26.4). Here I shall only very briefly outline ways of obtaining these working expressions.

The integrals (26.1) and (26.3) are special cases of Legendre’s reduced elliptic integral of the third kind \(\Pi(\varphi,\kappa,n)\):

\[ \Pi(\varphi,\kappa,n)=\int_{0}^{\varphi} \frac{d\varphi}{(1+n\sin^{2}\varphi)\sqrt{1-\kappa^{2}\sin^{2}\varphi}} \quad (n\geq 0,\; 0\leq \kappa^{2}\leq 1), \tag{29} \]

whose solution, according to [2], can be represented in the following finite form through Jacobi elliptic functions and theta functions:

\[ \Pi(\varphi,\kappa,n)=\int_{0}^{u}\frac{du}{1+n\,\operatorname{sn}^{2}u} =u+\frac{\operatorname{sn}\beta}{\operatorname{cn}\beta\,\operatorname{dn}\beta} \left[ \frac{\vartheta_1'(\beta)}{\vartheta_4(\beta)} +\frac{1}{2}\ln\frac{\vartheta_4(u-\beta)}{\vartheta_4(u+\beta)} \right], \tag{30} \]

where

\[ 1)\quad u=\int_{0}^{\varphi}\frac{d\varphi}{\sqrt{1-\kappa^{2}\operatorname{sn}^{2}\varphi}} =F(\varphi,\kappa); \qquad 2)\quad \operatorname{sn}^{2}\beta=-\,n:\kappa^{2}. \tag{31} \]

However, the application of expressions (30) and (31) to the solution of the integrals (26.1) and (26.3), for a very small value of \(\kappa^{2}\) (\(\kappa^{2}\leq e^{2}=0.0066934\) for the Krasovsky spheroid), proves extremely disadvantageous:

a) preliminary calculation of certain auxiliary quantities is required;

b) for the integral (26.1) the equality (30) assumes the indeterminate form \(\dfrac{0}{0}\), the resolution of which further complicates this equality;

c) for the integral (26.3), its solution according to (30) will be inaccurate, since for \(-n=m^{2}\gg\kappa^{2}\) the value \(\operatorname{sn}\beta\) is determined from (31) quite unreliably (\(\operatorname{sn}\beta\) in this case becomes very large in modulus).

Taking the foregoing into account, as well as the smallness of \(\kappa^{2}\) for the terrestrial spheroid, it will be more expedient to find the integral (26.1) by expansion in a series in powers of \(\kappa^{2}\).

For the same reasons, the integral (26.3), after expansion in a series in powers of \(\kappa^{2}\), will be solved by the method proposed in 1935 by Prof. V. P. Vetchinkin in [3].

As for the remaining two integrals (26.2) and (26.4), they are elliptic integrals of general form, and their transformation to an expression containing only Legendre’s elliptic integrals of the first to third kinds, \(F(\varphi,k)\), \(E(\varphi,k)\), and \(\Pi(\varphi,\kappa,n)\), will require a large expenditure of computational labor. Therefore, the integrals (26.2), (26.4) will be obtained by expanding the numerator \((\sin^{2}A-e^{2}\nu^{2})^{1/2}\) in a series in powers of the small quantity \(e^{2}\nu^{2}\), or else we shall find them by numerical integration according to Gauss.

Finally, let us note the important circumstance that, for a given \(e^{2}\), the integrals (26.1) and (26.3) contain only two parameters: the integral (26.1)—the parameters \(\varphi\) and \(\kappa^{2}\), and the integral (26.3)—the parameters \(\varphi\) and \(m^{2}\). Therefore the indicated integrals can be represented in the form of tables with two entries, like the reduced elliptic integral of the first kind \(F(\varphi,\kappa)\). The availability of such tables substantially facilitates the solution of problems in which these integrals are used (see below).

Div. 4. Solution of the direct problem for a rectified arc \(\Delta \Gamma_{1.2}\) on the spheroid

We shall give, with only very brief explanations, the sets of computational expressions for solving, by the proposed method, the direct problem (1) under various conditions concerning the values of the quantities entering into this problem.

Condition of the problem: Given \(B_1,\ L_1,\ A_{1.2},\ s_{1.2}\). Find \(B_2,\ A_{2.1},\ L_2\).

1. Determination of \(B_2\)

\[ \text{1) }\quad V_1=\sqrt{1+e'^2\cos^2 B_1}\quad \text{— from geodetic tables;} \]

\[ \text{2) }\quad \nu=\sqrt{1+e'^2\,\frac{\cos B_1\sin A_{1.2}}{V_1}}; \qquad \text{5) }\quad \kappa^2=\frac{e^2}{\tau^2}\ll e^2; \]

\[ \text{3) }\quad \rho=\frac{1-e^2}{\sqrt{1-e^2\nu^2}}; \qquad \text{4) }\quad \tau^2=\frac{1-e^2\nu^2}{1-\nu^2}>1; \qquad \text{6) }\quad \sin\varphi_1=\tau\sin B_1; \]

\[ \text{7) }\quad C_0=1+\sum_{\lambda=1}^{n}(-1)^\lambda \frac{\binom{-3/2}{\lambda}\cdot\binom{2\lambda}{\lambda}}{2^{2\lambda}}\, \kappa^{2\lambda} = 1+\sum_{\lambda=1}^{n} c_{0,2\lambda}\kappa^{2\lambda}; \]

\[ \text{8) }\quad C_{2u}=\sum_{\lambda=u}^{n}(-1)^{\lambda-u} \frac{\binom{-3/2}{\lambda}\cdot\binom{2\lambda}{\lambda-u}}{2^{2\lambda}\cdot u}\, \kappa^{2\lambda} = \sum_{\lambda=u}^{n} c_{2u,2\lambda}\kappa^{2\lambda}; \]

\[ \text{9) }\quad \frac{C_{2u}}{C_0}=D_{2u};\qquad (u=1,2,\ldots,n); \]

\[ \text{10) }\quad \varphi_2-\varphi_1=\Delta\varphi_{1.2} = \frac{\beta\cdot s_{1.2}}{C_0 a\mu} - \sum_{u=1}^{n}D_{2u}\bigl(\sin 2u\varphi_2-\sin 2u\varphi_1\bigr); \qquad \frac{\beta\cdot s_{1.2}}{C_0 a\mu}=Q; \]

\[ \text{11) }\quad \varphi_2=\varphi_1+\Delta\varphi_{1.2}; \qquad \text{12) }\quad \sin B_2=\frac{1}{\tau}\sin\varphi_2. \]

The computation of \(\Delta\varphi_{1.2}\) according to (10), owing to the unknown \(\varphi_2\), is performed by successive approximations, or still more rapidly in the following manner:

\[ \text{a) }\quad \varphi_2^{(0)}=\varphi_1+\frac{\beta s_{1.2}}{C_0a\mu}; \qquad \text{b) }\quad \Delta\varphi_{1.2}^{(0)} = \frac{\beta s_{1.2}}{C_0a\mu} - \sum_{u=1}^{n}D_{2u} \bigl(\sin 2u\varphi_2^{(0)}-\sin 2u\varphi_1\bigr); \]

\[ \text{c) }\quad \frac{\partial}{\partial\varphi_2}\Delta\varphi_{1.2}^{(0)} = \chi = \sum_{u=1}^{n}2uD_{2u}\cos 2u\varphi_2^{(0)}; \qquad \text{d) }\quad \Delta\varphi_{1.2} = \frac{\Delta\varphi_{1.2}^{(0)}}{1-\chi}. \]

If \(\Delta\varphi_{1.2}\) is small, then from \(\Delta\varphi_{1.2}\) we pass first to \(\Delta B_{1.2}\), and then to \(B_2\).

\[ \text{11a) }\quad \Delta B_{1.2}'' = (\tan B_1\cot\varphi_1)\Delta\varphi_{1.2}'' - \frac{\tan B_1}{2\rho}\bigl(\Delta\varphi_{1.2}^{2}-\Delta B_{1.2}^{2}\bigr)'' - \]

\[ -\frac{1}{6\rho^2} \left[(\tan B_1\cot\varphi_1)\Delta\varphi_{1.2}^{3} -\Delta B_{1.2}^{3}\right]''; \]

\[ \text{12a) }\quad B_2=B_1+\Delta B_{1.2}; \qquad \rho=206264.8. \]

For small \(\Delta\varphi_{1.2}\), \(\Delta\varphi_{1.2}\) may be computed otherwise:

\[ 10a)\quad \Delta\varphi''_{1.2} = \beta''\,\frac{s_{1.2}W_1^3}{a\mu} - \frac{3\kappa^2}{4W_1^2} \left[ \sin 2\varphi_1 + \frac{2}{3} \left( \cos 2\varphi_1 + \frac{5\kappa^2\sin^2\varphi_1}{4W_1^2} \right) \cdot \frac{\Delta\varphi''_{1.2}}{\rho} \right] \frac{(\Delta\varphi''_{1.2})^2}{\rho}, \]

where

\[ 13)\quad W_1=\sqrt{1-\kappa^2\sin^2\varphi_1} = \sqrt{1-e^2\sin^2 B_1} = \frac{V_1}{\sqrt{1+e^2}} \quad\text{from geodetic tables.} \]

2. Determination of \(A_{2.1}\)

\[ 1)\quad \sin A'_{2.1} = \sqrt{1-e^2}\, \frac{\nu V_2}{\cos B_2}; \]

\[ 2)\quad A_{2.1}=A'_{2.1}\pm 180^\circ = (A_{1.2}+\Delta A_{1.2})\pm 180^\circ. \]

a) If \(A'_{2.1}\) is close to \(90^\circ\) or \(270^\circ\), but \(B_2-B_1=\Delta B_{1.2}\) is large, then instead of \(A'_{2.1}\) one computes \(\Delta A_{1.2}\):

\[ 3)\quad g=\frac{V_2\cos B_1}{V_1\cos B_2}; \]

\[ 4)\quad 2\cdot\Delta A_{1.2} = \arc\sin(1-2\sin^2 A_{1.2}) - \arc\sin(1-2g^2\sin^2 A_{1.2}). \]

b) If \(B_2-B_1=\Delta B_{1.2}\) is small, then instead of \(A_{2.1}\) one may also compute \(\Delta A_{1.2}\):

\[ 4a)\quad 2\cdot\Delta A''_{1.2} = \rho''\, \frac{\varepsilon}{\sin 2A_{1.2}} \left[ 1+ \frac{\cos 2A_{1.2}}{2\sin^2 2A_{1.2}}\,\varepsilon + \frac{1+2\cos^2 2A_{1.2}}{6\sin^4 2A_{1.2}}\,\varepsilon^2 \right], \]

where

\[ 5)\quad \varepsilon=2(g^2-1)\sin^2 A_{1.2}. \]

3. Check of the computation of \(B_2\) and \(A_{2.1}\)

Since, according to (26.1) and (26.9), the back azimuth \(A_{2.1}=A'_{2.1}\pm 180^\circ=F(s_{1.2},\nu,B_2)\), the computed \(B_2\) and \(A_{2.1}\) are checked jointly by the equality (26.2)—the second expression for \(s_{1.2}\):

\[ s_{1.2} = a\nu \int_{A_{1.2}}^{A'_{2.1}} \sqrt{ \frac{\sin^2 A-e^2\nu^2}{\sin^2 A-\nu^2} } \,\operatorname{cosec}^2 A\,dA. \]

Evaluating this integral in three different ways:

a) by expanding the numerator \((\sin^2 A-e^2\nu^2)^{1/2}\) in a series in powers of the small quantity \(e^2\nu^2\),

b) by numerical integration by Gauss,

c) by expanding the integrand near the mean value

\[ \overline{A}_{1.2} = \frac{1}{2}(A_{1.2}+A'_{2.1}) \]

in a series in powers of the difference \(\Delta A_{1.2}=A'_{2.1}-A_{1.2}\),

we obtain three corresponding methods for checking \(B_2\) and \(A_{2.1}\).

4. Determination of \(L_2\)

\[ 1)\quad \sqrt{1-m^2}=p; \qquad 2)\quad \varphi_{1.2}=\frac{1}{2}(\varphi_1+\varphi_2); \qquad 3)\quad \Delta\varphi_{1.2}=\varphi_2-\varphi_1; \]

\[ 4)\ \operatorname{tg}\varphi=t;\qquad 5)\ \beta=\frac{\Delta\varphi_{1,2}}{|\Delta\varphi_{1,2}|};\qquad 6)\ \operatorname{arctg}(pt)=\vartheta; \]

\[ 7)\ \psi=\frac{\nu}{|\nu|}\,p\sqrt{1-e^2};\qquad 8)\ \frac{\nu}{|\nu|}\,p=\frac{\nu\sqrt{1-e^2}}{\sqrt{1-e^2\nu^2}};\qquad 9)\ \nu=\frac{\nu}{|\nu|}\frac{p}{\sqrt{(1-e^2)-p^2}}; \]

\[ 10)\ g(\lambda)=(-1)^\lambda\binom{-1/2}{\lambda}K^{2\lambda}; \qquad 11)\ \Phi(\lambda)=\int_{\varphi_1}^{\varphi_2} \frac{\sin^{2\lambda}\varphi\,d\varphi}{1-m^2\sin^2\varphi}; \]

\[ 12)\ F(\lambda)=g(\lambda)\Phi(\lambda); \]

\[ (\lambda=0,1,2,\ldots,n). \]

From (10)—(12) we find successively:

\[ 13)\ F(0)=\Phi(0)=\frac{1}{p}\,[\operatorname{arctg}(pt_2)-\operatorname{arctg}(pt_1)] =p[\vartheta_2-\vartheta_1]. \]

\[ 14)\ F(\lambda)=E(\lambda)[S(\lambda-1)-\Phi(\lambda-1)],\quad (\lambda=1,2,\ldots,n), \]

where it is denoted:

\[ 15)\ E(\lambda)=-\frac{g(\lambda)}{m^2} =(-1)^{\lambda+1}\binom{-1/2}{\lambda}\frac{K^{2\lambda}}{m^2}; \]

\[ 16)\ S(\lambda-1)=\int_{\varphi_1}^{\varphi_2}\sin^{2(\lambda-1)}\varphi\,d\varphi. \]

Integrating (16), we obtain for the function \(S(\lambda-1)\) entering into (14):

\[ 17)\ S(0)=\varphi_2-\varphi_1=\Delta\varphi_{1,2},\quad(\lambda=1); \]

\[ 18)\ S(\lambda-1)=S(\omega)= \frac{1}{2^{2\omega}} \left\{ \binom{2\omega}{\omega}(\varphi_2-\varphi_1) +(-1)^\omega\sum_{\chi=0}^{\omega-1} (-1)^\chi \frac{\binom{2\omega}{\omega}}{\omega-\chi} \right. \]

\[ \left. {}\times \left[ \sin 2(\lambda-\chi)\varphi_2-\sin 2(\lambda-\chi)\varphi_1 \right] \right\}, \quad (\lambda-1=\omega=1,2,\ldots,n). \]

After determining the functions \(F(\lambda)\), the required values \(\Delta L_{1,2}\) and \(L_2\) are computed as follows:

\[ 19)\ \beta\cdot\Delta L_{1,2} =\psi\sum_{\lambda=0}^{n}F(\lambda) =\sum_{\lambda=0}^{n}\psi F(\lambda) =\sum_{\lambda=0}^{n}R(\lambda); \qquad 20)\ L_2=L_1+\Delta L_{1,2}. \]

Below are given working expressions for the first five terms \(\psi F(\lambda)=R(\lambda)\) in equality (19):

\[ 20)\ \psi F(0)= \frac{\nu}{|\nu|}\sqrt{1-e^2}\,[\operatorname{arctg}(pt_2)-\operatorname{arctg}(pt_1)] = \]

\[ =\frac{\nu}{|\nu|}\sqrt{1-e^2}\,[\vartheta_2-\vartheta_1] =\frac{\nu}{|\nu|}\sqrt{1-e^2}\,\Delta\vartheta_{1,2} =R(0); \]

but if \(\varphi_2-\varphi_1=\Delta\varphi_{1,2}\) is small or if one of the quantities \(\varphi_1,\varphi_2\) is close to \(\dfrac{\pi}{2}\), then it is better to use

\[ 20a)\ \psi F(0)= \frac{\nu}{|\nu|}\sqrt{1-e^2}\operatorname{arctg} \frac{p\sin\Delta\varphi_{1,2}} {\cos\Delta\varphi_{1,2}-m^2\sin\varphi_1\sin\varphi_2}. \]

\[ \begin{aligned} 21)\quad \psi F(1) &=-\frac{e^2}{2}\,\psi(\varphi_2-\varphi_1)+\frac{1}{2}e^2 R(0)= \\ &=\frac{\nu}{|\nu|}\frac{e^2}{2}\sqrt{1-e^2}\,[(\vartheta_2-\vartheta_1)-p(\varphi_2-\varphi_1)]=R(1); \end{aligned} \]

\[ 22)\quad \psi F(2)=-\frac{3}{16}e^2\varkappa^2\psi\left[(\varphi_2-\varphi_1)-\frac{1}{2}(\sin 2\varphi_2-\sin 2\varphi_1)\right]+ \]

\[ +\frac{3}{4}e^2 R(1)=R(2); \]

\[ 23)\quad \psi F(3)=-\frac{5}{16}e^2\varkappa^4\psi\left[\frac{3}{8}(\varphi_2-\varphi_1)-\frac{1}{4}(\sin 2\varphi_2-\sin 2\varphi_1)+\right. \]

\[ \left.+\frac{1}{32}(\sin 4\varphi_2-\sin 4\varphi_1)\right]+\frac{5}{6}e^2 R(2)=R(3); \]

\[ 24)\quad \psi F(4)=-\frac{35}{128}e^2\varkappa^6\psi\left[\frac{5}{16}(\varphi_2-\varphi_1)-\frac{15}{64}(\sin 2\varphi_2-\sin 2\varphi_1)+\right. \]

\[ \left.+\frac{3}{64}(\sin 4\varphi_2-\sin 4\varphi_1)-\frac{1}{192}(\sin 6\varphi_2-\sin 6\varphi_1)\right]+\frac{7}{8}e^2 R(3)=R(4). \]

5. Final check

We perform the final check of the found \(B_2,\ A_{1.2}\), and \(\Delta L_{1.2}\) using the equality [Part 3; (26.4)]:

\[ \Delta L_{1.2}=\int_{A_{1.2}}^{A'_{2.1}}\sqrt{\frac{\sin^2 A-e^2\nu^2}{\sin^2 A-\nu^2}}\,dA. \]

Here, as above in Sec. 3 for \(s_{1.2}\), three methods of checking are possible.

Part 5. Solution of the inverse problem for the rectified arc \(\Delta \Gamma_{1.2}\) on the spheroid

As in the direct problem, here we shall give mainly only the sequences of working expressions that follow from the initial set [Part 3; (26)] and determine the totality of the required quantities in the inverse problem; the limits of validity of these sequences in various cases will also be indicated.

Statement of the problem: given \(B_1, L_1\) and \(B_2, L_2\); find \(A_{1.2}, A_{2.1}\), and \(s_{1.2}\).

1. First method of determining \(A_{1.2}\) and \(A_{2.1}\) (for \(s_{1.2}>1000\) km)

First of all, solve by 2 approximations the equation [Part 4, 4; (19)]

\[ 1)\quad \beta\cdot\Delta L_{1.2}=\psi\sum_{\lambda=0}^{n}F(\lambda)=\sum_{\lambda=0}^{n}R(\lambda)=R(0)+\sum_{\lambda=1}^{n}R(\lambda), \]

where

\[ R(0)=\frac{\nu}{|\nu|}\sqrt{1-e^2}\,[\operatorname{arc\,tg}(p\,\operatorname{tg}\varphi_2)-\operatorname{arc\,tg}(p\,\operatorname{tg}\varphi_1)]= \]

\[ =\frac{\nu}{|\nu|}\sqrt{1-e^2}\,[\vartheta_2-\vartheta_1]=\frac{\nu}{|\nu|}\sqrt{1-e^2}\,\Delta\vartheta_{1.2}; \]

\[ \begin{aligned} R(1) &= \frac{e^2}{2} R(0) - \frac{e^2}{2}\,\psi(\varphi_2-\varphi_1) = \frac{e^2}{2} R(0) - \frac{e^2}{2}\,\psi\,\Delta\varphi_{1.2} \\ &= \frac{\nu}{|\nu|}\frac{e^2}{2}\sqrt{1-e^2}\,[\Delta\vartheta_{1.2}-p\cdot\Delta\varphi_{1.2}]; \\[6pt] R(2) &= \frac{3}{4}e^2 R(1) - \frac{3}{16}e^2\kappa^2\psi \left[\Delta\varphi_{1.2}-\frac{1}{2}(\sin 2\varphi_2-\sin 2\varphi_1)\right]; \\[6pt] R(3) &= \frac{5}{6}e^2 R(2) - \frac{5}{16}e^2\kappa^4\psi \left[\frac{3}{8}\Delta\varphi_{1.2}\right. \\ &\qquad\left. -\frac{1}{4}(\sin 2\varphi_2-\sin 2\varphi_1) +\frac{1}{32}(\sin 4\varphi_2-\sin 4\varphi_1)\right] \end{aligned} \]

and so on,

with respect to the unknown \(p\), and the sum \(\sum_{\lambda=1}^{n} R(\lambda)\) is a quantity of order \(e^2\).

The initial, sufficiently accurate value \(A_{1.2}^{(0)}\) of the azimuth \(A_{1.2}\) entering into the computation of \(p\) is found from the relation

\[ 2)\quad \frac{\cos B_2 \operatorname{tg} B_1-\cos\Delta L_{1.2}\sin B_1}{\sin\Delta L_{1.2}} = \frac{\Pi_2-\Pi_1}{\sin\Delta L_{1.2}} = \operatorname{ctg}\alpha_{1.2} = \operatorname{ctg} A_{1.2}^{(0)}. \]

The corresponding approximate value \(p^{(0)}\) of the unknown \(p\) and the initial value \(R^{(0)}\) for \(R(0)\), as well as the successive values of the quantities \(\psi, m, \tau, \kappa^2, \varphi\) entering into \(R(\lambda)\), are then determined from the relations:

\[ 3)\quad p=\cos B_0 \approx \sin A_{1.2}^{(0)}\cos B_1=p^{(0)}; \]

\[ 4)\quad R(0)=\frac{\nu}{|\nu|}\sqrt{1-e^2}\,\Delta\vartheta_{1.2} \approx \frac{\nu}{|\nu|}\sqrt{1-e^2}\,\Delta L_{1.2} = R^{(0)}(0); \]

\[ 5)\quad \psi=\frac{\nu}{|\nu|}p\sqrt{1-e^2}; \qquad 6)\quad \sin B_0=m; \qquad 7)\quad \csc B_0=\tau; \]

\[ 8)\quad \kappa^2=e^2(1-p^2)=e^2\sin^2 B_0; \qquad 9)\quad \sin\varphi=\tau\sin B \]

for \(A_{1.2}=A_{1.2}^{(0)}\) and \(p=p^{(0)}, p^{(1)}, p^{(2)},\ldots\), where \(B_0\) is the latitude of the vertex \(O_{1.2}\) of the rectified curve \(\Gamma_{1.2}\).

The improved value \(p^{(s)}\) of the unknown \(p\) at the end of the \(s\)-th approximation is found as follows:

\[ 10)\quad w^{(s-1)} = \beta\sum_{\lambda=0}^{n} R^{(s-1)}(\lambda)-\Delta L_{1.2} = \Delta L_{1.2}^{(s-1)}-\Delta L_{1.2}, \]

\[ 11)\quad \frac{\partial w^{(s-1)}}{\partial p} = \sqrt{1-e^2}\,\tau^2[\operatorname{tg}\varphi_2-\operatorname{tg}\varphi_1] = \chi^{(s-1)}; \]

\[ 12)\quad p^{(s)} = p^{(s-1)}-\frac{w^{(s-1)}}{\chi^{(s-1)}} = p^{(s-1)}+\Delta p^{(s-1)}. \]

Having solved equation (1) by two or three approximations, we then compute the final values of the azimuths \(A_{1.2}, A_{2.1}\):

\[ 13)\quad \sin A_{1.2} = \frac{\nu}{|\nu|}\frac{\cos B_0}{\cos B_1} \sqrt{\frac{1-e^2\sin^2 B_1}{1-e^2\sin^2 B_0}} = \frac{\nu}{|\nu|}\frac{p}{\cos B_1}\frac{V_1}{V_0}; \]

\[ 14)\ \sin A'_{2.1} = \frac{\nu}{|\nu|}\frac{\cos B_0}{\cos B_2} \sqrt{\frac{1-e^2\sin^2 B_2}{1-e^2\sin^2 B_0}} = \frac{\nu}{|\nu|}\frac{p}{\cos B_2}\frac{V_2}{V_0}; \]

\[ 15)\ A_{2.1}=A'_{2.1}\pm 180^\circ. \]

The auxiliary quantity \(\mu\), needed later in computing the distance \(s_{1.2}\), may be expressed as follows through the quantity found above,
\(\kappa^2=e^2(1-p^2)=e^2\sin^2 B_0\):

\[ 16)\ \mu=\sqrt{(1-e^2)(1-\kappa^2)} =\sqrt{(1-e^2)(1-e^2\sin^2 B_0)} = \]

\[ =\sqrt{1-e^2}\,W_0=(1-e^2)V_0. \]

As for the quantity \(\nu\), it is related to the quantities \(p\) and \(\kappa\) by the relation

\[ 17)\ \nu = \frac{\nu}{|\nu|}\frac{p}{\sqrt{1-\kappa^2}} = \frac{\nu}{|\nu|}\frac{\cos B_0}{\sqrt{1-e^2\sin^2 B_0}} = \frac{\nu}{|\nu|}\frac{\cos B_0}{W_0} = \]

\[ = \frac{\nu}{|\nu|}\frac{\cos B_0}{\sqrt{1-e^2}\,V_0}. \]

2. Second method for determining \(A_{1.2}\) and \(A_{2.1}\) (for \(s_{1.2}\leqslant 1000\) km)

\[ 1)\ \operatorname{tg}\frac{1}{2}(\alpha'_{2.1}+\alpha_{1.2}) = \frac{\cos\frac{1}{2}(B_2+B_1)} {\sin\frac{1}{2}(B_2-B_1)} \operatorname{tg}\frac{\Delta L_{1.2}}{2} = \operatorname{tg}\alpha_{1.2}; \]

\[ 2)\ \operatorname{tg}\frac{1}{2}(\alpha'_{2.1}-\alpha_{1.2}) = \frac{\sin\frac{1}{2}(B_2+B_1)} {\cos\frac{1}{2}(B_2-B_1)} \operatorname{tg}\frac{\Delta L_{1.2}}{2} = \operatorname{tg}\frac{\Delta\alpha_{1.2}}{2}, \]

whence find \(\alpha_{1.2}\) and \(\alpha'_{2.1}\) separately.

\[ 3)\ Q=\frac{e^2}{\sin\Delta L_{1.2}}\left(V_1\sin B_2-V_2\sin B_1\right); \]

\[ 4)\ \alpha_{2.1}=\alpha'_{2.1}\pm 180^\circ=(\alpha_{1.2}+\Delta\alpha_{1.2})\pm 180^\circ; \]

\[ 5)\ \sin\sigma_{1.2} = \frac{\cos B_2}{\sin\alpha_{1.2}}\sin\Delta L_{1.2} = -\frac{\cos B_1}{\sin\alpha_{2.1}}\sin\Delta L_{1.2}; \]

\[ 6)\ \operatorname{ctg}\widetilde{A}_{1.2} = \operatorname{ctg}\alpha_{1.2} - Q\,\frac{\cos B_1}{V_1\cos B_2}; \]

\[ 7)\ \operatorname{ctg}\widetilde{A}_{2.1} = \operatorname{ctg}\alpha_{2.1} - Q\,\frac{\cos B_2}{V_2\cos B_1}; \qquad (\text{see }[4]). \]

\[ \left\{ \begin{aligned} 8)\ \eta''_{1.2} &= \frac{e^2}{6\rho''}(\sigma''_{1.2})^2 \sin\widetilde{A}_{1.2}\cos^2 B_1 \left( \cos\widetilde{A}_{1.2} - \frac{\sigma''_{1.2}}{4\rho''}\operatorname{tg} B_1 \right); \\[6pt] 9)\ \eta''_{2.1} &= \frac{e^2}{6\rho''}(\sigma''_{1.2})^2 \sin\widetilde{A}_{2.1}\cos^2 B_2 \left( \cos\widetilde{A}_{2.1} - \frac{\sigma''_{1.2}}{4\rho''}\operatorname{tg} B_2 \right). \end{aligned} \right. \qquad (\text{see }[5]). \]

\[ 10)\ A_{1.2}=\widetilde{A}_{1.2}-\eta''_{1.2}; \qquad 11)\ A_{2.1}=\widetilde{A}_{2.1}-\eta''_{2.1}. \]

3. Verification of the computation of \(A_{1.2}\) and \(A_{2.1}\)

The verification of the computed azimuths \(A_{1.2}\) and \(A_{2.1}\) is carried out by substituting them into one of the following equalities:

\[ 1)\quad \frac{\sin A_{1.2}}{\sin A'_{2.1}} = \frac{\cos B_2}{\cos B_1} \sqrt{\frac{1-e^2\sin^2 B_1}{1-e^2\sin^2 B_2}} = \frac{V_2\cos B_1}{V_1\cos B_2} = g; \]

\[ 2)\quad \Delta L_{1.2} = \int_{A_{1.2}}^{A'_{2.1}} \sqrt{\frac{\sin^2 A-e^2\nu^2}{\sin^2 A-\nu^2}}\,dA. \]

The actual evaluation of integral (2) is performed by one of the three methods applied above in the concluding part of the direct problem.

4. Computation of \(s_{1.2}\) by four methods

The distance \(s_{1.2}\) may be computed in four ways from the double equality

\[ s_{1.2} = \beta a\mu \int_{\varphi_1}^{\varphi_2} \frac{d\varphi}{(1-k^2\sin^2\varphi)^{3/2}} = a\nu \int_{A_{1.2}}^{A'_{2.1}} \sqrt{\frac{\sin^2 A-e^2\nu^2}{\sin^2 A-\nu^2}}\,dA. \]

From the first part of this double equality we have

\[ \beta s_{1.2} = C_0 a\mu(\varphi_2-\varphi_1) + a\mu\sum_{u=1}^{n} C_{2u}(\sin 2u\varphi_2-\sin 2u\varphi_1), \]

where \(\varphi_1,\ \varphi_2\), and \(k^2\) were found above in determining \(p=\cos B_0\), the quantity \(\mu\) is given in [Div. 5, 4; (14)], and the coefficients \(C_0,\ C_{2u}\) are indicated in [Div. 3, 1; (7), (8)].

The second part of the same double equality is solved by one of the three methods mentioned above in the direct problem.

Div. 6. Solution of the direct rectilinear intersection on the spheroid

By a direct rectilinear intersection on the spheroid we shall mean the problem of determining the geodetic coordinates \(B_3,\ L_3\) of point 3, if the geodetic coordinates \(B_1,\ L_1\) and \(B_2,\ L_2\) of the initial points 1, 2 are given, and also the geodetic azimuths \(A_{1.3},\ A_{2.3}\) of the rays \(\delta L_{1.3},\ \delta L_{2.3}\) intersecting at point 3, with vertices at the initial points 1, 2.

In addition, an additional requirement may be imposed: to find also the lengths \(s_{1.3},\ s_{2.3}\) of the intersecting sides 1.3, 2.3 and the reverse azimuths \(A_{3.1},\ A_{3.2}\) of these sides at the determined point 3. From what has been said there follows the following brief statement:

Condition of the problem. Given \(B_1,\ L_1,\ A_{1.3}\) and \(B_2,\ L_2,\ A_{2.3}\). Find \(B_3,\ L_3\), and also \(A_{3.1},\ s_{1.3}\) and \(A_{3.2},\ s_{2.3}\).

In the indicated formulation, this problem can be solved by two general methods, at the basis of which lies the obvious relation

\[ \Delta L_{1.2}=\Delta L_{1.3}-\Delta L_{2.3}. \tag{1} \]

We shall consider each of these methods separately.

A. First method for solving the direct intersection on the spheroid

This method is advisable to use when we need only the geodetic coordinates \(B_3, L_3\) of the point 3 to be determined. The essence of the method is as follows.

First of all, from the solution of the direct intersection on the sphere we find approximate values \(\tilde B_3^{(0)}, \tilde L_3^{(0)}\) of the geodetic coordinates \(B_3, L_3\) of the point 3 to be determined. For the indicated solution, which is carried out with a check, the following working expressions are used:

\[ \begin{gathered} 1)\quad \operatorname{tg}\Delta \tilde L_{1.3}^{(0)}= \\[3pt] = \frac{ \cos(90^\circ-B_2)\cos\Delta L_{1.2} -\operatorname{ctg} A_{2.3}\sin\Delta L_{1.2} -\operatorname{ctg}(90^\circ-B_1)\sin(90^\circ-B_2) }{ \operatorname{ctg} A_{1.3}\dfrac{\sin(90^\circ-B_2)}{\sin(90^\circ-B_1)} -\operatorname{ctg} A_{2.3}\cos\Delta L_{1.2} -\cos(90^\circ-B_2)\sin\Delta L_{1.2} }; \end{gathered} \]

\[ 2)\quad \tilde L_3^{(0)}=L_1+\Delta \tilde L_{1.3}^{(0)}; \qquad 3)\quad \Delta \tilde L_{2.3}^{(0)}=\tilde L_3^{(0)}-L_2; \]

\[ \begin{aligned} 4)\quad \operatorname{tg}\tilde B_3^{(0)} &=\bigl[\operatorname{ctg} A_{1.3}\sin\Delta \tilde L_{1.3}^{(0)} +\cos(90^\circ-B_1)\cos\Delta \tilde L_{1.3}^{(0)}\bigr]:\sin(90^\circ-B_1)\\ &=\bigl[\operatorname{ctg} A_{2.3}\sin\Delta \tilde L_{2.3}^{(0)} +\cos(90^\circ-B_2)\cos\Delta \tilde L_{2.3}^{(0)}\bigr]:\sin(90^\circ-B_2). \end{aligned} \]

The computations are performed with an accuracy up to \(0.00001\) for numbers and with an accuracy up to \(1''\) or up to \(0''.0001\) for angles.

Next, with final accuracy (for example, with accuracy up to \(10^{-8}\)), we compute the quantities \(\nu_{i3}, \kappa_{i3}^2, \tau_{i3}, p_{i3}, (\nu\psi)_{i3}, \varphi_i^3\) and \(\vartheta_i^3\) for both intersecting sides \(i3=1.3,\,2.3\). The corresponding computational expressions are given in Sec. 4 \((\sin\varphi_i^3=\tau_{i3}\sin B_i;\ \operatorname{tg}\vartheta_i^3=p_{i3}\operatorname{tg}\varphi_i^3)\).

We now proceed to the computation on the spheroid of successive approximations \(B_3^{(s)}, L_3^{(s)}\), \((s=1,2,\ldots)\), for the geodetic coordinates \(B_3, L_3\) of the point 3 to be determined.

Beginning the first approximation, we take as the initial value \(\tilde B_3^{(0)}=B_3^{(0)}\) for the latitude of point 3, obtained from the solution of the intersection on the sphere. Then, by solving inverse problems along the sides \(i3=1.3,\,2.3\), with known azimuths \(A_{1.3}, A_{2.3}\) of the rays \(\delta Л_{1.3}, \delta Л_{2.3}\), we compute, according to [Sec. 5, 1], the corresponding approximate values \(\Delta L_{1.3}^{(0)}, \Delta L_{2.3}^{(0)}\) on the spheroid of the longitude differences \(\Delta L_{1.3}, \Delta L_{2.3}\). In doing so, the calculation is carried out as follows:

\[ 1)\quad \tilde B_3^{(0)}=B_3^{(0)}\simeq B_3, \qquad 2)\quad \tau_{i3}\sin B_3=\sin\varphi_3^i, \qquad 3)\quad \varphi_3^i-\varphi_i^i=\Delta\varphi_{i3}, \]

\[ 4)\quad p_{i3}\operatorname{tg}\varphi_3^i=\operatorname{tg}\vartheta_3^i, \qquad 5)\quad \vartheta_3^i-\vartheta_i^3=\Delta\vartheta_{i3}, \qquad 6)\quad \frac{\nu_{i3}}{|\nu_{i3}|}=\omega_{i3}, \]

\[ 7)\quad \omega_{i3}\sqrt{1-e^2}\,\Delta\vartheta_{i3}=R_{i3}(0), \qquad 8)\quad (\nu\psi)_{i3}\Delta\varphi_{i3}=\Delta R_{i3}(0), \]

\[ 9)\quad \frac{e^2}{2}\,[R_{i3}(0)-\Delta R_{i3}(0)]=R_{i3}(1), \]

\[ 10)\quad \frac{1}{4}(\kappa^2\nu)_{i3} \bigl[\Delta\varphi_{i3}-(\sin2\varphi_3^i-\sin2\varphi_i^3)\bigr] =\Delta R_{i3}(1), \]

\[ 11)\quad \frac{3}{4}e^2\,[R_{i3}(1)-\Delta R_{i3}(1)]=R_{i3}(2), \qquad 12)\quad \sum_{\lambda=0}^{2} R_{i3}(\lambda)=\beta_{i3}\Delta L_{i3}^{(0)}. \]

These computations are performed retaining 6–7 digits after the decimal point.

Having computed \(\Delta L_{i3}^{(0)}=\Delta L_{1.3}^{(0)},\ \Delta L_{2.3}^{(0)}\), we compute the corresponding approximate values \(L_3^{(0)}, L_3^{(0)}\) of the longitude \(L_3\) of point 3 on the spheroid, and also compute the resulting misclosure \(w_L^{(0)}\) in longitude:

\[ 1)\ L_3^{(0)}=L_1+\Delta L_{1.3}^{(0)}, \qquad 2)\ L_3^{(0)}=L_2+\Delta L_{2.3}^{(0)}, \qquad 3)\ w_L^{(0)}=L_3^{(0)}-L_3^{(0)}, \tag{4.1} \]

i.e., we carry out the computations that follow from the fundamental relation (1): \(\Delta L_{1.2}=\Delta L_{1.3}-\Delta L_{2.3}\), if instead of the exact value \(B_3\) of the latitude of point 3 we take the approximate value \(\widetilde B_3^{(0)}=B_3^{(0)}\).

Having found the misclosure \(w_L^{(0)}\), we compute the corresponding corrections \(\delta B_3^{(0)}\) and \(\delta L_3^{(0)}, \delta L_3^{(0)}\), adding which to \(B_3^{(0)}, L_3^{(0)}, L_3^{(0)}\), we obtain the improved first-approximation values \(B_3^{(1)}, L_3^{(1)}\) of the latitude and longitude of point 3. This is done as follows:

\[ 1)\ \sin A_{i3}'=\frac{\sqrt{1-e^2}\,\nu_{i3}V_3}{\cos B_3}; \qquad 2)\ \frac{\partial L_3^i}{\partial B_3}=a_{i3}= \frac{\operatorname{tg} A_{3i}'}{V_3^2\cos B_3}; \]

\[ 3)\ \delta B_3^{(0)}=-\frac{w_L^{(0)}}{(a_{1.3}-a_{2.3})}; \tag{5.1} \]

\[ 4)\ a_{i3}\,\delta B_3^{(0)}=\delta L_3^{(0)}; \qquad 5)\ B_3^{(1)}=B_3^{(0)}+\delta B_3^{(0)}; \]

\[ 6)\ L_3^{(1)}=L_3^{(0)}+\delta L_3^{(0)}=L_3^{(0)}+\delta L_3^{(0)}. \]

This completes the first approximation.

Passing to the second approximation, as the initial value we take the value \(B_3^{(1)}\) of the latitude of point 3 obtained as the result of the first approximation. Here the computations are performed with the full number of digits, and the determination of \(\Delta L_{i3}^{(1)}\) is carried out with allowance for the correction term \(R_{i3}(3)\). Thus, in modification of and in addition to (3.1), we shall have for the second approximation:

\[ B_3^{(1)}\simeq B_3; \qquad 12)\ \frac{3}{8}(k^2\psi)_{i3} \left[ \frac{3}{8}\Delta\varphi_{i3} -\frac{1}{4}(\sin 2\varphi_3^i-\sin 2\varphi_i) + \right. \]

\[ \left. +\frac{1}{32}(\sin 4\varphi_3^i-\sin 4\varphi_i) \right] =\Delta R_{i3}(2); \]

\[ 13)\ \frac{5}{6}e^2\,[R_{i3}(2)-\Delta R_{i3}(2)]=R_{i3}(3); \qquad 14)\ \sum_{\lambda=0}^{3} R_{i3}(\lambda)=\beta_{i3}\cdot \Delta L_3^{(1)}. \tag{2.2} \]

Having computed \(\Delta L_{i3}^{(1)}=\Delta L_1^{(1)}, \Delta L_2^{(1)}\) in accordance with (2.1) and (2.2), we find, according to (4.1) (with replacement of the superscript \((0)\) by \((1)\)), the values \(L_3^{(1)}, L_3^{(1)}\) of the longitude of point 3 and the misclosure \(w_L^{(1)}\), after which, according to (5.1) (with replacement of \((x)\) by \((x+1)\)), we compute the latitude \(B_3^{(2)}\) and longitude \(L_3^{(2)}\) of point 3 as the result of the second approximation. In these computations the values \(\operatorname{tg} A_{3i}'\) and

\[ \frac{\partial L_3^i}{\partial B_3}=a_{i3} \]

may be taken from the first approximation.

After completion of the second approximation we carry out a checking third approximation, in which we confine ourselves to computing only the quantities \(R_{i3}(0)\) and \(R_{i3}(1)\) for \(B_3=B_3^{(2)}\). The remaining terms \(R_{i3}(2)\) and \(R_{i3}(3)\) are taken from the second approximation. Having then computed the corresponding

the corresponding longitude differences \(\Delta L_i^{(2)}\), we find the misclosure \(w_L^{(2)}\) of the third approximation. If the computations of the second and third approximations have been carried out correctly, then this misclosure \(w_L^{(2)} \simeq 0\) within the limits of the accuracy of the computations. Then, as the final coordinates \(B_3, L_3\) of the point 3 being determined, we take their values \(B_3^{(2)}, L_3^{(2)}\), obtained in the second approximation.

Having found the coordinates \(B_3, L_3\) of point 3, we also compute, if necessary, the back azimuths \(A_{3i}\) and the lengths \(s_{i3}\), which can be done according to [Del. 5].

Б. Second method for solving the direct intersection on the spheroid

The use of this method is advisable in the case when, in addition to the geodetic coordinates \(B_3, L_3\) of the point 3 being determined, it is necessary simultaneously to know the distances \(s_{1.3}, s_{2.3}\) and, possibly, also the back azimuths \(A_{3.1}, A_{3.2}\). The solution of the direct intersection by this method is carried out in the following order.

First of all, from the known coordinates \(B_1, L_1\) and \(B_2, L_2\) of the initial points 1, 2 and from the known azimuths \(A_{1.3}, A_{2.3}\) of the intersecting rays \(\delta L_{1.3}, \delta L_{2.3}\) at these points, we solve on the sphere the corresponding direct intersection, determining from this solution the spherical distances \(\sigma_{1.2}, \sigma_{1.3}, \sigma_{2.3}\) of the arcs \(1^\circ2^\circ, 1^\circ3^\circ, 2^\circ3^\circ\) and the spherical angles \(\gamma_1, \gamma_2, \gamma_3\) at the vertices \(1^\circ, 2^\circ, 3^\circ\) of the spherical triangle \(1^\circ2^\circ3^\circ\), representing the given spheroidal triangle 123. These computations are performed as follows:

\[ \begin{aligned} &1)\quad 90^\circ - B_1 = \Theta_1, &&2)\quad 90^\circ - B_2 = \Theta_2;\\[4pt] &3)\quad \frac{1}{2}(\Theta_1+\Theta_2)=\Theta_{1.2}, &&4)\quad \frac{1}{2}(\Theta_2-\Theta_1)=\delta\Theta_{1.2};\\[4pt] &5)\quad \frac{1}{2}\Delta L_{1.2}=\delta L_{1.2}, &&6)\quad \frac{\sin \delta\Theta_{1.2}\operatorname{ctg}\delta L_{1.2}}{\sin \Theta_{1.2}} =\operatorname{tg}\delta\alpha_{1.2};\\[4pt] &7)\quad \frac{\cos \Theta_{1.2}\operatorname{ctg}\delta L_{1.2}}{\cos \Theta_{1.2}} =\operatorname{tg}\alpha_{1.2}, &&8)\quad \alpha_{1.2}+\delta\alpha_{1.2}=\beta_1;\\[4pt] &9)\quad \alpha_{1.2}-\delta\alpha_{1.2}=\beta_2, &&10)\quad \beta_2+A_{2.3}=\gamma_1;\\[4pt] &11)\quad \beta_1-A_{1.3}=\gamma_2, &&12)\quad \frac{1}{2}(\gamma_1+\gamma_2)=\gamma_{1.2}; \end{aligned} \tag{1} \]

\[ 13)\quad \frac{1}{2}(\gamma_2-\gamma_1)=\delta\gamma_{1.2}; \]

\[ 14)\quad \frac{\sin \Theta_1}{\sin \beta_2}\sin \Delta L_{1.2} = \frac{\sin \Theta_2}{\sin \beta_1}\sin \Delta L_{1.2} = \sin \sigma_{1.2}; \]

\[ \begin{aligned} &15)\quad \frac{1}{2}\sigma_{1.2}=\delta\sigma_{1.2}; &&16)\quad \frac{\sin \delta\gamma_{1.2}}{\sin \gamma_{1.2}}\operatorname{tg}\delta\sigma_{1.2} =\operatorname{tg}\delta\sigma_0;\\[4pt] &17)\quad \frac{\cos \delta\gamma_{1.2}}{\cos \gamma_{1.2}}\operatorname{tg}\delta\sigma_{1.2} =\operatorname{tg}\sigma_0, &&18)\quad \sigma_0+\delta\sigma_0=\sigma_{1.3};\\[4pt] &19)\quad \sigma_0-\delta\sigma_0=\sigma_{2.3}, &&20)\quad \frac{\sin \gamma_1}{\sin \sigma_{2.3}}\sin \sigma_{1.2} = \frac{\sin \gamma_2}{\sin \sigma_{1.3}}\sin \sigma_{1.2} = \sin \gamma_3;\\[4pt] &21)\quad \gamma_1+\gamma_2+\gamma_3+180^\circ=\varepsilon. \end{aligned} \]

Computations on the sphere are performed with an accuracy of up to 0.00001 for numbers and with an accuracy of up to \(1''\), or up to \(0^\mathrm{g}.0001\), for angles.

Next we compute, with full accuracy, the auxiliary quantities \(\nu_{i3}\), \(K_{i3}^{2}\), \(p_{i3}\), \(C_{0}\), \(C_{2u}^{i}\), \(D_{2u}^{i}\), \(Q_{i3}\), \(\varphi_{i3}\), \(\vartheta_{i3}\), \((\psi\mu)_{i3}\) for both intersecting sides \(i3=1.3,\ 2.3\), which are needed for solving the corresponding direct problems according to [Del. 4].

In order then to begin solving the indicated direct problems along side 1.3 and along side 2.3, we must in some way find approximate lengths \(s_{1.3}^{(0)}\), \(s_{2.3}^{(0)}\) of these sides. The simplest and, perhaps, sufficiently reliable way to do this is the following. Starting from the solution, carried out above, of the direct intersection on the sphere, we construct the corresponding spherical triangle \(1^\circ 2^\circ 3^\circ\) on a globe, or construct an equiangular representation \(1'2'3'\) of this triangle on a map. Then, having divided the lengths of the intersecting sides \(1^\circ 3^\circ\), \(2^\circ 3^\circ\), or \(1'3'\), \(2'3'\), into an equal number of parts \((ek)_{i3}\), for example into 5, we determine for each such part \((ek)_{i3}\) the mean latitude \(B_{ek}^{(i3)}\). For each latitude \(B_{ek}^{(i3)}\) we compute the mean radius of curvature \(R_{ek}^{(i3)}\), and then calculate their mean value \(R^{(i3)}\) for each intersecting side \(i3\). Then we may take

\[ 1)\ s_{1.3}^{(0)}=\sigma_{1.3}R^{(1.3)}, \qquad 2)\ s_{2.3}^{(0)}=\sigma_{2.3}R^{(2.3)}, \tag{2} \]

where \(\sigma_{1.3}\), \(\sigma_{2.3}\) are the spherical distances found above.

We now pass to successive approximations, in which improved values of the distances \(s_{1.3}\), \(s_{2.3}\) and of the coordinates \(B_3,\ L_3\) of the point 3 being determined are computed jointly. As initial values for these approximations one takes the values \(s_{1.3}^{(0)}\), \(s_{2.3}^{(0)}\) of the distances \(s_{i3}\) obtained according to (2). Each \(x\)-th approximation is then divided into three stages:

a) finding the discrepancy \(w_{B}^{(x-1)}\) in the two computed values \(B_{3}^{(x-1)}\), \(B_{3}^{(x-1)}\) of the latitude of point 3, which was caused by the error of the values \(s_{1.3}^{(x-1)}\), \(s_{2.3}^{(x-1)}\), obtained in the \((x-1)\)-st approximation, for the distances \(s_{1.3}\), \(s_{2.3}\);

b) finding the discrepancy \(w_{L}^{(x-1)}\) in the two computed values \(L_{3}^{(x-1)}\), \(L_{3}^{(x-1)}\) of the longitude of point 3, which was caused by the same reason as in (a);

c) composing a system of two linear equations with corrections \(\delta s_{1.3}^{(x-1)}\), \(\delta s_{2.3}^{(x-1)}\) to the approximate values \(s_{1.3}^{(x-1)}\), \(s_{2.3}^{(x-1)}\) for the distances \(s_{i3}\), and solving this system; computing the corresponding corrections \(\delta B_{3}^{(x-1)}\), \(\delta B_{3}^{(x-1)}\) and \(\delta L_{3}^{(x-1)}\), \(\delta L_{3}^{(x-1)}\); computing, in the \(x\)-th approximation, the improved values \(s_{1.3}^{(x)}\), \(s_{2.3}^{(x)}\) for the sides \(s_{i3}\), and the improved values \(B_{3}^{(x)}\), \(L_{3}^{(x)}\) for the coordinates \(B_3,\ L_3\) of the point 3 being determined.

Let us consider in greater detail each of these principal operations performed in the \(x\)-th approximation.

Finding the discrepancy \(w_{B}^{(x-1)}\). Taking as initial values \(s_{1.3}^{(x-1)}\), \(s_{2.3}^{(x-1)}\) of the distances \(s_{i3}\), obtained in the preceding \((x-1)\)-st approximation, we compute, according to [Del. 4.1], the latitude \(B_3\) of point 3 twice—along side 1.3 and along side 2.3. In doing this we use the method indicated there of sharp acceleration of convergence in the computation of \(\Delta\varphi_{i3}^{(x-1)}\). As a result of solving these two problems we obtain two corresponding values \(B_{3}^{(x-1)}\), \(B_{3}^{(x-1)}\) for the latitude \(B_3\) of point 3, and then

\[ w_{B}^{(x-1)}=B_{3}^{(x-1)}-B_{3}^{(x-1)} . \tag{3} \]

Finding the misclosure $w_L^{(x-1)}$. Taking as initial values the values $\varphi_{3}^{\prime(x-1)}$, $\varphi_{3}^{\prime\prime(x-1)}$ of the transformed latitude $\varphi_3$ of point 3, which were obtained in computing the two values $B_{3}^{1(x-1)}$, $B_{3}^{2(x-1)}$ of the latitude $B_3$ of this point, we then determine, according to [Del. 4.4], the longitude $L_3$ of point 3 twice—along side 1.3 and along side 2.3. As a result of solving these two particular problems we obtain two values $L_{3}^{1(x-1)}$, $L_{3}^{2(x-1)}$ for the longitude $L_3$ of point 3, and then:

$$ w_L^{(x-1)} = L_{3}^{1(x-1)} - L_{3}^{2(x-1)} . \tag{4} $$

Computing $s_{3}^{(x)}$, $B_{3}^{(x)}$, $L_{3}^{(x)}$. Having found the misclosures $w_B^{(x-1)}$, $w_L^{(x-1)}$, we set up a system of two plane equations with the desired corrections $\delta s_{1.3}^{(x-1)}$, $\delta s_{2.3}^{(x-1)}$ to the distances $s_{1.3}^{(x-1)}$, $s_{2.3}^{(x-1)}$ and with the absolute terms $w_B^{(x-1)}$, $w_L^{(x-1)}$:

$$ \left\{ \begin{aligned} \frac{\partial B_{3}^{1}}{\partial s_{1.3}}\,\delta s_{1.3}^{(x-1)} - \frac{\partial B_{3}^{2}}{\partial s_{2.3}}\,\delta s_{2.3}^{(x-1)} + w_B^{(x-1)} &= 0;\\ \frac{\partial L_{3}^{1}}{\partial s_{1.3}}\,\delta s_{1.3}^{(x-1)} - \frac{\partial L_{3}^{2}}{\partial s_{2.3}}\,\delta s_{2.3}^{(x-1)} + w_L^{(x-1)} &= 0. \end{aligned} \right. \tag{5} $$

The coefficients entering here are computed as follows:

$$ \begin{aligned} 1)\quad \sin A_{3i}' &= \frac{\sqrt{1-e^2}\,\nu_{13}\,V_{3}^{i}}{\cos B_{3}^{i}}; &\qquad 2)\quad \frac{\partial B_{3}^{i}}{\partial s_{i3}} &= \frac{\rho''}{M_3}\cos A_{3i}';\\[6pt] 3)\quad \frac{\partial L_{3}^{i}}{\partial s_{i3}} &= \frac{\rho''}{N_3}\sin A_{3i}', \end{aligned} \tag{6} $$

where

$$ \begin{aligned} 1)\quad M &= \frac{a(1-e^2)}{(1-e^2\sin^2 B)^{3/2}}; &\qquad 2)\quad N &= \frac{a}{(1-e^2\sin^2 B)^{1/2}};\\[6pt] 3)\quad \rho'' &= 206264.8. \end{aligned} \tag{7} $$

Having determined from the solution of system (5) the corrections $\delta s_{i3}^{(x-1)}$, we compute the corresponding corrections $\delta B_{3}^{i(x-1)}$, $\delta L_{3}^{i(x-1)}$:

$$ 1)\quad \delta B_{3}^{i(x-1)} = \frac{\partial B_{3}^{i}}{\partial s_{i3}}\,\delta s_{i3}^{(x-1)}; \qquad 2)\quad \delta L_{3}^{i(x-1)} = \frac{\partial L_{3}^{i}}{\partial s_{i3}}\,\delta s_{i3}^{(x-1)} . \tag{8} $$

Now the improved values, as a result of the $x$-th approximation, of $s_{i3}^{(x)}$, $B_{3}^{(x)}$, $L_{3}^{(x)}$ of the distances $s_{i3}$ and of the coordinates $B_3$, $L_3$ will be found as follows:

$$ 1)\quad s_{i3}^{(x)} = s_{i3}^{(x-1)} + \delta s_{i3}^{(x-1)}; $$

$$ 2)\quad B_{3}^{(x)} = B_{3}^{1(x-1)} + \delta B_{3}^{1(x-1)} = B_{3}^{2(x-1)} + \delta B_{3}^{2(x-1)}; $$

$$ 3)\quad L_{3}^{(x)} = L_{3}^{1(x-1)} + \delta L_{3}^{1(x-1)} = L_{3}^{2(x-1)} + \delta L_{3}^{2(x-1)}, \tag{9} $$

These three basic operations complete the $x$-th approximation. Experience shows that, for lengths $s_{i3}$ of the intersecting sides up to 10000 km, two approximations are sufficient; in the third approximation, by a shortened computation, it is only necessary to verify that the new values $s_{i3}^{(2)}$, $B_{3}^{(2)}$, $L_{3}^{(2)}$ of the distances and coordinates give misclosures $w_B^{(2)} \approx 0$, $w_L^{(2)} \approx 0$ within the limits of computational accuracy.

If necessary, after determining \(s_{i3}, B_3, L_3\), the inverse azimuths \(A_{3i}\) of the rays \(3L_{3i}\) at the observed point 3 can also be found according to [Subsec. 4.2].

Subsec. 7. Some generalizations and additions

Let us consider some generalizations and additions relating to the solution of the first three problems on the terrestrial spheroid.

Addition 1. First of all, let us find the value of the transformed latitude \(\varphi_0=-\overline{\varphi}_0\) for the northern \(O^{(s)}\) and southern \(\overline{O}^{(s)}\) vertices of the rectified curve \(\Gamma\), proceeding from the notation \(\nu,\tau\) introduced in Subsec. 3 and from the substitution

\[ \sin\varphi=\tau\sin B. \]

Noting that at the vertices \(O^{(s)}, \overline{O}^{(s)}\) of the rectified curve \(\Gamma\) the corresponding azimuth is

\[ A_0=A_0\pm\pi=\frac{\pi}{2}, \]

we find:

\[ 1)\quad \nu^2=\frac{r^2}{a^2}\sin^2 A=\frac{r_0^2}{a^2}\sin^2 A_0 =\frac{\cos^2 B_0}{1-e^2\sin^2 B_0}; \]

\[ 2)\quad 1-e^2\nu^2=\frac{1-e^2}{1-e^2\sin^2 B_0}; \]

\[ 3)\quad 1-\nu^2=\frac{(1-e^2)\sin^2 B_0}{1-e^2\sin^2 B_0}; \qquad 4)\quad \tau^2=\frac{1-e^2\nu^2}{1-\nu^2}=\operatorname{cosec}^2 B_0; \qquad 5)\quad \sin\varphi_0=1. \]

It follows from this that, regardless of the value of the geodetic latitude \(B_0=-\overline{B}_0\), for the vertices \(O^{(s)}, \overline{O}^{(s)}\) of the rectified curve \(\Gamma\) we shall always obtain

\[ \varphi_0=-\overline{\varphi}_0=\frac{\pi}{2}. \tag{1} \]

Generalized expansions. We shall use equality (1) to obtain certain relations having a more general form than in Subsecs. 4–6.

The point is that the expansions which we used in solving the direct and inverse problems for the arc \(\Delta\Gamma_{1,2}\) of the rectified curve \(\Gamma_{1,2}\) on the terrestrial spheroid, and also for solving the direct spheroidal intersection, were obtained from the closed expressions (26) of Subsec. 3 under the assumption that the vertices \(O_{ij}^{(s)}, \overline{O}_{ij}^{(s)}, \overset{\circ}{O}_{ij}^{(s)}, \overset{\circ}{\overline{O}}_{ij}^{(s)}\) of the rectified curve \(\Gamma_{ij}\) are situated outside the corresponding arc \(\Delta\Gamma_{ij}\). In the same case, when on the rectified arc \(\Delta\Gamma_{ij}\) between its end points \(i,j\) there lies one of the vertices \(O_{ij}^{(s)}, \overline{O}_{ij}^{(s)}, \overset{\circ}{O}_{ij}^{(s)}, \overset{\circ}{\overline{O}}_{ij}^{(s)}\), or even several such vertices (Fig. 1), then the preceding expansions for solving the above-indicated problems must be generalized in the proper manner.

To obtain the corresponding generalizations, we shall start from the previously found expansions for computing the distance \(s_{ij}\) and the difference of longitudes \(\Delta L_{ij}\) between the end points \(i,j\) of the rectified arc \(\Delta\Gamma_{ij}\). Only now we shall divide the entire arc \(\Delta\Gamma_{ij}\) into a series of partial arcs \(\Delta\Gamma_{x,x+1}\), choosing as intermediate points \(x,x+1\) the most suitable points, which will be done somewhat later. In this case the expansions given in Subsecs. 4, 5 and also used in Subsec. 6, in solving the direct spheroidal intersection, may be represented in the following generalized form:

\[ \begin{aligned} 1)\quad \frac{s_{ij}}{a} &= \mu_{ij}\sum_{x=i}^{x+1=j}\beta_{x,x+1} \int_{\varphi_x}^{\varphi_{x+1}} \frac{d\varphi}{\left(1-k_{ij}^{2}\sin^{2}\varphi\right)^{3/2}} \\ &= \mu_{ij}\sum_{x=i}^{x+1=j}\beta_{x,x+1} \int_{\varphi_x}^{\varphi_{x+1}} \left[\sum_{\lambda=0}^{n}(-1)^\lambda \binom{-3/2}{\lambda} k_{ij}^{2\lambda}\sin^{2\lambda}\varphi\,d\varphi\right] \\ &= (\mu C_0)_{ij}\sum_{x=i}^{x+1=j}\beta_{x,x+1} (\varphi_{x+1}-\varphi_x) \\ &\quad +\mu_{ij}\sum_{x=i}^{x+1=j}\beta_{x,x+1} \left[\sum_{u=0}^{n} C_{2u} (\sin 2u\varphi_{x+1}-\sin 2u\varphi_x)\right]_{ij} \\ &= \mu_{ij} C_0^{(ij)} \sum_{x=i}^{x+1=j}\beta_{x,x+1}(\varphi_{x+1}-\varphi_x) \\ &\quad +\mu_{ij}\sum_{u=1}^{n} C_{2u}^{(ij)} \left[ \sum_{x=i}^{x+1=j}\beta_{x,x+1} (\sin 2u\varphi_{x+1}-\sin 2u\varphi_x) \right]. \end{aligned} \tag{2} \]

\[ \begin{aligned} 2)\quad \sum_{x=i}^{x+1=j}\beta_{x,x+1}(\varphi_{x+1}-\varphi_x) &= \frac{s_{ij}}{a(\mu C_0)_{ij}} \\ &\quad - \sum_{x=i}^{x+1=j}\beta_{x,x+1} \left[\sum_{u=1}^{n} D_{2u}^{(ij)} (\sin 2u\varphi_{x+1}-\sin 2u\varphi_x)\right] \\ &= \frac{s_{ij}}{a\mu_{ij}C_0^{(ij)}} -\sum_{u=1}^{n} D_{2u}^{(ij)} \left[ \sum_{x=i}^{x+1=j}\beta_{x,x+1} (\sin 2u\varphi_{x+1}-\sin 2u\varphi_x) \right]. \end{aligned} \]

\[ \begin{aligned} 3)\quad \Delta L_{ij} &= (\nu\psi)_{ij} \sum_{x=i}^{x+1=j}\beta_{x,x+1} \int_{\varphi_x}^{\varphi_{x+1}} \frac{d\varphi}{ (1-m_{ij}^{2}\sin^{2}\varphi)\sqrt{1-k_{ij}^{2}\sin^{2}\varphi} } \\ &= \sum_{x=i}^{x+1=j}\beta_{x,x+1} \left[\sum_{\lambda=0}^{n}\nu_{ij}\psi_{ij}F_{x,x+1}(\lambda)\right] = \sum_{x=i}^{x+1=j}\beta_{x,x+1} \left[\sum_{\lambda=0}^{n}R_{x,x+1}(\lambda)\right] \\ &= \sum_{\lambda=0}^{n} \left[ \sum_{x=i}^{x+1=j}\beta_{x,x+1}R_{x,x+1}(\lambda) \right] = \sum_{\lambda=0}^{n}R_{ij}(\lambda), \end{aligned} \tag{2} \]

where

\[ \begin{aligned} a)\quad R_{ij}(0) &= \frac{\nu_{ij}}{|\nu_{ij}|}\sqrt{1-e^{2}}\, \sum_{x=i}^{x+1=j}\beta_{x,x+1} \left[ \operatorname{arc\,tg}(\rho_{ij}\operatorname{tg}\varphi_{x+1}) -\operatorname{arc\,tg}(\rho_{ij}\operatorname{tg}\varphi_x) \right] \\ &= \frac{\nu_{ij}}{|\nu_{ij}|}\sqrt{1-e^{2}}\, \sum_{x=i}^{x+1=j}\beta_{x,x+1} [\vartheta_{x+1}-\vartheta_x]; \end{aligned} \]

\[ b)\quad R_{ij}(1) = \frac{1}{2}e^{2}R_{ij}(0) -\frac{e^{2}}{2}(\nu\psi)_{ij} \sum_{x=i}^{x+1=j}\beta_{x,x+1}(\varphi_{x+1}-\varphi_x); \]

\[ \text{в)}\quad R_{ij}(2)=\frac{3}{4}e^2 R_{ij}(1)-\frac{3}{16}e^2(k^2\psi)_{ij} \sum_{x=i}^{x+1=j}\left[\beta_{x,x+1}(\varphi_{x+1}-\varphi_x)- \frac{1}{2}\beta_{x,x+1}(\sin 2\varphi_{x+1}-\sin 2\varphi_x)\right], \]

and so on (see Sec. 4.4). Here

\[ \beta_{x,x+1}=\frac{\cos A_{x,x+1}}{|\cos A_{x,x+1}|} =\frac{\varphi_{x+1}-\varphi_x}{|\varphi_{x+1}-\varphi_x|}=\pm 1 . \]

Let us now assume that, in the expansions (2), the points \(x,\ x+1\) are arranged, depending on the limits of variation of the azimuth \(A_{ij}\), in one of the following sequences (see Sec. 1.3 and Fig. 1):

\[ \text{а)}\quad x=i,\ O_{ij}^{(1)},\ \Э_{ij}^{(1)},\ \overline{O}_{ij}^{(1)},\ \Э_{ij}^{(2)},\ldots,\ x+1=j \quad \text{for } 0\leq A_{ij}<\frac{\pi}{2}\ \text{(Fig. 1,a);} \]

\[ \text{б)}\quad x=i,\ \Э_{ij}^{(1)},\ \overline{O}_{ij}^{(1)},\ \Э_{ij}^{(2)},\ldots,\ x+1=j \quad \text{for } \frac{\pi}{2}\leq A_{ij}<\pi\ \text{(Fig. 1,б);} \]

\[ \tag{3} \text{в)}\quad x=i,\ \Э_{ij}^{(1)},\ \overline{O}_{ij}^{(2)},\ldots,\ x+1=j \quad \text{for } \pi\leq A_{ij}<\frac{3}{2}\pi\ \text{(Fig. 1,в);} \]

\[ \text{г)}\quad x=i,\ O_{ij}^{(1)},\ \Э_{ij}^{(1)},\ \overline{O}_{ij}^{(1)},\ \Э_{ij}^{(2)},\ldots,\ x+1=j \quad \text{for } \frac{3}{2}\pi\leq A_{ij}<2\pi\ \text{(Fig. 1,г).} \]

Then it is not difficult, first of all, to calculate that with such an arrangement of the points \(x,\ x+1\) we shall always have:

\[ 1)\quad \sum_{x=i}^{x+1=j}\beta_{x,x+1}\bigl(\sin 2u\varphi_{x+1}-\sin 2u\varphi_x\bigr) = \beta'_{ji}\sin 2u\varphi_j-\beta_{ij}\sin 2u\varphi_i, \]

where

\[ \tag{4} 2)\quad \beta_{ij}=\frac{\cos A_{ij}}{|\cos A_{ij}|},\qquad 3)\quad \beta'_{ji}=\frac{\cos A'_{ji}}{|\cos A'_{ji}|},\qquad 4)\quad A'_{ji}=A_{ji}\pm 180^\circ . \]

What has been said follows from the fact that if \(x,\ x+1=\Э_i^{(s)},\ \Э_{ij}^{(s)}\), then \(\varphi_x=\varphi_{x+1}=0\); whereas if \(x,\ x+1=O_{ij}^{(s)},\ \overset{\circ}{O}_{ij}^{(s)},\ \overline{O}_{ij}^{(s)},\ \overset{\circ}{\overline{O}}_{ij}^{(s)}\), then \(\varphi_x,\varphi_{x+1}=\pm \dfrac{\pi}{2}\). Thus, in all these cases

\[ \sin 2u\varphi_{x+1}-\sin 2u\varphi_x=0,\qquad (x\ne i,\ x+1\ne j). \]

As for the sums

\[ \tag{5} 1)\quad \sum_{x=i}^{x+1=j}\beta_{x,x+1}(\varphi_{x+1}-\varphi_x)=\Delta\varphi_{ij}, \qquad 2)\quad \sum_{x=i}^{x+1=j}\beta_{x,x+1}(\vartheta_{x+1}-\vartheta_x)=\Delta\vartheta_{ij}, \]

then, on the basis of (1) and the meaning of the factor \(\beta_{x,x+1}\), we always have:

\[ \tag{6} 1)\quad \Delta\varphi_{ij}=q\pi+(\beta'_{ji}\varphi_j-\beta_{ij}\varphi_i); \qquad 2)\quad \Delta\vartheta_{ij}=q\pi+(\beta'_{ji}\vartheta_j-\beta_{ij}\vartheta_i), \]

where \(q\) is the number of vertices \(O_{ij}^{(s)},\ \overline{O}_{ij}^{(s)}\), or \(\overset{\circ}{O}_{ij}^{(s)},\ \overset{\circ}{\overline{O}}_{ij}^{(s)}\) lying between the endpoints \(i,\ j\) of the arc \(\Delta\Gamma_{ij}\); moreover, in (6) the transformed latitudes \(\varphi_i,\ \varphi_j\) and the arcs \(\vartheta_i,\ \vartheta_j\) are taken with their signs \(\pm\). Thus, for example, for the arcs \(\Delta\Gamma_{ij}\) shown in Fig. \(1a,\ 1b\), we have:

Fig. 1,a: 1) \(\Delta\varphi_{ij}=3\pi-(\varphi_j+\varphi_i)\), 2) \(\Delta\vartheta_{ij}=3\pi-(\vartheta_j+\vartheta_i)\),

\[ (\varphi_i,\vartheta_i>0,\ \varphi_j,\vartheta_j<0); \]

Fig. 1,b: 1) \(\Delta\varphi_{ij}=2\pi-(\overline{\varphi}_j-\varphi_i)\), 2) \(\Delta\vartheta_{ij}=2\pi-(\overline{\vartheta}_j-\vartheta_i)\),

\[ (\varphi_i,\vartheta_i>0,\ \overline{\varphi}_j,\overline{\vartheta}_j<0). \]

Equalities (4)—(6) are a generalization of the corresponding particular equalities in section 4, which there were written without the factor \(\beta_{ij}=\beta'_{ji}\), transferred to the left-hand side. Substituting equalities (4)—(6) into the expansions of the general form (2), we obtain the following final expressions for these expansions for any length \(s_{ij}\) of the adjusted arc \(\Delta\Gamma_{ij}\):

\[ 1)\quad \frac{s_{ij}}{a} = v_{ij}C^{(ij)}_0\,\Delta\varphi_{ij} + v_{ij}\sum_{u=1}^{n} C^{(ij)}_{2u} \left(\beta'_{ji}\sin 2u\varphi_j-\beta_{ij}\sin 2u\varphi_i\right); \]

\[ 2)\quad \Delta\varphi_{ij} = \frac{s_{ij}}{a v_{ij} C^{(ij)}_0} - \sum_{u=1}^{n} D^{(ij)}_{2u} \left(\beta'_{ji}\sin 2u\varphi_j-\beta_{ij}\sin 2u\varphi_i\right); \]

\[ 3)\quad \Delta L_{ij}=\sum_{\lambda=0}^{n} R_{ij}(\lambda), \tag{2a} \]

where

\[ a)\quad R_{ij}(0)=\frac{\gamma_{ij}}{|\gamma_{ij}|}\sqrt{1-e^2}\,\Delta\vartheta_{ij}; \qquad b)\quad R_{ij}(1)=\frac{e^2}{2}R_{ij}(0)-\frac{e^2}{2}(\gamma\psi)_{ij}\Delta\varphi_{ij}; \]

\[ c)\quad R_{ij}(2)=\frac{3}{4}e^2R_{ij}(1)- \]

\[ -\frac{3}{16}e^2(k^2\gamma\psi)_{ij} \left[ \Delta\varphi_{ij} - \frac{1}{2} \left(\beta'_{ji}\sin 2\varphi_j-\beta_{ij}\sin 2\varphi_i\right) \right]; \]

\[ d)\quad R_{ij}(3)=\frac{5}{6}e^2R_{ij}(2) - \frac{5}{16}e^2(k^4\gamma\psi)_{ij} \left[ \frac{3}{8}\Delta\varphi_{ij} -\right. \]

\[ \left. -\frac{1}{4} \left(\beta'_{ji}\sin 2\varphi_j-\beta_{ij}\sin 2\varphi_i\right) + \frac{1}{32} \left(\beta'_{ji}\sin 4\varphi_j-\beta_{ij}\sin 4\varphi_i\right) \right], \]

and so on.

Supplement 2. Starting from the general expansions (2a), we shall show that the azimuth \(A_{1.2}\) of the adjusted arc \(\Delta\Gamma_{1.2}\), going from point 1 to point 2 along the shortest path, is not equal to the azimuth \(\breve A_{1.2}\) of the adjusted arc \(\Delta\breve\Gamma_{1.2}\), connecting the same points 1, 2 but drawn in the opposite direction and therefore, in general, not being the shortest on the spheroid between the indicated points.

Let us suppose, for definiteness, that \(0<A_{1.2}<\dfrac{\pi}{2}\) and that between the end points 1, 2 of the arc \(\Delta\Gamma_{1.2}\) there is no vertex of the adjusted curve \(\Gamma_{1.2}\). Then the azimuth \(\breve A_{1.2}\) of the opposite path \(\Delta\breve\Gamma_{1.2}\) will satisfy the condition \(\pi<\breve A_{1.2}<\dfrac{3}{2}\pi\), and between the end points 1, 2 of this arc there will be contained two vertices of the curve \(\breve\Gamma_{1.2}\): the southern \(\breve O_{1.2}\) and the northern \(\breve O_{1.2}\). Taking into account the indicated features of the disposition of the adjusted arcs \(\Delta\Gamma_{1.2}\) and \(\Delta\breve\Gamma_{1.2}\) on the spheroid, let us write for them,

restricting ourselves to 2 terms, the general expressions for the differences of longitudes \(\Delta L_{1.2}\), \(\Delta \check L_{1.2}\) according to (2a.3), (6), and [Sect. 5; (5)]:

\[ \Delta L_{1.2} = \sqrt{1-e^2}\,(\vartheta_2-\vartheta_1) + \frac{e^2}{2}\sqrt{1-e^2}\,(\vartheta_2-\vartheta_1) - \frac{e^2}{2}\,p_{1.2}\sqrt{1-e^2}\,(\varphi_2-\varphi_1); \]

\[ \Delta \check L_{1.2} = -\sqrt{1-e^2}\left[2\pi-(\check\vartheta_2-\check\vartheta_1)\right] - \frac{e^2}{2}\sqrt{1-e^2}\left[2\pi-(\check\vartheta_2-\check\vartheta_1)\right] + \]

\[ \qquad\qquad + \frac{e^2}{2}\,\check p_{1.2}\sqrt{1-e^2}\left[2\pi-(\check\varphi_2-\check\varphi_1)\right], \]

where these expansions will be accurate up to quantities of order \(e^4\) [see (2a)]. Since \(\Delta L_{1.2}>0\), while \(\Delta \check L_{1.2}<0\), and the endpoints of the arcs \(\Delta \Gamma_{1.2}\), \(\Delta \check \Gamma_{1.2}\) are the same, it follows that \(\Delta L_{1.2}-\Delta \check L_{1.2}=2\pi\), and, after dividing both parts by \(\sqrt{1-e^2}\), we shall have:

\[ \frac{\Delta L_{1.2}-\Delta \check L_{1.2}}{\sqrt{1-e^2}} = 2\pi\left(1+\frac{e^2}{2}+\frac{3}{8}e^4\right) = 2\pi\left(1+\frac{e^2}{2}\right) + \]

\[ \qquad + \left(1+\frac{e^2}{2}\right) \left[(\vartheta_2-\vartheta_1)-(\check\vartheta_2-\check\vartheta_1)\right] - 2\pi\frac{e^2}{2}\,\check p_{1.2} - \]

\[ \qquad - \frac{e^2}{2} \left[p_{1.2}(\varphi_2-\varphi_1)-\check p_{1.2}(\check\varphi_2-\check\varphi_1)\right]. \]

Dividing then the last equality by \(2\pi\), we obtain further, again with accuracy up to \(e^4\):

\[ \frac{3}{8}e^4 \approx 0 \approx \frac{1}{2\pi} \left[(\vartheta_2-\vartheta_1)-(\check\vartheta_2-\check\vartheta_1)\right] - \frac{e^2}{2}\,\check p_{1.2} - \]

\[ \qquad - \frac{e^2}{4\pi}\,\check p_{1.2} \left[(\varphi_2-\varphi_1)-(\check\varphi_2-\check\varphi_1)\right]. \]

Hence it follows that

\[ \frac{1}{2\pi} \left[(\vartheta_2-\vartheta_1)-(\check\vartheta_2-\check\vartheta_1)\right] = \frac{e^2}{2}\,\check p_{1.2} + \operatorname{rem.}(\hat e^4) = \frac{e^2}{2}\cos B_0 + \operatorname{rem.}(\hat e^4), \tag{7} \]

and, consequently, in the present case

\[ \check\vartheta_2-\check\vartheta_1 \ne \vartheta_2-\vartheta_1. \tag{8} \]

But, according to section 5:

\[ 1)\quad \vartheta_x=\operatorname{arc\,tg}(p\,\operatorname{tg}\varphi_x) = \operatorname{arc\,tg} \frac{p\sin B_x}{\sqrt{\cos^2 B_x-p^2}} = \]

\[ = \operatorname{arc\,tg} \frac{\cos B_0\sin B_x}{\sqrt{\cos^2 B_x-\cos^2 B_0}} = f(B_0,B_x); \]

\[ 2)\quad \sin A_{1.2} = \frac{\check v_{1.2}}{|v_{1.2}|}\, \frac{\cos B_0}{\cos B_1} \sqrt{ \frac{1-e^2\sin^2 B_1}{1-e^2\sin^2 B_0} } = \Psi(B_0,B_1). \]

Therefore, from inequality (8) it follows that

\[ f(\breve B_0,B_2)-f(\breve B_0,B_1)\ne f(B_0,B_2)-f(B_0,B_1), \tag{9} \]

i.e.

\[ \breve B_0\ne B_0. \]

Hence we conclude that

\[ \sin A_{1,2}=\Psi(B_0,B_1)\ne \Psi(\breve B_0,B_1)=\sin A_{1.2}. \tag{10} \]

Our assertion is proved.

From consideration of the transformations carried out in deriving inequalities (9), (10), it follows that these inequalities will also be preserved in the case when, for example, the northern vertex \(O_{1.2}\) of the adjusted curve \(\Gamma_{1.2}\) lies on the arc \(\Delta \Gamma_{1.2}\), while on the opposite arc \(\Delta\breve\Gamma_{1.2}\) lies the southern vertex \(\breve O_{1.2}\) of the corresponding adjusted curve \(\breve\Gamma_{1.2}\).

Inequalities (9) and (10) are strengthened still further if the arc \(\Delta\breve\Gamma_{1.2}\) contains \(q>2\) vertices \(\breve O_{1.2}, \breve{\bar O}_{1.2}\) of the adjusted curve \(\breve\Gamma_{1.2}\).

Supplement 3. Let us compute the longitude difference \(\Delta L_{1.2}\) for the case when the beginning 1 of the arc \(\Delta \Gamma_{1.2}\) is the southern vertex \(\bar O_{1.2}\) of the adjusted curve \(\Gamma_{1.2}\), and the end 2 of the arc \(\Delta \Gamma_{1.2}\) is the northern vertex \(O_{1.2}\) of the curve \(\Gamma_{1.2}\), while the arc \(\Delta \Gamma_{1.2}\) intersects the equator at the azimuth \(A_{\mathrm e}^{(1.2)}\), which satisfies the condition \(0<A_{\mathrm e}^{(1.2)}<\dfrac{\pi}{2}\).

With such a formulation of the problem we shall have, first of all:

\[ \begin{aligned} &1)\ \frac{\nu_{1.2}}{|\nu_{1.2}|}=+1; \qquad 2)\ \varphi_1-\varphi_0=-\frac{\pi}{2}, \qquad \varphi_2-\varphi_0=+\frac{\pi}{2};\\[4pt] &3)\ \vartheta_1=-\frac{\pi}{2}, \qquad \vartheta_2=+\frac{\pi}{2}; \qquad 4)\ A'_{1.2}=A'_{2.1}=\frac{\pi}{2};\\[4pt] &5)\ \beta_{1.2}=\frac{\varphi_2-\varphi_1}{|\varphi_2-\varphi_1|}=+1=\beta_{2.1}; \qquad 6)\ \beta'_{2.1}\varphi_2-\beta_{1.2}\varphi_1=\Delta\varphi_{1.2}=\pi;\\[4pt] &7)\ \beta'_{2.1}\vartheta_2-\beta_{1.2}\vartheta_1=\Delta\vartheta_{1.2}=\pi; \qquad 8)\ \beta'_{2.1}\sin 2u\varphi_2-\beta_{1.2}\sin 2u\varphi_1=0. \end{aligned} \]

Substituting the found values of the auxiliary quantities into the expansion of the general form (2a.3) and also using equalities (3)—(8) of Appendix 5, we express the desired longitude difference \(\Delta L_{1.2}\) for the arc \(\Delta \Gamma_{1.2}\) in terms of the vertex latitude \(B_0\):

\[ \Delta L_{1.2} = \sqrt{1-e^2}\,\pi + \frac{e^2}{2}\sqrt{1-e^2}\,\pi(1-\cos B_0) + \]

\[ +\left[ \frac{3}{8}e^4\sqrt{1-e^2}\,\pi(1-\cos B_0) - \frac{3}{16}e^4\sqrt{1-e^2}\,\pi\sin B_0\cos B_0 \right] = \]

\[ = \sqrt{1-e^2}\,\pi \left[ 1-e^2\sin^2\frac{B_0}{2} + \frac{3}{16}e^4\sin^2\frac{B_0}{2} - \frac{3}{32}e^4\sin 2B_0 \right] = \]

\[ = \sqrt{1-e^2}\,\pi \left[ 1-e^2\sin^2 B_0\left(1-\frac{3}{16}e^2\right) -\frac{3}{32}e^4\sin 2B_0 \right] <\pi. \]

Thus, if \(0<A_{\mathrm{e}}^{(1,2)}<\dfrac{\pi}{2}\), then for the given adjusted arc \(\Delta \Gamma_{1.2}\) the longitude difference \(\Delta L_{1.2}<\pi\). Hence it follows that for the opposite arc \(\check{\Delta}\Gamma_{1.2}\) the corresponding longitude difference is

\[ \check{\Delta}L_{1.2}=2\pi-\Delta L_{1.2}>\pi . \]

From the relations found,

\[ \Delta L_{1.2}<0,\qquad \check{\Delta}L_{1.2}>0 \quad \text{for} \quad A_{1.2}=A'_{2.1}=\frac{\pi}{2},\qquad B_1=\bar B_0=-B_2, \]

we further conclude that the lengths \(s_{1.2}\), \(\check{s}_{1.2}\) of the arcs \(\Delta \Gamma_{1.2}\), \(\check{\Delta}\Gamma_{1.2}\) are also not equal to one another, namely \(s_{1.2}<\check{s}_{1.2}\).

Finally, since for the equatorial azimuths \(A_{\mathrm{e}}^{(1,2)}\), \(\check A_{\mathrm{e}}^{(1,2)}\) of the arcs \(\Delta \Gamma_{1.2}\), \(\check{\Delta}\Gamma_{1.2}\) we have

\[ 1)\quad A_{\mathrm{e}}^{(1,2)}=F(B_1,B_2,\Delta L_{1.2}),\qquad 2)\quad \check A_{\mathrm{e}}^{(1,2)}=F(B_1,B_2,\check{\Delta}L_{1.2}), \]

and, as proved above, \(\Delta L_{1.2}\ne \check{\Delta}L_{1.2}\), it follows that \(A_{\mathrm{e}}^{(1,2)}\ne \check A_{\mathrm{e}}^{(1,2)}\).

§ 8. Examples of solving the first three basic problems on the terrestrial spheroid by the new methods

In conclusion let us consider examples of solving, by the proposed new methods, the first three basic problems on the terrestrial spheroid.

Example 1. (Appendix 1, Fig. 2a). A solution is given of the direct problem for the adjusted arc \(\Delta \Gamma'_{1.2}\) for the distance \(s_{1.2}=25649\) km and the azimuth \(A_{1.2}=229^\circ03'\), where \(B_1>0\), \(B_2=\bar B_2<0\). Laying off the arc \(\Delta \Gamma_{1.2}\)

Fig. 2.

of length \(s_{1.2}\) and with azimuth \(A_{1.2}\) on the globe, we find that between the end points \(1\) and \(2=\bar 2\) of this arc lies the southern vertex \(\bar O_{1.2}\) of the curve \(\Gamma_{1.2}\), and the azimuth \(A'_{2.1}\) at the point \(2=\bar 2\) lies in the interval \(\dfrac{3}{2}\pi<A'_{2.1}<2\pi\).

Therefore, applying for the solution of the given problem the expansions of the general form [§ 7; (2a)], we shall have the following working expressions for the auxiliary quantities \(\beta_{1.2}\), \(\beta'_{2.1}\),

\[ \Delta \varphi_{1.2},\quad \Delta \vartheta_{1.2}\quad \text{and}\quad (\beta'_{2.1}\sin 2u\bar{\varphi}_2-\beta_{1.2}\sin 2u\varphi_1): \]

\[ \begin{aligned} 1)\quad &\beta_{1.2}=\frac{\cos A_{1.2}}{|\cos A_{1.2}|}=-1,\qquad \beta'_{2.1}=\frac{\cos A'_{2.1}}{|\cos A'_{2.1}|}=+1;\\ 2)\quad &\Delta\varphi_{1.2}=\pi+\beta'_{2.1}\bar{\varphi}_2-\beta_{1.2}\varphi_1 =\pi+\varphi_1-|\bar{\varphi}_2|;\\ 3)\quad &\Delta\vartheta_{1.2}=\pi+\beta'_{2.1}\bar{\vartheta}_2-\beta_{1.2}\vartheta_1 =\pi+\vartheta_1-|\bar{\vartheta}_2|;\\ 4)\quad &\beta'_{2.1}\sin 2u\bar{\varphi}_2-\beta_{1.2}\sin 2u\varphi_1 =\sin 2u\varphi_1-\sin 2u|\bar{\varphi}_2|. \end{aligned} \]

We divide the solution of the problem into four parts: 1) computation of the quantities \(\gamma, k^2, \tau^2, \varphi_1, C_0, D_{2u}\); 2) computation of \(B_2\); 3) computation of \(L_2\); 4) computation of \(A_{2.1}\). In finding the numbers \(C_0, D_{2u}\) we use the prepared values of the auxiliary coefficients \(c_{2u.2\lambda}\), placed in Appendix 3. In computing the transformed latitude \(\varphi_2\) we apply the accelerated method for calculating the correction \(\Delta\varphi_{1.2}\), indicated in the note to [Sec. 4; (10)].

Example 2. (Appendix 2, Fig. 2, б). A solution is given for the inverse problem for the straightened arc \(\Delta \Gamma_{1.2}\), whose length is \(s_{1.2}=24447\) km, and whose azimuth is \(A_{1.2}=147^\circ27'\), with \(B_1>0,\ B_2=\bar B_2<0\). After laying off the arc \(\Delta \Gamma_{1.2}\) on the globe, it turned out that between the endpoints 1, 2 of this arc lies the southern vertex \(\bar O_{1.2}\) of the curve \(\Gamma_{1.2}\), and the azimuth \(A'_{2.1}\) lies within

\[ 0<A'_{2.1}<\frac{\pi}{2}. \]

Hence it follows that

\[ \begin{aligned} 1)\quad &\beta_{1.2}=-1,\quad \beta'_{2.1}=+1; \qquad &2)\quad &\Delta\varphi_{1.2}=\pi+\varphi_1-|\bar{\varphi}_2|;\\ 3)\quad &\Delta\vartheta_{1.2}=\pi+\vartheta_1-|\bar{\vartheta}_2|; \qquad &4)\quad &\beta'_{2.1}\sin 2u\bar{\varphi}_2-\beta_{1.2}\sin 2u\varphi_1=\\ &&&=\sin 2u\varphi_1-\sin 2u|\bar{\varphi}_2|. \end{aligned} \]

All the essential questions of solving this problem have been set forth in sufficient detail in Section 5. Here we shall only note that, if in the direct problem the principal working quantity is \(v_{1.2}=\sin A^{(1.2)}_0\), then here such a quantity will be \(p_{1.2}=\cos B^{(1.2)}_0\), which is found by successive approximation from the equation

\[ \Delta L_{1.2}=\sum_{\lambda=0}^{n} R_{1.2}(\lambda). \]

To determine the quantity \(p_{1.2}\) with an accuracy of 8–9 figures, two complete approximations and one incomplete, checking approximation are sufficient even for \(s_{1.2}\simeq 25000\) km.

After computing the principal quantity \(p_{1.2}\), and incidentally also the difference \(\Delta\varphi_{1.2}\), we find the azimuths \(A_{1.2}, A_{2.1}\) and the distance \(s_{1.2}\) from equalities in which \(v_{1.2}\) and \(\mu_{1.2}\) are expressed through \(p_{1.2}=\cos B^{(1.2)}_0\).

Examples 3 and 4. (Appendices 4, 5; Fig. 3). In these examples a solution is given for the direct spheroidal intersection in two ways: a) with computation only of the coordinates \(B_3, L_3\) of the point 3 being determined, b) with simultaneous computation of the distances \(s_{1.3}, s_{2.3}\) and the coordinates \(B_3, L_3\).

In both examples one and the same direct intersection is solved: its control points 1, 2 are taken near Murmansk and Khabarovsk, and the determined point 3 is near San Francisco in the USA, so that the distances \(s_{1.3}, s_{2.3}\) from the control points to the point determined turned out to be almost equal: \(s_{1.3}=8073\) km, \(s_{2.3}=7947\) km. The azimuths of the intersecting rays are respectively: \(A_{1.3}=341^\circ13'\), \(A_{2.3}=53^\circ06'\), with the north pole \(P\) of the spheroid falling inside triangle 123 (Fig. 3).

The sequence of computation of the direct spheroidal intersection by both of the indicated methods is set forth in sufficient detail in Sec-

Table 6, but only in the given examples, instead of particular expansions, expansions of the general form [Case 7; (2a)] were taken, since on the arcs \(\Delta \Gamma_{1.3}\) and \(\Delta \Gamma_{2.3}\) lie the northern vertices \(O_{1.3}\) and \(O_{2.3}\) of the rectified curves \(\Gamma_{1.3}, \Gamma_{2.3}\). Here we shall confine ourselves only to a few remarks.

1. In computing the direct intersection by the first method, the initial value \(B_3^{(0)}\) of the latitude of the point being determined, point 3, was obtained by solving the intersection on the sphere with 5 decimal places. For computing the coordinates \(B_3, L_3\) of point 3 on the spheroid with 8–9 significant figures, two complete approximations were required (one with 6 figures, the other with 8–9 figures) and one incomplete check approximation.

Fig. 3.

1. In computing the direct intersection by the second method, the initial values \(\sigma_{1.3}^{(0)}, \sigma_{2.3}^{(0)}\) of the intersecting sides 1.3, 2.3 in angular measure were obtained by a 5-figure solution of the intersection on the sphere. But next it was necessary to find the corresponding initial values \(s_{1.3}^{(0)}, s_{2.3}^{(0)}\) of the lengths of these sides on the spheroid. To this end, a small cartographic graticule of the northern hemisphere was drawn in polar stereographic projection (Fig. 3), on which there was then constructed an equiangular representation \(1'2'3'\) of the corresponding spherical triangle \(1^\circ2^\circ3^\circ\). Dividing each intersecting side \(1'3' = 1^\circ3^\circ\), \(2'3'\) of triangle \(1'2'3'\) into four parts, from the cartographic graticule, with an accuracy of

Appendix 1

Solution of the direct problem for the reduced arc \(\Delta \Gamma_{1.2}\)

1. Initial data.

Quantity	Value
\(B_1\)	\(68^\circ34'15''.739\)
\(L_1\)	\(29^\circ42'16''.347\)
\(A_{1.2}\)	\(229^\circ03'15''.460\)
\(s_{1.2}\)	\(25\ 648\ 923.7\)

2. Computation of the quantities:

\[ \nu,\ \kappa^2,\ \tau^2,\ \varphi,\ C_0,\ D_{2u} \]

Quantity	Value	Quantity	Value
\(\lg \sin A_{1.2}\)	\(9.878\ 13727n\)	\(\kappa^6\)	\(0.000\ 000236\)
\(\lg \cos B_1\)	\(9.562\ 70597\)	\(\kappa^8\)	\(0.000\ 000001_5\)
\(-\lg V_1\)	\(-0.000\ 19522\)
\(-\lg \sqrt{1-e^2}\)	\(9.998\ 54166\)	\(\lg \sin B_1\)	\(9.968\ 88962\)
\(\lg \nu\)	\(9.442\ 10636n\)	\(\lg \tau\)	\(0.017\ 19304\)
\(\lg \nu^2\)	\(8.884\ 21272\)	\(\lg \sin \varphi_1\)	\(9.986\ 08266\)
\(\lg e^2\)	\(7.825\ 64818\)	\(\varphi_1\)	\(75^\circ34'19''.741\)
\(\lg e^2\nu^2\)	\(6.709\ 86090\)
\(e^2\)	\(0.006\ 693422\)	\(1\)	\(1.000\ 000000\)
\(e^2\nu^2\)	\(0.000\ 512697\)	\({}^{*})\ c_{02}\kappa^2\)	\(0.004\ 637921\)
\(\nu^2\)	\(0.076\ 59717\)	\(c_{04}\kappa^4\)	\(0.000\ 026888\)
\(e^2-e^2\nu^2\)	\(0.006\ 180725\)	\(c_{06}\kappa^6\)	\(161\)
\(1-e^2\nu^2\)	\(0.999\ 48730\)	\(c_{08}\kappa^8\)	\(1\)
\(1-\nu^2\)	\(0.923\ 40283\)	\(C_0\)	\(+1.004\ 664971\)
\(\lg(e^2-e^2\nu^2)\)	\(7.791\ 03942\)	\(c_{22}\kappa^2\)	\(-0.002\ 318961\)
\(\lg(1-e^2\nu^2)\)	\(9.999\ 77728\)	\(c_{24}\kappa^4\)	\(-0.000\ 017925\)
\(\lg(1-\nu^2)\)	\(9.965\ 39120\)	\(c_{26}\kappa^6\)	\(-121\)
\(\lg \kappa^2\)	\(7.791\ 26214\)	\(c_{28}\kappa^8\)	\(-1\)
\(\lg \tau^2\)	\(0.034\ 38608\)	\(C_2\)	\(-0.002\ 337008\)
\(\kappa^2\)	\(0.006\ 183895\)	\(c_{44}\kappa^4\)	\(+0.000\ 002242\)
\(\kappa^4\)	\(0.000\ 038241\)	\(c_{46}\kappa^6\)	\(+21\)
		\(C_4\)	\(+0.000\ 002266\)
		\(C_6=-c_{66}\kappa^6\)	\(-0.000\ 000027\)
		\(C_2:C_0=D_2\)	\(-0.002\ 326156\)
		\(C_4:C_0=D_4\)	\(+0.000\ 002255\)
		\(C_6:C_0=D_6\)	\(-0.000\ 000027\)

3. Computation of \(B_2\).

Quantity	Value
\(\lg s_{1.2}\)	\(7.4090\ 69146\)
\(\lg\sqrt{1-e^2\nu^2}\)	\(9.9998\ 88836\)
\(-\lg(1-e^2)a^2\)	\(6.8017\ 84509\)
\(-\lg C_0\)	\(0.0020\ 21260\)
\(\lg Q\)	\(0.6051\ 52013\)
\(Q\)	\(4.028\ 58019\)
\(-(\pi+\varphi_1)\)	\(4.460\ 57550\)
\(\varphi_2^{(0)}\)	\(-0.431\ 99531\)
\(=\)	\(-24^\circ45'05''.4\)
\(\sin 2\varphi_2^{(0)}\)	\(-0.760\ 440\)
\(+\ \sin 2\varphi_1\)	\(+0.482\ 605\)
\(\sin 4\varphi_2^{(0)}\)	\(-0.9877\)
\(+\ \sin 4\varphi_1\)	\(-0.8454\)
\(\sin 6\varphi_2^{(0)}\)	\(-0.52\)
\(+\ \sin 6\varphi_1\)	\(+1.00\)
\(\varepsilon_2^{(0)}\)	\(-0.277\ 835\)
\(\varepsilon_4^{(0)}\)	\(-1.8331\)
\(\varepsilon_6^{(0)}\)	\(+0.48\)
\(-D_2\varepsilon_2^{(0)}\)	\(-0.000\ 646289\)
\(-D_4\varepsilon_4^{(0)}\)	\(+0.000\ 004134\)
\(-D_6\varepsilon_6^{(0)}\)	\(+13\)
\(\delta\varphi_2^{(0)}\)	\(-0.000\ 642142\)
\(\cos 2\varphi_2^{(0)}\)	\(+0.649\ 408\)
\(\cos 4\varphi_2^{(0)}\)	\(-0.156\ 54\)
\(\cos 6\varphi_2^{(0)}\)	\(-0.060\)
\(-2D_2\cos 2\varphi_2^{(0)}\)	\(+0.003\ 02125\)
\(-4D_4\cos 4\varphi_2^{(0)}\)	\(+0.000\ 00141\)
\(-6D_6\cos 6\varphi_2^{(0)}\)	\(+1\)
\(\chi\)	\(+0.003\ 02267\)

\({}^{*})\) The values of the numbers \(c_{2u\cdot 2\lambda}\) are given in Appendix 3.

Continuation of Appendix 1

$1-x=\chi_1$	$+0.996\ 97733$	$p$	$0.275\ 90490$
$\delta\varphi_2^{(0)}:\chi_1=\Delta\varphi_2^{(0)}$	$-0.000\ 644089$	$\times\ \dfrac{p}{\omega\sqrt{1-e^2}}$	$\dfrac{-0.996\ 6477}{-0.274\ 9800}$	$\varepsilon_2=\varepsilon_2^{(1)}$	$\dfrac{-0.2787}{+0.1394}$
$\varphi_2^{(0)}$	$\dfrac{-0.431\ 99531}{-0.432\ 63940}$	$\psi_\mu$		$-1/2\varepsilon_2$
$\varphi_2^{(1)}$				$+\Delta\varphi_{1,2}$	$\dfrac{+4.0279}{+4.1673}$
$=$	$-24^\circ47'18'',282$	$+$		$\sigma_2$
		$\pi^{-1}\varphi_1$	$4.460\ 576$	$\times$
		$\varphi_2$	$\dfrac{-0.432\ 639}{+4.027\ 937}$	$3/16e^2\kappa^3\psi_\mu$	$\dfrac{-0.00000\ 21341}{-0.00000\ 8893}$
$\sin 2\varphi_2^{(1)}$	$-0.761\ 276$	$\Delta\varphi_{1,2}$		$\Delta R(1)$
$\sin 4\varphi_2^{(1)}$	$-0.9873$	$\times$
$\sin 6\varphi_2^{(1)}$	$-0.52$	$\psi_\mu$	$\dfrac{-0.274\ 9800}{-1.107\ 602}$
$\varepsilon_2^{(1)}$	$-0.278\ 671$	$\delta$		$3/4e^2$	$0.005\ 0201$
$\varepsilon_4^{(1)}$	$-1.8327$			$\times$
$\varepsilon_6^{(1)}$	$+0.48$	$\lg\operatorname{tg}\varphi_2$	$9.664\ 47223n$	$R(1)$	$\dfrac{-0.009\ 0854}{-0.000\ 045610}$
$-D_2\varepsilon_2^{(1)}$	$-0.000\ 648232$	$\lg p$	$9.440\ 75941$	$3/4e^2R(1)$
$-D_4\varepsilon_4^{(1)}$	$+0.000\ 004133$	$\lg\operatorname{tg}\varphi_1$	$\dfrac{0.589\ 60696}{9.105\ 23164n}$	$-\Delta R(1)$	$\dfrac{+0.000\ 008893}{-0.000\ 037717}$
$-D_6\varepsilon_6^{(1)}$	$\dfrac{+\qquad 13}{-0.000\ 644086}$	$\lg\operatorname{tg}(pt_2)$		$\psi_\mu F(2)=-R(2)$
$\Delta\varphi_2^{(0)}$		$\lg\operatorname{tg}(pt_1)$	$0.030\ 36637$
$\varphi_2^{(0)}$	$\dfrac{-0.431\ 99531}{-0.432\ 63940}$	$pt_2=\vartheta_2$	$-7^\circ15'41'',038$	$\varepsilon_4=\varepsilon_4^{(1)}$	$\dfrac{-1.8327}{+1.5105}$
$\varphi_2^{(1)}=\varphi_2$		$pt_1=\vartheta_1$	$\dfrac{+47^\circ00'05'',268}{219^\circ44'24'',230}$	$3/8\Delta\varphi_{1,2}$
$=$	$-24^\circ47'18'',282$	$180^\circ+(\vartheta_1+\vartheta_2)$		$-1/4\varepsilon_2$	$+0.0697$
		$\Delta\vartheta_{1,2}$	$3.835\ 18761$	$+1/32\varepsilon_4$	$\dfrac{-0.0573}{+1.5229}$
$\lg\sin\varphi_2$	$9.622\ 49221n$	$\times$		$\varepsilon_4$
$-\lg \overline{\nu}$	$\dfrac{0.017\ 19304}{9.605\ 29917n}$	$\omega\sqrt{1-e^2}$	$\dfrac{-0.996\ 64767}{-3.822\ 33080}$	$\times$	$\dfrac{-0.00000\ 00220}{-0.000\ 000034}$
$\lg\sin B_2$		$\psi_\mu F(0)=-R(0)$		$5/16e^2\kappa^4\psi_\mu$
$\overline{B}_2$	$-23^\circ45'55'',858$	$-\delta$	$\dfrac{+1.107\ 602}{-2.714\ 729}$	$\Delta R(2)$
		$\sigma_0$
		$\times$		$5/6e^2$	$0.005\ 578$
3. Computation of $L_2$		$e^2/2$	$\dfrac{0.003\ 346711}{-0.009\ 085413}$	$\times$
		$\psi_\mu F(1)=R(1)$		$R(2)$	$\dfrac{-0.000\ 037717}{-0.000\ 000210}$
		$3/16e^2$	$0.031\ 2350$	$5/6e^2R(2)$
		$5/16e^2$	$\dfrac{0.002\ 2092}{0.00000\ 77608}$	$-\Delta R(2)$	$\dfrac{+\qquad 34}{-0.000\ 000176}$
$\dfrac{\nu}{|\nu|}=\omega$	$-1$	$3/16e^2\kappa^3$		$\psi_\mu F(3)=-R(3)$
		$\psi_\mu$	$-0.27\ 498$
		$5/16e^2\kappa^4$	$\dfrac{0.00000\ 00800}{-0.00000\ 21341}$	$R(2)$	$-0.000\ 037717$
$\lg\dfrac{\omega}{\nu}$		$3/16e^2\kappa^3\psi_\mu$		$R(1)$	$-0.009\ 085413$
$=\lg\sin B_0$	$9.982\ 80696n$	$5/16e^2\kappa^4\psi_\mu$	$-0.00000\ 00220$	$R(0)$	$\dfrac{-3.822\ 33080}{-3.831\ 45411}$
$B_0$	$-73^\circ59'02'',590$			$\Delta L_{1,2}$
$\lg\cos B_0:\lg p$	$9.440\ 75941$

Continuation of Appendix 1

$\Delta L_{1.2}$	$-219^\circ31'34''.139$
$+$
$L_1$	$29^\circ42'16''.347$
$L_2$	$170^\circ10'42''.208$

4. Calculation of $A_{2.1}$

$\lg\sqrt{1-e^2}$	$9.998\ 54166$
$\lg \nu$	$9.442\ 10636n$
$\lg V_2$	$0.001\ 22217$
$-\lg\cos B_2$	$-9.961\ 51714$
$\lg\sin A'_{2.1}$	$9.480\ 35305n$
$A'_{2.1}$	$342^\circ24'27''.940$
$A_{2.1}$	$162^\circ24'27''.940$

Appendix 2

Solution of the inverse problem for the adjusted arc $\Delta\Gamma_{1.2}$

Initial data	Initial data
$B_2$	$-31^\circ13'27''.653$	$n_1-n_2$	$+0.63\ 169$	$e^2$	$0.00\ 6693$
$B_1$	$+68^\circ34'15''.739$	$\sin\Delta L_{1.2}$	$-0.39\ 998$	$\tau^2$	$1.03\ 973$
$L_2$	$233^\circ16'53''.814$	$\operatorname{ctg}\alpha_{1.2}$	$-1.57\ 930$	$e^2:\tau^2=\kappa^2$	$0.00\ 6438$
$L_1$	$29^\circ42'16''.347$	$\alpha_{1.2}=A^{(0)}_{1.2}$	$147^\circ39'30''$
$\Delta L_{1.2}$	$203^\circ34'37''.467$			$\sin B_2$	$-0.51\ 839$
$=$	$3.553\ 09023$			$\times\ \tau$	$+1.01\ 967$
				$\times\ \sin B_1$	$+0.93\ 087$
		3. Calculation of $p=\cos B_0$		$\sin\varphi_2$	$-0.52\ 859$
		a) Approximation 1		$\sin\varphi_1$	$+0.94\ 918$
		$\sin A^{(0)}_{1.2}$	$+0.53\ 496$	$\varphi_2$	$-31^\circ54'.36''$
2. Calculation of $A^{(0)}_{1.2}$		$\cos B_1$	$+0.36\ 535$	$\varphi_1$	$+71^\circ39'18''$
$\operatorname{tg} B_2$	$-0.60\ 621$	$\cos B^{(0)}_0=p^{(0)}$	$+0.19\ 545$	$180^\circ+(\varphi_1+\varphi_2)$	$219^\circ45'42''$
$\times$		$\times\ \dfrac{\nu}{|\nu|}\sqrt{1-e^2}$	$+0.99\ 665$	$=\Delta\varphi_{1,2}$	$+3.83\ 527$
$\cos B_1$	$+0.36\ 535$	$\psi$	$+0.19\ 480$	$\times\ \psi$	$+0.19\ 480$
$\cos\Delta L_{1.2}$	$-0.91\ 653$	$B_0$	$+78^\circ43'44''$	$\delta$	$+0.74\ 171$
$\times$		$\csc B_0=\tau$	$1.01\ 967$
$\sin B_1$	$+0.93\ 087$
$n_1$	$-0.22\ 148$
$n_2$	$-0.85317$

Continuation of Appendix 2

$\Delta L_{1.2}$	$+3.553\ 090$	$\operatorname{tg}\varphi_2=t_2$	$-0.62\ 269$	$\lg\operatorname{tg}\varphi_2$	$9.794\ 39292n$
$\times$		$\operatorname{tg}\varphi_1=t_1$	$+3.01\ 58$	$\lg p$	$9.293\ 28519$
$\sqrt{1-e^2}$	$+0.996\ 648$	$(t_2+t_1)$	$+2.3931$	$\lg\operatorname{tg}\varphi_1$	$0.480\ 30178$
$\psi F(0)=R(0)$	$+3.541\ 180$	$\times$		$\lg\operatorname{tg}(pt_2)$	$9.08767811n$
$-\delta$	$-0.747\ 11$	$\dfrac{\gamma}{|\gamma|}\tau^2\sqrt{1-e^2}$	$+1.03\ 623$	$\lg\operatorname{tg}(pt_1)$	$9.773\ 58697$
$\varepsilon_0$	$+2.794\ 07$	$\dfrac{d\omega^{(0)}}{dp}=\chi^{(0)}$	$+2.4798$	$pt_2=\theta_2$	$-6^\circ58'35''.936$
$\times$		$p^{(0)}$	$+0.195\ 450$	$pt_1=\theta_1$	$+30^\circ41'55''.529$
$e^2/2$	$0.003\ 3467$	$-w^{(0)}:\chi=\Delta p^{(0)}$	$+\ \ \ \ 1\ 015$	$180^\circ+(\theta_1+\theta_2)$	$203^\circ43'19''.593$
$\psi F(1)=R(1)$	$+0.009\ 351$	$p^{(1)}$	$+0.196\ 465$	$=\Delta\beta_{1.2}$	$+3.555\ 62156$
$\sin 2\varphi_2$	$-0.8974$	b) Approximation 2		$\times$
$+$		$p^{(1)}=\cos B_0$	$+0.196\ 46500$	$\dfrac{\gamma}{|\gamma|}\sqrt{1-e^2}$	$+0.996\ 64767$
$\sin 2\varphi_1$	$+0.59\ 75$	$\lg p^{(1)}=\lg p$	$9.293\ 28519$	$\psi F(0)=R(0)$	$+3.543\ 70194$
$\varepsilon_2$	$-0.2999$	$B_0$	$+78^\circ40'10''.857$	$-\delta$	$-0.751\ 066$
$-\dfrac12\varepsilon_2$	$+0.1500$	$\lg\operatorname{cosec}B_0=\lg\tau$	$0.008\ 54760$	$\varepsilon_0$	$+2.792\ 636$
$+$		$\dfrac{\gamma}{|\gamma|}\sqrt{1-e^2}$	$+0.996\ 648$	$\times$
$\Delta\varphi_{1.2}$	$+3.8353$	$\times$		$e^2/2$	$0.003\ 34671$
$\varepsilon_2$	$+3.9853$	$p$	$+0.196\ 465$	$\psi F(1)=R(1)$	$+0.009\ 34614$
$3/16\,e^2$	$0.001\ 255$	$\psi$	$+0.195\ 806$	$e^2:\varepsilon_2^2=\kappa^2$	$0.036\ 435$
$3/16\,e^2\kappa^2$	$0.0000\ 08079$	$\lg\sin B_2$	$9.714\ 65700n$	$\kappa^4$	$0.0000\ 4141$
$3/16\,e^2\kappa^2\psi$	$+0.0000\ 01574$	$\lg\tau$	$0.008\ 54760$	$3/16\,e^2$	$0.001\ 255$
$\times$		$\lg\sin B_1$	$9.968\ 88962$	$5/16\,e^2$	$0.002\ 092$
$\varepsilon_2$	$+3.9853$	$\lg\sin\varphi_2$	$9.723\ 20460n$	$3/16\,e^2\kappa^2$	$0.0000\ 08076$
$\Delta R(1)$	$0.0000\ 0627$	$\lg\sin\varphi_1$	$9.977\ 43722$	$\times\ \psi$	$+0.19\ 581$
$3/4\,e^2$	$0.005\ 020$	$\varphi_2$	$-31^\circ55'02''.186$	$\times\ 5/16\,e^2\kappa^4$	$0.0000\ 000866$
$\times$		$\varphi_1$	$+71^\circ41'26''.029$	see below
$R(1)$	$+0.009\ 351$	$180^\circ+(\varphi_1+\varphi_2)$	$219^\circ46'23''.843$	$\sin 2\varphi_2$	$-0.8975$
$3/4\,e^2R(1)$	$+0.000\ 047$	$=\Delta\varphi_{1.2}$	$+3.835\ 76731$	$+$
$-\Delta R(1)$	$-0.000\ 006$	$\times$		$\sin 2\varphi_1$	$+0.5965$
$\psi F(2)=R(2)$	$+0.000\ 041$	$\psi$	$+0.195\ 806$	$\varepsilon_2$	$-0.3010$
$R(0)$	$+3.541\ 180$	$\delta$	$+0.751\ 066$	$-\dfrac12\varepsilon_2$	$+0.1505$
$R(1)$	$+0.009\ 351$			$+$
$R(2)$	$+\ \ \ \ \ \ \ 41$			$\Delta\varphi_{1.2}$	$+3.8358$
$\Delta L_{1.2}^{(0)}$	$+3.550\ 572$			$\varepsilon_2$	$+3.9863$
$-\Delta L_{1.2}$	$+3.553\ 090$			$\times$
$\omega^{(0)}$	$-\ \ \ \ \ 2\ 518$			$3/16\,e^2\kappa^2\psi$	$+0.0000\ 01581$
				$\Delta R(1)$	$+0.0000\ 06302$

Continuation of Appendix 2

\(3/4\,e^2\)	\(0.005\ 020\)
\(\times\ R(1)\)	\(+0.009\ 346\)
\(3/4\,e^2R(1)\)	\(+0.0000\ 4692\)
\(-\Delta R(1)\)	\(-0.0000\ 0630\)
\(\psi\mu F(2)=R(2)\)	\(+0.0000\ 4062\)
\(\sin 4\varphi_2\)	\(-0.7916\)
\(\sin 4\varphi_1\)	\(-0.9575\)
\(\sigma_4\)	\(-1.7491\)
\(3/8\,\Delta\varphi_{1.2}\)	\(+1.4384\)
\(-1/4\,\varepsilon_2\)	\(+0.0753\)
\(+1/32\,\varepsilon_4\)	\(-0.0547\)
\(\sigma_4\)	\(+1.4590\)
\(\times\ 5/16\,e^2\kappa'\psi\mu\)	\(+0.0000\ 000170\)
\(\Delta R(2)\)	\(+0.0000\ 000248\)
\(5/6\,e^2\)	\(0.005\ 578\)
\(\times\ R(2)\)	\(+0.0000\ 4062\)
\(5/6\,e^2R(2)\)	\(+0.0000\ 00227\)
\(-\Delta R(2)\)	\(-\quad 025\)
\(\psi\mu F(3)=R(3)\)	\(+0.0000\ 0020\)
\(R(0)\)	\(+3.543\ 70194\)
\(R(1)\)	\(+0.009\ 34614\)
\(R(2)\)	\(+\quad 4062\)
\(R(3)\)	\(+\quad 20\)
\(\Delta L^{(1)}_{1.2}\)	\(+3.553\ 08890\)
\(\Delta L_{1.2}\)	\(+3.553\ 09023\)
\(\omega^{(1)}\)	\(-\quad 133\)

\(\operatorname{tg}\varphi_2=t_2\)	\(-0.62\ 286\)
\(\operatorname{tg}\varphi_1=t_1\)	\(+3\ 02\ 188\)
\((t_1+t_2)\times \dfrac{\nu}{|\nu|}\tau^2\sqrt{1-e^2}\)	\(+2.39\ 902\)
	\(+1.03\ 666\)
\(\dfrac{d\omega^{(1)}}{dp}=\chi^{(1)}\)	\(+2.48\ 70\)
\(-\omega^{(1)}:\chi^{(1)}=\Delta p^{(1)}\)	\(+\quad 53\)
\(p^{(1)}\)	\(+0.196\ 46500\)
\(p^{(2)}=p\)	\(+0.196\ 46553\)

4. Check computation for \(p\)

\(p=\cos B_0\)	\(+0.196\ 46553\)
\(\lg p\)	\(9.293\ 28636\)
\(B_0\)	\(78^\circ40'10''.746\)
\(\lg\sin B_2\)	\(9.714\ 65700n\)
\(\lg\csc B_0=\lg\tau\)	\(0.008\ 54765\)
\(\lg\sin B_1\)	\(9.968\ 88962\)
\(\lg\sin\varphi_2\)	\(9.723\ 20465n\)
\(\lg\sin\varphi_1\)	\(9.977\ 43727\)
\(\varphi_2\)	\(-31^\circ55'02''.201\)
\(\varphi_1\)	\(+71^\circ41'26''.100\)
\(180^\circ+(\varphi_1+\varphi_2)\)	\(219^\circ46'23''.899\)
\(=\Delta\varphi_{1.2}\)	\(+3.835\ 76799\)
\(\times\ \psi\mu\)	\(+0.195\ 807\)
\(\delta\)	\(+0.750\ 070\)
\(\lg\operatorname{tg}\varphi_2\)	\(9.794\ 39299\)
\(\lg p\)	\(9.293\ 28636\)
\(\lg\operatorname{tg}\varphi_1\)	\(0.480\ 30229\)
\(\lg pt_2\)	\(9.087\ 67935\)
\(\lg pt_1\)	\(9.773\ 58865\)
\(pt_2=\vartheta_2\)	\(-6^\circ58'36''.038\)
\(pt_1=\vartheta_1\)	\(+30^\circ41'55''.879\)

\(180^\circ+(\vartheta_1+\vartheta_2)\)	\(203^\circ43'19''.371\)
\(=\Delta\vartheta_{1.2}\)	\(+3.555\ 62291\)
\(\times\ \dfrac{\nu}{|\nu|}\sqrt{1-e^2}\)	\(+0.996\ 64767\)
\(\psi\mu F(0)=R(0)\)	\(+3.543\ 70328\)
\(-\delta\)	\(-0.751\ 070\)
\(\times\ \sigma_0\)	\(+2.792\ 638\)
\(e^2/2\)	\(0.003\ 34671\)
\(\psi\mu F(1)=R(1)\)	\(+0.009\ 34613\)
\(R(0)\)	\(+3.543\ 70328\)
\(R(1)\)	\(+0.009\ 34613\)
\(R(2)\)	\(+\quad 04062\)
\(R(3)\)	\(+\quad 20\)
\(\Delta L^{(2)}_{1.2}\)	\(+3.553\ 09023\)
\(\Delta L_{1.2}\)	\(+3.553\ 09023\)
\(\omega^{(2)}\)	\(0\)

5. Computation of \(A_{1.2}\) and \(A_{2.1}^{\prime}\)

\(\lg\dfrac{\nu}{|\nu|}p\)	\(9.923\ 28636\)
\(\lg V_1\)	\(0.000\ 19522\)
\(-\lg V_0\)	\(-0.000\ 05647\)
\(-\lg\cos B_1\)	\(-9.562\ 70597\)
\(\lg\sin A_{1.2}\)	\(9.730\ 71914\)
\(A_{1.2}\)	\(147^\circ27'27''.800\)
\(\lg\dfrac{\nu}{|\nu|}p\)	\(9.923\ 28636\)
\(\lg V_2\)	\(0.001\ 06741\)
\(-\lg V_0\)	\(-0.000\ 05647\)
\(-\lg\cos B_2\)	\(-9.932\ 03927\)
\(\lg\sin A'_{2.1}\)	\(9.362\ 25803\)
\(A'_{2.1}\)	\(13^\circ18'49''.009\)
\(A_{2.1}\)	\(193^\circ18'49''.009\)

Continuation of Appendix 2.

6. Computation of $s_{1,2}$.		$\sin 2\varphi_2$	$-0.897\ 524$
$\lg(1-e^2)$	$9.997\ 083312$	$+$ $\sin 2\varphi_1$	$+0.596\ 488$
$+$ $\lg V_0$	$0.000\ 056473$	$\sin 4\varphi_2$	$-0.7916$
$\lg \gamma$	$9.997\ 139785$	$+$ $\sin 4\varphi_1$	$-0.9575$
$\lg a$	$6.804\ 701197$	$\sin 6\varphi_2$	$+0.21$
$\lg a\mu$	$6.801\ 840982$	$+$ $\sin 6\varphi_1$	$+0.87$
		$\varepsilon_2$	$-0.301\ 036$
$\lg e^2$	$7.825\ 64818$	$\varepsilon_4$	$-1.7491$
$-$ $\lg \tau^2$	$0.017\ 09530$	$\varepsilon_6$	$+1.08$
$\lg k^2$	$7.808\ 55288$	$C_2\varepsilon_2$	$+0.000\ 732330$
$k^2$	$0.00643\ 50641$	$+C_4\varepsilon_4$	$-\quad 4291$
$k^4$	$0.00004\ 14100$	$C_6\varepsilon_6$	$-\quad 34$
$k^6$	$0.00000\ 02665$	$\Delta s_{1,2}$	$+0.000\ 72802$
$k^8$	$0.00000\ 00017$	$\lg \Delta s_{1,2}$	$6.86214_{-10}$
		$\lg a\mu$	$6.80\ 184$
$1$	$1.000\ 000000$	$\lg \Delta s^{(0)}_{1,2}$	$3.66\ 398$
*) $c_{02}k^2$	$0.004\ 826298$	$\Delta s^{(0)}_{1,2}$	$+4613.0\ \text{m}$
$c_{04}k^4$	$0.000\ 029116$
$c_{06}k^6$	$0.000\ 000\ 187$	$\lg C_0$	$0.002\ 103657$
$c_{08}k^8$	$01$	$\lg \Delta \varphi_{1,2}$	$0.583\ 852331$
$C_0$	$+1.004\ 85560$	$\lg a\mu$	$6.801\ 840982$
		$\lg s^{(0)}_{1,2}$	$7.387\ 796970$
$c_{22}k^2$	$-0.002\ 413149$	$s^{(0)}_{1,2}$	$24\ 422\ 885.1$
$c_{24}k^4$	$-\quad 019411$	$\Delta s^{(0)}_{1,2}$	$4613.0$
$c_{26}k^6$	$-\quad 137$	$s_{1,2}$	$24\ 427\ 488.1$
$C_2$	$-0.002\ 43270$
$c_{44}k^4$	$+0.000\ 002426$
$c_{46}k^6$	$+\quad 27$
$C_4$	$+0.000\ 002453$
$c_{66}k^6=C_6$	$-0.000\ 000031$

*) See Appendix 3.

Appendix 3

Values of the numbers $C_{0.2\lambda}, C_{2u.2\lambda}$

\[ \text{1) }\quad c_{0.2\lambda} = -(-1)^\lambda \frac{ \binom{-3/2}{\lambda} \binom{2\lambda}{\lambda} }{2^{2\lambda}} . \]

\[ \text{2) }\quad c_{2u.2\lambda} = (-1)^{\lambda-u} \cdot \frac{ \binom{-3/2}{\lambda} \binom{2\lambda}{\lambda-u} }{2^{2\lambda}\cdot u} \]

\[ (u,\lambda=1,2,\ldots,n;\ u\leq \lambda) \]

$c_{02}=+0.75$	$c_{22}=-0.375$
$c_{04}=+0.703125$	$c_{24}=-0.468750$
$c_{06}=+0.6836$	$c_{26}=-0.5127$
$c_{08}=+0.63$	$c_{28}=-0.54$
$c_{44}=+0.058594$	$c_{66}=-0.116$
$c_{4.6}=+0.1025$	$c_{68}=-0.03$
$c_{4.8}=+0.13$
$c_{88}=+0.0517$

Appendix 4

Solution of the direct aligned-ray resection (first method)

1. Initial data.

\(B_1\)	\(67^\circ28'52''.763\)
\(L_1\)	\(36^\circ54'39''.412\)
\(=\)	\(0.644\ 21757\)
\(A_{1.3}\)	\(341^\circ13'15''.376\)
\(B_2\)	\(46^\circ12'34''.548\)
\(L_2\)	\(136^\circ07'13''.693\)
\(=\)	\(2.375\ 75039\)
\(A_{2.3}\)	\(53^\circ05'34''.727\)

2. Computation of \(B_3^{(0)},\ L_3^{(0)}\)

Operation	Value
\(90^\circ-B_1=\Theta_1\)	\(22^\circ31'07''\)
\(90^\circ-B_2=\Theta_2\)	\(43^\circ47'25''\)
\(\Delta L_{1.2}\)	\(99^\circ12'34''\)
\(\cos\Theta_2\)	\(+0.72\ 188\)
\(\times\ \cos\Delta L_{1.2}\)	\(-0.16\ 004\)
\(\operatorname{ctg}A_{2.3}\)	\(+0.75\ 101\)
\(\times\ \sin\Delta L_{1.2}\)	\(+0.98\ 711\)
\(\operatorname{ctg}\Theta_1\)	\(+2.41\ 200\)
\(\times\ \sin\Theta_2\)	\(+0.69\ 202\)
\(+\ p_1\)	\(-0.11\ 553\)
\(-\ p_2\)	\(-0.74\ 133\)
\(-\ p_3\)	\(-1.66\ 915\)
\(a\)	\(-2.52\ 601\)

Operation	Value
\(\operatorname{ctg}A_{1.3}\)	\(-2.9410\)
\(\times\ \sin\Theta_2\)	\(+0.69\ 202\)
\(:\ \sin\Theta_1\)	\(+0.38\ 298\)
\(\operatorname{ctg}A_{2.3}\)	\(+0.75\ 101\)
\(\times\ \cos\Delta L_{1.2}\)	\(-0.16\ 004\)
\(\cos\Theta_2\)	\(+0.72\ 188\)
\(\times\ \sin\Delta L_{1.2}\)	\(+0.98\ 711\)
\(+\ p_1\)	\(-5.31\ 420\)
\(-\ p_2\)	\(+0.12\ 019\)
\(-\ p_3\)	\(-0.71\ 257\)
\(b\)	\(-5.90\ 658\)
\(\dfrac{a}{b}=\operatorname{tg}\Delta L_{1.3}^{(0)}\)	\(+0.42\ 766\)
\(\Delta L_{1.3}^{(0)}\)	\(203^\circ09'16''\)
\(+\,L_1\)	\(36^\circ54'39''\)
\(L_3^{(0)}\)	\(240^\circ03'55''\)
\(-\,L_2\)	\(136^\circ07'14''\)
\(\Delta L_{2.3}^{(0)}\)	\(103^\circ56'41''\)
\(\operatorname{ctg}A_{1.3}\)	\(-2.9410\)
\(\times\ \sin\Delta L_{1.3}^{(0)}\)	\(-0.39\ 321\)
\(\cos\Theta_1\)	\(+0.92\ 376\)
\(\times\ \cos\Delta L_{1.3}^{(0)}\)	\(-0.91\ 945\)
\(r_1\)	\(+1.15\ 643\)
\(r_2\)	\(-0.81\ 935\)
\(r_1+r_2\)	\(+0.30\ 708\)
\(:\ \sin\Theta_1\)	\(+0.38\ 298\)
\(\operatorname{tg}B_3^{(0)}\)	\(+0.80\ 182\)

Operation	Value
\(\operatorname{ctg}A_{2.3}\)	\(+0.75\ 101\)
\(\times\ \sin\Delta L_{2.3}\)	\(+0.97\ 053\)
\(\cos\Theta_2\)	\(+0.72\ 188\)
\(\times\ \cos\Delta L_{2.3}\)	\(-0.24\ 099\)
\(ж_1\)	\(+0.72\ 888\)
\(ж_2\)	\(-0.17\ 397\)
\(ж_1+ж_2\)	\(+0.55\ 491\)
\(:\ \sin\Theta_2\)	\(+0.69\ 202\)
\(\operatorname{tg}B_3^{(0)}\)	\(+0.80\ 187\)
\(B_3^{(0)}\simeq B_3\)	\(38^\circ43'27''\)

Continuation of Appendix 4

3. Calculation of the quantities \(\nu_{i3},\ k_{i3}^{2},\ \tau_{i3},\ p_{i3},\ \nu_{i3}p_{i3},\ \varphi_i^3,\ \vartheta_i^3;\ (i=1,2)\)

Quantity	\(i=1\)	\(i=2\)
\(\lg\sin A_{i3}\)	\(9.507\ 74760\ n\)	\(9.902\ 87881\)
\(\lg\cos B_i\)	\(9.583\ 18128\)	\(9.840\ 12007\)
\(-\lg V_i\)	\(-0.000\ 21452\)	\(-0.000\ 69961\)
\(-\lg\sqrt{1-e^2}\)	\(-9.998\ 54166\)	\(-9.998\ 54166\)
\(\lg\nu_{i3}\)	\(9.092\ 17270\ n\)	\(9.743\ 75761\)
\(\lg\nu^2\)	\(8.184\ 34540\)	\(9.487\ 51522\)
\(\lg e^2\)	\(7.825\ 64818\)	\(7.82564818\)
\(\lg e^2\nu^2\)	\(6.009\ 99358\)	\(7.313\ 16340\)
\(e^2\)	\(0.006\ 693422\)	\(0.006\ 693422\)
\(e^2\nu^2\)	\(0.000\ 102328\)	\(0.002\ 056664\)
\(\nu^2\)	\(0.015\ 28781\)	\(0.307\ 26650\)
\(e^2-e^2\nu^2\)	\(0.006\ 591094\)	\(0.004\ 636758\)
\(1-e^2\nu^2\)	\(0.999\ 89767\)	\(0.997\ 94334\)
\(1-\nu^2\)	\(0.984\ 71219\)	\(0.692\ 73350\)
\(\lg(\nu^2-e^2\nu^2)\)	\(7.818\ 95750\)	\(7.666\ 21443\)
\(\lg(1-e^2\nu^2)\)	\(9.999\ 95556\)	\(9.999\ 10588\)
\(\lg(1-\nu^2)\)	\(9.993\ 30932\)	\(9.840\ 56619\)
\(\lg k^2\)	\(7.819\ 00194\)	\(7.667\ 10855\)
\(\lg\tau^2\)	\(0.006\ 64624\)	\(0.158\ 53969\)
\(k^2\)	\(0.006\ 591768\)	\(0.004\ 646314\)
\(k^4\)	\(0.000\ 043451\)	\(0.000\ 021588\)
\(k^6\)	\(0.000\ 000286\)	\(0.000\ 000100\)
\(k^8\)	\(0.000\ 0000019\)	\(0.000\ 0000005\)
\(\lg\dfrac{1}{\tau}=\lg\sin B_0\)	\(9.996\ 67688\)	\(9.920\ 73016\)
\(B_0\)	\(82^\circ55'16''.038\)	\(56^\circ25'30''.272\)
\(\lg\cos B_0=\lg p_0\)	\(9.090\ 73637\)	\(9.742\ 74631\)
\(\dfrac{\nu}{|\nu|}=\omega\)	\(-1\)	\(+1\)

Quantity	\(i=1\)	\(i=2\)
\(p\)	\(0.123\ 23565\)	\(0.553\ 02697\)
\(\omega\sqrt{1-e^2}\)	\(-0.996\ 64767\)	\(+0.996\ 64767\)
\(\nu\)	\(-0.122\ 8226\)	\(+0.551\ 1730\)
\(\lg\sin B_3\)	\(9.965\ 55668\)	\(9.858\ 46269\)
\(+\ \lg\tau_{i3}\)	\(0.003\ 32312\)	\(0.079\ 26984\)
\(\lg\sin\varphi_i^3\)	\(9.968\ 87980\)	\(9.937\ 73253\)
\(\varphi_i^3\)	\(68^\circ34'03''.855\)	\(60^\circ02'46''.239\)
\(\lg\tg\varphi_i^3\)	\(0.406\ 11008\)	\(0.239\ 36934\)
\(+\ \lg p_{i3}\)	\(9.090\ 73637\)	\(9.742\ 74631\)
\(\lg\tg\vartheta_i^3\)	\(9.496\ 84645\)	\(9.982\ 11565\)
\(\vartheta_i^3\)	\(17^\circ25'44''.940\)	\(43^\circ49'14''.183\)

4. First approximation for \(B_3, L_3\)

Quantity	\(i=1\)	\(i=2\)
\(B_3^{(0)}\simeq B_1\)	\(38^\circ43'27''\)
\(\lg\sin B_3\)	\(9.796\ 277\)	\(9.796\ 277\)
\(\lg\tau_{i3}\)	\(0.003\ 323\)	\(0.079\ 270\)
\(\lg\sin\varphi_3^i\)	\(9.799\ 600\)	\(9.875\ 547\)
\(\varphi_3^i\)	\(39^\circ04'40''.4\)	\(48^\circ39'47''.2\)
\(\varphi_i^3\)	\(68^\circ34'03''.9\)	\(60^\circ02'46''.2\)
\(180^\circ-(\varphi_i^3+\varphi_3^i)\)	\(+72^\circ21'15''.7\)	\(+71^\circ17'26''.6\)
\(\pi-(\varphi_i^3+\varphi_3^i)\)	\(+1.262\ 822\)	\(+1.244\ 258\)
\(\times\ \nu\)	\(-0.122\ 82\)	\(+0.551\ 17\)
\(\delta\)	\(-0.15\ 510\)	\(+0.68\ 580\)
\(\lg\tg\varphi_3^i\)	\(9.909\ 576\)	\(0.055\ 684\)
\(+\ \lg p_{i3}\)	\(9.090\ 736\)	\(9.742\ 746\)
\(\lg\tg\vartheta_3^i\)	\(9.000\ 312\)	\(9.798\ 430\)
\(\vartheta_3^i\)	\(5^\circ42'52''.8\)	\(32^\circ09'24''.5\)
\(\vartheta_i^3\)	\(17^\circ25'44''.9\)	\(43^\circ49'14''.2\)
\(180^\circ-(\vartheta_i^3+\vartheta_3^i)\)	\(156^\circ51'22''.3\)	\(104^\circ01'21''.3\)

Continuation of Appendix 4

$\bar z-\left(\vartheta_i^3+\vartheta_3^i\right)$	$2.737\ 6571$	$1.815\ 5367$
$\times\ \omega\sqrt{1-e^2}$	$-0.996\ 64767$	$+0.996\ 64767$
$\nu\omega F(0)=R(0)$	$-2.728\ 4796$	$+1.809\ 4504$
$-\bar\delta$	$+0.155\ 10$	$-0.685\ 80$
$\bar\delta_0$	$-2.57\ 338$	$+1.123\ 65$
$\times\ e^2/2$	$0.003\ 3467$	$0.003\ 3467$
$\nu\omega F(1)=R(1)$	$-0.008\ 6123$	$+0.003\ 7605$
$3/16\,e^2$	$0.001\ 2550$	$0.001\ 2550$
$3/16\,e^2k^2$	$0.000\ 00827$	$0.000\ 00583$
$-\begin{bmatrix}\sin 2\varphi_3^i\\+\sin 2\varphi_i^3\end{bmatrix}$	$+0.979$ $+0.680$	$+0.992$ $+0.865$
$\varepsilon_2$	$-1.659$	$-1.857$
$-\dfrac12\,\varepsilon_2$	$+0.830$	$+0.928$
$+\ \dfrac z2-\left(\zeta_i^3+\varphi_3^i\right)$	$+1.263$	$+1.244$
$\varepsilon_2$	$+2.093$	$+2.172$
$\times\ 3/16\,e^2k^2\nu\mu$	$-0.000\ 00102$	$+0.000\ 00321$
$\Delta R(1)$	$-0.000\ 0021$	$+0.000\ 0070$
$3/4\,e^2$	$+0.005\ 020$	$+0.005\ 020$
$\times\ R(1)$	$-0.008\ 612$	$+0.003\ 761$
$3/4\,e^2R(1)$	$-0.000\ 0432$	$+0.000\ 0189$
$-\Delta R(1)$	$+\phantom{0}21$	$-\phantom{0}70$
$\nu\omega F(2)=R(2)$	$-0.000\ 0411$	$+0.000\ 0119$
$R(1)$	$-0.008\ 6123$	$+0.003\ 7605$
$R(0)$	$-2.728\ 4796$	$+1.809\ 4504$
$\nabla L_{1.3}^{(0)},\ \Delta L_{2.3}^{(0)}$	$-2.737\ 1330$	$+1.813\ 2228$
$L_1-2\pi,\ L_2$	$+6.927\ 4029$	$+2.375\ 7504$
$L_3^{(0)}$	$4.190\ 2699$	$4.188\ 9732$
$w_L^{(0)}$	$0.001\ 2967$	$+267''.46$

$\lg\sqrt{1-e^2}$	$9.99\ 854$	$9.99\ 854$
$\lg\nu_{13}$	$9.09\ 217\ n$	$9.74\ 376$
$\lg V_3$	$0.00\ 089$	$0.00\ 089$
$-\lg\cos B_3$	$-9.89\ 219$	$-9.89\ 219$
$\lg\sin A'_{3i}$	$9.19\ 941\ n$	$9.85\ 100$
$A'_{3i}$	$189^\circ06'24''$	$134^\circ48'00''$
$\lg\tg A'_{3i}$	$9.20\ 491$	$0.00\ 303\ n$
$-\lg V_3^2$	$-0.00\ 178$	$-0.00\ 178$
$-\lg\cos B_3$	$-9.89\ 219$	$-9.89\ 219$
$\lg\dfrac{\partial L_3^i}{\partial B_3}=\lg a_{13}$	$9.31\ 094$	$0.10\ 906\ n$
$a_{i3}$	$+0.20461$	$-1.28\ 541$
$b=a_{1.3}-a_{2.3}$	$+1.4901$

\[ \left(\frac{\partial L_3^1}{\partial B_3}-\frac{\partial L_3^2}{\partial B_3}\right)\delta B_3^{(0)}+w_L^{(0)}=0 \]

\[ +1.49001\,\delta B_3^{(0)}+267''.46=0 \]

$\delta B_3^{(0)}$	$-0.000\ 87024$	$-02'59''.50$
$B_3^{(0)}$		$38^\circ43'27''.0$
$B_3^{(1)}$		$38^\circ40'27''.5$
$a_{i3}$	$+0.20461$	$-1.28\ 541$
$\times\ \delta B_3^{(0)}$	$-0.000\ 87024$	$-0.000\ 87024$
$\delta L_i^{(0)}$	$-0.000\ 1781$	$+0.001\ 1186$
$+\,L_i^{(0)}$	$4.190\ 2699$	$4.188\ 9732$
$L_3^{(1)}\simeq L_3$	$4.190\ 0918$	$4.190\ 0918$

Continuation of Appendix 4

5. Second approximation for \(B_3,L_3\)

Quantity	Value	Value
\(B_3^{(1)} \approx B_3\)	\(38^\circ40'27''.5\)
\(\lg \sin B_3\)	\(9.795\ 80539\)	\(9.795\ 80539\)
\(\lg \tau_{i3}\)	\(0.003\ 32312\)	\(0.079\ 26984\)
\(\lg \sin \varphi_3^i\)	\(9.799\ 12851\)	\(9.875\ 07523\)
\(\varphi_3^i\)	\(39^\circ01'38''.777\)	\(48^\circ35'32''.882\)
\(\varphi_i^3\)	\(68^\circ34'03''.855\)	\(60^\circ02'46''.239\)
\(180^\circ-(\varphi_i^3+\varphi_3^i)\)	\(72^\circ24'17''.368\)	\(71^\circ21'40''.879\)
\(\pi-(\varphi_i^3+\varphi_3^i)\)	\(1.263\ 70258\)	\(1.24549061\)
\(\times\ \psi\)	\(-0.122\ 8226\)	\(+0.551\ 1730\)
\(\delta\)	\(-0.155\ 2113\)	\(+0.686\ 4808\)
\(\lg \tg \varphi_3^i\)	\(9.908\ 79441\)	\(0.054\ 60416\)
\(\lg \rho_{i3}\)	\(9.090\ 73637\)	\(9.742\ 74631\)
\(\lg \tg \vartheta_3^i\)	\(8.999\ 53078\)	\(9.797\ 35047\)
\(\vartheta_3^i\)	\(5^\circ42'16''.082\)	\(32^\circ05'33''.491\)
\(\vartheta_i^3\)	\(17^\circ25'44''.940\)	\(43^\circ49'14''.183\)
\(180^\circ-(\vartheta_i^3+\vartheta_3^i)\)	\(156^\circ51'58''.968\)	\(104^\circ05'12''.326\)
\(\pi-(\vartheta_i^3+\vartheta_3^i)\)	\(2.737\ 834817\)	\(1.816\ 656621\)
\(\times\ \omega\sqrt{1-e^2}\)	\(-0.996\ 647670\)	\(+0.996\ 647670\)
\(\psi F(0)=R(0)\)	\(-2.728\ 656691\)	\(+1.810\ 566589\)
\(-\delta_0\)	\(-0.155\ 2113\)	\(-0.686\ 4808\)
\(\delta_0\)	\(-2.573\ 4454\)	\(+1.124\ 0856\)
\(\times\ e_2^2\)	\(0.003\ 346711\)	\(0.003\ 346711\)
\(\psi F(1)=R(1)\)	\(-0.008\ 612578\)	\(+0.003\ 761989\)
\(3/16\,e^2,\ 5/16\,e^2\)	\(0.000\ 12550\)	\(0.000\ 20917\)
\(3/16\,e^2 k^2\)	\(0.000\ 008273\)	\(0.000\ 005831\)
\(\dfrac{5}{16}e^2 k^4\)	\(0.000\ 000091\)	\(0.000\ 000045\)

Quantity	Value	Value
\(-\left[\sin 2\varphi_3^i+\sin 2\varphi_i^3\right]\)	\(+0.9783\)	\(+0.9922\)
\(-\left[\sin 2\varphi_3^i+\sin 2\varphi_i^3\right]\)	\(+0.6803\)	\(+0.8652\)
\(-\left[\sin 4\varphi_3^i+\sin 4\varphi_i^3\right]\)	\(+0.405\)	\(-0.248\)
\(-\left[\sin 4\varphi_3^i+\sin 4\varphi_i^3\right]\)	\(-0.997\)	\(-0.868\)
\(\varepsilon_2\)	\(-1.6586\)	\(-1.8574\)
\(\varepsilon_4\)	\(+0.592\)	\(+1116\)
\(-\dfrac{1}{2}\varepsilon_2\)	\(+0.8293\)	\(+0.9287\)
\(\pi-(\varphi_i^3+\varphi_3^i)\)	\(+1.2637\)	\(+1.2455\)
\(\varepsilon_0\)	\(+2.0930\)	\(+2.1742\)
\(\times\ 3/16\,e^2k^2\psi\)	\(-0.000\ 001016\)	\(+0.000\ 003214\)
\(\Delta R(1)\)	\(0.000\ 002126\)	\(+0.000\ 006988\)
\(3/4\,e^2\)	\(0.005\ 0200\)	\(0.005\ 0200\)
\(R(1)\)	\(-0.008\ 61258\)	\(0.003\ 76199\)
\(3/4\,e^2R(1)\)	\(-0.000\ 043235\)	\(+0.000\ 018885\)
\(-\Delta R(1)\)	\(+0.000\ 002126\)	\(-0.000\ 006988\)
\(\psi F(2)=R(2)\)	\(-0.000\ 041109\)	\(+0.000\ 011897\)
\(\dfrac{3}{8}\left[\pi-(\varphi_i^3+\varphi_3^i)\right]\)	\(+0.474\)	\(+0.467\)
\(-\dfrac{1}{4}\varepsilon_2\)	\(+0415\)	\(+0.464\)
\(+\dfrac{1}{32}\varepsilon_4\)	\(+0.019\)	\(+0.035\)
\(\varepsilon_4\)	\(+0.908\)	\(+0.966\)
\(\times\ 5/16\,e^2k^4\psi\)	\(-0.000\ 000011\)	\(+0.000\ 000025\)
\(\Delta R(2)\)	\(-0.000\ 000010\)	\(+0.000\ 000024\)
\(\times\ 5/6\,e^2\)	\(0.005\ 578\)	\(0.005\ 578\)
\(R(2)\)	\(-0.000\ 04111\)	\(+0.000\ 01190\)
\(5/6\,e^2R(2)\)	\(-0.000\ 000229\)	\(+0.000\ 000066\)
\(-\Delta R(2)\)	\(+10\)	\(-24\)
\(\psi F(3)=R(3)\)	\(-0.000\ 000219\)	\(+0.000\ 000042\)

Continuation of Appendix 4

$R(0)$	$-2.728\ 656691$	$+1.810\ 566389$
$R(1)$	$-0.008\ 612578$	$+0.003\ 761989$
$R(2)$	$-\phantom{0}41109$	$+\phantom{0}11897$
$R(3)$	$-\phantom{0}219$	$+\phantom{0}42$
$\nabla L_{1.3},\ \nabla L_{2.3}$	$-2.737\ 310597$	$+1.814340517$
$L_1+2\pi,\ L_2$	$+6.927\ 402876$	$+2.375750387$
$L^{(1)i}_3$	$+4.190\ 092279$	$+4.190090904$
$w_L^{(1)}$	$+0.000\ 001375$	$+0''.2836$
$\lg \nu_{i3}V_3\sqrt{1-e^2}$	$9.09\ 060\ n$	$9.74\ 319$
$-\lg\cos B_3$	$-9.89\ 239$	$-9.89\ 239$
$\lg\sin A'_{3i}$	$9.19\ 921\ n$	$9.85\ 080$
$A'_{3i}$	$189^\circ06'09''$	$134^\circ49'32''$
$\lg\tg A'_{3i}$	$9.20\ 471$	$0.00\ 264\ n$
$-\lg V_3^2$	$-0.00\ 178$	$-0.00\ 178$
$-\lg\cos B_3$	$-9.89\ 239$	$-9.89\ 239$
$\lg\dfrac{\partial L^i_3}{\partial B_3}=\lg a_{i3}$	$9.31\ 054$	$0.10\ 847\ n$
$a_{i3}$	$+0.20443$	$-1.2837$
$b=a_{1.3}-a_{2.3}$	$+1.4881$
$1.4881\,\delta B^{(1)}_3+0''.2836=0$
$\delta B^{(1)}_3$	$-0.000\ 000924$	$-0''.1905$
$B^{(1)}_3$		$38^\circ40'27''.500$
$B^{(2)}_3$		$38^\circ40'27''.310$
$a_{i3}$	$+0.20443$	$-1.2837$
$\times\ \delta B^{(1)}_3$	$-0.000\ 000924$	$-0.000000924$
$\delta L^{(1)i}_3$	$-0.000\ 000189$	$+0.000001186$
$L^{(1)i}_3$	$4.190\ 09\ 2279$	$4.190\ 090\ 904$
$L^{(2)i}_3$	$4.190\ 092\ 090$	$4.190\ 092\ 090$
$=$		$240^\circ04'28''.533$

6. Check computation for $B_3,\ L_3$

$B_3$	$38^\circ40'27''.310$
$\lg\sin B_3$	$9.795\ 80489$	$9.795\ 80489$
$\lg\tau_{i3}$	$0.003\ 32312$	$0.079\ 26984$
$\lg\sin\varphi^i_3$	$9.799\ 12801$	$9.875\ 07473$
$\varphi^i_3$	$39^\circ01'38''.585$	$48^\circ35'32''.613$
$\varphi^3_i$	$68^\circ34'03''.855$	$60^\circ02'46''.239$
$180^\circ-(\varphi^3_i+\varphi^i_3)$	$72^\circ24'17''.560$	$71^\circ21'41''.148$
$\pi-(\varphi^3_i+\varphi^i_3)$	$1.263\ 703511$	$1.245\ 491913$
$\times\ \varkappa\mu$	$-0.122\ 8226$	$+0.551\ 1730$
$\delta$	$-0.155\ 2114$	$+0.686\ 4815$
$\lg\tg\vartheta^i_3$	$9.908\ 79359$	$0.054\ 60302$
$+\ \lg p_{i3}$	$9.090\ 73637$	$9.742\ 74631$
$\lg\tg\vartheta^i_3$	$8.999\ 52996$	$9.797\ 34933$
$\vartheta^i_3$	$5^\circ42'16''.044$	$32^\circ05'33''.248$
$\vartheta^3_i$	$17^\circ25'44''.946$	$43^\circ49'14''.183$
$180^\circ-(\vartheta^3_i+\vartheta^i_3)$	$156^\circ51'59''.010$	$104^\circ05'12''.569$
$\pi-(\vartheta^3_i+\vartheta^i_3)$	$2.737\ 835020$	$1.816\ 657\ 800$
$\times\ \omega\sqrt{1-e^2}$	$-0.996\ 647670$	$+0.996\ 647670$
$\psi F(0)=R(0)$	$-2.728\ 656894$	$+1.810\ 567\ 764$
$-\delta$	$+0.155\ 2114$	$-0.686\ 4815$
$\xi_0$	$-2.573\ 4455$	$+1.1240863$
$\times\ e^2/2$	$0.003\ 346711$	$0.003\ 346711$
$\psi F(1)=R(1)$	$-0.008\ 612576$	$+0.003\ 761991$
$R(0)$	$-2.728\ 656894$	$+1.810\ 567764$
$R(1)$	$-0.008\ 612576$	$+0.003\ 761\ 991$
$R(2)$	$-\phantom{0}41109$	$+\phantom{0}11897$
$R(3)$	$-\phantom{0}219$	$+\phantom{0}42$
$\Delta L^{(2)}_1,\ \Delta L^{(2)}_2$	$-2.737\ 310798$	$+1.814\ 341694$
$L_1+2\pi,\ L_2$	$+6.927\ 402876$	$+2.375\ 750387$
$L^{(2)i}_3$	$4.190\ 092078$	$4.190\ 092\ 081$
$w_L^{(2)}$	$+0.000\ 000003$	$+0''.0006$
$L_3$	$4.190\ 092\ 080$	$240^\circ04'28''.531$

Appendix 5

Solution of the direct conformal-ray intersection problem (second method)

1. Initial data

$B_1$	$67^\circ28'52''.763$	$\alpha_{1.2}$	$44^\circ58'14''$	$\lg\cos\delta\gamma_{1.2}$	$9.99\ 992$
$L_1$	$34^\circ54'39''.412$	$\delta\alpha_{1.2}$	$16^\circ01'22''$	$-\lg\cos\gamma_{1.2}$	$9.19\ 877$
	$0.644\ 217568$	$\alpha_1$	$60^\circ59'36''$	$\lg\operatorname{ctg}\delta\zeta_{1.2}$	$9.68\ 198$
$A_{1.3}$	$341^\circ13'15''.376$	$\alpha_2$	$28^\circ56'52''$	$\lg\operatorname{tg}\zeta_{1.3}$	$0.48\ 313$
$B_2$	$46^\circ12'34''.548$	$\alpha_2$	$28^\circ56'52''$	$\zeta_{i3}$	$71^\circ48'06''$
$L_2$	$136^\circ07'13''.693$	$A_{2.3}$	$53^\circ05'35''$	$\delta\zeta_{i3}$	$0^\circ33'08''$
	$2.375\ 750387$	$\alpha_1$	$60^\circ59'36''$	$\zeta_{1.3}$	$72^\circ21'14''$
$A_{2.3}$	$53^\circ05'34''.727$	$-\bar A_{1.3}$	$341^\circ13'15''$		$1.262\ 8136$
		$\gamma_1$	$82^\circ02'27''$	$\zeta_{2.3}$	$71^\circ14'58''$
		$\gamma_2$	$79^\circ46'21''$		$1.243\ 5374$
		$\dfrac12(\gamma_1+\gamma_2)=\gamma_{1.2}$	$80^\circ54'24''$
		$\dfrac12(\gamma_1-\gamma_2)=\gamma_{1-2}$	$1^\circ08'03''$

2. Computation of $\zeta_{1.3},\ \zeta_{2.3}$

$90^\circ-B_2=\Theta_2$	$43^\circ47'25''$	$\lg\sin\Theta_1$	$9.58\ 318$	$\lg\sin\gamma_1$	$9.99\ 305$
$90^\circ-B_1=\Theta_1$	$22^\circ31'07''$	$\lg\sin\alpha_2$	$9.68\ 486$	$\lg\sin\zeta_{2.3}$	$9.97\ 632$
$\dfrac12(\Theta_2+\Theta_1)=\Theta_{1.2}$	$33^\circ09'16''$	$\lg\sin\Theta_2$	$9.84\ 012$	$\lg\sin\gamma_2$	$9.99\ 580$
$\dfrac12(\Theta_2-\Theta_1)=\delta\Theta_{1.2}$	$10^\circ38'09''$	$\lg\sin\alpha_1$	$9.94\ 179$	$\lg\sin\zeta_{1.3}$	$9.97\ 907$
$\Delta L_{1.2}$	$99^\circ12'34''$	$\lg m_1$	$9.89\ 832$	$\lg n_1$	$0.01\ 673$
$\dfrac12\Delta L_{1.2}=\delta L_{1.2}$	$49^\circ36'17''$	$\lg\sin\Delta L_{1.2}$	$9.99\ 437$	$\lg\sin\zeta_{1.2}$	$9.89\ 269$
		$\lg m_2$	$9.89\ 833$	$\lg n_2$	$0.01\ 673$
$\lg\sin\delta\Theta_{1.2}$	$9.26\ 615$	$\lg\sin\zeta_{1.2}$	$9.89\ 269$	$\lg\sin\gamma_{i3}$	$9.90\ 942$
$-\lg\sin\Theta_{1.2}$	$-9.73\ 788$	$\zeta_{1.2}$	$51^\circ21'30''$	$\gamma_3$	$54^\circ16'00''$
$\lg\operatorname{ctg}\delta L_{1.2}$	$9.92\ 989$	$\dfrac12\zeta_{1.2}=\delta\zeta_{1.2}$	$25^\circ40'45''$	$\gamma_2$	$82^\circ02'27''$
$\lg\operatorname{tg}\delta\alpha_{1.2}$	$9.45\ 816$			$\gamma_1$	$79^\circ46'21''$
				$\displaystyle\sum_{i=1}^{3}\gamma_i$	$216^\circ04'48''$
$\lg\cos\delta\Theta_{1.2}$	$9.99\ 248$			$\varepsilon$	$36^\circ04'48''$
$-\lg\cos\Theta_{1.2}$	$-9.92\ 283$
$\lg\operatorname{ctg}\delta L_{1.2}$	$9.92\ 989$
$\lg\operatorname{tg}\alpha_{1.2}$	$9.99\ 954$
		$\lg\sin\delta\gamma_{1.2}$	$8.29\ 653$
		$-\lg\sin\gamma_{1.2}$	$9.99\ 451$
		$\lg\operatorname{tg}\delta\zeta_{1.2}$	$9.68\ 198$
		$\lg\operatorname{tg}\delta\zeta_{i3}$	$7.98\ 400$

Continuation of Appendix 5

3. Calculation of the quantities \(\nu_{i3},\ k_{i3}^{3},\ \tau_{i3},\ p_{i3}\)
\(C_0^i,\ D_{2u}^i,\ Q_{i3},\ \varphi_i^3,\ \vartheta_i^3,\ \nu_{i3}\mu_{i3}\).

Quantity	Value	Value	Quantity	Value	Value
\(\lg \sin A_{i3}\)	\(9.507\ 74760\,n\)	\(9.902\ 87881\)	\(1\)	\(1.000\ 000000\)	\(1.000\ 000000\)
\(\lg \cos B_i\)	\(9.583\ 18128\)	\(9.840\ 12007\)	\({}^{*})\ c_{02}k^2\)	\(0.004\ 943826\)	\(0.003\ 484736\)
\(-\lg V_i\)	\(-9.998\ 54166\)	\(-9.998\ 54166\)	\(c_{04}k^4\)	\(0.000\ 030551\)	\(0.000\ 015179\)
\(-\lg \sqrt{1-e^2}\)	\(-0.000\ 21452\)	\(-0.000\ 69961\)	\(c_{06}k^6\)	\(196\)	\(68\)
\(\lg \nu_{i3}\)	\(9.092\ 17270\,n\)	\(9.743\ 75761\)	\(c_{08}k^8\)	\(1\)	\(0\)
\(\lg \nu_{i3}^{2}\)	\(8.184\ 34540\)	\(9.487\ 51522\)	\(C_0^i\)	\(+1.004\ 974574\)	\(+1.003\ 499983\)
\(\lg e^2\)	\(7.825\ 64818\)	\(7.825\ 64818\)
\(\lg e^2\nu_{i3}^{2}\)	\(6.009\ 99358\)	\(7.313\ 16340\)	\(c_{22}k^2\)	\(-0.002\ 471913\)	\(-0.001\ 742368\)
			\(c_{24}k^4\)	\(-0.000\ 020368\)	\(-0.000\ 010119\)
\(e^2\)	\(0.006\ 693422\)	\(0.006\ 693422\)	\(c_{26}k^6\)	\(-147\)	\(-51\)
\(e^2\nu^2\)	\(0.000\ 102328\)	\(0.002\ 056664\)	\(c_{28}k^8\)	\(-1\)	\(-0\)
\(\nu^2\)	\(0.015\ 28781\)	\(0.307\ 26650\)	\(C_2^i\)	\(-0.002\ 492429\)	\(-0.001\ 752538\)
\(e^2-e^2\nu^2\)	\(0.006\ 591094\)	\(0.004\ 636758\)
\(1-e^2\nu^2\)	\(0.999\ 89767\)	\(0.997\ 94334\)	\(c_{44}k^4\)	\(+0.000\ 002546\)	\(-0.000\ 001265\)
\(\lg(e^2-e^2\nu^2)\)	\(7.818\ 95750\)	\(7.666\ 21443\)	\(c_{46}k^6\)	\(+29\)	\(+10\)
\(\lg(1-e^2\nu^2)\)	\(9.999\ 95556\)	\(9.999\ 10588\)	\(c_{48}k^8\)	\(+0\)	\(+0\)
\(\lg(1-\nu^2)\)	\(9.993\ 30932\)	\(9.840\ 56619\)	\(C_4^i\)	\(+0.000\ 002575\)	\(+0.000\ 001275\)
\(\lg k^2\)	\(7.819\ 00194\)	\(7.667\ 10855\)
\(\lg z^2\)	\(0.006\ 64624\)	\(0.158\ 53969\)	\(c_{66}k^6\)	\(-0.000\ 000033\)	\(-0.000\ 000012\)
\(k^2\)	\(0.006\ 591768\)	\(0.004\ 646314\)	\(c_{68}k^8\)	\(-0\)	\(0\)
\(k^4\)	\(0.000\ 043451\)	\(0.000\ 021588\)	\(C_6^i\)	\(-0.000\ 000033\)	\(-0.000\ 000012\)
\(k^6\)	\(0.000\ 000286\)	\(0.000\ 000100\)
\(k^8\)	\(0.000\ 000019\)	\(0.000\ 000005\)	\(C_2^i:C_0^i=D_2^i\)	\(-0.002\ 480092\)	\(-0.001\ 746426\)
\(\lg \dfrac{1}{z}=\lg \sin B_0\)	\(9.996\ 67688\)	\(9.920\ 73016\)	\(C_4^i:C_0^i=D_4^i\)	\(+0.000\ 002562\)	\(+0.000\ 001271\)
\(B_0\)	\(82^\circ55'16''.038\)	\(56^\circ25'30''.272\)	\(C_6^i:C_0^i=D_6^i\)	\(-0.000\ 000033\)	\(-0.000\ 000012\)
\(\lg \cos B_0=\lg p\)	\(9.090\ 73637\)	\(9.742\ 74631\)
\(p\)	\(0.123\ 23565\)	\(0.553\ 02697\)	\(\lg(1-e^2)a\)	\(6.8017\ 84509\)	\(6.8017\ 84509\)
			\(\lg V_0^i\)	\(0.0000\ 22221\)	\(0.0004\ 47059\)
			\(\lg C_0^i\)	\(0.0021\ 55074\)	\(0.0015\ 17369\)
			\(\lg Q_{i3}\)	\(6.8039\ 61804\)	\(6.8037\ 48937\)
			\(\dfrac{\nu}{|\nu|}=\omega\)	\(-1\)	\(+1\)

\({}^{*})\) See Appendix 3.

Continuation of Appendix 5

\(p_{i3}\)	0.123 23565	0.553 02697
\({}^{\omega}\sqrt{1-e^2}\)	\(-0.996 64767\)	\(+0.996 64767\)
\(\gamma_{i3}\mu_{i3}\)	\(-0.122 8226\)	\(+0.551 1730\)
\(\lg \sin B_i\)	9.965 55668	9.858 46269
\(\lg \tau_{i3}\)	0.003 32312	0.079 26984
\(\lg \sin \varphi_i^3\)	9.968 87980	9.937 73253
\(\varphi_i^3\)	\(68^\circ34'03''.855\)	\(60^\circ02'46''.239\)
\(+\ \lg \operatorname{tg}\varphi_i^3\)	0.406 11008	0.239 36934
\(\lg p_{i3}\)	9.090 73637	9.742 74631
\(\lg \operatorname{tg}\vartheta_i^3\)	9.496 84645	9.982 11565
\(\vartheta_i^3\)	\(17^\circ25'44''.946\)	\(43^\circ49'14''.183\)

4. First approximation for \(s_{i3}, B_3, L_3\).

a) Computation of \(w_B^{(0)}\)

\(B_{ik}\)—from the drawing	\(75^\circ, 72^\circ, 62^\circ, 46^\circ\)	\(49^\circ, 53^\circ, 50^\circ, 43^\circ\)
\(R_{i1}, R_{i2}\)	6397, 6397	6381, 6384
\(R_{i3}, R_{i4}\)	6390, 6378	6383, 6377
\(\overline{R}_{i3}=\dfrac14\sum_k R_{ik}\)	6390500 m	6381 300 m
\(\times\ \sigma_{i3}\)	1.262 8136	1.243 5374
\(s_{i3}^{(0)}\)	80700 010 m	7935 385 m
\(\lg s_{i3}^{(0)}\)	6.906 87407	6.899 56800
\(-\ \lg Q_{i3}\)	6.803 96180	6.803 74894
\(\lg \widetilde{\tau}_{i3}^{(0)}\)	0.102 91227	0.095 81906
\(-\ \widetilde{\tau}_{i3}^{(0)}\)	\(-1.267 3958\)	\(-1.246 8639\)
\(+\ \widetilde{\tau}-\varphi_i^3\)	\(+1.944 8599\)	\(+2.093 5892\)
\(\varphi_3^{l(0)}\approx \varphi_3^i\)	\(+0.677 4641\)	\(+0.846 7253\)
\(=\)	\(+38^\circ48'57''\)	\(+48^\circ30'50''\)
\(-\left[\begin{array}{c}\sin 2\varphi_3^i\\+\\\sin 2\varphi_i^3\end{array}\right.\)	\(+0.97 679\) \(+0.68 026\)	\(+0.99 249\) \(+0.86 522\)

\(-\left[\begin{array}{c}\sin 4\varphi_3^i\\+\\\sin 4\varphi_i^3\end{array}\right.\)	\(+0.418\) \(-0.997\)	\(-0.243\) \(-0.868\)
\(\varepsilon_2^{(00)}\)	\(-1.65705\)	\(-1.85711\)
\(\varepsilon_4^{(00)}\)	\(+0.579\)	\(+1.111\)
\(+\ D_2\varepsilon_2^{(00)}\)	\(+0.004 1096\)	\(+0.003 2444\)
\(D_4\varepsilon_4^{(00)}\)	\(+\phantom{0}15\)	\(+\phantom{0}14\)
\(\delta\varphi_3^{(00)}\)	\(+0.004 1111\)	\(+0.003 2458\)
\(\cos 2\varphi_3^i\)	\(+0.21 420\)	\(-0.12 235\)
\(\cos 4\varphi_3^i\)	\(-0.908\)	\(-0.970\)
\(-2D_2\cos 2\varphi_3^i\)	\(+0.001 0625\)	\(-0.000 4274\)
\(4D_4\cos 4\varphi_3^i\)	\(+\phantom{0}93\)	\(+\phantom{0}49\)
\(\chi\)	\(+0.001 0718\)	\(-0.000 4225\)
\(1-\chi=\chi_1\)	\(+0.998 9282\)	\(+1.000 4225\)
\(\delta\varphi_3^{(00)}:\chi_1=\Delta\varphi_3^{(00)}\)	\(+0.004 1155\)	\(+0.003 2444\)
\(\varphi_3^{(00)}\)	\(+0.677 4641\)	\(+0.846 7253\)
\(\varphi_3^{(0)}\approx\varphi_{i3}\)	\(+0.681 5796\)	\(+0.849 9697\)
\(=\)	\(+39^\circ03'05''.9\)	\(+48^\circ41'58''.8\)
\(\sin 2\varphi_3^i\)	\(+0.97 852\)	\(+0.99 167\)
\(\sin 4\varphi_3^i\)	\(+0.404\)	\(-0.255\)
\(\varepsilon_2^{(0)}\)	\(+1.65 878\)	\(+1.85 689\)
\(\varepsilon_4^{(0)}\)	\(+0.593\)	\(+1.123\)
\(+\ D_2\varepsilon_2^{(0)}\)	\(+0.004 1139\)	\(+0.003 2429\)
\(D_4\varepsilon_4^{(0)}\)	\(+\phantom{0}15\)	\(+\phantom{0}14\)
\(\Delta\varphi_3^{(0)}\approx\Delta\varphi_3^{(00)}\)	\(+0.004 1154\)	\(+0.003 2443\)
\(\lg \sin\varphi_3^{(0)}\)	9.799 354	9.875 791
\(-\ \lg \tau_{i3}\)	0.003 323	0.079 270
\(\lg \sin B_i^{(0)}\)	9.796 031	9.796 521
\(B_i^{(0)}\)	\(38^\circ41'53''.3\)	\(38^\circ45'00''.0\)
\(w_B^{(0)}\)	\(-0.000 9051\)	\(-3'06''.7\)

Continuation of Appendix 5

b) Computation of \(w_L^{(0)}\)

\(\pi-\varphi_1^3\)	1.944 8599	2.093 5892
\(-\varphi_3^i\)	0.681 5795	0.849 9696
\(\pi-(\varphi_1^i+\varphi_3^i)\)	\(-1.263\ 2804\)	1.243 6196
\(\times\)
\(\gamma\)	\(-0.122\ 8226\)	\(+0.551\ 1730\)
\(\delta\)	\(-0.155\ 159\)	\(+0.685\ 449\)
\(\lg \operatorname{tg}\varphi_3^i\)	9.909 169	0.056 243
\(\lg p_{i3}\)	9.090 736	9.742 746
\(\lg \operatorname{tg}\vartheta_3^i\)	8.999 905	9.798 989
\(\vartheta_3^i\)	\(5^\circ42'33''.7\)	\(32^\circ11'24''.0\)
\(\vartheta_1^i\)	\(17^\circ25'44''.9\)	\(43^\circ49'14''.2\)
\(180^\circ=(\vartheta_1^i+\vartheta_3^i)\)	\(156^\circ51'41''.4\)	\(103^\circ59'21''.8\)
\(\pi-(\vartheta_1^i+\vartheta_3^i)\)	2.737 7497	1.814 9573
\(\times\)
\(\omega\sqrt{1-e^2}\)	\(-0.996\ 64767\)	\(+0.996\ 64767\)
\(\gamma F(0)=R(0)\)	\(-2.728\ 5719\)	\(+1.808\ 8730\)
\(-\delta\)	\(+0.155\ 159\)	\(-0.685\ 449\)
\(\xi_0\)	\(-2.573\ 413\)	\(+1.123\ 424\)
\(\times\)
\(e^2:2\)	0.003 3467	0.003 3467
\(\gamma F(1)=R(1)\)	\(-0.008\ 6124\)	\(+0.003\ 7598\)
\(3/16\,e^2\)	0.001 255	0.001 255
\(3/16\,e^2 k^2\)	0.000 00827	0.000 00583
\(\varepsilon_2^{(0)}\)	\(-1.657\)	\(-1.858\)
\(-\dfrac{1}{2}\varepsilon_2^{(0)}\)	\(+0.828\)	\(+0.929\)
\(\pi-(\varphi_1^i+\varphi_3^i)\)	\(+1.263\)	\(+1.244\)
\(\xi_2\)	\(+2.091\)	\(+2.173\)
\(\times\)
\(3/16\,e^2 k^2 \gamma\)	\(-0.000\ 00102\)	\(+0.000\ 00321\)
\(\Delta R(1)\)	\(-0.000\ 0021\)	\(+0.000\ 0070\)

\(3/4\,e^2\)	0.005 020	0.005 020
\(\times\)
\(R(1)\)	\(-0.008\ 612\)	\(+0.003\ 760\)
\(3/4\,e^2R(1)\)	\(-0.000\ 0432\)	\(+0.000\ 0189\)
\(-\Delta R(1)\)	\(+\quad21\)	\(-\quad70\)
\(\gamma F(2)=R(2)\)	\(-0.000\ 0411\)	\(+0.000\ 0119\)
\(R(0)\)	\(-2.728\ 5719\)	\(+1.808\ 8730\)
\(R(1)\)	\(-0.008\ 6124\)	\(+0.003\ 7598\)
\(R(2)\)	\(-0.000\ 0411\)	\(+0.000\ 0119\)
\(\Delta L_{1.3}^{(0)},\ \Delta L_{2.3}^{(0)}\)	\(-2.737\ 2254\)	\(+1.812\ 6447\)
\(L_1+2\pi,\ L_2\)	\(+6.927\ 4029\)	\(+2.375\ 7504\)
\(L_3^{(0)}\)	\(+4.190\ 1775\)	\(+4.188\ 3951\)
\(w_L^{(0)}\)	\(+0.001\ 7824\)	\(+\quad6'07''.65\)

c) Computation of the quantities \(\dfrac{\partial B_3^i}{\partial s_{i3}},\ \dfrac{\partial L_3^i}{\partial s_{i3}},\ \delta s_{i3}^{(0)},\ \delta B_3^{(0)},\ \delta L_3^{(0)},\ s_{i3}^{(1)},\ B_3^{(1)},\ L_3^{(1)}\)

\(B_3^{(0)}\approx B_3^i\)	\(38^\circ41'53''.3\)	\(38^\circ45'00''.0\)
\(\lg \sqrt{1-e^2}\)	9.99 854	9.99 854
\(\lg \gamma_{i3}\)	\(9.09\ 217\,n\)	9.74 376
\(\lg V_3^i\)	0.00089	0.00089
\(-\lg\cos B_3^i\)	\(-9.89\ 234\)	\(-9.89\ 203\)
\(\lg\sin A'_{3i}\)	\(9.19\ 926\,n\)	9.85 116
\(A'_{3i}\)	\(189^\circ06'13''\)	\(134^\circ46'42''\)
\(\lg\cos A'_{3i}\)	\(9.99\ 450\,n\)	\(9.84\ 780\,n\)
\(+\)
\(\lg (1)_3^i\)	8.51 093	8.51 093
\(\lg\dfrac{\partial B_3^i}{\partial s_{i3}}\ \dfrac{\text{sec}}{\text{km}}\)	\(1.50\ 543\,n\)	\(1.35\ 873\,n\)
\(a_{i3}=\dfrac{\partial B_3^i}{\partial s_{i3}}\)	\(-32.021\)	\(-22.842\)

Continuation of Appendix 5

$\sin A_{3i}'$	$9.19\ 926\ n$	$9.85\ 116$
$\lg (2)_3^i$	$8.50\ 915$	$8.50\ 915$
$-\lg \cos B_3^i$	$-9.89\ 234$	$-9.89\ 203$
$\lg \dfrac{\partial L_3^i}{\partial s_{i3}}\ \dfrac{\text{sec}}{\text{km}}$	$0.81\ 607\ n$	$1.46\ 828$
$b_{i3}=-\dfrac{\partial L_3^i}{\partial s_{i3}}$	$-6.547$	$+29.395$

\[ \begin{cases} \dfrac{\partial B_3^1}{\partial s_{1.3}}\delta s_{1.3}^{(0)} -\dfrac{\partial B_3^2}{\partial s_{2.3}}\delta s_{2.3}^{(0)} +w_B^{(0)}=0,\\[6pt] \dfrac{\partial L_3^1}{\partial s_{1.3}}\delta s_{1.3}^{(0)} -\dfrac{\partial L_3^2}{\partial s_{2.3}}\delta s_{2.3}^{(0)} +w_L^{(0)}=0, \end{cases} \]

\[ -32.021\,\delta s_{1.3}^{(0)}+22.842\,\delta s_{2.3}^{(0)}=+186''.7 \]

\[ -6.547\,\delta s_{1.3}^{(0)}-29.395\,\delta s_{2.3}^{(0)}=-367''.7 \]

\[ \Delta=+1090.804;\qquad \Delta_1=+2910.96;\qquad \Delta_2=+12996.45 \]

\[ \delta s_{1.3}^{(0)}=+2.6686\ \text{km};\qquad \delta s_{2.3}^{(0)}=+11.9145\ \text{km}. \]

\[ \delta B_3^{1(0)}=-85''.5;\qquad \delta B_3^{2(0)}=-272''.2. \]

\[ \delta L_3^{1(0)}=-17''.471;\qquad \delta L_3^{2(0)}=+350''.227. \]

$s_{i3}^{(0)}$	$8070\ 010\ \text{m}$	$7.935\ 385\ \text{m}$
$\delta s_{i3}^{(0)}$	$+2\ 669\ \text{m}$	$+11\ 915\ \text{m}$
$s_{i3}^{(1)}$	$8.072\ 679\ \text{m}$	$7.947\ 300\ \text{m}$
$B_3^{i(0)}$	$38^\circ41'53''.3$	$38^\circ45'00''.0$
$\delta B_3^{i(0)}$	$-1'25''.5$	$-4'32''.2$
$B_3^{(1)}$	$38^\circ40'27''.8$	$38^\circ40'27''.8$
$L_3^{i(0)}$	$4.190\ 1775$	$4.1883951$
$\delta L_3^{i(0)}$	$-0.000\ 0847$	$+0.0016979$
$L_3^{(1)}$	$4.190\ 0928$	$4.190\ 0930$

5. Second approximation

a) Computation of $w_B^{(1)}$

$\lg s_{i3}^{(1)}$	$6.907\ 017684$	$6.900\ 219607$
$\lg Q_{i3}$	$6.803\ 961804$	$6.803\ 748937$
$\lg \tilde{s}_{i3}^{(1)}$	$0.103\ 055880$	$0.096\ 470670$
$-\tilde{s}_{i3}$	$-1.267\ 814983$	$-1.248\ 736109$
$\pi-\varphi_i^3$	$+1.944\ 859873$	$+2.093\ 589153$
$\varphi_3^{i(10)}$	$+0.677\ 044890$	$+0.844\ 853044$
$\lg\sin B_3^{(1)}$	$9.795\ 807$	$9.795\ 807$
$\lg \tau_{i3}$	$0.003\ 323$	$0.079\ 270$
$\lg\sin\varphi_i^{(1)}$	$9.799\ 130$	$0.875\ 077$
$\varphi_3^{i(1)}\approx\varphi_3^i$	$39^\circ01'39.2$	$48^\circ35'35''.3$
$-\left[\sin 2\varphi_3^i+\sin 2\varphi_i^3\right]$	$+0.978\ 347;\ +0.680\ 267$	$+0.992\ 150;\ +0.865\ 218$
$-\left[\sin 4\varphi_3^i+\sin 4\varphi_i^3\right]$	$+0.4050;\ -0.9972$	$-0.2482;\ -0.8676$
$-\left[\sin 6\varphi_3^i+\sin 6\varphi_i^3\right]$	$-0.81;\ +0.78$	$-0.93;\ +0.00$
$\varepsilon_2^{(1)}$	$-1.658\ 614$	$-1.857\ 363$
$\varepsilon_4^{(1)}$	$+0.5922$	$+1.1158$
$\varepsilon_6^{(1)}$	$+0.03$	$+0.93$
$D_2\varepsilon_2^{(1)}$	$+0.004\ 113515$	$+0.003\ 243756$
$D_4\varepsilon_4^{(1)}$	$+0.000\ 001517$	$+0.000\ 001418$
$D_6\varepsilon_6^{(1)}$	$-1$	$-11$
$\Delta\varphi_3^{i(1)}$	$+0.004\ 115031$	$+0.003\ 245163$
$\varphi_3^{i(10)}$	$+0.677\ 044890$	$+0.844\ 853044$
$\varphi_3^{i(1)}\approx\varphi_3^i$	$+0.681159921$	$+0.848\ 098207$
$=$	$39^\circ01'39''.319$	$48^\circ35'32''.813$

Continuation of Appendix 5

Quantity	I	II
$\sin 2\varphi_3^i$	$+0.978\ 348$	$+0.992\ 150$
$\varepsilon_2^{(1)}$	$-1.658\ 615$	$-1.857\ 368$
$D_2\varepsilon_2^{(1)}$	$+0.004\ 113518$	$+0.003\ 243756$
$\Delta\varphi_3^{(1)}$	$+0.004\ 115034$	$+0.003\ 245163$
$\varphi_3^{(1)}\approx \varphi_3^i$	$0.681\ 159\ 924$	$\pm 0.848\ 098207$
$=$	$39^\circ01'39''.320$	$48^\circ35'32''.813$
$\lg\sin\varphi_3^{(1)}$	$9.799\ 12992$	$9.875\ 07510$
$-\lg r_{i3}$	$0.003\ 32312$	$0.079\ 26984$
$\lg\sin B_3^{(1)}$	$9.795\ 80680$	$9.795\ 80526$
$B_3^{(1)}$	$38^\circ40'28''.038$	$38^\circ40'27''.452$
$w_B^{(1)}$	$+0.000\ 002841$	$+0''.586$

b) Computation of $w_L^{(1)}$

Quantity	I	II
$\pi-\varphi_i^3$	$1.944\ 8599$	$2.093\ 5892$
$-\varphi_3^i$	$0.681\ 1599$	$-0.848\ 0982$
$\pi-(\varphi_i^3+\varphi_3^i)$	$1.263\ 7000$	$1.245\ 4910$
$\times\ \psi$	$-0.122\ 8226$	$+0.551\ 1730$
$\delta$	$-0.155\ 2109$	$+0.686\ 4810$
$\lg\tan\varphi_3^i$	$9.908\ 79675$	$0.054\ 60387$
$\lg p_i$	$9.090\ 73637$	$9.742\ 74631$
$\lg\tan\vartheta_3^i$	$9.999\ 53312$	$9.797\ 35018$
$\vartheta_3^i$	$5^\circ42'16''.192$	$32^\circ05'33''.430$
$\vartheta_i^3$	$17^\circ25'44''.946$	$43^\circ49'14''.183$
$180^\circ-(\vartheta_i^3+\vartheta_3^i)$	$156^\circ51'58''.862$	$104^\circ05'12''.387$
$\pi-(\vartheta_i^3+\vartheta_3^i)$	$2.737\ 834304$	$4.816\ 656917$
$\times\ \omega\sqrt{1-e^2}$	$-0.996\ 647\ 670$	$+0.996\ 647670$
$\psi F(0)=R(0)$	$-2.728\ 656180$	$+1.810\ 566884$
$-\delta$	$+0.155\ 2109$	$-0.686\ 4810$
$\zeta_0$	$-2.573\ 4453$	$+1.124\ 0859$
$\times\ e^2/2$	$0.003\ 346711$	$0.003\ 346711$
$\psi F(1)=R(1)$	$-0.008\ 612578$	$+0.003\ 764991$

Quantity	I	II
$3/16\,e^2,\ 5/16\,e^2$	$0.000\ 12550$	$0.000\ 20917$
$3/16\,e^2 k^2$	$0.000\ 008273$	$0.000\ 005831$
$5/16\,e^2 k^4$	$0.000\ 000091$	$0.000\ 000045$
$-\dfrac{1}{2}\varepsilon_2^{(1)}$	$+0.8293$	$+0.9287$
$\pi-(\varphi_i^3+\varphi_3^i)$	$+1.2637$	$+1.2455$
$\zeta_2$	$+2.0930$	$+2.1742$
$\times\ 3/16\,e^2 k^2\,\psi$	$-0.000\ 001016$	$+0.000\ 003214$
$\Delta R(1)$	$-0.000\ 002128$	$+0.000\ 006988$
$3/4\,e^2$	$0.005\ 0200$	$0.005\ 0200$
$\times\ R(1)$	$-0.008\ 61258$	$+0.003\ 76199$
$3/4\,e^2R(1)$	$-0.000\ 043235$	$+0.000\ 018885$
$-\Delta R(1)$	$+0.000\ 002126$	$-0.000\ 006988$
$\psi FR(2)=R(2)$	$-0.000\ 041109$	$+0.000\ 011897$
$3/8[\pi-(\varphi_i^3+\varphi_3^i)]$	$+0.474$	$+0.467$
$-1/4\,\varepsilon_2^{(1)}$	$+0.415$	$+0.464$
$+\dfrac{1}{32}\varepsilon_4^{(1)}$	$+0.019$	$+0.035$
$\zeta_4$	$+0.908$	$+0.966$
$\times\ 5/16\,e^2 k^4\,\psi$	$-0.000\ 000011$	$+0.000\ 000025$
$\Delta R(2)$	$-0.000\ 000010$	$+0.000\ 000024$
$5/6\,e^2$	$0.005\ 578$	$0.005\ 578$
$\times\ R(2)$	$-0.000\ 04111$	$+0.000\ 01190$
$5/6\,e^2R(2)$	$-0.000\ 000229$	$+0.000\ 000066$
$-\Delta R(2)$	$+\phantom{0}10$	$-\phantom{0}24$
$\psi FR(3)=R(3)$	$-0.000\ 000219$	$+0.000\ 000042$

$R(0)$	$-2.728\ 656180$	$+1.810\ 566884$
$R(1)$	$-0.008\ 612578$	$+0.003\ 761991$
$R(2)$	$-\quad 41109$	$+\quad 11897$
$R(3)$	$-\quad 219$	$+\quad 42$
$\nabla L^{(1)}_{1.3},\ \Delta L^{(1)}_{2.3}$	$-2.737\ 310086$	$+1.814\ 340814$
$L_1+2\pi,\ L_2$	$+6.927\ 402876$	$+3.375\ 75387$
$L^{(1)}_3$	$4.190\ 092790$	$4.190\ 091201$
$w^{(1)}_L$	$+0.000\ 001591$	$+0''.328$

b) Calculation of the quantities $\dfrac{\partial B^i_3}{\partial s_{i3}},\ \dfrac{\partial L^i_3}{\partial s_{i3}},\ \delta s^{(1)}_{i3},\ \delta B^i_3,\ \delta L^i_3,\ s^{(2)}_{i3},\ B^{(2)}_3,\ L^{(2)}_3$.

$\lg \nu_{i3} V^i_3 \cdot \sqrt{1-e^2}$	$9.09\ 160\ n$	$9.74\ 319$
$-\lg \cos B^i_3$	$9.89\ 239$	$9.89\ 239$
$\lg \sin A'_{3i}$	$9.19\ 921\ n$	$9.85\ 080$
$A'_{3i}$	$189^\circ06'09''$	$134^\circ49'32''$
$\lg \cos A'_{3i}$	$9.99\ 450\ n$	$9.84\ 816\ n$
$+\lg (1)^i_3$	$8.51\ 093$	$8.51\ 093$
$\lg\dfrac{\partial B^i_3}{\partial s_{i3}}\ \dfrac{\sec}{100\,m}$	$0.50\ 543\ n$	$0.35\ 909\ n$
$a_{i3}=\dfrac{\partial B^i_3}{\partial s_{i3}}$	$-3.2021$	$-2.2860$
$\lg \sin A'_{3i}$	$9.19\ 921\ n$	$9.85\ 080$
$\lg (2)^i_3$	$8.50\ 915$	$8.50\ 915$
$-\lg \cos B^i_3$	$9.89\ 239$	$9.89\ 239$
$\lg\dfrac{\partial L^i_3}{\partial s_{i3}}\ \dfrac{\sec}{100\,m}$	$7.81\ 597\ n$	$8.46\ 756$
$b_{i3}=\dfrac{\partial L^i_3}{\partial s_{i3}}$	$0.6546$	$+2.9347$

\[ 3.2021\,\delta s^{(1)}_{1.3}+2.2860\,\delta s^{(1)}_{2.3}+0''.586=0 \]

\[ -0.6546\,\delta s^{(1)}_{1.3}-2.9347\,\delta s^{(1)}_{2.3}+0''.328=0 \]

\[ \Delta=+10.8936;\qquad \Delta_1=+2.4695;\qquad \Delta_2=+0.66669 \]

Continuation of Appendix 5

\[ \delta s^{(1)}_{1.3}=+22.669 \qquad \delta s^{(1)}_{2.3}=+6.120 \]

\[ \delta B^{(1)}_3=-0''.726;\qquad \delta B^{(1)}_3=-0''.140 \]

\[ \delta L^{(1)}_3=-0''.1484 \qquad \delta L^{(1)}_3=+0''.1796 \]

$s^{(1)}_{i3}$	$8\ 072\ 679.0$	$7\ 917\ 300.0$
$\delta s^{(1)}_{i3}$	$+22.7$	$+6.1$
$s^{(2)}_{i3}$	$8\ 072\ 701.7$	$7\ 917\ 306.1$
$B^{(1)}_3$	$38^\circ40'28''.038$	$38^\circ40'27''.452$
$\delta B^{(1)}_3$	$-0''.726$	$-0.140$
$B^{(2)}_3$	$38^\circ40'27''.312$	$38^\circ40'27''.312$
$L^{(1)}_3$	$4.190\ 092790$	$4.190\ 091201$
$\delta L^{(1)}_3$	$-0.000\ 000719$	$+0.000\ 000571$
$L^{(2)}_3$	$4.190\ 092071$	$4.190\ 09272$
		$240^\circ04'28''.529$

6. Check calculation

a) Determination of $w^{(2)}_B$

$\lg s^{(2)}_{i3}$	$6.907\ 018905$	$6.900\ 219941$
$-\lg Q_{i3}$	$6.803\ 961804$	$6.803\ 748937$
$\lg \tilde{\sigma}^{(2)}_{i3}$	$0.103\ 057101$	$0.096\ 471004$
$-\tilde{\sigma}^{(2)}_{i3}$	$-1.267818547$	$-1.248\ 737069$
$\pi-\varphi^3_i$	$+1.944\ 859873$	$+2.093\ 589153$
$\varphi^{(20)}_3$	$0.677\ 041326$	$0.844\ 852084$
$+\Delta\varphi^{(2)}_3$	$+0.004\ 115034$	$+0.003\ 245163$
$\varphi^{(2)}_3=\varphi^i_3$	$+681\ 156360$	$+0.848\ 097247$
$=$	$39^\circ01'38''.583$	$48^\circ35'32''.614$
$\lg \sin \varphi^{(2)}_3$	$9.799\ 12801$	$9.875\ 07473$
$\lg \tau_{i3}$	$0.003\ 32312$	$0.079\ 26984$
$\lg \sin B^{(2)}_3$	$9.795\ 80489$	$9.795\ 80489$

Continuation of Appendix 5

Quantity	Value	Value	Quantity	Value	Value
$B^{(2)}_3$	$38^\circ40'27''.312$	$38^\circ40'27''.312$	$\pi-(\vartheta_i^3+\vartheta_3^i)$	$2.737\ 835025$	$1.816\ 657799$
$w^{(2)}_B$		$0''.000$	$\times\ \omega\sqrt{1-e^2}$	$-0.996\ 647670$	$0.996\ 647670$
			$\psi F(0)=R(0)$	$-2.728\ 656899$	$+1.810\ 567763$
			$-\delta$	$+0.155\ 2114$	$-0.686\ 4816$
b) Determination of $w^{(2)}_L$			$\sigma_0$	$-2.573\ 4455$	$+1.124\ 0862$
$\pi-\varphi_1^3$	$1.944\ 8599$	$2.093\ 5892$	$\times\ \dfrac{e^2}{2}$	$0.003\ 346711$	$0.003\ 346711$
$-\varphi_3^i$	$-0.681\ 1564$	$-0.848\ 0972$	$\psi F(1)=R(1)$	$-0.008612578$	$+0.003\ 761992$
$\pi-(\varphi_i^3+\varphi_3^i)$	$1.263\ 7035$	$1.245\ 4920$
$\times\ \psi$	$0.122\ 8226$	$+0.551\ 1730$
$\delta$	$-0.155\ 2114$	$+0.686\ 4816$
$\lg \tg \varphi_3^i$	$9.908\ 79358$	$0.054\ 60302$	$R(0)$	$-2.728656899$	$+1.810\ 567763$
$\lg p_{i3}$	$9.090\ 73637$	$9.742\ 74631$	$R(1)$	$-0.008\ 612578$	$+0.003\ 761992$
$\lg \tg \vartheta_3^i$	$9.999\ 52995$	$9.797\ 34933$	$R(2)$	$-\phantom{0}41109$	$+\phantom{0}11897$
$\vartheta_3^i$	$5^\circ42'16''.043$	$32^\circ05'33''.248$	$R(3)$	$-\phantom{0}219$	$+\phantom{0}42$
$\vartheta_i^3$	$17^\circ25'44''.946$	$43^\circ49'14''.183$	$\nabla L^{(2)}_{1,3},\ \Delta L^{(2)}_{2,3}$	$-2.737\ 310805$	$+1.814\ 341692$
$180^\circ-(\vartheta_i^3+$			$L_1+2\pi,\ L_2$	$+6.927402876$	$+2.375\ 750387$
$+\vartheta_3^i)$	$156^\circ51'59''.011$	$104^\circ05'12''.569$	$L^{(2)}_3$	$4.190\ 092071$	$4.190\ 092081$
			$w^{(2)}_L$	$-0.000\ 000010$	$-0''.0021$

to \(1\)–\(3^\circ\), the mean latitude \(B_{ik}\) of each such part was found and the corresponding value \(R_{ik}\) of the mean radius of curvature (to an accuracy of \(1\) km) was written out. It was then possible to compute the required initial values \(s_{i3}^{(0)}\) of the lengths of the secant sides \(i3\) on the spheroid, taking

\[ s_{i3}^{(0)}=\left(\frac{1}{4}\sum_k R_{ik}\right)\delta_{i3}^{(0)}, \qquad (i=1,2). \]

The subsequent computations were carried out as described in [Div. 6, B], and it turned out that, despite the apparent crudity of the above calculation of the lengths \(s_{i3}^{(0)}\), the errors \(\nabla s_{i3}^{(0)}\) of these lengths proved to be comparatively small:

a) \(\nabla s_{1.3}^{(0)}=-2.7\) km for \(s_{1.3}=8072.7\) km;

b) \(\nabla s_{2.3}^{(0)}=-11.9\) km for \(s_{2.3}=7947.3\) km.

To determine on the spheroid the lengths \(s_{i.3}\) with an accuracy of \(0.2\) m and the coordinates \(B_3, L_3\) with an accuracy of \(0''.001\)–\(0''.002\), two full approximations were required (one with 6 digits, the other with 8–9 digits) and one incomplete, checking approximation.

1. Comparing the values of the coordinates \(B_3, L_3\) of the point 3 being determined, found by both methods, we are convinced of their good agreement:

1st method: \(B_3=48^\circ40'27''.310;\quad L_3=240^\circ04'28''.531.\)

2nd method: \(B_3=48^\circ40'27''.312;\quad L_3=240^\circ04'28''.529.\)

The discrepancies arising here do not exceed the limits of the accuracy of computation by 8-digit tables of logarithms.

LITERATURE

1. F. W. Bessel. Ueber die Berechnung der geographischen Längen und Breiten aus geodetischen Vermessungen. Astr. Nachr., Bd. 4, № 86, 1826.
2. Yu. S. Sikorskii. Elements of the Theory of Elliptic Functions. ONTI, 1936.
3. V. P. Vetchinkin. New Formulas and Tables of Elliptic Integrals and Functions. VVA RKKA Publishing House, Moscow, 1935.
4. F. A. Sludskii. Lectures on Higher Geodesy. Moscow, 1894.
5. F. N. Krasovskii. Course of Higher Geodesy. Part 2, Moscow, 1942.
6. B. F. Krutoi. General methods for solving the basic computational problems on the terrestrial spheroid (brief communication). Izvestiya TPI, vol. 118, 1963.