CLASSIFICATION AND ANALYSIS OF METHODS FOR ELIMINATING TEMPERATURE EFFECTS ON MEASURING WIRES IN FIELD PRODUCTION CONDITIONS

G. N. Gushchin

The construction of all engineering structures requires the performance of geodetic work.

A special place among these works is occupied by urban and engineering polygonometry of medium accuracy (on the order of 1 : 100000—1 : 200000), necessary both for large-scale surveys and for all kinds of geodetic setting-out during the construction of structures. At the same time, a characteristic feature of departmental (especially underground) polygonometry is the required high accuracy with very short traverse sides and unfavorable external conditions for measuring lengths.

At present, the required accuracy of linear measurements in departmental polygonometry can be achieved in two ways:
1) with the aid of measuring wires (the geometric method);
2) with the aid of light and radio rangefinders (the wave method).

I. Comparative Characteristics and Accuracy of the Two Basic Methods of Measuring Lengths—the Geometric and the Wave Methods

Recently, new methods for determining line lengths, developed on the basis of radio electronics, have been intensively introduced into production. This is facilitated by a number of organizational and technical advantages of the wave method of measuring distances in comparison with the geometric method. This series of advantages becomes especially clear when measuring the lengths of long lines, several tens of kilometers in extent. Of all wave rangefinders, the “Tellurometer” is of greatest interest. The high accuracy and portability of this rangefinder promise it widespread application in the construction of state geodetic networks.

The use of the “Tellurometer” and other rangefinders in departmental polygonometry, however, will remain very limited and, in most cases, impossible. There are a number of reasons for this circumstance.

1. The technical imperfection of the equipment of modern wave rangefinders.

2. The high lower limit of the shortest distance measured by wave rangefinders—on the order of 150–200 m for favorable terrain and external conditions.

1. Low absolute and relative accuracy in measuring short distances with electromagnetic distance meters (about $\pm 10$ cm).

When measuring short distances, the electromagnetic method, compared with the geometric one, is less economical because of the following shortcomings:

1) the duration of the process of measuring short distances;
2) the complexity and high cost of electromagnetic-distance-meter equipment;
3) the comparatively great complexity of office processing of the measurement results;
4) high requirements for the qualifications of the operating personnel.

From what has been said it is clear that the conditions of production work in departmental polygonometry do not permit abandonment of precise methods of directly measuring lines with measuring wires or tapes. The wire as a measuring instrument is indispensable in underground polygonometry, in tunnel construction, in measuring the depth of a mine shaft, in solving special problems where precise measurement of short line lengths is required, in setting out specified lengths, etc.

To determine line lengths with the accuracy indicated above when measuring them with wires, it is necessary to take into account a number of corrections and, first of all, those for the inclination of the line, temperature, and comparison of the wire.

The listed corrections, except for the temperature correction, can be determined fairly easily and accurately. Accounting for the temperature correction, however, remains approximate, mainly because of the crude measurement of the wire temperature.

The problem of accounting for temperature in linear measurements has always been a subject of special attention for geodesists. Many specialists have studied and continue to study it.

With the appearance of the Ederin–Guillaume base apparatus, two principal directions emerged in the search for ways to eliminate the effect of temperature on measuring wires:

a) development and improvement of methods for determining the temperature of wires during the measurement process;
b) creation and improvement of methods for manufacturing such wires that they are almost entirely not subject to linear change under the influence of temperature changes.

II. Comparative characteristics of invar and steel measuring wires

At present, in the performance of high-precision linear measurements in all countries, exclusively invar wires or tapes are used.

In measuring line lengths of low and medium accuracy, however, steel and invar measuring instruments are used. In addition, in solving a number of special problems—for example, measuring the depth of a mine shaft, etc.—exclusively steel wires are used.

Invar wires possess a number of excellent metrological properties: they scarcely oxidize, have a high modulus of elasticity, and a very small coefficient of linear thermal expansion $\alpha$. At the same time, invar as an alloy has a number of negative properties. The principal drawback of invar wires is their low stability, arising from the instability of the microstructure of invar. This is manifested primarily in the fact that:

a) an invar wire, over time, continuously and irregularly lengthens;

b) the temperature coefficients $\alpha$ and $\beta$ of invar wires are not constant quantities; they must be determined for each wire separately and at known intervals of time;

c) the technology for manufacturing invar wires is complex, lengthy, and costly.

Taking into account the noted shortcomings of invar wires, it would be economically more advantageous, for medium-accuracy line measurements, to carry them out with steel wires, just as, for example, the measurement of angles in theodolite traverses of ordinary accuracy is carried out with ordinary rather than precision theodolites. In addition, steel wires, unlike invar wires, have greater molecular stability and therefore retain their length considerably better during the measurement process. Finally, and most importantly, steel wires have a stable temperature-expansion coefficient $\alpha$ close to the tabulated value.

The value of $\alpha$ for the grades of steel used in the manufacture of measuring instruments ranges from 0.0000115 to 0.0000125 deg$^{-1}$ [8]. Table 1 gives the permissible limiting errors in determining the coefficient $\alpha$ as a function of the accuracy of length measurement and of the difference between the temperatures of wire comparison and of its measurement.

Table 1

A study of Table 1 shows that, when measuring lines with steel wires, in most cases of departmental traverse surveying the coefficient $\alpha$ may be taken directly from a handbook. A very small portion of the errors indicated in Table 1 exceeds 10 units of the seventh decimal place, i.e., the values within whose limits the tabulated values of the coefficient $\alpha$ differ from one another. This circumstance significantly simplifies and reduces the cost of carrying out linear measurements with steel wires. Further, it is quite clear that if we indicate in the certificate of a measuring wire the grade of steel from which it is made, then, by selecting the coefficient $\alpha$ directly from the handbook for the specified steel grade, we thereby gain the possibility of dispensing altogether with special or repeated investigations for determining $\alpha$. The results of comparing a considerable number of steel wires on the MIIGAiK comparator confirm this proposition.

But steel measuring wires, having a large value of the coefficient $\alpha$, are subject to considerable temperature influence. For example, a change in the temperature of a 24-meter steel wire by $\pm 1.0^\circ\text{C}$ causes a change in its length by $\pm 0.3$ mm.

Thus, the development of methods for accurately determining the temperature of steel measuring wires, using for this purpose the current state of thermometric technology, is of great importance. The question of increasing the accuracy with which the temperature of a measuring instrument is taken into account is of practical significance also in work carried out with invar

wires, for example, in solving special problems when it is necessary to obtain very high accuracy in linear measurements.

III. Reasons for the Difficulty of Determining the Temperature of Measuring Wires

Temperature is a parameter of the state of a body and characterizes the degree to which it is heated. In the general case, the temperature \(t\) at a given point \(M\) of a body is expressed by some function of the coordinates \(x, y, z\) of the point \(M\) and the instant of time \(\tau\):

\[ t = F(x, y, z, \tau). \tag{1} \]

A measuring wire, having a length of 24 meters and an extremely small cross-sectional area in comparison with its length, equal to \(3.14 \times 10^{-6}\) sq. m, may be regarded as a thread. Taking into account also the circumstance that the measuring wire has a large surface of contact with the surrounding medium, equal to 0.15072 sq. m, we have every reason to consider its temperature \(t_{\text{pr}}\) a function of the distance \(x\) along the wire and of the instant of time \(\tau\):

\[ t_{\text{pr}} = f(x, \tau), \tag{2} \]

where the form of the function \(f\) depends on the thermal conductivity of the metal from which the wire is made.

Equation (1) may also serve to express the temperature field of the medium surrounding the wire. Denoting the temperature of this medium by \(t_{\text{sr}}\), we shall have:

\[ t_{\text{sr}} = \varphi(x, y, z, \tau). \tag{3} \]

Equations (2) and (3) show that, in determining the temperature of a wire located under production conditions, we are dealing with a one-dimensional nonstationary temperature field of the measuring wire and a three-dimensional nonstationary temperature field of the surrounding medium.

The difficulty of determining the temperature of the wire \(t_{\text{pr}}\) is explained by a number of reasons.

First of all, the circumstance has an effect that the measurement of the wire temperature is carried out while the state of the surrounding air is continuously changing: in cloudy, variable, sunny, calm, and windy weather.

Further, the accuracy of measuring the temperature of the wire depends on the type of temperature sensor and on the method of its application. Here the following point is most essential: whatever principle the thermometer may be based on, it indicates its own temperature. Therefore, it is necessary first of all to implement such a method of its application as would ensure equality of the temperatures of the wire and the heat-sensitive part (sensor) of the thermometer. At the same time it is very important to exclude the effect of the surrounding medium (sun rays, wind) directly on the heat-sensitive part of the thermometer. The only exception is the case when the sensor is the measuring wire itself (for example, Berleev’s method).

IV. Various Methods for Eliminating the Temperature Effect on Measuring Wires

In 1884, Ederin proposed that line lengths be measured with measuring wires. In doing so, to determine the temperature of the wires he used the idea of Bessel’s metallic thermometer, but in a new form.

Around 1898, through specially organized work at the International Bureau of Weights and Measures in Breteuil, under the direction first of Benoît and later of Guillaume, it proved possible to obtain invar.

The improved Yederin instrument with invar wires was first tested in 1898 during the Swedish-Russian expedition to Spitsbergen. Later, Russian geodesists used the Yederin-Guillaume instrument in 1903–1904 when measuring the Termez and Samarkand base lines. The work was directed by the military geodesist D. D. Gedeonov. In 1907, under the direction of the same D. D. Gedeonov, the Kazalinsk base line was measured. Subsequently, until 1917, Russian military geodesists Osipov, Seliverstov, Pavlov, and O. G. Dits were engaged in the application and improvement of this instrument. After 1921 and until 1935, these questions were taken up by the Main Geodetic Administration. Beginning in 1935, a group of staff members of the Department of Geodesy of MIIGAiK and of TsNIIGAiK worked on mastering and studying measuring wires and tapes made of Soviet invar, as well as on developing the thermoelectric method for determining the coefficient of expansion of tapes and wires. The overall supervisor of this work was Professor A. S. Chebotarev. The most active participants in the work were Professor A. S. Yurkevich and Docents D. S. Shenn and E. N. Krasotkin. The work they carried out constituted a significant stage in the development of the country’s geodetic metrology service. As a result of this work, by 1940 Soviet geodesists had obtained domestically produced invar wires that, in their main physical and metrological characteristics, were not inferior to foreign ones.

Determination of the temperature of the measuring instrument directly at the moment of measurement was carried out by the following methods.

Thus, in 1929 F. Pavlov [6] determined the temperature of the wire with a mercury thermometer having a scale division value of 0.2°C.

In 1937 B. Sheitles, in his work [9], proposed fitting onto the working mercury thermometer a holder made from a spiral made of the same wire as the measuring instrument.

In 1936 G. N. Lyudt was the first to propose using a thermocouple for measuring the temperatures of a measuring instrument [2].

In 1944, the problem of determining the temperature of a measuring instrument was one of the central topics discussed at TsNIIGAiK at a metrological conference [12].

Subsequently, up to the present time, a number of other authors devoted attention to the development and improvement of methods for determining the temperature of measuring wires. Thus, in 1948 V. K. Mitorev carried out a study on the topic “Determination of the Temperature of Steel Tapes and Wires by the Thermoelectric Method” [3]. Somewhat later, in 1950–1951, research associates of the metrological laboratory of TsNIIGAiK, M. V. Shabat and M. E. Krasotkin [10], used a “combined instrument” to determine the temperature of invar wires; it consisted of a resistance thermometer and differential thermocouples. In this arrangement, the resistance thermometer was used to read the air temperature, and the thermocouples the difference between the temperatures of the air and the wire.

In 1954 Professor A. S. Yurkevich again published an article on accounting for the temperature factor in precise linear measurements [13].

In 1956 G. I. Berleev proposed, for determining the temperature of a measuring instrument, measuring the change in the electrical resistance of the measuring instrument itself as a function of the change in its temperature. In this case the resistance of the instrument is measured by a direct-current differential bridge constructed by G. I. Berleev [1].

In 1956 M. V. Postnikov proposed using a semiconductor micro-thermoresistance (thermistor) to measure the temperature of wires. By his order, a thermistor of the first model was manufactured in Novosibirsk [4]. In 1957, at the Department of Physics of TISI, a second model of the thermistor was manufactured [5].

In 1959 the author of the present work proposed a thermistor of the third model, which was manufactured at the Department of Physics of TISI [14].

V. Classification of Methods for Eliminating the Temperature Effect on Measuring Wires

On the basis of a study of the designs of domestic and foreign thermometers and of methods for determining the temperature of measuring wires, the author has developed a classification of methods for eliminating the temperature effect on measuring wires, which sums up to some extent the centuries-long work of specialists in creating these methods.

The proposed classification is based on:

1. The operating principle of the thermometer used.
2. The thermometric basis used for taking into account the temperature of the measuring wire.
3. The method of contact between the temperature sensor and the wire under investigation.
4. The method of reducing the linear change of the wire under the influence of temperature changes.

In their historical development, the methods for eliminating the temperature effect on measuring wires, according to the first classification criterion, have been divided into the following methods:

a) the bimetallic method, which records the temperature of wires with a metallic thermometer;
b) the thermoliquid method, comprising all methods of measuring the temperature of a measuring instrument with mercury thermometers;
c) the thermoelectric method, in which the temperature of the wire is measured by a thermocouple;
d) the method of temperature-sensitive resistances, based on the dependence of the electrical resistance of the sensor on its temperature.

According to the thermometric basis used for recording the temperature of wires at the moment when they determine the length of a line, the methods under consideration are based on measuring:

a) the relative thermal elongation of wires;
b) the thermal elongation of mercury;
c) the thermoelectromotive force of a thermocouple;
d) the electrical resistance of the temperature sensor.

According to the third criterion taken as the basis of the classification, methods for determining the temperature of wires may be represented as:

a) the method of direct contact, and
b) the method of indirect contact.

Finally, for the purpose of reducing the temperature effect on the length of wires, we have two methods:

a) night measurements of line lengths, and
b) the creation of thermostable alloys used for manufacturing measuring wires.

The proposed classification of methods for eliminating the temperature effect on suspended measuring wires may be presented in the form of a diagram.

VI. Evaluation of Various Methods for Determining the Temperature of Measuring Wires under Field Conditions

Having considered the known methods for determining the temperature of measuring wires, we see that, for measuring it under field conditions, various authors have recommended four types of thermometers: metallic, mercury, thermoelectric, and resistance thermometers.

To compare the indicated methods for determining the temperature of wires, we carried out numerous special investigations. These investigations were conducted at the geodetic test ranges of the TPI and TISI under the supervision of Associate Professor B. F. Krutoy. To determine the temperature in these experiments, the following were used: wire thermometers in all variants of their application; mercury thermometers suspended at the level of the measuring wire, both open and shaded; a copper-constantan thermocouple and thermistors of the second and third models.

The results of our investigations, as well as of investigations published by other authors, make it possible to evaluate the various methods for determining the temperature of measuring wires under field conditions.

A. Metallic thermometers

The method for determining the temperature of a wire proposed by Ederin and based on the principle of bimetallism has not found practical application because of substantial shortcomings:

1. The comparatively high values of the coefficients of thermal expansion of steel and brass wires may lead to inaccurate allowance for temperature if even small fluctuations in it are not taken into account, especially at the moment of changing wires.

2. This method is based on the assertion that, at each separately taken instant of time, the temperatures of the steel and brass wires are exactly equal to one another. In reality, however, the rate at which a wire assumes a temperature during field measurements depends on the specific heat capacity and density of the metal, on its capacity for absorption, and on its thermal conductivity. But the indicated thermal parameters of different metals are not equal in magnitude. Consequently, such an assumption leads to the appearance of systematic errors, which may amount, over the entire measured length, to several millimeters.

3. The method of temperature allowance under consideration complicates the process of linear measurements.

B. Mercury thermometers

The principal and only advantage of the thermometers under consideration consists in the simplicity of determining the temperature. But mercury thermometers, in all variants of their application, have substantial shortcomings. Let us identify the main sources of error when measuring temperature with mercury thermometers:

1. The lack of any possibility of ensuring reliable thermal contact between the thermometer and the measuring wire.

2. The considerable inertia of the thermometer.

3. The absence of clear and well-founded recommendations regarding the speed and duration of rotation of the sling thermometer.

4. Not all thermometers respond equally quickly to the temperature of the surrounding medium.

Investigations by many authors show that wire temperatures differ from the readings of a sling thermometer on average by $\pm 1.0$—

Classification of Methods for Eliminating Temperature Effects on Measuring Wires in Field Conditions

Methods for eliminating temperature effects on measuring wires

By the principle of contact

Method of direct contact

• Bimetallic principle
• Sensors as a measure of wire length

Method of indirect contact

• Sensor touches the wire near the measuring wire
• Sensor touches a section of wire near the measuring wire
• Sensor mounted in a steel sleeve, arranged alongside the wire
• Sensor mounted in a steel sleeve, arranged alongside, with an insulating [[unclear: battery/pack]]

By the principle of attenuation of temperature effects

Method of night measurements

Method of creating thermostable alloys

• Iridium platinum
• Platinum
• Invar
• Severinvar
• Neoxylvia invar

By the principle of operation of a thermometer

Use of a metallic thermometer

Bimetallic method

• Bessel thermometer
• Edrina instrument

Use of a mercury thermometer

Thermo-liquid method

• Thermostat scale
• Thermostat scale on a wooden bar
• Thermostat scale with a circumferential graduation
• Two thermometer scales
• Wire-length measuring instrument

Use of an electrothermometer

Thermoelectric method

• Thermocouple

Method of temperature-sensitive resistances

• Resistance thermometer
• Differential bridge, direct-current
• Thermistor

On a thermometric basis

Method of measuring thermal elongation of a wire

Method of measuring thermal expansion of mercury

Method of measuring EMF

Method of measuring resistance

• Chemically pure metals
• Black wires
• Semiconductors

$\pm 1.5^\circ\mathrm{C}$, reaching at the surface of the ground a maximum value of $\pm 5^\circ\mathrm{C}$ (for thermometers attached to a wooden board).

On the basis of the foregoing, we come to the conclusion that if the required limiting accuracy of measuring (or setting out) a line must be less than $1:20000$ of the length of a 24-meter span, then the method of determining the temperature of a steel wire by means of a mercury thermometer, in one or another form of its application, proves unsuitable for field production conditions.

B. Thermocouples

The principal advantages of the thermoelectric method over the thermoliquid method just considered are as follows.

First of all, there is no inertia in the readings of a thermocouple. Temperature changes are registered by the galvanometer with the speed of an electric current.

Further, it becomes possible to measure temperature while being at some distance from the measuring instrument.

Finally, unlike mercury thermometers, it is possible to achieve more reliable contact of the temperature sensor with the wire.

But the thermoelectric method has a whole series of substantial disadvantages:

1. The thermocouple measures the temperature of the wire only at the single point that it touches.

2. In addition, transition processes take place between the wire and the hot junction of the thermocouple, caused by imperfect contact (when a thermocouple is used, the working junction touches the wire).

3. Under field conditions it is almost impossible to deal with ice, and it is not always possible to obtain it.

4. The difficulty of taking into account the change in resistance of the lead wires, and the considerable volume of computational work involved in converting the galvanometer readings into degrees Celsius.

5. The necessity of introducing into the measurement result a correction for the scale divisions of the working galvanometer, corrections for the change in temperature of the cold junction and for the difference between its temperature and the calibration temperature of the thermocouple, and others. Under field conditions, some of the indicated corrections are difficult to detect.

6. The complexity and bulkiness of the apparatus.

The indicated shortcomings of the thermoelectric method of determining the temperature of a wire have led to the fact that the thermocouple, which proved excellent for measuring the temperature of a wire under laboratory conditions, has not found application in field production conditions.

C. Berleev’s Electrothermometer

The method for determining the temperature of a measuring wire proposed by G. I. Berleev has, from the theoretical point of view, an unquestionable advantage over all the methods considered earlier; namely:

1. The temperature sensor is the measuring wire itself.

2. When using the differential direct-current bridge according to the circuit proposed by Berleev, the mean temperature of the entire measuring wire is determined.

But the method under consideration for determining temperatures is not without shortcomings, the chief of which are:

1. The presence of very long lead wires. This makes the instrument bulky and inconvenient for use in the field. In addition, changes in the resistance of the lead wires due to changes in external conditions will significantly affect the accuracy of the bridge readings.

2. The unstable and unreliable contact of the lead wires with the measuring wire.

3. The differential bridge is, after all, several tens of times less sensitive than, for example, semiconductor resistances.

The presence of the noted shortcomings in Berleev’s method makes its application possible only in laboratory or experimental studies. This method has not, for the most part, found application under field production conditions, as already noted, because of the bulkiness of the instrument and the inconvenience of using it.

D. Semiconductor thermometer—thermistor.

The advantages of semiconductor thermal resistances (TS) over all the thermometers considered are:

1. High temperature sensitivity. As an example, the following data may be cited: when the temperature is raised from 0 to 100°C, the resistance of platinum increases by 35%, whereas in semiconductors it decreases by 20–70 times (depending on the temperature coefficient of TS) [7].

2. Small dimensions and, consequently, small thermal inertia of TS.

3. Extreme simplicity of construction.

4. High stability of TS characteristics over time.

5. Durability and the complete absence of any need for special maintenance during operation.

6. The large magnitude of the TS resistance makes it possible to neglect completely the resistance of the lead wires, contact resistances, and contact EMFs.

In field tests of thermistors, applying the principle of bimetallism (invar and steel wire), we were able to precalculate the actual temperatures of the steel wires that they had at the moment when the span was measured with them. The precalculated wire temperatures were then compared with the readings of the instruments under study (sling thermometers, thermistors, etc.).

Studying the discrepancies obtained between the precalculated temperatures and those indicated by the thermistors, we arrive at the following conclusions:

1. The temperature of a 24-meter wire located in a field production environment is practically identical at all its points.

2. The temperature of a suspended measuring wire in practice differs very little, if at all, from the temperature of the surrounding air. B. F. Krutoy was the first to point this out. The investigations carried out by us confirmed this proposition. The maximum differences between the precalculated wire temperatures and those indicated by the thermistor when the sensor was shaded were observed within the limits of ±0.4°C—±0.6°C; the root-mean-square deviation from 50 series of observations was ±0.2°C.

3. At the moment of determining the temperature there is no need for the sensor to touch directly the measuring wire or a nearby segment of wire; moreover, when touched, the sensor makes poor contact with the wire, and the sun’s rays falling on it overstate its readings relative to the actual temperature of the measure.

A temperature sensor placed in a casing, made in the form of a wire of steel or invar and protected from the sun’s rays by shade, gives readings close to the actual temperature of the wire.

1. The third-model thermistor is an instrument that makes it possible to measure the actual temperature of the measuring wire with an error of no more than ±0.2°C.

VII. Conclusion

Summarizing the questions considered above, we arrive at the following conclusion:

1. The conditions under which departmental polygonometry is carried out do not permit abandonment of precise methods of direct measurement of lines by means of measuring wires or tapes.

2. It is economically more expedient to measure lines with an accuracy up to 1:250,000 using steel wires. Replacing invar wires with steel ones significantly reduces the cost of measurements of the indicated accuracy.

3. Under field conditions, the use of mercury thermometers to determine the temperature of a steel measuring wire leads to reliable results only when measuring lines with an accuracy averaging 1:60,000–1:50,000. In this case it is advisable to use sling thermometers, either bare or wrapped with a spiral of wire of the same grade of steel as the measuring wire.

4. Methods for determining wire temperature based on the use of a thermocouple, a resistance thermometer, and a differential direct-current bridge, which have proved excellent under stationary laboratory conditions, are inconvenient and insufficiently reliable for use in field production conditions because of the complexity and bulkiness of the apparatus and the poor contact of the sensor with the wire.

5. The semiconductor thermometer-thermistor has serious advantages in comparison with other types of sensors. The use of a thermistor for measuring temperature will make it possible to increase the accuracy of linear measurements with steel wires by a factor of 4–5 and thereby to expand the range of their application.

6. Taking account of the temperature of the measuring instrument by means of a thermistor is of practical importance also in work performed with invar wires when solving special problems where it is necessary to obtain very high accuracy of linear measurements.

LITERATURE

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2. Liоdt G. Determination of the temperature of a steel tape by a thermoelectric thermometer. Geodezist, No. 2, 1936.

3. Mityurev V. K. Report at the first conference of young scientists of Western Siberia, 1948.

4. Postnikov M. V. Certain questions of increasing the accuracy of line measurement in geodetic work. Collection of Works of TISI, vol. II, 1957.

5. Postnikov M. V. and Shvartsman Ya. V. Instrument for determining the temperatures of measuring wires (second model). Collection of Works of TISI, vol. IV, 1958.

6. Pavlov F. Measurement of a base with steel wires. Geodezist, No. 1, 1929.

7. Thermal resistances. Semiconductor resistances with a large temperature coefficient. L., 1957.

8. Chebotarev A. S. Geodesy, part II, Geodezizdat, Moscow, 1949.

9. Sheytles B. On field determination of the linear coefficient of expansion of steel wires. Geodezist, No. 3, 1937.

1. Shabat M. V. Investigation of errors in accounting for the temperature of invar wires. Collection of Articles on Geodesy, issue III, Moscow, 1953.

2. Yurkevich A. S. Proceedings of TsNIIGAиK, issue 34, Geodezizdat, Moscow, 1941.

3. Yurkevich A. S. Metrological Conference of TsNIIGAиK. Collection of Articles on Geodesy, issue III, Geodezizdat, Moscow, 1944.

4. Yurkevich A. S. Accounting for the temperature factor in precise linear measurements. Proceedings of NIIGAиK, vol. VI, 1954.

5. Postnikov M. V. and Gushchin G. N. Determination of the temperature of measuring wires by semiconductor micro-thermoresistance. Izvestiya Vuzov, section “Geodesy and Aerial Photography,” issue 5, Moscow, 1964.