MAIN TYPES OF CONDITIONS ARISING IN SURFACE RAY NETWORKS

B. F. Krutoi

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

In the present article we shall consider, with the necessary completeness, the main types of those original relations of connection in surface ray networks which are used as initial ones in the mixed and direct adjustment of these networks [2].

§ 1. Basic and additional network conditions, their nature and expression

Further investigation shows that in surface ray networks of general structure the following four main types of mixed original relations (1) may occur:

1) ray,
2) lateral,
3) vertex,
4) additional.

The relations of connection of the first three types arise in ray networks of any structure, provided only that the composition of their initial data is extended. Therefore we shall call the indicated relations basic.

As for the additional relations of connection, they arise only in ray networks of genus \(s > 0\) [1]. The additional relations have the remarkable feature that they appear in ray networks of genus \(s > 0\) not only when the composition of the initial data is extended, but even when the initial data in these networks are strictly necessary.

By means of the corresponding transformation indicated in [2], the mixed relations of connection (1) are converted into direct relations (3), which do not contain additional data \(\bar{\gamma}_g\). Hence it follows that the direct relations (3) split only into the first three types: 1) ray, 2) lateral, 3) vertex.

In accordance with established custom, in what follows we shall call the original relations of connection (1)—(4) more briefly network conditions, respectively mixed or direct.

Without yet considering in greater detail the four main types of network conditions indicated above, let us first clarify the reason for the occurrence of these conditions in surface ray networks.

From a number of particular examples it is not difficult to observe that such a cause is the presence, in the surface ray network, of non-unique-

... particles $\breve P$—rays $\breve L_{kм}$, sides $\breve B_{kм}$, or vertices $\breve K$. These ambiguous network particles $\breve P$ have the feature that, for each of them, at least two values can be established:

1) $\dot P$—the value adopted by us,

2) $\ddot P$—the value obtained by transmission along the network from the necessary initial and additional data, starting from some initial network particle $\dot И$ of the same nature as $\dot P$.

In what follows, we shall call an ambiguous network particle $\breve P$ a split one, and the two values of it indicated above, $\dot P$ and $\ddot P$, respectively the supporting and the constructed values. Let us consider these two values of the particle $\breve P$ in greater detail.

It is not difficult to see that the supporting value $\dot P$ adopted by us for a split network particle $\breve P$ may be: a) fixed $\bar P$, b) specified within narrow limits close to its possible value, c) specified completely arbitrarily. Case (a) occurs when $\dot P \ne \dot И$, and then, obviously, the value $\dot И$ must also be fixed: $\dot И=\bar И$. The last two cases arise only when $\dot P=\dot И$, and requirement (b) is desirable for side, vertex, and additional conditions, while requirement (c) is quite acceptable for ray conditions.

As for the second, constructed value $\ddot P$ of the split network particle $\breve P$, it may be: a) either necessary, and then we shall denote it by $\tilde P$, b) or redundant, and in that case we shall denote it by $\mathring P$. Let us note that $\ddot P$ is a necessary value only in additional conditions, and there $\breve P$ will be a split vertex $\breve K$ for the residual ray $\sigma \breve L_{nk}$ associated with it (see § 2).

Each pair of corresponding values $\dot P$, $\ddot P$ of a split network particle $\breve P$ yields some network condition. Depending on the nature of the network particle taken, this may be either a ray condition, or a side condition, or a vertex condition, or one of the additional conditions.

As for the essence of the four principal kinds of network conditions listed above, each of them consists in the requirement that, for an adjusted (internally consistent) ray network, the corresponding values $\dot P$ and $\ddot P$ of the split network particle $\breve P$ coincide with one another. Thus this requirement may be expressed by the following brief equality

\[ \ddot P-\dot P=0. \tag{13} \]

Now suppose that in condition (13), instead of the adjusted, consistent values $\dot P$, $\ddot P$ of the split network particle $\breve P$, we insert respectively its unadjusted (mutually inconsistent) values $\dot P$ and $\ddot P$, obtained by the method indicated above. Then, owing to the errors of the necessary initial and additional data, the condition

(13), when the values \(\dot P\) and \(\ddot P\) are substituted into it, will not be strictly satisfied. Therefore, instead of (13), in this case one obtains

\[ \ddot P-\dot P=w_p\ne 0, \tag{13a} \]

where \(w_p\) is the residual of the corresponding network condition.

Depending on whether \(\dot I\) and \(\dot P\) are different network particles of one and the same nature, or whether these particles coincide with one another, we shall distinguish between external and internal conditions in a ray network.

If \(\dot I\) does not coincide with \(\dot P\):

\[ \dot I\ne \dot P, \tag{14} \]

and, consequently, \(\dot I,\dot P\) are necessarily fixed \(\bar I,\bar P\), then such a condition will be called external.

If, however, \(\dot I\) and \(\dot P\) are one and the same network particle:

\[ \dot I=\dot P, \tag{14a} \]

then the corresponding network condition will be called internal.

The division of conditions in surface ray networks into external and internal ones is essential. This follows from the fact that the formation of residuals \(w_p\) of internal conditions is affected only by the errors of the observed \(\zeta_i\) and of the approximate additional quantities \(\nu_g^{(0)}\),\(^{1)}\) whereas the residuals \(w_p\) of external conditions also reflect errors in the adopted positions \(\dot I\) and \(\dot P\) of the initial and terminal split network particles.

The generalized conception given here of the conditions arising in surface ray networks will be developed in more detail below, by considering in all essential features the main types of such conditions listed above. We shall begin this consideration with conditions of the last type—additional ones, which, despite their unusual properties, have not been noted by anyone until now.

§ 2. Additional Conditions

As indicated above, additional conditions arise in a surface ray network of genus \(s\) for any composition of its initial data (extended or necessary), provided only that the genus of the network is \(s>0\). By this property, additional conditions differ sharply from principal conditions, which may appear in any ray network (i.e., for \(s\ge 0\)), but only when the composition of the initial data is extended.

The additional conditions of a surface ray network of genus \(s\) consist in the requirement that each of its \(s\) residual rays

\[ \delta \tilde L_{pk}\quad (\text{or } \delta \tilde{\bar L}_{pk}), \]

constructed from the necessary initial and additional data (or from the necessary equalized data), should pass with complete exactness through the specified position \(\dot K(\bar K)\) of the corresponding split vertex \(\dot K\). For example, in an unoriented

\(^{1)}\) If the approximate values \(\nu_g^{(0)}\) of the additional data are consistent with the corresponding necessary set of initial data \(\zeta_i,\bar\chi_\mu\), then the residuals \(w_p\) of internal conditions will be affected only by the errors of the observed quantities \(\zeta_i\).

of the ray network of genus \(s=3\), shown in Fig. 1, the three residual rays \(\delta \widetilde{L}_{5\cdot18}\), \(\delta \widetilde{L}_{9\cdot19}\), and \(\delta \widetilde{L}_{8\cdot20}\) must pass directly through their respective hard vertices 18, 19, and 20. Hence it follows that in a surface ray network of genus \(s\) exactly \(s\) additional conditions arise.

The additional conditions of a ray network of genus \(s\) can also be expressed somewhat differently, if one takes into account that on each constructed residual ray \(\delta \widetilde{L}_{nk}\) (or \(\delta \overline{\widetilde{L}}_{nk}\)) there is a second position, obtained by transfer along the network, \( \widetilde K_n\) (\(\overline{\widetilde K}_n\)) of the corresponding split vertex \(\dot K\) (the first position \(\dot K\) (\(\dot{\overline K}\)) is given). In view of the indeterminacy of the location of the indicated point \(\widetilde K_n\) (\(\overline{\widetilde K}_n\)) on the ray \(\delta \widetilde L_{nk}\) (\(\delta \overline{\widetilde L}_{nk}\)), nothing prevents us from regarding this point as lying at the base of the cross-section dropped from the point \(\dot K\) (\(\dot{\overline K}\)) onto the ray \(\delta \widetilde L_{nk}\) (\(\delta \overline{\widetilde L}_{nk}\)).

Fig. 1. Examples of nonfulfillment of additional conditions.

Denoting then by \(\uparrow \widetilde\lambda_{nk}\) (\(\uparrow \overline{\widetilde\lambda}_{nk}\)) the directed segment—the arrow \(\dot K\widetilde K_n\) (\(\dot{\overline K}\,\overline{\widetilde K}_n\)), i.e., the transverse displacement of the ray \(\delta\widetilde L_{nk}\) (\(\delta\overline{\widetilde L}_{nk}\)) relative to the point \(\dot K\) (\(\dot{\overline K}\)), we can write the additional conditions in the following concise form:

a) when constructing the ray network from the necessary initial data,

\[ \widetilde K_n-\dot K-\uparrow\widetilde\lambda_{nk}=0. \tag{15′} \]

b) when constructing the ray network from the extended initial data,

\[ \overline{\widetilde K}_n-\dot{\overline K}-\uparrow\overline{\widetilde\lambda}_{nk}=0. \tag{15} \]

Let us now consider in greater detail the question of writing the additional conditions in both of the indicated cases, and also analyze the methods of composing these conditions.

1. Suppose first that the set of initial data \(\zeta_i,\overline\chi_\mu\) in a surface ray network of genus \(s\) is the necessary set \(\sigma\)

\[ \sigma=(\zeta_1,\zeta_2,\ldots,\zeta_\alpha;\ \overline\chi_1,\overline\chi_2,\ldots,\overline\chi_\rho). \tag{16} \]

and therefore the bundle \(\tilde{\zeta}_1,\zeta_2,\ldots,\zeta_s\) in (16) must satisfy the necessary condition for constructibility of a ray network indicated in [1]. Then, in such a network, for \(s>0\) there arise \(s\) mixed connection relations (1) of the following particular form:

\[ D_{nk}(\gamma_1,\gamma_2,\ldots,\gamma_s;\sigma')= D_{nk}(\sigma'')=\tilde{\lambda}_{nk}=0. \tag{15°} \]

Here \(\sigma'\) is obtained from \(\sigma\) by discarding the \(s\) angular quantities \(\zeta_h\) that define the directions of the \(s\) residual rays \(\delta\tilde{\Pi}_{nk}\). As for \(\sigma''\), then

\[ \sigma''=(\gamma_1,\gamma_2,\ldots,\gamma_s;\sigma'). \tag{16.a} \]

In addition:

\[ \begin{aligned} \gamma_g&\text{ are the }s\text{ additional data of a ray network of type }s,\text{ mentioned in [1];}\\ \zeta_i&\text{ are the observed angular quantities, the directions }\beta_j\text{ or the angles }\gamma_k;\\ \chi_\mu&\text{ are fixed quantities;}\\ \tilde{\lambda}_{nk}&\text{ is the directed length of the corresponding arrow } \uparrow\,\tilde{\lambda}_{nk}. \end{aligned} \]

It is essential to note here that the values \(\gamma_g\) of the additional data entering (15°) are regarded as strictly consistent with the corresponding set \(\sigma\) of necessary initial data [equality (16)].

Replacing, in condition (15°), the additional data \(\gamma_g\) consistent with \(\sigma\) by their approximate values \(\gamma_g^{(0)}\) leads to a violation of the indicated condition and to the replacement of the rays \(\delta\tilde{\Pi}_{nk}\) by displaced residual rays \(\delta\tilde{\Pi}_{nk}^{(0)}\). In this case, instead of (15°) we shall have

\[ D_{nk}\bigl(\gamma_1^{(0)},\gamma_2^{(0)},\ldots,\gamma_s^{(0)};\sigma'\bigr) =\tilde{\lambda}_{nk}^{(0)}=w_{\tilde{\Pi}_{nk}}\ne 0, \tag{15°.a} \]

where \(w_{\tilde{\Pi}_{nk}}\) is the residual of condition (15°) (see Fig. 1б, on which the residuals \(w_{\tilde{\Pi}_{nk}}=\tilde{\lambda}_{nk}^{(0)}\) are shown in their actual magnitude). We shall assign the sign of the residual \(w_{\tilde{\Pi}_{nk}}=\tilde{\lambda}_{nk}^{(0)}\) in accordance with the rule indicated in [1]:

\[ \tag{17} \begin{cases} \tilde{\lambda}_{nk}^{(0)}>0, & \text{if the ray } \delta\tilde{\Pi}_{nk}^{(0)} \text{ is to the right of the point } K,\\[4pt] \tilde{\lambda}_{nk}^{(0)}<0, & \text{if the ray } \delta\tilde{\Pi}_{nk}^{(0)} \text{ is to the left of the point } K. \end{cases} \]

The simultaneous elimination of all \(s\) residuals \(w_{\tilde{\Pi}_{nk}}\) in the ray network is carried out, for the present case, by the general method set forth in [1].

1. Let us now suppose that the set of initial data in a surface ray network of type \(s\) is an extended set \(\Sigma\). Then, in such a network, in addition to a series of principal conditions (ray, lateral, or vertex conditions), there arise another \(s\) additional conditions of a form close to (15°),

\[ D_{nk}(\bar{\sigma}'')= D_{nk}(\bar{\gamma}_1,\bar{\gamma}_2,\ldots,\bar{\gamma}_s;\sigma') =\tilde{\bar{\lambda}}_{nk}=0, \tag{15} \]

where the barred quantities denote adjusted quantities. These additional relations express the requirement that the transverse displacements \(\tilde{\bar{\lambda}}_{nk}\) of all \(s\) residual rays \(\delta\tilde{\Pi}_{nk}\), constructed by means of the adjusted additional data, be equal to 0.

over some necessary set \(\sigma''\) of balanced \(\nu_g,\bar{\xi}_i\) and fixed \(\bar{\chi}_\mu\) extended initial data, where the unbalanced \(\sigma''\) is determined according to (16a).

For unbalanced extended initial data, conditions (15) will be satisfied with any degree of accuracy if the adopted values \(\nu_g\) of the additional data are sufficiently strictly coordinated with the corresponding set \(\sigma\) of necessary initial data \(\zeta_i,\bar{\chi}_\mu\) [see (16)]. In this case, instead of (15) we shall have

\[ D_{nk}(\nu_1,\nu_2,\ldots,\nu_z;\sigma')=\tilde{\lambda}_{nk}=w_{D_{nk}}\simeq 0, \tag{15a} \]

where \(w_{D_{nk}}\) is the residual of the additional condition under consideration.

The elimination of the residuals \(w_{D_{nk}}\) of all \(s\) additional conditions in a ray network of the \(s\)-th kind is carried out, in the given case, jointly with the residuals of the basic conditions of this network.

1. The writing of the additional conditions \((15^\circ)\) or (15) in expanded form can be done as follows.

a) First we choose a suitable necessary set
\(\sigma''=(\nu_1,\nu_2,\ldots,\nu_s;\sigma')\) of initial data \(\zeta_i,\bar{\chi}_\mu\) and additional data \(\nu_g\). Next we form the expanded literal expressions, in terms of \(\nu_g,\zeta_i,\bar{\chi}_\mu\), for the lengths \(d_{ch}\) of the connecting sides \(CH\) in the chain of cells joining the original vertices \(\dot{A},\dot{B}\) with the corresponding split vertex \(K_\partial\) \((\partial=1,2,\ldots,s)\). Then, using the literal representations found for the lengths \(d_{ch}\) and the directions \(\uparrow_{ch}^{\,2}\) of the connecting sides \(CH\) in the chain of cells under consideration, we successively express, in general form, the positions of all determined vertices \(C,H\) of this chain, including the adjacent determined vertex \(P_\partial\). Finally, we write the desired expanded general expression for the transverse displacement \(\tilde{\lambda}_{(nk)_\partial}\) of the residual ray \(\delta \tilde{L}_{(nk)_\partial}\) relative to the reference position \(\dot{K}_\partial\) of the corresponding split vertex \(\dot{K}_\partial\). The problem is solved.

b) In contrast to (a), with the aid of successive direct intersections we at once form the expanded literal expression for the constructed positions of all determined vertices \(C,H\) in the connecting chain of cells from \(\dot{A},\dot{B}\) to \(\dot{K}_\partial\). The rest is as in (a). This method is convenient only for plane ray networks.

The degree of complexity of the expanded expressions \((15^\circ)\) or (15) for the additional conditions depends on the type of cells of the connecting chain—triangular or mixed. This question will be considered in greater detail below for vertex conditions, which have much in common with additional conditions.

1. Let us note one more remarkable feature of additional conditions in surface ray networks. Namely, from [1] and from equalities \((15^\circ a)\), (15a) it follows that the residuals \(w_{D_{nk}}\) of these conditions are always directed lengths \(\tilde{\lambda}_{(nk)_\partial}\) of certain arrows \(\uparrow \tilde{\lambda}_{(nk)_\partial}\)—transverse displacements of the constructed residual rays \(\delta \tilde{L}_{(nk)_\partial}\). Meanwhile, the corresponding additional data \(\nu_g\) may in this case be not

\(^2\) The direction \(\uparrow_{ch}\) of some ray \(\uparrow L_{ch}\) is the angle formed by this ray with the first axis \(\uparrow U^1\) of the adopted reference frame.

only by segments, but also by angular quantities. In contrast to what has been said, such diversity never occurs in the basic conditions. There the discrepancies of angular conditions are always angular quantities, and the discrepancies of side and vertex conditions are length quantities.

§ 3. Ray conditions

We now turn to the basic network conditions and, among them, consider first of all the ray conditions, as the simplest in their structure.

Ray conditions in a surface ray network of the \(s\)-th kind connect only angular quantities that determine the position of its rays \(\bar L_{cn}\) on the underlying surface. These conditions arise in a ray network when it contains split rays \(\delta \bar L_{km}\). For such rays at least two positions can always be established: 1) \(\delta \bar L_{km}\)—the reference one, adopted by us; 2) \(\delta \bar L_{km}\)—the constructed one, obtained by transfer along the network, according to the required initial data, from some ray \(\delta \bar L_{ab}\) taken as the source ray.

Each of the ray conditions expresses the requirement that, after adjustment, the directions \(\overset{\circ}{\uparrow}_{km}\) and \(\dot{\uparrow}_{km}\) of the two positions indicated above, \(\delta \bar L_{km}\) and \(\delta \bar L_{km}\), of the given split ray \(\delta \bar L_{km}\) should prove to be equal to one another. In other words, any ray condition may be written briefly as

\[ \overset{\circ}{\uparrow}_{km} - \dot{\uparrow}_{km} = 0. \tag{18} \]

To present this condition further in a more expanded form, suppose that, in the adjusted network, the transfer of direction from the source ray \(\delta \bar L_{ab}\) to the constructed ray \(\delta \bar L_{km}\) is performed along a path in which the beginning of each succeeding arrow is combined with the end of the preceding arrow (Fig. 2). Then every \(f\)-th angular condition in the ray network is written in the following expanded form:

Fig. 2. Examples of nonfulfillment of ray conditions.

\[ \uparrow_{ab} + \sum_{e=1}^{n} \eta_{f,e}\bigl(\gamma_e \pm 180^\circ\bigr) + i \cdot 360^\circ - \dot{\uparrow}_{km} = 0 \tag{18'} \]

Here:

a) \(\gamma_e\) are the adjusted angles of the ray network, \(n\) in number, obtained directly or through the adjusted directions \(\beta_i, \beta_k\);

b) \(\eta_{f.e}\) are coefficients equal to \(+1\) or \(0\), depending on whether or not the angle \(\gamma_e\) enters the given \(f\)-th ray condition \((\eta_{f.e}=+1\) for angles \(\gamma_e\) to the left, and \(\eta_{f.e}=-1\) for those to the right along the course);

c) \(i\) is an integer chosen so as to satisfy the relation

\[ 360^\circ > \left[ \sum_{e=1}^{n} \eta_{f.e}\,(\bar{\gamma}_e \pm 180^\circ) + i\cdot 360^\circ \right] \geq 0; \tag{*} \]

d) the sign \(\pm\) is taken arbitrarily.

Owing to errors of observation and to the possible erroneousness of the adopted positions \(\delta\dot{J}_{ab}, \delta\dot{J}_{km}\) for the initial and terminal rays, in an unadjusted network the ray conditions (18) or \((18^\circ)\) will not be satisfied exactly. Instead of these equalities we shall then have

\[ \overset{\circ}{\uparrow}_{km} - \dot{\uparrow}_{km} = \dot{\uparrow}_{ab} + \sum_{e=1}^{n}\eta_{f.e}(\bar{\gamma}_e \pm 180^\circ) + i\cdot 360^\circ - \overset{\circ}{\uparrow}_{km} = w_{J_{km}}\ne 0. \]

Here \(w_{J_{km}}=w_f\) is the misclosure of the given \(f\)-th condition, while \(\dot{\uparrow}_{ab}\) and \(\dot{\uparrow}_{km}\) are the directions adopted by us for the initial \(\delta\dot{J}_{ab}\) and terminal \(\delta\dot{J}_{km}\) rays in the transmission course. It is assumed, moreover, that the values \(\dot{\uparrow}_{km}\) and \(\overset{\circ}{\uparrow}_{km}\) have been obtained along one and the same transmission course.

In the general case the initial ray \(\delta\dot{J}_{ab}\) does not coincide with the terminal control ray \(\delta\dot{J}_{km}\):

\[ \delta\dot{J}_{ab}\ne \delta\dot{J}_{km}, \]

and therefore in relation (18a), in general, \(\dot{\uparrow}_{ab}\ne \dot{\uparrow}_{km}\) (see Fig. 2, where \(\delta\dot{J}_{ab}=\delta\dot{J}_{15-11}\), and \(\delta\dot{J}_{km}=\delta\dot{J}_{16-17}\)). In this case, in accordance with what was said in § 1, the corresponding angular condition will be called external.

More often, however, it happens that both indicated rays \(\delta\dot{J}_{ab}\) and \(\delta\dot{J}_{km}\) are one and the same ray:

\[ \delta\dot{J}_{ab}=\delta\dot{J}_{km}, \]

and then, of course, \(\dot{\uparrow}_{ab}=\dot{\uparrow}_{km}\). Precisely this case occurs in the angular relations arising in closed cells of a ray network (Fig. 2). On this basis, the ray conditions \((18^\circ)\) for coinciding rays \(\delta\dot{J}_{ab}\) and \(\delta\dot{J}_{km}\) will be called cellular, and also internal (according to § 1).

From what has been said it follows that the cellular \(f\)-th ray condition and its misclosure \(w_f\) are written in the following form:

\[ \sum_{e=1}^{n}\eta_{f.e}\,(\bar{\gamma}_e \pm 180^\circ) + i\cdot 360^\circ = 0 \tag{18^{\circ\circ}} \]

\[ \sum_{e=1}^{n} \eta_{f,e}(\gamma_e \pm 180^\circ) + i \cdot 360^\circ = w_f \tag{18б} \]

It is not difficult to see that the ray conditions for surface ray networks of genus \(s\), directed and undirected, do not contain additional data \(\gamma_g\). Therefore the ray conditions have one and the same form both in mixed and in direct adjustment of ray networks.

§ 4. Side conditions

We now turn to those relations of type (1) and (3) that arise in ray networks of original links and that were called side conditions in § 1.

Side conditions relate, in a surface ray network, the lengths \(\bar d_{c\ell}\) of its adjusted sides \(\bar B_{c\ell}\) and arise when there are split sides \(\bar B_{k\ell}\) in the network. Split sides \(\bar B_{k\ell}\) have the special feature that for them at least two values can always be established: 1) \(\bar B_{k\ell}\)—the reference value, accepted by us; 2) \(\dot B_{k\ell}\)—the constructed value, obtained by transmission along the network using the necessary initial and additional data from some side \(B_{ab}\) taken as the initial one.

Each of the side conditions expresses the requirement that, after adjustment of the ray network, the lengths \(\dot d_{k\ell}\) and \(\bar d_{k\ell}\) of the corresponding values \(\dot B_{k\ell}\) and \(\bar B_{k\ell}\) of the given split side \(\bar B_{k\ell}\) should coincide with one another. Hence follows the following general form of the side conditions (otherwise, side conditions \(C_{k\ell}\)) in a ray network:

\[ \dot d_{k\ell} - \bar d_{k\ell} = C_{k\ell}(\gamma_1,\gamma_2,\ldots,\gamma_s;\bar\gamma_1,\bar\gamma_2,\ldots,\bar\gamma_n;\dot d_{ab}) - \bar d_{k\ell} =0, \tag{19} \]

where \(\bar d_{ab}, \bar d_{k\ell}\) are the lengths of the initial \(\bar B_{ab}\) and final \(\bar B_{k\ell}\) sides (edges) of the network, adjusted or fixed;

\(\gamma_g\) are the adjusted additional data, whose number is equal to \(s\) in a ray network of genus \(s\);

\(\bar\gamma_e\) are the adjusted angles of the ray network, \(n\) in number, obtained directly or through pairs of directions \(\bar\beta_j,\bar\beta_k\).

For an unadjusted ray network this condition, owing to errors of observation and of the adopted quantities, will not be exactly satisfied, and instead of (19) we obtain

\[ \dot d_{k\ell} - \bar d_{k\ell} = C_{k\ell}(\gamma_1,\gamma_2,\ldots,\gamma_s;\gamma_1,\gamma_2,\ldots,\gamma_n;\dot d_{ab}) - \dot d_{k\ell} = w_{C_{k\ell}} \ne 0, \tag{19а} \]

where \(\gamma_g\) are the values of the additional data coordinated with one of the necessary sets \(\sigma\) of initial data [see (16)], and \(w_{C_{k\ell}}\) is the misclosure of the corresponding side condition. From the preceding it follows that the indicated misclosure \(w_{C_{k\ell}}\) is obtained by its successive computation along the chain of cells connecting the initial side \(\dot B_{ab}\) with the final side \(\dot B_{k\ell}\) of this chain.

In the general case the initial side \(\dot B\) does not coincide with the final side \(\dot B_{k\ell}\):

\[ \dot B_{ab} \ne \dot B_{k\ell}. \]

(see Figs. 3a and 3b; in Fig. 3b the side \(\dot B_{ab}=B_{14,15}\), and the side \(\dot B_{k m}=B_{16,17}\)), and then we speak of an external lateral condition.

However, it more often happens that the initial side \(\dot B_{ab}\) and the final side \(\dot B_{k m}\) coincide with one another:

\[ \dot B_{ab}=\dot B_{k m} \]

(see Fig. 3c, where as \(\dot B_{ab}=\dot B_{k m}\) one may take any connecting side \(B_{c h}\) of the given chain of cells; for example, one may take the side \(B_{2,3}\)). In this case the corresponding lateral condition will be called internal.

Fig. 3. Examples of nonfulfillment of lateral conditions.

§ 5. Expanded representation of lateral conditions in surface ray networks. Difficulty of solving this problem in the general case

We shall next set as our aim to give an expanded expression of the lateral conditions in surface ray networks of different structure. This problem evidently reduces to an expanded representation of the function \(C_{k m}=\dot d_{k m}\) in equality (19a). The indicated function \(C_{k m}=\dot d_{k m}\) is obtained by its successive computation along the corresponding chain of cells. Therefore, first of all, we need to have general expressions for the lengths \(d_{c h}\) of the determined sides \(B_{c h}\) in some intermediate cell of the network.

In order that the corresponding expressions for the lengths \(d_{c h}\) be sufficiently general, we shall accompany the angles \(\gamma\) and the lengths \(d\) of sides in the cells by two indices \(e,j\): \(\gamma_{e j}, d_{e j}\). We shall agree that the first index \(e\) determines the fixed place of the quantity \(\gamma\) or \(d\) in a cell of the given type under some uniform method of recalculating the quantities \(\gamma,d\). The second index \(j\) determines the place of the cell taken in the given chain.

Suppose now that the given ray network is constructed only from triangular and quadrilateral cells, and that the methods of recalculation of the quanti-

values of \(\gamma, d\) in the cell chains selected from this network are established as shown in Fig. 4. Then, depending on the type of the underlying surface—plane or spheroidal—we shall have the following expressions for the lengths of the sides to be determined in cells of the stated type.

Underlying surface—a plane. In this case

a) If the \(j\)-th cell of the chain taken is a triangle (Fig. 4a), then the calculation of the lengths \(d_{2j}, d_{3j}\) of its unknown sides \(AB, BB\) is performed by means of the equalities

Fig. 4. Designation of angles and sides in triangles and quadrilaterals

\[ d_{2j}=d_{1j}\frac{\sin\gamma_{2j}}{\sin\gamma_{1j}}, \qquad d_{3j}=d_{1j}\frac{\sin\gamma_{3j}}{\sin\gamma_{1j}}, \tag{20} \]

where the length \(d_{1j}\) of the initial side \(AB\) is assumed to be known: it is given or has been found from the calculation of the preceding cell.

b) If, however, the given \(j\)-th cell of the chain is a quadrilateral (Fig. 4b), then the calculation of the lengths \(d_{3j}, d_{4j}\) of its unknown sides \(AG, VG\) is performed according to the equalities

\[ d_{3j}=\cosec\gamma_{2j}\left[d_{1j}\sin\gamma_{3j} +d_{2j}\sin(\gamma_{1j}+\gamma_{2j})\right], \]

\[ d_{4j}=\cosec\gamma_{2j}\left[d_{2j}\sin\gamma_{1j} +d_{1j}\sin(\gamma_{2j}+\gamma_{3j})\right], \tag{21} \]

where the lengths \(d_{1j}, d_{2j}\) of the initial sides \(AB, BV\) are assumed to be known: they are fixed or have been found from the calculation of the preceding cells. Equalities (21) were first indicated by I. V. Zubritskii [3]. These equalities can be obtained on the basis of the auxiliary constructions in Fig. 4b, where \(DG \parallel EB\), \(GK \parallel AB\), \(BE \parallel GZh\), \(DB \parallel GV\), and the points \(Zh\) and \(E\) lie respectively on the sides \(GA, GV\) or on their extensions.

Underlying surface—a spheroid. In this case, to determine the lengths of the sides in surface ray networks, their cells are first subjected to a preliminary plane transformation according to Legendre.

For spheroidal triangular cells such a transformation is well known.

For spheroidal quadrilateral cells, the plane transformation according to Legendre will consist in the fact that we first

we solve them approximately, as plane ones, using the same equalities (21) as above. Next, having divided each quadrilateral cell cut by a diagonal (for example, by the diagonal \(BG\) in Fig. 4b) into two triangles, we find, by Legendre’s theorem, the spheroidal corrections to the angles in these auxiliary non-plane triangles. Then the angles \(\gamma'_{ij}\) of the plane quadrilateral \(A'B'V'G'\), with the same side lengths as those of the given spheroidal quadrilateral \(ABVG\), will evidently be found as follows—when divided by the diagonal \(BG\):

\[ \begin{aligned} 1)\quad &\gamma'_{1j}=\gamma_{1j}-\frac{1}{3}\varepsilon_1, &\qquad 3)\quad &\gamma'_{3j}=\gamma_{3j}-\frac{1}{3}\varepsilon_2,\\[6pt] 2)\quad &\gamma'_{2j}=\gamma_{2j}-\frac{1}{3}(\varepsilon_1+\varepsilon_2), &\qquad 4)\quad &\gamma'_{4j}=\gamma_{4j}-\frac{1}{3}(\varepsilon_1+\varepsilon_2), \end{aligned} \tag{22} \]

where \(\varepsilon_1,\varepsilon_2\) are the spherical excesses of the auxiliary triangles \(BAG\), \(BVG\). Now, regarding the lengths \(d_{1j}, d_{2j}\) of two sides in each such plane quadrilateral \(A'B'V'G'\) as known, we find, according to (21), the lengths \(d_{3j}, d_{4j}\) of its remaining two sides. The problem is solved.

Form of the side conditions in surface ray networks of various structures. A comparison of the equalities (20), (21) written above shows that the expressions for the sides being determined in quadrilateral cells have a more complicated form than in triangular cells. It is therefore desirable to determine how the nonuniformity of the equalities (21) affects the final expressions of the side conditions in ray networks of various structures. For definiteness, we shall restrict ourselves here to plane ray networks, since the more general case of spheroidal ray networks will not differ in any essential way from the one indicated.

Suppose first that the chain of cells between the given sides \(\breve{B}_{ab}\) and \(\breve{B}_{km}\) of the ray network consists only of triangles (see Fig. 2a, where \(\breve{B}_{ab}=9\cdot 10\), \(\breve{B}_{km}=11\cdot 12\)). Then the length \(\overset{\circ}{d}_{km}\), entering into (19a), of the value \(\overset{\circ}{B}_{km}\) found along the chain for the split side \(\breve{B}_{km}\) is expressed through the observed angles \(\gamma_{1j},\gamma_{2j}\) very simply:

\[ \overset{\circ}{d}_{km}=d_{2,n}=\bar d_{1,1}\prod_{j=1}^{n}\frac{\sin\gamma_{2,j}}{\sin\gamma_{1,j}}, \tag{23} \]

where \(\bar d_{1,1}=d_{ab}\) is the length of the initial side \(\breve{B}_{ab}\), and \(n\) is the number of triangles in the chain. For example, in the chain of triangles shown in Fig. 3a, the length \(\overset{\circ}{d}_{11\cdot 12}\) corresponding to the value \(\overset{\circ}{B}_{11\cdot 12}\) of the split side \(\breve{B}_{11\cdot 12}\) is written, in expanded form, as follows:

\[ \overset{\circ}{d}_{11\cdot 12} = d_{9\cdot 12} \frac{ \sin 2 \sin 5 \sin 8 \sin 11 \sin 14 \sin 17 \sin 20 \sin 23 \sin 26 \sin 29 }{ \sin 1 \sin 4 \sin 7 \sin 10 \sin 13 \sin 16 \sin 19 \sin 22 \sin 25 \sin 28 }. \tag{a} \]

Let us now suppose that, in the chain of cells connecting the initial side \(\breve{B}_{ab}\) and the terminal split side \(\breve{B}_{km}\) of the ray network, there are not only triangles but also quadrilaterals (Figs. 3b and 3c). Then from (20) and (21) it follows that, in this case, the length \(\overset{\circ}{d}_{km}\) of the value \(\overset{\circ}{B}_{km}\) found along the chain for the split side \(\breve{B}_{km}\) is determined

through the observed angles \(\gamma_{ij}\) by an expression consisting of several terms of type (a). For example, in the chain of mixed cells shown in Fig. 3b, we obtain the following expression for the computed chain length \(\overset{\circ}{d}_{16.17}\) of the constructed terminal side \(\overset{\circ}{B}_{16.17}\), found from the observed angles \(\gamma_{ij}\):

\[ \begin{aligned} \overset{\circ}{d}_{16.17} &= d_{14.15}\, \frac{\sin 28 \sin 32 \sin 36 \sin 40 \sin 44} {\sin 29 \sin 33 \sin 37 \sin 41 \sin 45} + \\[4pt] &\quad + d_{14.2}\left[ \frac{\sin(29+30)\sin 32 \sin 36 \sin 40 \sin 44} {\sin 29 \sin 33 \sin 37 \sin 41 \sin 45} +\right. \\[4pt] &\quad + \frac{\sin 2 \sin 5 \sin 9 \sin(33+34)\sin 36 \sin 40 \sin 44} {\sin 1 \sin 4 \sin 7 \sin 33 \sin 37 \sin 41 \sin 45} + \\[4pt] &\quad + \frac{\sin 2 \sin 5 \sin 8 \sin 11 \sin 15 \sin(37+38)\sin 40 \sin 44} {\sin 1 \sin 4 \sin 7 \sin 10 \sin 13 \sin 37 \sin 41 \sin 45} + \tag{б}\\[4pt] &\quad + \frac{\sin 2 \sin 5 \sin 8 \sin 11 \sin 14 \sin 17 \sin 21 \sin(41+42)\sin 44} {\sin 1 \sin 4 \sin 7 \sin 10 \sin 13 \sin 16 \sin 19 \sin 41 \sin 45} + \\[4pt] &\quad \left. + \frac{\sin 2 \sin 5 \sin 8 \sin 11 \sin 14 \sin 17 \sin 20 \sin 23 \sin 27 \sin(45+46)} {\sin 1 \sin 4 \sin 7 \sin 10 \sin 13 \sin 16 \sin 19 \sin 22 \sin 25 \sin 45} \right] = \\[4pt] &= d_{14.15}\cdot \varphi_1 + d_{14.2}\cdot \varphi_2 . \end{aligned} \]

Here \(d_{14.15}\) is the length of the initial side \(B_{14.15}\), and \(d_{14.2}\) is the additional datum \(\nu_1\) of the directed ray network of the first kind under consideration, found from the necessary initial data. Thus, the expression we have obtained for \(\overset{\circ}{d}_{16.17}\) contains fixed, observed, and additional data; i.e., it will have the general form of the function \(C_{k\ell}(\bar{\nu}_1,\bar{\nu}_2,\ldots,\bar{\nu}_s;\bar{\gamma}_1,\bar{\gamma}_2,\ldots,\bar{\gamma}_n;\bar{d}_{ab})\) in (19), if the observed and adopted quantities are replaced by their adjusted values.

At the same time, it is not difficult to establish that if the initial \(B_{ab}\) and terminal \(B_{k\ell}\) sides in the chain of cells coincide (Fig. 3c), then the fixed and additional data do not, in the final count, enter into the lateral relations (19).

Conclusion. Summing up what has been said above and relying on a comparison of equalities (a) and (b), we arrive at the conclusion that the lateral conditions (19) for chains of mixed cells are formed more complicatedly and have a considerably more cumbersome form than for chains consisting of triangles alone.

§ 6. Vertex Conditions

Finally, let us consider the last type of primary relations of connection (1), (3) in surface ray networks, namely, the vertex conditions.

Vertex conditions connect in a ray network individual sets of its adjusted vertices. These conditions arise when there are split vertices \(K\) in the ray network, connected with redundant rays \(\overset{\circ}{L}_{pk}\) observed at them. Such split…

vertices \(\breve K\) have the property that, for each of them, at least two positions can always be established: 1) \(\dot K\)—the control position, specified by us; 2) \(\mathring K_n\)—the constructed position. At the same time, with respect to the second, constructed position \(\mathring K_n\) of the split vertex \(\breve K\), it is known only that the point \(\mathring K_n\) lies on the redundant ray \(\delta \mathring L_{nk}\), which is obtained by transmission along the network according to the necessary initial \(\zeta_i, \bar\chi_\mu\) and supplementary \(\chi_\nu\) data, starting from a pair of some initial vertices \(A, B\). Therefore, as in the case of supplementary conditions, we shall take as \(\mathring K_n\) the base of the cross-section—the arrow \(K\mathring K_n = \uparrow \lambda_{nk}\), dropped from \(\breve K\) onto the redundantly constructed ray \(\delta \mathring L_{nk}\).

In the general case, several redundant rays \(L_{n_h k}\) from neighboring determining vertices \(P_h\) may be observed at the split vertex \(K\). Then each pair consisting of the control position \(\dot K\) of the split vertex \(\breve K\) and the constructed position \(\delta \mathring L_{n_h k}\) for one of these rays \(\mathring L_{n_h k}\) will correspond to its own vertex condition of a form close to (15):

\[ \overline{K}_n-\overline{K}=\overline{l}_{nk}=0, \tag{24} \]

where the overbar denotes adjusted data.

What has been said may be explained by the example of the ray chain shown in Fig. 5b. Here the control position \(\dot K\) of the split vertex \(\breve K\) is the fixed vertex \(\overline{12}\), while the redundant rays \(L_{n_h k}\) observed to it will be the rays \(\mathring L_{14,\overline{20}}\), \(\mathring L_{15,\overline{20}}\), \(\mathring L_{16,\overline{20}}\). Thus, in this network there are three vertex conditions associated with the indicated fixed vertex \(\overline{20}\).

In surface ray networks, cases are also possible in which the control positions \(\dot K\) of individual split vertices \(\breve K\) are redundant for the network in combination with some adjoining rays \(L_{n_h k}\), while in combination with other adjoining rays \(L_{n_j k}\) these points \(\dot K\) must be regarded as necessary. Such a dual significance is possessed, for example, by the fixed vertex \(\overline{12}\) in Fig. 5a, which depicts a directed ray network of genus \(s=1\). It is easy to see that this vertex \(\dot K=\overline{12}\) is, for the given network, both necessary and redundant at the same time. If we take the ray \(\delta \widetilde L_{10,\overline{12}}\) as the residual \(\delta \widetilde L_{nk}\), and the ray \(\delta \mathring L_{8,\overline{12}}\) as the redundant \(\delta \mathring L_{nk}\), then in combination with the first of these rays the vertex \(\overline{12}\) must be recognized as necessary, and together with the second ray—as redundant.

Let us now clarify the essence of vertex conditions in surface ray networks.

Denote by \(\Sigma\) the complete expanded set of initial data \(\zeta_i, \bar\chi_\mu\) in a ray network. Let \(\omega\) be some selected partial set of initial data \(\zeta_i, \bar\chi_\mu\) from \(\Sigma\), with the help of which the control positions \(\dot K\) of the split vertices may be specified in the ray network.

split vertices \(\breve K\), and all excess rays \(\delta\bar L_{nk}\) connected with these vertices \(\breve K\) have been constructed uniquely. Next, let \(\omega'\) denote the set of data \(\varsigma_i,\bar\gamma_\mu\) that is obtained from \(\omega\) by discarding the \(s\) angular quantities \(\varsigma_h\) that specify the directions of all \(s\) residual rays \(\delta\bar L_{nk}\) in the ray network of order \(s\) under consideration. Finally, let \(\omega''\) denote a derived set of the form

Fig. 5. Examples of failure of the vertex conditions.

\[ \omega''=(\nu_1,\nu_2,\ldots,\nu_s;\omega'), \tag{25} \]

where \(\nu_1,\nu_2,\ldots,\nu_s\) are the supplementary data of our network. Then the vertex conditions of this network are expressed by the requirement that, after adjustment of the network, all excess rays \(\delta\bar L_{nk}\), constructed from the adjusted value \(\omega''\) of the above-indicated subset \(\omega''\) of the extended data, should pass exactly through the adjusted reference positions \(\bar K\) of the corresponding split vertices \(\breve K\). Hence there follows the following general form of the vertex conditions of a ray network:

\[ \overset{\circ}{\lambda}_{nk}=B_{nk}(\bar\omega'')=B_{nk}(\bar\nu_1,\bar\nu_2,\ldots,\bar\nu_s;\bar\omega')=0, \tag{26} \]

where \(\overset{\circ}{\lambda}_{nk}\) is the directed transverse displacement, relative to the \(\bar K\) connected with \(\bar K\), of the balanced redundant ray \(\delta\overset{\circ}{\Pi}_{nk}\).

Thus we see that, according to (26), the transverse displacement \(\overset{\circ}{\lambda}_{nk}\) can be represented by a certain function \(B_{nk}\) of the corresponding necessary set \(\omega''\) of balanced values \(\nu_g, \zeta_i, \bar\chi_\mu\) of the additional and initial extended data of the ray network. It follows from this that the functions \(B_{nk}\), in their structure, are related to the functions \(D_{nk}\), which express the transverse displacements \(\overset{\sim}{\lambda}_{nk}\) of the balanced residual rays \(\delta\overset{\sim}{\Pi}_{nk}\) [see (15)]. Therefore the expanded representation of the vertex conditions can be carried out by the same two general methods that were indicated in § 2 for the additional conditions (15) or (15a). The degree of complexity of the vertex relations (26) depends on the particular features of the structure of the corresponding ray network. We shall consider this question in greater detail in the following paragraph.

Owing to the errors of the initial data, of the observed \(\zeta_i\) and the fixed \(\bar\chi_\mu\), the transverse displacements \(\overset{\circ}{\lambda}_{nk}\) of the redundant rays \(\delta\overset{\circ}{\Pi}_{nk}\), constructed from the necessary set \(\omega''\) of unbalanced extended data, will not strictly satisfy condition (26). Therefore, instead of the indicated equality, we shall then obtain

\[ \overset{\circ}{\lambda}_{nk} = B_{nk}(\omega'') = B_{nk}(\nu_1,\nu_2,\ldots,\nu_s;\omega'') = w_{B_{nk}} \ne 0, \tag{26a} \]

where \(\nu_g\) are the values of the additional data, coordinated with a certain necessary set \(\sigma\) of extended initial data [see (16)], and \(w_{B_{nk}}\) is the misclosure of the vertex condition under consideration. These misclosures are shown together with the corresponding ray network in Fig. 5c, and separately as actual quantities in Fig. 5d; moreover, the sign of the misclosure \(w_{B_{nk}}=\overset{\circ}{\lambda}_{nk}\) is established according to rule (17). It follows from the preceding that each misclosure of a vertex condition \(w_{B_{nk}}\) is found by its successive computation along the chain of cells connecting the initial pair of vertices \(\dot A,\dot B\) with the corresponding split vertex \(\dot K\).

In the general case, the adopted position \(\dot B\) of the initial vertex \(B\) does not coincide with the adopted position \(\dot K\) for the split vertex \(K\):

\[ \dot B \ne \dot K, \]

and then we shall speak of an external vertex condition. Just such a case occurs in open chains of cells (see Fig. 3a, where \(\dot A=9,\ \dot B=10\), and \(K \supset K_1,K_2\) will be, respectively, 11 and 12).

Occasionally, however, cases are possible in which the adopted positions \(\dot A,\dot B\) of the initial vertices \(A,B\) are combined with the adopted positions \(\dot K_1,\dot K_2\) of the split vertices \(K_1,K_2\) of the network:

\[ \dot A=\dot K_1,\qquad \dot B=\dot K_2. \]

(see Fig. 5d, where $\dot A=\dot K_1=1$, and $\dot B=\dot K_2=2$). Ray networks with coincident initial $\dot B$ and final supporting $\dot K$ vertices are also sometimes encountered:

\[ \dot B=\dot K \]

(see Fig. 5a, where $\dot A=\bar{1}$, $\dot B=\dot K=\bar{12}$). The vertex conditions arising in all such cases will be called internal.

§ 7. Interchangeability of side and vertex conditions

In the preceding section it was indicated that each split vertex $\breve K$, together with the corresponding redundant ray $\mathring L_{n_h,k}$, gives one vertex condition. It follows that if, in a given ray network, there are $e$ pairs $\breve K$, $\mathring L_{n_h,k}$ of such interconnected split vertices and redundant rays, then $e$ vertex conditions arise in this network.

On the other hand, in § 5 it was established that each split side $\breve B_{\kappa\mu}$ of the network leads to one side condition. Thus, with $a$ split sides $\breve B_{\kappa\mu}$, a total of $a$ side conditions will appear in the ray network.

Suppose now that, in the given ray network, there are $a$ split sides $\breve B_{\kappa\mu}$ and $e$ corresponding pairs $\breve K$, $\mathring L_{n_h,k}$ of split vertices and redundant adjacent rays. Then in such a network, besides the ray and additional conditions, there will arise only $e$ independent conditions of some kind—in number, $e$ adjacent-redundant rays $\mathring L_{n_h,k}$. This follows quite obviously from the consideration that if we equalize all the ray and additional conditions of the given network and at the same time ensure the passage of the equalized redundant rays $\mathring L_{n_h,k}$ through the corresponding equalized supporting positions $\dot K$ of the split vertices $K$, then the network is completely equalized.

The question arises how to reconcile the last assertion about the number of remaining independent conditions with the circumstance that the network has $a$ split sides $\breve B_{\kappa\mu}$ and $e$ pairs $\breve K$, $\mathring L_{n_h,k}$ of split vertices and redundant adjacent rays, with each value of a split quantity giving one condition.

The way out of the contradiction thus arising is contained in two easily proved propositions on the interchangeability of side and vertex conditions in surface ray networks.

Proposition one (direct). Any side condition of a ray network, supplied by some split side $\breve B_{\kappa\mu}$, can always be expressed through one or two vertex conditions corresponding to the adopted position of one or both ends of the given side $\breve B_{\kappa\mu}$.

Examples: a) the side condition in Fig. 3c for the split side $\breve B_{2.3}$ can be expressed through the condition that the ray $\mathring L_{7.3}$ pass through the supporting position $\dot 3$ of the split vertex $\breve 3$;

b) the side condition in Fig. 3a for the split side \(\check{B}_{\overline{12}.\overline{11}}\) can be expressed through the condition that the rays \(\delta \mathring{L}_{8,\overline{12}}\), \(\delta \mathring{L}_{8,\overline{11}}\), or \(\delta \mathring{L}_{8.12}\), \(\delta \mathring{L}_{7.11}\) pass through the solid vertices \(\overline{11}\), \(\overline{12}\).

Second proposition (converse). A vertex condition, supplied by the accepted position of one of the ends of the split side \(\check{B}_{km}\) of the ray network, can always be expressed either through a single side condition for \(\check{B}_{km}\), or else is expressed by this side condition together with the vertex condition with respect to the accepted position for the second end of the side \(\check{B}_{km}\).

Examples. a) The vertex condition in Fig. 3c, corresponding to the passage of the ray \(\delta \mathring{L}_{7.3}\) through the reference position \(\dot{3}\) of the split vertex \(3\), can be expressed through the side condition for \(\check{B}_{2.3}\);

b) the vertex condition in Fig. 3a, corresponding to the passage of the ray \(\delta \mathring{L}_{7.\overline{11}}\) or \(\delta \mathring{L}_{8.\overline{11}}\) through the solid vertex \(\overline{11}\), is expressed jointly by the side condition for \(\check{B}_{\overline{11}.\overline{12}}\) and the condition that the ray \(\delta \mathring{L}_{8.\overline{12}}\) pass through the solid vertex \(\overline{12}\).

Relying on the two almost obvious propositions stated here, we now arrive at the conclusion that the contradiction noted above does not in fact exist. It turns out that everything depends on which of the conditions of the given ray network—side conditions or the corresponding vertex conditions—are to be regarded as independent.

The choice of \(e\) independent conditions corresponding to \(e\) incident-redundant rays \(\delta \mathring{L}_{n_h k}\) can be made in several ways when the ray network contains \(a\) split sides \(\check{B}_{k.m}\) and \(e\) corresponding pairs \(\check{K}\), \(\mathring{L}_{n_h k}\) of split vertices and redundant incident rays. The following three methods will be basic here:

1) Regarding the \(a\) side conditions as dependent, take as independent all \(e\) vertex conditions of the given network.

2) Regarding as dependent \(a\) vertex conditions expressible through side and certain other vertex conditions, take as independent the \(a\) side conditions and the \((e-a)\) remaining vertex conditions.

3) Regarding as dependent only \(f\) side conditions out of their total number \(a\), take as independent \((a-f)\) side conditions and \([e-(a-f)]\) remaining vertex conditions not connected with the selected \(f\) side conditions.

As for the choice from among these three methods of the one most suitable for the given ray network, here the decision is determined by the peculiarities of the internal structure of the network under consideration. One may, however, hold that if the network has only side conditions and consists solely of triangular cells, then method 2 should be followed. If, on the other hand, in the network, in addition to side conditions, vertex and additional conditions also arise and, especially, if the network is formed by mixed cells, then it is more advantageous to apply method 1 to such a network.

§ 8. Degree of complexity of vertex conditions in various surface ray networks

Let us now determine the degree of complexity of the expanded representation of the vertex conditions (26)

\[ \overset{\circ}{\lambda}_{nk} = B_{nk}(\bar{\omega}'') = B_{nk}(\bar{\nu}_1,\bar{\nu}_2,\ldots,\bar{\nu}_s;\bar{\omega}') = 0 \]

for various surface ray networks. As was already noted above, the degree of this complexity depends entirely on the structural features of the corresponding ray network.

The simplest form is possessed by the vertex conditions \(\overset{\circ}{\lambda}_{nk}=0\) for chains of triangles of genus \(s=0\) (Figs. 3a, 5a). For such chains, papers [3], [4] indicate rapid procedures for composing vertex conditions of a form somewhat different from (23), namely:

\[ \overset{\circ}{\bar u}{}^{1}_{k}-\bar u^{1}_{k}=0 \qquad \overset{\circ}{\bar u}{}^{2}_{k}-\bar u^{2}_{k}=0. \tag{27} \]

Here \(\overset{\circ}{\bar u}{}^{1}_{k}\), \(\overset{\circ}{\bar u}{}^{2}_{k}\) are the surface coordinates of the constructed position \(\overset{\circ}{\bar K}\) of the split vertex \(K\), which is obtained from the necessary equations, starting from some pair \(A,\dot B\) of initial vertices. The quantities \(\bar u^{1}_{k}\) and \(\bar u^{2}_{k}\) are the surface coordinates of the accepted adjusted position \(\bar K\) of the same split vertex \(\breve K\).

Nor will it be especially difficult to compose the vertex condition \(\overset{\circ}{\lambda}_{nk}=0\) for a chain of triangles laid between fixed vertices \(\bar A=A\) and \(\bar B=B=\bar K\) (shown in Fig. 5a, where \(\bar A=11\), \(\bar K=12\)). For a chain of this type, book [5] proposes taking, as the vertex condition, the second of equalities (27),

\[ \overset{\circ}{\bar u}{}^{2}_{k}-\bar u^{2}_{k}=0, \]

but in a conditional reference support with the axis \(\uparrow U^{1}\) along the straight line \(\dot A\dot K\) and the axis \(\uparrow U^{2}\) transverse to \(\dot A\dot K\).

However, the complexity of the expanded representation of the vertex conditions \(\overset{\circ}{\lambda}_{kn}=0\) increases sharply as soon as we pass from chains of triangles to chains of mixed cells (Figs. 5b and 5c). This follows from the fact that, in networks of triangles, neighboring connecting chains are constructed independently of one another, whereas in networks of mixed cells each subsequent connecting chain usually rests on all preceding connecting chains. Therefore, regardless of the method of composing them, vertex conditions in networks of mixed cells will contain a significantly larger number of angular quantities \(\varepsilon_i\) than in networks of triangular cells. Hence follows the cumbersomeness of vertex conditions in ray networks of general form. This conclusion also follows indirectly from a comparison of the particular expressions (a) and (b) on pp. 93 and 94.

§ 9. Comparison of supplementary and vertex conditions in surface ray networks

It was indicated above that, in their significance for a ray network, supplementary and vertex conditions differ sharply from one another.

A set of \(s\) supplementary conditions in a surface ray network of the \(s\)-th kind reflects fundamental features in the internal structure of this network. According to [1], the determinant of the indicated set establishes the degree of inaccuracy in constructing the network from the chosen aggregate of necessary initial data, taken as error-free. Therefore, supplementary conditions for \(s>0\) arise in a ray network even in the case when the composition of the initial data is strictly necessary.

In contrast to what has been said, vertex conditions arise in a surface ray network only when the composition of its initial data is enlarged, namely when redundant adjoining rays \(\delta \overset{\circ}{L}_{nk}\), associated with the corresponding split vertices \(\breve{K}\), appear in the network. Therefore, for \(s>0\), vertex conditions serve as an additional means for estimating the quality of the construction of the network from the specified initial data. In the case \(s=0\), however, the vertex conditions are the only means of such an estimate.

Despite the rather substantial internal differences noted, these two types of network conditions are externally very similar to one another. They express one and the same requirement: the passage of some constructed adjoining ray \(\delta \tilde{L}_{nk}\) through the reference position \(\dot{K}\) of the corresponding split vertex \(\breve{K}\). Hence it follows that, when the composition of the initial data is enlarged, there is no point in somehow specially separating the residual rays \(\delta \tilde{L}_{nk}\) from the redundant \(\delta \overset{\circ}{L}_{nk}\). All the more so since such a division in the indicated case cannot be carried out unambiguously even when the network contains only one residual and one redundant adjoining ray (unless one takes into account the sufficient condition, established in [1], for the constructibility of a ray network of the \(s\)-th kind). Therefore, in what follows, when the composition of the initial data in a ray network is enlarged, we shall denote the aggregate belonging to it of residual \(\delta \tilde{L}_{nk}\) and redundant adjoining \(\delta \overset{\circ}{L}_{nk}\) rays uniformly by \(\delta \tilde{L}_{nk}\). The indicated rays \(\delta \tilde{L}_{nk}\) themselves will be called, in generalized fashion, check rays. Thus,

\[ \delta \tilde{L}_{nk} \supset \delta \tilde{L}_{nk},\ \delta \overset{\circ}{L}_{nk}. \]

Let us also note that the aggregate of data \(\omega''\) from which the residual rays \(\delta \tilde{L}_{nk}\) are constructed constitutes only a part of the aggregate of data \(\omega''\) specifying the redundant adjoining rays \(\delta \overset{\circ}{L}_{nk}\). Therefore, the set of check rays \(\delta \tilde{L}_{nk}\) is completely determined by one and the same general aggregate \(\omega''\) of initial data \(\gamma_g,\ \zeta_i,\ \bar{\chi}_\mu\). We shall use this circumstance below when compiling the set of mixed finite-difference correction equations corresponding, in a surface ray network, to the initial set of supplementary and vertex conditions.

§ 10. Comparative evaluation of the principal types of initial conditions and the conclusions following from it

In concluding the article, we shall give a final comparative evaluation of the four principal types of initial conditions in surface ray networks considered above and, on this basis, determine the most important task of our further investigations.

First of all, let us note that the free terms of the indicated conditions are quantities of only two kinds—either an angle or a length. Namely, the free terms of the ray conditions are angles, while lengths are the free terms of the other three conditions. On this basis the ray conditions may also be called angular, while the supplementary, side, and vertex conditions may naturally be combined under the general name length conditions.

Next, let us emphasize the essential circumstance that, in order to formulate the ray conditions, it is not at all necessary to know the actual outline of the ray network being adjusted, whereas the formation of the other three conditions is impossible without a concomitant construction of this network in true or reduced dimensions.

Finally, let us draw attention to the fact that the ray conditions directly relate the angular quantities themselves and therefore have a very simple form (18). In contrast to this, the other three conditions, called above collectively length conditions and represented by the equalities (15), (19), and (26), have the feature that each of them expresses some segment \(\tilde{\lambda}_{nk}\), \(\overset{\circ}{d}_{nk}\), or \(\overset{\circ}{\lambda}_{nk}\) as a function of the initial and supplementary adjusted data of the network. Therefore the length conditions, unlike the ray conditions, always have a complicated form, especially in networks with mixed cells.

Thus, from the comparison of ray and length conditions carried out here, it follows that the expanded expression of conditions of the latter type is a rather laborious and, in a number of cases, even simply impracticable task. Meanwhile, for all that follows, what we ultimately need is not the initial conditions of the ray network themselves, but their finite-difference representation in the form of planar conditional equations of corrections. At the same time, the free terms of these correction equations, and above all the length ones, could be computed without even resorting to the expansion of the initial relations. Hence it follows that, in view of the ease of formulating the angular conditions, our further task will be to find simple ways of directly obtaining the length conditional equations of corrections, bypassing the preceding laborious stage of expanding the initial length conditions. This task will be solved in my next article.

REFERENCES

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3. Urmaev N. A. Adjustment of polygons in geographic and rectangular coordinates. Trudy TsNIIGAiK, issue 1, Goskartgeodeziya, Leningrad, 1931.

4. Nuvar’ev V. S. A new method for the adjustment of triangulations. OGIZ, Moscow, Irkutsk, 1932.

5. Kell N. G. and Barinov V. A. Higher geodesy and geodetic works. Part II, State Mining Publishing House, Leningrad, Moscow, Novosibirsk, 1933.