Precise Time Measurement¹

A. Scheibe, Berlin

Contents

Introduction

I. Astronomical day
    a. The Earth as an astronomical time standard
    b. Transfer of time
    c. Errors in the determination of time
    d. Time service

II. Measurement of time by means of pendulum clocks
    a. General remarks
    b. Astronomical pendulum clocks
    c. Properties of pendulum clocks

III. Measurement of time by means of quartz clocks
    a. General remarks
    b. Construction and operation of quartz clocks
    c. Technical details of the construction of quartz clocks
    d. Factors affecting the rate of quartz clocks
    e. Rate of quartz clocks

IV. Constancy of the rate of quartz clocks and constancy of the duration of the astronomical day

Conclusion

Introduction

1. Of the three fundamental units of the absolute system—length, mass, and time—only the units of length and mass have been established by special international agreements and are preserved in the form of prototypes, which as such are in no way dependent on any quantities encountered in nature; whereas, with respect to the unit of time, the tradition has so far still been maintained of taking as such \(1/86400\) part of the mean solar day. Thanks to this we find ourselves compelled to relate every physical measurement connected with time to a time scale determined by a single rotation of the terrestrial globe about its axis. In this way the necessity was avoided of establishing a time standard independent of the rotation of the Earth, but in return there arose the inconvenience that, even with the most precise measurements, we do not

¹ Ergebn. d. Exakt. Naturwiss., vol. XV, 1936. Translated by N. S. Khlebnikova.

we can directly refer to some convenient standard of time and obtain a time scale only through the medium of an auxiliary instrument—the clock.

The fact that until now people have been satisfied with such a state of affairs cannot be considered creditable to anyone, since it has long been known that the establishment, for large intervals of time, of an astronomical measure of time embodied in the motion of the Earth can provide only an approximately constant prototype of the unit of time and is associated with exceptional difficulties, and perhaps is altogether unattainable.

Generally speaking, any instrument that with the greatest accuracy repeats some periodic process could be used as a terrestrial measure of time. Unfortunately, however, in the requirement of great accuracy of repetition there lies a very serious difficulty. As practice shows, the time course of every periodic process, beginning from a certain limit of accuracy of measurement, depends perceptibly on external influences. Thus, in the search for a terrestrial standard of time, all those periodically repeating processes are excluded from consideration for which the boundary of susceptibility to external influences is such that these influences cause a disturbance of the uniformity of the course in time (as determined by the astronomical time standard).

Only two instruments, namely: astronomical pendulum clocks of the newest construction and piezoelectric quartz clocks, may at present be regarded as satisfying the requirement that the boundary of susceptibility to external influences of the periodic processes taking place in them is such that these processes may be used as a terrestrial standard of time.

1. Having raised the question of creating a terrestrial standard of time, one can appreciate the scope and difficulty of this problem only by recalling that the question of the applicability of modern pendulum clocks as a terrestrial standard of time could be posed only as the result of centuries of work by the best and most capable physicists and designers. Likewise, one should remember that this question of applicability is inevitably connected with another question: that of the invariability of the periodic process occurring in the astronomical standard (the rotation of the Earth). The comparison of the terrestrial and astronomical time standards widens the limits of investigation, putting forward the requirement that the terrestrial standard surpass the astronomical one in accuracy not only over small but also over large intervals of time. In accordance with the degree to which this condition is fulfilled, the terrestrial standard will make it possible either merely to note fluctuations in the uniformity of the Earth’s rotation, or else will serve as an instrument for studying possible gradual changes in the rate of the Earth’s revolution about its axis. Using the terminology of clockmakers, one may say that in the first of the cases mentioned above it is a matter of “random fluctuations

the rate,” and in the second—of a “systematic change in the rate” of the Earth. This determines the suitability of the Earth’s motion as a time standard.

The present article is devoted mainly to questions concerning the rate of astronomical pendulum clocks and quartz clocks and to a comparison of the results obtained with these and other results. We shall show that today the quartz clocks of the Physikalisch-Technische Reichsanstalt in Berlin1 have the best rate, and that only they make it possible to investigate the constancy of the astronomical time standard.

1. Astronomical day

a. The Earth as an astronomical time standard

1. We are accustomed to use as the basic unit of time the mean second, taking it as something self-evident. In reality, however, the need to establish the mean second, determined indirectly from the motion of the Earth, arose because of the inconstancy of the length of the solar day. The interval of time elapsing between two successive transits of the Sun across the meridian, called the apparent solar day, changes from day to day owing to the inclination of the ecliptic relative to the celestial equator and the elliptical form of the Earth’s orbit. The differences in duration between apparent solar days may reach 31 min. In addition, owing to the inclination of the ecliptic to the equator and to the inclined position of the plane of the Moon’s orbit, the direction of the Earth’s axis does not remain unchanged. The Earth’s axis describes a cone about the axis of the ecliptic (precession), completing one full revolution in 26,000 years. Moreover, superposed on this motion is a periodic elliptical motion of the Earth’s axis (nutation) with a principal period of 18.5 years. As a result of precession the line of intersection of the planes of the ecliptic and the equator, which determines the points of the vernal and autumnal equinoxes, turns annually by approximately 50 seconds of arc, as a consequence of which the vernal equinox point is displaced. The result of this is a change in the duration of the apparent revolution of the Sun around the Earth, and therefore it is customary to define the year as the interval of time between two true vernal equinoxes, for the points of which fixed points in the sky of the fixed stars are accepted once and for all. The year determined by the first method (the displacement of the vernal equinox point), called the tropical year, will be shorter than that determined by the second method and called sidereal.

To avoid, in the reckoning of time, coming into conflict with the seasons existing in nature, the tropical year, referred to the true points of the vernal equinox, was adopted as the unit of chronological reckoning. This year, by introducing the “mean sun,” i.e. a sun which would go around the Earth along the celestial equator with constant velocity, was divided into 365.242198 . . . parts—the so-called mean solar days. Mean solar days, thus, in contrast to true solar days, have equal duration. The difference between the true solar days and the mean ones, which as such are not observable and are no more than a fiction, is equal to the “equation of time,” whose magnitude, on the basis of knowledge of the forces determining the times of revolution, can be computed with sufficient accuracy.

Fig. 1. Curve of the equation of time

Figure 1 shows the course of this equation of time, constructed from the data of the Berlin Astronomical Yearbook¹ for 1936. The equation of time is the interval of time by which the true Sun is ahead of or behind the “mean sun” in its passage across the meridian. A positive sign in the equation of time means that the mean sun passes through the meridian by the corresponding interval of time earlier than the true one, i.e. that the true sun is on the meridian, for example, only at \(12^h 14^m 23^s\).

1. The mean second, serving as the principal unit of time, is \(1/86400\) part of the mean solar day. The mean second is, therefore,

\[ \frac{1}{86\,400 \cdot 242\,198}\ldots \]

part of the tropical year. As such it could be derived directly from the tropical year if there existed clocks that would run perfectly uniformly throughout an entire year and that, with a sufficient degree of accuracy, could be set according to the mean sun at the moment of the true vernal equinox.

Unfortunately, with the present state of clockmaking technology, the direct establishment of a measure of time according to the tropical year does not appear possible. Likewise, a sufficiently accurate determination of time according to true solar days does not appear possible. Therefore, for the purposes of measuring time there remain only the sidereal days, which stand in a definite relation to the mean solar days.

Sidereal days are the interval of time elapsing between two successive transits across the meridian of one and the same fixed star. In this connection one speaks of an upper or lower culmination depending on whether the transit across the meridian takes place above the pole of the world or below it. Sidereal days and mean solar days, owing to the motion of the Earth around the Sun, differ from one another by a constant amount: a sidereal day is \(86164.0905 \ldots\) mean seconds. With the aid of this conversion coefficient we obtain a unit for measuring times—despite its origin from mean solar days—from observations of the stars.

The duration of the mean solar day, determined from observations of the stars, will henceforth be called the astronomical day. The astronomical day is the time scale which we apply to terrestrial clocks in order to investigate their suitability.

b. Transfer of time

5. One of the most honorable tasks of practical astronomy has always been the transfer of the astronomical day to terrestrial clocks and the preservation of this astronomical measure for terrestrial measurements of time over as long an interval as possible. In Europe, several institutes specially intended for this purpose are occupied with this task: the German Naval Observatory (H), the Potsdam Geodetic Observatory (P), the Greenwich Observatory (G), and the Paris International Time Bureau (Pa).

In the transfer of time, certain stars serve as points of reference in the sky (“time stars”). These are fixed stars whose positions, as a result of continuous observations, are precisely known and entered in special catalogues published by astronomical institutes. Such catalogues include, for example, the Third Fundamental Catalogue of the Berlin Astronomical Society (FK3) or the Eichelberger catalogue (EK).

As these time stars there usually serve only fixed stars with upper culmination, whose transit across the meridian takes place not too far to the south or to the north of the zenith. Experience has shown that the transfer of time by stars with lower culmination, i.e. those whose transit across the meridian occurs between the pole and the north point of the horizon, can be carried out only with insufficient accuracy. The German Naval Observatory² selects stars for the purposes of time transfer in such a way that the mean position of all observed stars falls, as far as possible, at the zenith.

6. For transferring the time from a time star to a recording device, which simultaneously serves for marking the checked clocks, a special “transit instrument” is used. This apparatus is a telescope, the optical axis of which is set precisely in the plane of the corresponding meridian ...

of the point of the earth’s surface and can rotate about a horizontal axis. This axis, as is self-evident, must meet the strictest requirements with respect to rigidity and the correctness of its position. A vertically stretched thread, placed in the plane of the image, plays the role of a mark on the clock dial. The instant at which the thread coincides with the image of a time star moving through the instrument’s field of view is the time instant fixed by these astronomical clocks. The absolute value of this instant of time is called the right ascension of the corresponding star.

Transferring the instant of time at which the images of the star and the thread coincide, and actuating the pen that makes a mark on the tape of the recording device (chronograph), is carried out most accurately by means of the Repsold “impersonal” micrometer connected with the transit instrument; it marks the coincidence of the vertical thread, situated in the plane of the instrument’s image, with the image of the star moving in this plane. By means of this impersonal micrometer, the personal errors of the observer are greatly reduced or even eliminated altogether; such errors, when the passage of the star is observed simultaneously and the chronograph pen is actuated (by hand), may reach a considerable magnitude.

The transfer of the time instant and the actuation of the chronograph pen are carried out in such a way that, during the alignment of the star’s image with the thread, a drum connected with a micrometric screw and bearing contact plates, by means of that same micrometric screw, closes at equal intervals of time the electric circuit controlling the chronograph. In this way, during the passage of only one star, as many time marks are obtained on the chronograph tape as the drum has contact plates. By finding the mean of these marks, errors connected with the properties of the entire installation (telescope—chronograph), as well as errors caused by the observer’s perception, are eliminated.

1. As recording devices in the instruments under consideration, both ordinary pens with points and, in part, modern rapid recorders with rotating coils are used. On the basis of the author’s experience, it may be asserted that, under present-day heightened requirements for accuracy, only the use of pens of the second type is appropriate. For this reason, it is necessary to dwell on their properties in somewhat greater detail, especially since, when working with quartz clocks, only they can be used.

Rapid recorders with rotating coils make it possible to operate with a very small current consumption (a few milliamperes) in the recording coil and permit the use of a constant tape speed of up to 300 mm/sec. The constancy of the tape speed can be substantially increased by linking

of the motor pulling the tape, with a synchronous motor maintaining the constancy of the rotational speed of the first. In the Physico-Technical Institute the synchronizing alternating current is generated by the quartz clocks themselves. With a proper choice of the frequency of this current it is possible to achieve, for example, that the speed of motion of the chronograph tape will be exactly 100 mm/sec. In this case the distance measured in millimeters between two marks on the tape will correspond to the interval of time elapsed between them, expressed in hundredths of a second. Thus, under these conditions, measurement with the aid of a millimeter scale will directly, without any calculations, give the corresponding interval of time.

A high-speed recorder of the type under consideration, when at rest, produces a continuous line on the tape. Depending on the direction of the current impulse in the coil, the pen is deflected to the right or to the left (if one looks in the direction of motion of the tape). Thus, when several electrical circuits of different polarity and with different currents are used, with the aid of a recording device of this type it is possible simultaneously to record many phenomena and to be able to distinguish the marks belonging to each of them.

Fig. 2. Electrical circuit of the recording device

1. The accuracy with which it is possible to measure the distances between marks on the chronograph tape depends on the speed of the tape, and also on the angle at which the pen traces a mark near the zero line. As the speed of the tape increases, the accuracy increases up to the point at which the aforementioned angle does not differ greatly from 90°. At high speeds, however, this is no longer the case, unless the current impulses that set in motion the coil with the pen have such a shape that the coil, even in this case (at high speeds of motion of the tape), leaves its position of rest as sharply as possible.

These measures were adopted by Scheibe and Adelsberger³ in determining time with the aid of quartz clocks, in receiving time signals, and also in determining the difference of frequencies of a synchronous motor and quartz clocks. The method consisted in the following: (Fig. 2) the phenomenon to be recorded, for example a time signal, removes the charge from the grid of an amplifying tube, as a result of which a capacitor connected in the anode circuit discharges through the coil that sets the pen in motion. In view of the short duration of the discharge, one may, without fear of damaging the coil, choose the voltage on the capacitor and its capacitance so large that the discharge current is many times greater than the current permissible for prolonged

of passage through the coil. As a result, even at high speeds of motion of the tape, a sharp deflection of the pen from the zero line is obtained.

In this way it is possible to achieve the result that, even with a tape speed of 100 mm/sec and the use of a glass millimeter scale, the error in measuring the segment does not exceed 0.1 mm, which corresponds to an error in determining the time interval of no more than 0.001 sec.

If still greater accuracy is required, a loop oscillograph may be successfully used.

c. Errors in determining time when using a transit instrument

9. The instant of time obtained as a result of marking a signal on the chronograph tape, it goes without saying, cannot be assumed to coincide with the instant at which the star was actually in the meridian plane of the optical axis of the tube of the transit instrument. The difference between the true and the marked instants of time is the error in the determination of time. This resultant error is made up of numerous partial errors, the chief of which are: 1) errors due to refraction, arising as a result of the deviation of the ray coming from the star in the upper layers of the atmosphere, and also because of local deviations in the lower layers; 2) errors in the setting of the visual tube and errors caused by the inertia of the recording apparatus; 3) errors due to the personal characteristics of the observer.

10. Errors caused by refraction, since they are due to the layered structure of the atmosphere, cannot be eliminated even with every effort on the part of the observer. They can perhaps be estimated on the basis of meteorological data. Theoretically, according to Schott’s data⁴, these errors, with a total inclination of the layers throughout the entire thickness of the atmosphere amounting to 1° at the zenith, in the worst case lead to a displacement of the instant of time by 0.07 sec (this, however, must be a rarely occurring extreme case). Disturbances caused by refractions of a local character—for example, those due to heated parts of buildings past which the continuation of the optical axis of the tube passes—are much more difficult to estimate. With the aid of good clocks they may, however, be determined experimentally. A similar case was investigated by Freiesleben and Lange⁵ at the German Naval Observatory, whose new building is situated very close to the meridian plane of the transit instrument. They established that the systematic discrepancies between the signal current impulses transmitted in Hamburg from the quartz clocks III, manufactured by PTR, and the determinations of sidereal time were due to refraction caused by this new building, and that these phenomena manifested themselves especially strongly after hot sunny days.

and on sultry summer evenings. Deviations reached 0.015 sec. Reliable elimination of these and similar atmospheric disturbances is possible only by installing the transit instrument on an elevation, in terrain having a steppe character and a moderate temperature.

1. The circumstances are more favorable with respect to eliminating errors caused by the properties of the instrument itself. Noncoincidence of the axis of the telescope of the transit instrument with the plane of the meridian, inequality of the pivots of the horizontal axis of rotation of the telescope, etc.—these factors can to a considerable extent be checked, and the errors caused by them can be eliminated by taking definite measures during observations. In a similar way the errors caused by the impersonal micrometer and the recording device can be studied. Subsequently they can be eliminated from the result by appropriate computations.

2. The matter is less simple with the elimination of errors introduced by the observer himself when timing a star, and determined by the physical and psychic properties and condition of the observer. These errors, which we have called the observer’s personal errors, were investigated in detail by Stoiko ⁶ and others. Personal errors reach values of several hundredths of a second and differ very greatly among different observers, changing, moreover, with time. Repsold ², in connection with the international determination of longitudes in 1933, carried out, by means of quartz clocks, III comparative investigations of the personal errors of three observers. He found that differences in the determination of moments of time by two observers may reach several hundredths of a second.

3. In order that the total error, made up of those enumerated, be small, in determining the moment of time one uses not one star, but several; moreover, for each of the stars, with the aid of an impersonal micrometer, several marks are made. At the German Naval Observatory ten stars are usually used, with twenty marks for each. The mean error in determining the moment of time under these conditions is about $\pm 0.02$ sec.

In addition to this mean error, systematic errors of unknown origin may also appear in the determination of the moment of time. Their magnitude can be determined only on condition that the determination of moments of time is carried out simultaneously by two transit instruments. These systematic errors, determined from the difference of the readings of two different instruments, may, according to Schütte’s calculations, reach 0.1 sec. These errors may be partly personal, partly instrumental, and their existence indicates at least that, when discussing questions of checking time by astronomical methods, data obtained by observing stars should be treated with caution.

1. At the present time it is not yet expedient from day...

to check the astronomical time scale during the day by means of quartz clocks. What would be found in this way would be no more than experimental errors of the measurement itself, and by no means errors of the scale. The exceptional constancy of the rate of quartz clocks over intervals of a day’s duration makes it possible to mark the instant of time with an accuracy of several ten-thousandths of a second.^7 This high accuracy of determining time by means of quartz clocks can be used only on the condition that the error in determining time from the stars is approximately 100 times smaller than the present 0.02 sec. Thus the need to seek ways and means of reducing the error in determining time by the astronomical method is a real one.

As has already been indicated, in order to eliminate atmospheric disturbances it is necessary to install transit instruments in suitable places. On the other hand, in order to avoid personal errors it is necessary to replace the impersonal micrometer by a corresponding objective receiving device. Attempts have been made to construct such devices operating with the aid of photoelectric cells.^1 Here, however, one should not forget the serious difficulties that arise in connection with fixing the passage of a stellar disk.

d. Time Service

15. The climate of Central Europe permits only a small number of time determinations to be made during a month. In late autumn, after one determination of time, two weeks may easily pass before a cloudless night sky again makes it possible to carry out measurements. During these intervals the institutes of the time service must indicate the correct time on the basis of clock readings. The quality of these clocks determines the degree of accuracy with which, on the basis of a previously derived formula for their rate (from data on the determination of the moments of time), the time will be indicated.

For our purposes it is now sufficient to know that there exists the possibility of a more or less inaccurate extrapolation of time. In this connection the institutes of the time service are faced with the task of making, with greater or lesser accuracy, the measure of time obtained by observing the stars common property. This is accomplished by means of the so-called time signals, transmitted by some of the largest radio broadcasting stations. Time signals, originally intended for the needs of mariners—for determining the position of ships—have acquired a much broader significance in time measurement, since in many cases the requirements for the accuracy of time make it necessary constantly to check the clocks in use.

^1 On this, see, for example, the book by Simon and Zurmarn ^45 (translator’s note).

16. The principal radio stations in Central Europe transmitting exact-time signals are the following: Nauen (wavelength 18,130 m, times \(1^{h}00^{m}\) and \(13^{h}00^{m}\)), Bordeaux (wavelength 19,100 m, times \(9^{h}06^{m}\) and \(21^{h}06^{m}\)), Rugby (wavelength 18,740 m, times \(11^{h}00^{m}\) and \(19^{h}00^{m}\)). These radio stations transmit Central European time (MEZ — mitteleuropische Zeit)\(^{1}\).

In Germany the time service is carried out chiefly by the German Naval Observatory in Hamburg. Working jointly with this observatory is the Geodetic Institute in Potsdam. The German Naval Observatory transmits by cable to Nauen the main signal, with the supplementary signals adjoining it (the form of the signal is shown in Fig. 3). The main signal begins at \(0^{h}55^{m}00^{s}\) and correspondingly at \(12^{h}55^{m}00^{s}\) (Central European time). It begins with a dash serving as the call signal, and ends at \(1^{h}00^{m}00^{s}\), or correspondingly at \(13^{h}00^{m}00^{s}\), with the end of the third dash of the third three-dash signal. At 30.5 sec. later there begins, lasting until \(1^{h}06^{m}00^{s}\), or correspondingly until

Fig. 3. Diagram of the Nauen time signal

\(13^{h}06^{m}00^{s}\), the supplementary signal, consisting of a series of dashes. These dashes divide the interval of 60 sec. into 61 parts, so that from the beginning of one dash to the beginning of the next \(0.98361\) sec. elapses. Each dash ending a minute, for convenience of orientation, is somewhat longer than the others. For many measuring purposes it is sufficient to use the supplementary signal as a time scale with a vernier having divisions of \(1/60\).

The starting of the signal apparatus emitting the main signal is carried out at the German Naval Observatory by means of a pendulum clock (signal clock)\(^{8}\). The signal apparatus emitting the supplementary signal is connected with the apparatus for the main signal.

The beginnings and ends of the time signals transmitted by radio stations give true time only in the rarest cases. Usually errors occur here, composed of errors of the transmitting mechanism—from the signal clocks to the antenna—and errors connected with the signal clocks.

\(^{1}\) In the USSR the time service is carried out by the Moscow Observatory at the Sternberg Astronomical Institute; the transmission of time signals is made at \(12^{h}00^{m}\) and \(18^{h}00^{m}\) through the Comintern radio station. (Translator’s note.)

1. In view of this, time signals are received by the time-service institutes themselves and are compared with the time scales at their disposal. If there is a discrepancy between the signals and the scale, this discrepancy is measured and published as a correction, with its sign indicated. These corrections, however, are in most cases published with a delay of many months, since, owing to the unreliability of the time scale provided by pendulum clocks, in order to check the correctness of the corrections it is necessary to resort to observations of stars after the signal has already been transmitted.

The corrections indicate those intervals of time by which the time signal must be corrected in order that it coincide with the corresponding instant of true time. In this case the correction in no way applies to the entire signal as a whole. It concerns only one definite mark of the signal. Thus, for example, for the Nauen supplementary signal the plus sign, referring to the beginning of the first minute signal, means that it began too early, while the minus sign means that it began too late.

Fig. 4. Fluctuations of the Nauen time signal before and after the connection of the German Naval Observatory to quartz clocks III

The corrections given by the various time-service institutes for time signals constitute a measure of the correctness of the given signal. At the same time they are a criterion for checking the perfection of the conduct of the time service by the corresponding institute, as well as the quality of the time scale at its disposal. In recent years the time-service institutes^10 have made considerable efforts to reduce the magnitude of the corrections. To this end, on the one hand, the signaling apparatuses were improved, and on the other, the inaccuracies of the time scales were reduced. According to Freiesleben and Repsold, the mean value of the correction for the Nauen time signals in 1930 was from 0.05 to 0.08 sec. (not counting gross errors), and the day-to-day fluctuations likewise amounted to almost 0.05 sec., whereas in 1931 and 1932 the mean correction was already 0.04 sec. with day-to-day fluctuations of 0.03 sec. During this time the German Naval Observatory also made the connection to quartz clocks III PTR, thereby reducing the errors in its time signals. This can be seen from the curves in Fig. 4, where the corrections are plotted for the first minute mark of the Nauen supplementary signal \(13^h01^m00^s\) for one and the same month (July), before the connection to the quartz clocks (1933) and after it (1934). According to these data, the mean error in 1934 proves to be no greater than \(\pm 0.02\) sec.

The technical possibility of further reducing errors in time signals by simple means is doubtful. It is also doubtful that especially great benefit could be derived from such a reduction. It would be more useful to find means for reducing errors in the corrections themselves and to shorten the interval of time elapsing between the time signal and the publication of the corrections.

1. In the event that the time scales of the various time-service institutes coincided and errors in the reception of time signals could be neglected, then all time-service institutes would find identical values of the corrections for one and the same signal. In reality, however, even now the data of different institutes show considerable discrepancies in estimating the magnitude of the corrections.

Fig. 5. Corrections to the Nauen time signal transmitted at \(13^h 01^m\), according to data from Potsdam (○○○), Hamburg (+++), Paris (···), quartz clock III (—○—).

In Fig. 5 are shown the correction curves for the Nauen supplementary signal at \(13^h 01^m 00^s\), obtained in Hamburg (+++), Potsdam (○○○), and Paris (···) for one month of 1936. In Fig. 6 the same curves are given for the time signals transmitted by the observatory in Rugby. From these figures it is evident that, in general and as a whole, both observatories give daily fluctuations of identical magnitude and identical deviations from true time. Thus both these time-service stations are equally perfect. The difference between them consists only in the fact that, at the end of the month, Nauen gives larger fluctuations of the deviations. This must be attributed to disturbances arising in the system signal apparatus—radio transmitter.

In estimating the errors of the same time signals, the data of different institutes differ considerably. A strikingly noticeable

is, for example, the increase in the discrepancy between Hamburg and Paris with respect to the Nauen signal in the transition from 10/I to 11/I 1936 (Fig. 5). At the very same time both institutes give an approximate agreement in estimating the change in the error of the Rugby signal (Fig. 6). The corrections to the Rugby signal, given for 7—8/I by Potsdam and Paris, differ extremely strongly from one another. Especially striking is the discrepancy between Hamburg, on the one hand, and Potsdam and Paris, on the other, with respect to the Nauen and Rugby signals, beginning with 23/I. This should probably be attributed to a change in the inertia of registration9, 31 of the receiving apparatus in Hamburg. Whereas for 23/I all three institutes give corrections differing from one another within \(+0.01\) sec., by 29/I the discrepancy already reaches (for Nauen) \(+0.035\) sec. This means that the computation of the rate of a clock between 1/I and 29/I will differ over twenty-four hours by 0.002 sec., depending on whether this computation is made according to Hamburg or according to Paris.

Fig. 6. Corrections to the Rugby time signal, transmitted at \(11^{h}00^{m}\), according to the data of Potsdam (...), Hamburg (+++), Paris (...).

It is easy to establish which of the institutes gives an incorrect value of the correction. The magnitude of the correction found with the aid of the III PTR quartz clocks shows that the German Naval Observatory, beginning with 24/I, gave values of this correction that were too small. The data of the Geodetic Institute in Potsdam, which also had quartz clocks at its disposal, agree in general with the data of the III quartz clocks.

From the example considered, the desirability becomes clear of better agreement among the values of the corrections given by the various institutes of the time service. In fact, changes in the values of the corrections from day to day by 0.01 sec. permit determination of the rate over 24 hours only with an error of the same magnitude. This magnitude considerably exceeds the actual daily fluctuations of the rate of quartz clocks, as well as of Shortt clocks.

PRECISE MEASUREMENT OF TIME

1. In view of all that has been set forth, at present there is only one method for the precise checking of the rate of clocks by the corrections given by institutes of the time service. This method consists in computing, from the data of several institutes, the mean astronomical time and determining the regularity of the rate according to this mean astronomical time. From the deviations of the values given by the individual institutes from this mean, one can determine the mean deviations of these mean astronomical clocks and at the same time find the mean errors inherent in the various institutes. In this way one finds the degree of accuracy of the time determinations made both by the mean clocks and by individual institutes, for example, the determinations of the accuracy of the rate of quartz clocks. At the International Time Bureau in Paris^10, for this purpose the data of nine institutes are used.

N. Stoyko,^11 who works at the bureau mentioned, carried out computations of the random fluctuations in the rate of such mean clocks, as well as the random fluctuations in the rate of the clocks of individual time-service institutes, the rate of these clocks being established with allowance for corrections. According to these computations, the mean random daily deviation of the mean clocks, computed from the corrections of four institutes, is $\pm 0.003$ sec., whereas for the Hamburg Institute of Time this quantity is $\pm 0.007$ sec., for Potsdam $\pm 0.010$ sec., and for Paris $\pm 0.003$ sec.

When these results are compared, the large difference between Paris and the German institutes is striking. This difference, however, at the present time belongs only to the realm of history, as may be seen without any computations from Figs. 5 and 6. The materials for Stoyko’s indicated computations relate to 1933. According to the measurements of Scheibe and Adelsberger^7 in 1934–1935, for all three institutes mentioned the mean monthly fluctuation of the rate relative to the mean clocks was the same and amounted to $\pm 0.00068$ sec.

1. Stoyko’s computations were made under the assumption that there were no systematic changes in the rate of the clocks of the time-service institutes slipping away from observation. That such changes may in fact exist is shown by the example of the Hamburg institute—the end of the monthly period of observations shown in Fig. 6. As a result, the value of the mean random daily fluctuation of the rate of astronomical clocks, determined as $0.003$ sec., is, in considering the degree of accuracy, only conditional. Therefore, if one wishes not to be dependent on the random and systematic fluctuations in the rate of the clocks of individual institutes and on the corrections reported by the institutes, it is necessary to choose time intervals in which, from time signals, for example, the rate of quartz clocks is determined, sufficiently large. Scheibe and Adelsberger usually use intervals from 10 to 30 days.

With an interval of 30 days, as the measurements with the quartz clocks III PTR have shown, the mean error of the rate of the mean clocks may be considered,

obtained from the data of the three institutes, equal to \(\pm 0.0009\) sec. Thus, absolutely continuously running clocks, when checked by time signals at intervals of 30 days and when using the corrections of the three institutes mentioned, can be checked only with an accuracy up to 0.001 sec. It should be noted here that in this result the systematic changes of the astronomical day, as they are at present measured with the aid of quartz clocks, are reflected only in the form of their mean oscillations over a yearly period. This result also contains the errors made by the time-service institutes in receiving time signals, although they almost do not show up.

The mean error of 0.0009 sec. may prove to be many times smaller than the true one during a month of a large fluctuation in the length of the astronomical day (according to the PTR quartz clocks, such an interval was, for example, the time from June 1934 to July 1935\(^7\)).

On the other hand, as a consequence of this, comparison of the data of different institutes with one another should give too small a value of the mean random fluctuation, as was indeed the case in Stoyko’s computations.

II. Measurement of time with the aid of pendulum clocks

a. General remarks

21. For the determination of time with the highest possible accuracy only astronomical pendulum clocks and quartz clocks can be used. Therefore other instruments serving for the measurement of time will not be considered in the present article.

Clocks of both types, by means of their mechanisms—the pendulum in the case of pendulum clocks and the synchronous motor in the case of quartz clocks—set off intervals of definite duration, the succession of which forms the time scale. Knowledge of the duration of these intervals of time makes it possible to indicate the instant of time corresponding to any phenomenon occurring at one of the instants lying within the time scale and registered in some way simultaneously with the time intervals of the scale. For convenience of orientation on the time scale, astronomical pendulum clocks are usually provided with hands and a dial. The PTR quartz clocks have nothing resembling hands, since these latter are, generally speaking, wholly unnecessary for the measurement of time. In view of this last circumstance, we shall not dwell below on any questions connected with hands or other indicating devices.

In pendulum clocks the length of the interval of time separated off by the pendulum is chosen so that the intervals of the time scale should be as close as possible to 1 sec. In the PTR quartz clocks\(^3,13\) syn-

chronous motors in various types of clocks mark off intervals of time with durations from 1 to 9 sec. In designing these clocks, no special importance was attached to making the intervals of time equal to 1 sec or to an integer multiple of this value. This is a matter of secondary importance, since precision measurement of time does not depend on the size of the interval of time adopted as the unit of the time scale. Setting quartz clocks with the greatest possible accuracy to a one-second interval does not, however, present any constructive difficulties.

1. It is not technically possible to start clocks in such a way that their indications coincide exactly with the indications of true clocks, i.e., so that, for example, at the instant of true time \(13^h00^m00^s\) the clocks also indicate \(13^h00^m00^s\). The difference between true time and the indications of the clocks is considered positive when the clocks indicate a time earlier than the true time, and negative in the opposite case. If, for example, at the instant of true time \(13^h00^m00^s\) the clocks indicate \(13^h00^m03^s\), this difference is equal to \(-3\) sec. To obtain the true time, this difference, called the error of the clock and denoted by \(U\), must be added, with the appropriate sign, to the indications of the clock.

Since the intervals of the clock’s time scale can never be made to coincide perfectly accurately with the unit of time—the second—the indicated difference will change from day to day even for the best clocks. As is said, the error of the clock shifts. The magnitude of the displacement of the clock error in a day, i.e., in 86,400 sec, is called the daily rate of the clock and is denoted by the letter \(g\).

\[ g = U_2 - U_1, \]

where \(U_1\) and \(U_2\) denote the clock errors corresponding to two successive days. \(g\) has a positive sign when the clock is slow, i.e., when the interval of its time scale is greater than 1 sec, and a negative sign when the clock is fast, i.e., when this interval is less than 1 sec. In the first case the pendulum or synchronous motor of the clock moves too slowly; in the second, too fast.

Neither the error \(U\) nor the daily rate \(g\) is a measure of the perfection of a clock, since these quantities in no way limit the possibilities of its use. Excessively large values of \(U\) and \(g\), in the worst case, merely make the conversion of the clock indications into true time somewhat inconvenient. The measure of the quality of a clock is only the fluctuations of the value of \(g\) over time. As it turns out, \(g\) cannot, generally speaking, be regarded as a constant quantity. Depending on the construction, this quantity proves to be a function of various and most diverse influences.

The task of research in the field of time measurement is to find the law according to which \(g\) changes. Since influences causing short-term changes in \(g\) cannot be introduced into the computations and, consequently, are not manifested as syste-

matical influences; they manifest themselves in random variations of \(g\). The magnitudes of these random errors are obtained from the differences between the observed and computed rates.

If the change of rate is a function only of the time \(t\), then the rate formula may be written in the form

\[ g_t = g_0 + \Delta g t \pm \delta g, \]

where \(g_0\) denotes the initial value of \(g\), \(\Delta g\) is the change of rate per day, and \(\delta g\) is a random daily fluctuation of the rate, caused by influences not subject to control, for example by errors of measurement.

If the daily change of rate \(\Delta g\) has a positive sign, one speaks of a retardation of the clock—the clock runs more slowly; if it has a positive sign, the clock gains.

The formula given above is applicable to quartz clocks. In the case of pendulum clocks the relations turn out to be much more complicated.

b. Astronomical pendulum clocks

1. The semiperiod of oscillation of the pendulum of pendulum clocks, and at the same time the daily rate of astronomical pendulum clocks, for an infinitely small angle of deviation of the pendulum from the position of equilibrium, i.e. under the condition \(\varphi \approx 0\), is given in the first approximation by the expression

\[ T_0 = 2\pi \sqrt{\frac{l}{g}} = 2\pi \sqrt{\frac{K + s^2 m}{gsm}}, \tag{1} \]

where

\[ l = \frac{K + s^2 m}{sm} = \frac{\rho_0^2 + s^2}{s} \]

is the reduced length of the pendulum, \(s\) is the distance between its center of gravity and the axis of rotation, \(K\) is the moment of inertia, \(\rho_0\) is the radius of inertia relative to the center of gravity, and \(m\) is the total mass of the pendulum.

For large values of the angle \(\varphi\), which is the case in real clocks, the semiperiod of oscillation is not \(T_0\), but \(T_\varphi\). The transition from \(T_\varphi\) to \(T_0\) is effected by means of the expression

\[ T_\varphi = T_0 \left( 1 + \frac{1}{4}\sin^2 \frac{\varphi}{2} + \frac{9}{64}\sin^4 \frac{\varphi}{2} + \cdots \right). \tag{2} \]

Thus we see that changes in the length of the pendulum \(l\), the angle of deviation \(\varphi\), and the acceleration of gravity \(g\) lead to a change in the rate of pendulum clocks. The causes producing these changes, insofar as they concern the acceleration of gravity \(g\), must be sought outside the mechanism of the clock. In other respects these causes may be of either external or internal origin. Among the external factors, the following play a role: temperature, air pressure, its humidity, shaking-

sion, as well as geological and cosmic forces; among the internal ones—impulses imparted to the pendulum by the mechanism, lubrication of the bearings, and deformations occurring in the material of the pendulum and the spring.

Measures for eliminating all such influences and ensuring constancy of rate are therefore either constructive measures, whose aim is, by proper design of the parts and by introducing compensating devices, to render ineffective the causes of variation in the rate, or else measurement of the influence of these factors on the variations of the rate, so as to be able to reduce the observed rate to the rate under a certain normal condition.

1. The simplest from the experimental point of view is the elimination of the influence of changes in atmospheric pressure and humidity. For this purpose Riefler¹⁴ and Shortt placed: the former—the clock as a whole, the latter—the pendulum in an airtight cylinder, in which a reduced pressure is created. To what pressure the vessel containing the clock may be evacuated depends on the oil by means of which lubrication is carried out. In the case of the Riefler clocks, which are entirely, with all their gear wheels and hands, in a vacuum, for this reason the pressure is not reduced beyond 650 mm Hg, whereas for the Shortt clocks, which practically do not need lubrication, the pressure may be no higher than 30 mm Hg. The latter clocks therefore have, over Riefler clocks, the advantage that, simultaneously with the reduction of pressure, friction is also reduced. Following Shortt, Schüler¹⁵ constructed a pendulum (Schüler pendulum) swinging in a copper cylinder, the pressure in which is 100 mm Hg.

2. From the experimental side, maintaining for a long time a low pressure (30 mm and below) in a vessel of large volume, as is necessary, for example, for the Shortt pendulum, presents rather considerable difficulties. A consequence of this is that in Shortt clocks, both those installed at the Greenwich Observatory and those at the Paris Institute of Time, changes of pressure occur, caused by insufficient sealing. Jackson and Bowyer¹⁶ carried out very detailed investigations of the dependence of the change in the rate of the Shortt clocks on changes of pressure. They found, for example, that when the pressure is lowered by 9 mm Hg (from 34 to 25 mm Hg), the daily rate of the clock decreases by 0.06 sec. A further lowering of the pressure (from 27 to 10 mm Hg) gives a decrease of the rate of only 0.04 sec. Thus the rate of the clock does not vary in any simple dependence on the air pressure. This result should be explained by the fact that a decrease in pressure acts simultaneously in two ways: by reducing friction and by increasing the amplitude of oscillation, the two effects influencing the rate in opposite ways.

As a result of their investigations, Jackson and Bowyer¹⁷ obtained the dependence of the daily rate of the clock on the air pressure. In Fig. 7 a curve is given expressing this dependence. As for the amplitude of oscillation, it changes with change of pressure almost

linear; the change in the daily rate has a minimum and proves to be equal to zero at a pressure of about 19 mm Hg (0.74 inch of mercury). According to Jackson and Bowyer, at this value of the pressure the influence of changes in the damping and in the amplitude of the oscillations on the rate of the clock compensate one another. Therefore, in practice it is desirable that the Shortt clock operate at a pressure of 15–20 mm Hg; under these conditions small fluctuations of pressure will not affect its rate.

1. A change of temperature in the clock case leads to a change in its rate as a result of a change in the length of the pendulum. The most radical means of combating the influence of temperature changes is to place the clock in a thermostat maintained at constant temperature. Because of the large dimensions of the clock in the direction of the pendulum, maintaining the constancy of the temperature, which, obviously, must have one and the same value along the entire length of the pendulum, proves to be a very difficult matter. This circumstance is the reason why, up to the present time, measures for maintaining the constancy of the temperature of the pendulum of astronomical clocks have been limited to placing them in special cellars, which in themselves are rooms with constant temperature.

Fig. 7. Influence of pressure changes on the amplitude of the pendulum and the rate of Shortt clock No. 3

To eliminate the influence of temperature, attempts were also made to use, for making pendulums, materials possessing low values of the coefficient of expansion, and to construct the pendulum from materials possessing different coefficients of expansion. The latter had as its aim (it is used, for example, in Riefler clocks) the compensation of temperature changes. Shortt, Schuler, and also Riefler used invar as a material for pendulums, an alloy remarkable for its small coefficient of expansion. Jackson and Bowyer⁶ investigated a Shortt clock with an invar pendulum and found that a change in temperature by 1°C changes its daily rate by 0.006 sec., which corresponds to a change in the clock’s state by 1.1 sec. per year. On the basis of experience in the use of thermostats for PTR quartz clocks, it should be considered possible to construct thermostats also for pendulum clocks that, at such low temperature coefficients of rate as that indicated above, would make it possible to eliminate all temperature effects.

1. At present, the influence of temperature fluctuations is eliminated by recalculating (reducing) the observed rate of the clock by means of an experimentally established temperature coefficient. In doing so it is found that the reduced rate of the clock

does not remain constant from day to day, as one would have expected, but that it is subject to changes corresponding to a continuous lengthening of the pendulum. Schuler[^18] investigated this phenomenon in detail on the invar pendulum of his clock and, on the basis of his measurements, as well as of analogous phenomena observed in the clocks of Shortt and Riefler, came to the conclusion that the irreversible lengthening of the pendulum is connected with temperature fluctuations. The influence of this irreversible lengthening on the change in the rate of the clock, of course, cannot be allowed for by the reduction formula. By precisely measuring the rate of his clock against the quartz clocks of the PTR, Schuler showed that stepwise irreversible changes in the rate always occur when, for purposes of control, the thermal protection of the clock was removed. The abrupt temperature fluctuations of the pendulum associated with this lead to changes in the crystalline structure of invar. Thus it turns out that, despite the low value of the coefficient of expansion of invar, its use as a material for astronomical clocks, from which the highest accuracy is required, proves disadvantageous because of the instability of its crystalline structure. At present experiments are being made on the use, for pendulums, of rods of fused quartz instead of invar. The use of quartz may make it possible to improve the quality of astronomical clocks, since quartz is characterized by greater homogeneity, elasticity, and a coefficient of expansion even smaller than that of invar.

1. Alongside changes in the structure of the material of the pendulum, analogous changes occur in the materials of the spring and of the knife supporting the pendulum. These parts undergo both gradual and stepwise changes which, on the average, increase the length of the pendulum. In the Shortt and Riefler clocks a thin plate-shaped spring serves as the suspension spring. A heavy pendulum hangs on this spring, and in the course of a day it is subjected to bending 86,400 times. It is quite natural that these bendings, under strong pressure, do not pass without trace for the material of the spring, and the center of rotation of the pendulum, located inside the spring, is displaced. Therefore the author does not consider justified Schuler’s rejection of his former view of the role of deformation of the spring in increasing the length of the pendulum and the explanation of this phenomenon solely on the basis of the assumption of changes occurring in the pendulum rod. It should be considered more correct to assume that both phenomena affect changes in the rate.

2. In his clock design Schuler did not use a spring suspension of the pendulum, but, in contrast to the usual method, provided the pendulum with a steel knife resting on steel pads (A in Fig. 9). Undoubtedly, in this way that factor of uncertainty which is due to the bending of the spring is eliminated, but at the same time the possibility is introduced of wear of the knife as a result of its prolonged friction against the pads. Wear of the knife leads, on the one hand, to a decrease in its length and, on the other, to an increase in the width of the blade. The decrease in the length of the knife is associated with an increase in the distance

\(s\) from the knife-edge to the center of gravity of the pendulum, and this, according to formula (1), causes an increase in the period of oscillation. Schuler avoids this unpleasant property of his design by giving the pendulum the dimensions of a minimal pendulum, under which condition small changes in \(s\) cannot lead to a change in rate. The mathematical expression of this condition is the equality of the reduced length of the pendulum \(l\) to the quantity \(\rho_0\) and \(s\) 19:

\[ l=\frac{K+s^2m}{sm}=2s=2\rho_0 . \]

Oscillations \(s\) that arise as a consequence of a change in the knife-edge, under these conditions, have no influence. Oscillations in \(s\), however, caused by a change in temperature, which in this case also determine the radius of inertia, which in turn determines the duration of the oscillation, still play a role.

The effect of increasing the width of the blade on the change in rate, when the pendulum is given the dimensions of the minimal one, is not only not diminished, but even increases. Concerning the role of this shortcoming, a broad discussion developed in its time, in which Repsold 20, Schmervitz 21, Gebelein 22, Gentler 19, and Graff 23 took part. The results of the various works prove to be highly contradictory, since the authors based themselves on different initial assumptions. Graff attempted to clarify the question experimentally with the aid of a Schuler pendulum. By plotting a curve expressing the dependence between the amplitude and the period of oscillation, he found that there really exists a change in the radius of curvature of the blade, the effect of which is not corrected by giving the pendulum the dimensions of the minimal one. According to his data, the influence of this change is only negligible in the duration of oscillation: after three years the value of the ratio

\[ \frac{\Delta T}{T} \]

will be only \(3\cdot 10^{-8}\). If these data are confirmed, this will be a great success for Schuler clocks. However, this question can be resolved only by prolonged systematic measurements of the rate.

1. Changes in the amplitude of the pendulum’s oscillations are as essential for the constancy of the rate as changes in temperature and length. We have already spoken of the change in amplitude that arises under the action of changes in air pressure. In addition, the amplitude of oscillation is affected by the impulses imparted to the pendulum and by shocks. All clocks, in order to maintain the oscillations of the pendulum, need to impart energy to it. The weaker the connection can be made between the pendulum and the source of energy, the less influence the properties of this source and of the connecting mechanism exert on the duration of oscillation, and the more freely the pendulum oscillates.

Riefler used for these purposes the transmission of impulses through the suspension spring. A further major step forward was made by Ferrié and Joua 24, Shortt and Schuler, who transferred

all operations connected with the interaction of the pendulum and the mechanism to other auxiliary clocks, and in this way removed the hands and gear wheels from the main clocks.

In Shortt’s clocks (Fig. 8) the auxiliary clocks every 30 sec. release, by means of an electromagnet, a lever which, in falling, transmits through a roller an impulse to the pendulum of the main clocks. This lever, connected during its fall for the duration of one instant with the pendulum of the main clocks, at the same time synchronizes, by means of another electromagnet, the pendulum of the auxiliary clocks and thereby creates the necessary coincidence of phases.

Fig. 8. Diagram of Shortt’s clocks

Fig. 9. Diagram of Schuler’s clocks

Still somewhat freer is the pendulum of Schuler’s clocks (Fig. 9). The auxiliary clocks act electromagnetically on a magnet \(M\), located in the upper part of the main pendulum. The synchronization of the switchings is carried out by the main pendulum by means of a light beam. In this way Schuler eliminated all mechanical influences on the main pendulum and, in any case, destroyed the connection between the two pendulums during synchronization.

1. The amplitudes of oscillation of astronomical clocks lie within the limits \(2—3^\circ\). A change in amplitude by \(0.1'\), at an amplitude of \(2^\circ\), leads to a change in the daily rate by \(0.011\) sec., and at an amplitude equal to \(3^\circ\), by \(0.016\) sec. Jackson and Bowser\({}^{17}\) showed that many oscillations in the rate of Shortt’s clocks that occurred in 1927–1929 must undoubtedly be attributed to oscillations of amplitude. They rightly indicate that constancy of the daily rate to \(0.0015\) sec. requires constancy of amplitude to approximately \(0.03\) (which corresponds to \(0.01\) mm).

For these reasons, Schuler carried out a long-term recording of the amplitude and, from the data obtained, reduced the observed value of the daily rate of the clocks to a constant amplitude. Jackson and Bowyer also monitor the amplitude for the purpose of recalculation. These recalculations are associated with laborious computations and therefore are not applicable for time determinations with the great speed and accuracy that are necessary in many cases. Loomis^25, in order to avoid this difficulty, improved Shortt’s clock by introducing a special contact which automatically limits any excess over a certain preset amplitude, switching off the energy-supplying device as soon as the amplitude exceeds the established value by more than 0.0025 mm.

Graff^23 showed that considerable amplitude oscillations in Schuler’s clocks arise from insufficient constancy of the current acting on the magnet. By stabilizing the voltage feeding the electromagnet with an accuracy of 0.002–0.003 V, he succeeded in reducing amplitude fluctuations to a value of the order of several seconds of arc.

1. Changes in amplitude as a result of shocks caused by ground vibrations common in cities, or by seismic factors, are just as undesirable as those caused by the mechanism of the clock. According to Hebelein’s investigations^26, not all disturbances caused by shocks can be determined by recording the amplitude, so that the introduction of a subsequent recalculation proves impossible.

Rickman^27 points to the possibility of eliminating the influence of short-term shocks having a horizontal direction by suspending the main pendulum on a special heavy “protective pendulum.” For such a system (when a mathematical pendulum is used) there exists a definite condition of compensation, consisting in the requirement that the centers of gravity of both pendulums must be at the same height. Oscillations having a vertical direction must be damped by the suspension of the protective pendulum.

1. If there is some possibility of hoping that the influence of shocks can be eliminated by appropriate design measures, then one cannot count on anything similar with regard to eliminating the influence of changes in the acceleration of gravity. Changes in gravity occur, on the one hand, as a result of the redistribution of masses on the Earth, and, on the other hand, are caused by changes in the positions of the Sun and the Moon (tidal forces). The causes of mass redistribution are changes in atmospheric pressure, the movement of water as a result of evaporation and precipitation, seismic processes inside the terrestrial globe, and changes in the load on the Earth’s surface due to the movement of water masses during tides and ebbs.

According to Tomaschek^28, ordinarily the relative changes in the acceleration of gravity do not exceed \(\dfrac{\Delta g}{g} = 10^{-6}\), while the share of tidal forces caused by the Sun and the Moon accounts for a part amounting to

amounting to approximately \(10^{-7}\). The attractive force of the Moon, expressed by the so-called principal lunar term \(M_2\), over a period of 12.42 hours changes the acceleration of gravity by \(\pm 4.3\cdot 10^{-8}\) of its value. Therefore, if it were possible rapidly to measure the frequency of oscillation of the clock pendulum, its values at two corresponding moments would differ by \(8.6\cdot 10^{-8}\). By experimentally comparing three Shortt clocks with Morrison–Loomis quartz clocks\(^{29}\), and also Brown and Brouwer\(^{30}\), the influence of the principal lunar term on the rate of pendulum clocks was indeed detected. The influence of each of the ten terms of the equation expressing the tidal forces on the amplitude, period, and phase of the pendulum oscillation can theoretically be eliminated by calculating the dependence of the clock rate on the tidal forces. For rapid and at the same time precise determinations of time, however, this method is unsuitable.

Nonperiodic changes in the acceleration of gravity caused by the redistribution of masses naturally cannot be allowed for in advance. Since these changes are detected by gravimetric measurements, they can be taken into account as corrections to the rate of the clock. To accomplish this it is necessary that a recording gravimeter be installed next to the clock\(^{28}\).

Independence from fluctuations of the acceleration of gravity, which in part are altogether beyond control and in part are only with difficulty allowed for, can be obtained only by using a time-measuring instrument whose time intervals do not depend on the acceleration of gravity. Such instruments are, for example, quartz clocks.

c. Rate of pendulum clocks

1. The preceding discussion has shown how numerous and varied are the factors influencing the rate of clocks. In order to make use of quartz clocks, it is necessary to know the law according to which these factors exert their influence. If, for example, the function expressing the dependence of the clock rate on temperature is known, the influence of temperature on the result of measurements can be eliminated. In this case, in accordance with Section 22, one obtains a formula containing on the right-hand side a term expressing the dependence on temperature. The random fluctuations of the rate, which may be regarded as an indicator of the quality of the clock, will then be the smaller, the more fully the influence of all factors on the clock rate has been taken into account in the formula. This tendency—to establish as well as possible the laws governing changes in rate, in order to separate random deviations from systematic ones and thereby make them sufficiently small—if it goes beyond permissible limits, leads to dubious results. The better time-measuring instrument is always the one which, with a simple formula for the rate, gives small random fluctuations of the rate, and not the one which, with a very complicated formula, gives still smaller random fluctuations. Moreover, it is not immaterial over what interval of time the random ...

fluctuations of the rate. For example, over a period two hours long, clocks may have identical random fluctuations of rate and nominally be of the same quality, while in reality having a very different rate. This will occur when, for one of these clocks, the rate formula consists of separate parts with different coefficients. Consequently, the criterion proposed everywhere in the literature for judging clocks—the mean random fluctuation of the rate, computed from random fluctuations over intervals of different duration—cannot serve as a criterion for a correct judgment about clocks.

1. Mankopf and Kinle carried out a detailed investigation of the Riefler clocks. They found, as the mean daily fluctuation of the rate, \(\pm 0.002\) and \(0.004\) sec.; these are very favorable results. Less favorable data were obtained by Rentsold\(^2\). His results are of great importance for judging these clocks, since they were obtained by daily comparison of the Riefler clocks with PTR quartz clocks. Rentsold, for the interval from September 20, 1933, to January 31, 1934, gives for two clocks a mean daily fluctuation of the rate of \(0.010\) and \(0.016\) sec. These quantities, as is evident, are much greater than the values cited above, and they correspond more closely to reality.

2. With regard to Schuler clocks, at present there is still too little numerical material to make it possible to judge their rate seriously. In any case, the Riefler clocks surpass the Schuler clocks.

3. The material concerning the rate of Shortt clocks is very extensive. Jackson and Bowyer\(^ {16}\) showed that, for both Shortt clocks installed at Greenwich—No. 3 and No. 11—the formula for the state has the form

\[ U_t = U_0 + gt + \frac{\Delta g}{2} t^2 + \Delta' g \int (T - T_0)\,dt + \text{nutation} \]

and that it permits \(U_t\) to be computed without changing the coefficients for long intervals of time. In this formula the coefficient \(\Delta' g\) takes into account the influence of temperature fluctuations (\(T_0\) is the temperature of the normal state). Differentiating this equation with respect to time \(t\) gives the rate formula.

The differences between the measured values of the state and those computed by this formula for Shortt clock No. 11 for 1929 are shown in Fig. 10. Computations for an interval of 8 months (August) give an error of 2 sec., which retains its magnitude for several weeks and then decreases again. At the end of 1930 the difference proves to have the opposite sign and the same absolute value: it is equal to \(-2\) sec. These deviations, obtained as a result of the computations, indicate that in the Shortt clocks there are factors influencing the rate and not taken into account by the formula. Nevertheless, it should be considered that the quality of these Shortt clocks is very high, since the difference of states of 4 sec. over the period from October

from 1929 to December 1930 could have been reduced to zero by changing the acceleration coefficient \(\Delta g\) of the quadratic term, equal to approximately 0.0004 sec., by about 0.00006 sec. The acceleration coefficient, i.e. the daily rate of the clock of 0.0004 sec., is very large by modern standards. For comparison it may be noted that for the PTR III quartz clocks, according to the rate formula for 1934, it amounted to only 0.00002 sec. That it is possible to obtain still better results follows from the work of Stoyko \(^{32}\), who, for Shortt clock No. 44, installed at the International Time Bureau, found for the period from May 1934 to April 1936 an extraordinarily low acceleration coefficient, equal to only 0.0000056 sec.

Fig. 10. Differences between the measured and computed positions of Shortt clock No. 11

According to all the data presently available, the Shortt clocks must be recognized as the best of the existing astronomical pendulum clocks.

III. Measurement of Time by Means of Quartz Clocks

a. General remarks

1. In 1922 W. G. Cady published his studies on the use of quartz rods as piezoelectric oscillators and resonators for high frequencies. Since then, piezoelectric phenomena, which until that time had played almost no role in technology, have proved to be very fruitfully included in the field of high-frequency oscillation engineering. At present piezoelectric devices have become an indispensable auxiliary means in all fields of the use of oscillations, since with their aid it is possible not only conveniently to excite oscillations of any frequencies, but also to detect such oscillations. The most substantial scientific application of piezoelectricity proved to be

“quartz clocks,” by means of which it became possible to measure time independently of pendulum clocks. Even if quartz clocks were merely not inferior to the best astronomical pendulum clocks, then already, on the basis of the fact that their operation rests on an entirely different principle, they would make it possible to obtain considerably greater confidence in solving problems connected with the determination of time.

The direct piezoelectric effect was first discovered by Pierre and Jacques Curie in 1880 in tourmaline. As often happens, in this case too several decades passed before the physical phenomenon found application in technology.

The direct piezoelectric effect consists in the fact that, when a piezoelectric crystal is compressed or stretched in definite directions, electric charges arise in it. The inverse piezoelectric effect manifests itself in a change of the dimensions of a piezoelectric crystal in definite directions, occurring when it is placed in an electric field.

Fig. 11. Glowing resonator for 60,000 Hz

Cady showed that quartz rods, cut in a definite way from a quartz crystal, when placed in a high-frequency alternating electric field, begin to perform elastic vibrations in the direction of their length. The amplitude of these vibrations reaches its greatest value at resonance, i.e., when the frequency of the exciting oscillations is equal to the frequency of the rod’s natural vibrations.

39. E. Giebe and A. Scheibe[^35] showed that in quartz rods and rings it is possible to excite not only longitudinal vibrations, but also torsional vibrations connected with bending. They constructed the so-called glowing resonator, consisting of a quartz rod placed in a bulb filled with a rarefied gas. At resonance, the potential difference arising on the rod causes the gas to glow. Investigations of the constancy of the frequency of the glowing resonator showed invariability of the frequency down to one millionth of its value and less. The limit on the side of the lower bound was apparently set not by insufficient constancy of frequency, but by the impossibility, by observing the glow, of measuring the frequency with an accuracy better than several ten-millionths of its value. The high constancy of the frequency was achieved by the fact that the quartz rod, by means of rigid fastening at two nodal points, was held motionless between two exciting electrodes (Fig. 11).

40. In contrast to this, American investigators, in creating the quartz frequency stabilizer, took a different path. Already Cady had proposed a connection circuit containing piezo-quartz and a tube amplifier, serving for the generation of undamped oscillations with a frequency equal to the natural frequency of the quartz

rod. D. W. Pierce^36 greatly simplified this circuit. The circuit, named in his honor the Pierce circuit, differs in that the piezo-quartz is connected between the grid and the anode of the amplifier tube, while an oscillatory circuit with a suitable natural period is connected between its anode and cathode. The frequency of oscillation in such a circuit is determined chiefly by the natural frequency of the controlling quartz. But since, when the quartz rod is connected to the tube, it does not remain completely free, the tube itself and the oscillatory circuit also exert some influence on the frequency. Therefore the problem arose of making these influences so small that the frequency changes caused by changes in the tube and in the circuit would become entirely negligible.

This goal was very successfully achieved by W. A. Marrison^37, who in 1929 published the first work on crystal clocks (Cristal Clock), entitled “A high-precision frequency standard.” As the controlling quartz he used a quartz ring, cut in a definite manner from crystalline quartz and having definite dimensions^1). The valuable property of quartz rings, consisting in the weak damping of longitudinal oscillations, had in its time been noted by Giebe and Scheibe^39.

In 1930 the author of the present survey initiated work at the PTR aimed at creating quartz clocks. Since the investigations preceding it showed that with the aid of the luminous resonator described above one could hope to obtain still better results than those achieved by Marrison with quartz rings, in developing clocks for the PTR there was no reason to follow the path taken by Marrison. Therefore the designers of the quartz clocks—Scheibe and Adelsberger^3—used the controlling quartz in the form of a quartz rod. They also used a different construction and a different connection circuit, so that quartz clocks and crystal clocks have no external resemblance to one another (Figs. 24 and 28).

b. Properties and operation of quartz clocks

41. In their construction, quartz clocks are an alternating-current generator with the highest constancy of frequency, the alternating current from which feeds a special synchronous motor. The undamped oscillations excited by the generator provided with piezo-quartz have, generally speaking, too high a frequency (for example, 60,000 Hz) and too little energy for them to be able directly to set a synchronous motor in motion. Therefore it becomes necessary to amplify this alternating current by means of tube amplifiers and then, with the aid of frequency transformers, to convert it into alternating current of low

^1) This question was investigated by Lack^38.

frequencies (for example, 250 Hz). In the quartz clocks of Scheibe and Adelsberger the frequency reduction proceeds from 60,000 Hz to 10,000, then to 1000 and to 333, 33333… or to 250 Hz.

1. Quartz clocks equipped with a frequency transformer constitute a frequency standard, and, when a synchronous motor is connected, serve as a time scale. The correspondence between the rate of the quartz clock and astronomical time is established by means of the time contact of the synchronous motor. From the fluctuations of the time intervals elapsing between two closures of the contact, one can compute the fluctuations of the daily rate. Comparison of quartz clocks with one another makes it possible to determine the frequencies of the alternating currents supplied by the 10,000 Hz or 1000 Hz frequency transformers. In this comparison, which consists in measuring the difference of the frequencies of two alternating currents, possibilities are opened for applying the most accurate of the measurement methods existing in high-frequency technology; with an expenditure of several minutes on the measurement they make it possible to establish a change of frequency of the order of \(10^{-10}\)—\(10^{-9}\) of the measured quantity. From the change in the frequency difference it proves quite simple to calculate how much the daily rate of the clocks changes in 24 hours, since, for example, a frequency change of \(10^{-9}\) corresponds to a change of the daily rate by 0.0001 sec. The changes in the rate of quartz clocks determined in this way are called by Scheibe and Adelsberger\(^{13}\) the instantaneous values of the variations of the daily rate. The possibility of rapid, convenient, and systematic control of quartz clocks by the instantaneous values of the daily rate is their great advantage over pendulum clocks, in whose case such comparisons over short intervals of time are impossible. It is clear that, with continuous observation of the rate of several quartz clocks, it is possible to detect in good time a change in the rate of any of the clocks and to introduce corrections in good time. Finally, with these clocks it is convenient to study the influence of various factors on the rate of the clocks.

2. The situation with the inclusion of quartz clocks in astronomical time proves to be considerably less favorable. As was already indicated in detail in section 20, the mean daily rate of quartz clocks, owing to errors connected with the transmission of time signals, even for a period of 30 days cannot be determined with an accuracy greater than 0.001 sec. In view of this it is quite impossible to measure the daily rate from day to day with an accuracy even approaching 0.001 sec. Therefore, in determining the random and true daily fluctuations of the rate, in the case of quartz clocks just as in the case of pendulum clocks, it would be necessary to resort to the aid of highly speculative calculations, if, by continuously measuring the instantaneous values of the daily changes of rate in combination with data on the absolute rate, obtained from measurements over very large inter-

of the time intervals, no experimental material had been collected that would make it possible to determine the absolute daily rate with great accuracy, without using any computational tricks.

c. Technical details of the construction of quartz clocks

The quartz clocks of Scheibe and Adelsberger[^3]

1. Fig. 12 shows the basic circuit for connecting the quartz clocks developed at the PTR. The piezoquartz \(Q\), connected with the high-frequency generator \(RS\), through two amplifiers \(VI\) and \(VII\)

Fig. 12. Basic circuit of the quartz clocks

and three frequency transformers 10,000, 1000, and 333 drive a synchronous motor \(S\), which, by means of its time contact, acts on a recording apparatus that places time marks on a rotating drum. The supply of energy to the clocks is from the alternating-current mains through rectifiers, chokes, and buffer batteries.

Fig. 13. Arrangement of the electrodes and general appearance of the controlling quartz of quartz clocks I and II

At the PTR four clocks have been installed. The first two specimens—the quartz clocks I and II—are furnished with quartzes that are completely identi-

... with the luminous resonator shown in Fig. 11 for 60,000 Hz. The arrangement of the electrodes \(b\), the orientation of the quartz rod \(a\) with respect to the electrical \(X\) and neutral \(Y\) axes of crystalline quartz, and also the general appearance of the entire instrument are shown in Fig. 13. The length of the rod is 91 mm.

The temperature coefficient of the daily rate of the quartz clocks is \(0.4\) sec/\(1^\circ\mathrm{C}\). Therefore, the requirement that the daily rate remain constant to an accuracy of \(0.001\) sec makes it necessary to maintain the constancy of the temperature to an accuracy of \(0.002^\circ\mathrm{C}\). This is accomplished by placing the controlling quartz in an inner thermostat, whose transverse section, longitudinal section, and general appearance are shown in Figs. 14, 15, and 16, and then in an outer one, which is a wooden box insulated with down. This

Fig. 14. Transverse section of the inner thermostat

Fig. 15. Longitudinal section of the inner thermostat

thermostat can be seen in Fig. 24, on the right. Of especially great importance for maintaining the constancy of the temperature is the inner thermostat, which, by means of the contact thermometer \(KTi\) and the heating winding \(Hi\), maintains a constant temperature of \(36^\circ\mathrm{C}\). This thermostat consists of 11 alternating well-insulating and heat-conducting layers, shown in Fig. 15: air (1), copper (2), air (3), copper (4), air (5), copper (6), air (7), cork (8), aluminum (9), down (10), aluminum (11). The outer thermostat, whose purpose is to soften sharp fluctuations of room temperature, is kept at a constant temperature by means of the contact thermometer \(KTa\), inserted into layer 11, and the outer heating winding \(Ha\).

Work with these first two quartz clocks gave good results, but at the same time it soon showed that it was necessary to achieve still greater independence of the daily rate of the clocks from temperature. In view of this, the construction followed of two new clocks—quartz clocks III and IV. The controlling quartzes of these clocks \((a)\) are arranged with respect to the electrodes \((b)\) and the axes \((X,Y)\) as shown in Fig. 17, where the general appearance of the device is also presented...

Studies by Giebe and Scheibe[^35] showed that, with such an orientation of the rods, in the interval from 35 to 37°C the temperature coefficient of the rate becomes vanishingly small, provided that the ratio of one side of the square cross-section to the length of the elastic half-wave is chosen equal to 0.25. In the operation of these quartz clocks it was found that, indeed, at a thermostat temperature of 36°C the effect of temperature fluctuations becomes immeasurable.

Fig. 16. Internal thermostat with an external heating winding at 36°C

Fig. 17. Arrangement of the electrodes and general view of the control quartz of quartz clocks III and IV

In view of this, quartz clocks of types I and II are no longer being built. We shall see, however, that with these clocks too one can obtain very valuable results if, by means of clocks III and IV, systematic observation of temperature disturbances is carried out.

1. In Fig. 18 a detailed circuit of the generator and amplifier is given; within the limits of the present survey, however, there is no need to dwell on them. The circuit of the frequency transformer is shown

in Fig. 19. Each of the transformers consists essentially of a low-power tube generator with inductive feedback, tightly coupled to the grid circuit of the next generator of lower frequency. In this way it is achieved that each of

Fig. 18. Circuit of the quartz clock (thermostat, generator, amplifier)

Fig. 19. Circuit of the frequency transformer

the three generators oscillates freely only when their periods are integral multiples of the period of the high-frequency generator, which oscillates synchronously with a frequency of 60,000 Hz.

The cessation of oscillations of a generator owing to the failure of tubes occurs very rarely and is analogous to the stopping of pendulum clocks. Such a stop, however, is connected only with a change in the reading of the clock, which, by means of other quartz clocks, can be determined with the greatest accuracy, and it has absolutely no effect on the rate of the clock.

Alternating current from the last frequency transformer sets in motion a synchronous motor, the simplest design of which is shown in Fig. 20. The contact pin of this motor, at

Fig. 20. Synchronous motor

constant intervals of time, short-circuits the grid circuit of the relay shown in Fig. 2, so that capacitor 2 of the anode circuit is discharged through the coil of the recording instrument, which makes the corresponding mark on the tape.

1. Measurements of the frequency of the clocks, necessary for determining the instantaneous values of rate changes, are carried out by means of coupling coils \(L4\) or \(L6\) of Fig. 19. The oscillations of the frequency being investigated, together with the frequency of other quartz clocks, are applied to terminals \(N_2\) and \(N'_2\) of the device, the circuit of which is shown in Fig. 21. Both frequencies are amplified by three REN 904 tubes and rectified by two RE 114 tubes. The extraction of the frequency of the resulting beats

Fig. 21. Circuit for obtaining beats

comes from terminals \(A_1\), whence it is fed to terminals \(A\) of a special low-frequency amplifier, shown in Fig. 22. In this latter instrument the sinusoidal beat oscillations are transformed into sharp current pulses, which are fed from terminals \(R\) to the coil of the recording device. The measurement of the beat frequency, necessary for determining the instantaneous values of the diurnal variations of rate, is carried out by means of a glass scale.

Fig. 22. Special low-frequency amplifier

When the requirements as to accuracy are not especially high, an example of the beat chronogram need not be made, since the sinusoidal beats obtained in the circuit of Fig. 21 act on the counter \(SZ\), which registers their number. From this number of beats and the corresponding time interval, which is registered by the counter \(ZZ\) under the action of signals from the quartz clocks fed to terminals \(K\) (Fig. 21), the instantaneous value of the difference in rate of the two quartz clocks being compared can be calculated by simple division.

Fig. 23. General view of the device for obtaining beats

In Fig. 23 a general view of the instrument is shown, the diagram of which was given in Fig. 21. Automatic frequency control by means of this device makes it possible to leave the clock without observation for entire days, since under this system of control the data necessary for finding the mean diurnal rate are automatically obtained.

Fig. 24. Front view of the open quartz clock

47. Fig. 24 shows a front view of the open quartz clock. On the right stands the external thermostat with the internal thermostat placed inside it, setting the generator and the amplifier; on the left are the frequency transformers. The control measuring instruments make it possible to observe all the oscillatory circuits of the device.

Fig. 25. Rear view of the quartz clock

Fig. 25 shows the rear view of the quartz clock, where the rheostats, ammeters, and voltmeters necessary for maintaining the constancy of the supply are mounted.

The crystal clocks of Marrison^37

48. Fig. 26 shows a general view of the quartz ring used in Marrison’s clocks, and Fig. 27 shows a transverse section of the assembled ring. The mounting of such a ring between the corresponding electrodes is a rather complicated matter. It is ar-

[[unclear: beginning of sentence]] is connected to another electrode, fastened to the supporting [[unclear: part]] of the DFT structure; another small block is made of Pyrex glass and rests on a metal pin.

In order to reduce friction, the inner surface of the quartz ring is ground so that its cross-section has a V-shaped form (Fig. 27). The quartz ring lies completely freely, as a result of which, when shaken, it can change its position relative to the electrodes, which leads to variations in the daily

Fig. 26. Quartz ring of the “crystal clock”

Fig. 27. Method of fastening the quartz ring

Diagram labels: glass; support; quartz ring; cylinder; electrodes.

Fig. 28. General view of the “crystal clock”

Fig. 29. Synchronous motor of the “crystal clock”

rate. Mac Ilroy\(^{40}\) especially emphasizes this shortcoming of the quartz-ring mounting. He attempts to eliminate this defect by improving the damping of the unit containing the controlling quartz.

In Marrison’s clock the controlling quartz is placed in a small metal thermostat, the constancy of temperature in which is maintained by means of a contact thermometer. Since the quartz ...

the ring is not in a vacuum; to protect it from fluctuations of pressure and humidity, it, together with the thermostat, is placed under a glass bell jar, in which a reduced pressure is created. Fig. 28 shows the external appearance of the thermostat under the glass bell jar. The stand for the bell jar is a metal box in which the generator is located; between the grid and the cathode of its tube, according to Pierce’s circuit,^36 a piezoquartz is connected.

The natural frequency of the controlling quartz in these clocks is 100,000 Hz. After preliminary amplification the generator frequency is reduced in two stages—to 10,000 Hz and then to 1,000 Hz. The 1,000-period current drives a synchronous motor, which, for indicating the time, is connected with a clock mechanism and hands. These parts of the crystal clocks are shown in Fig. 29. By means of the clock mechanism the clocks are brought into agreement with astronomical time. The instantaneous values of the fluctuations of the daily rate are measured automatically by the beat method in a circuit with a frequency of 10,000 Hz.

1. Marrison states, on the basis of a 10-hour series of measurements with two clocks of his own design, that the mean fluctuations of the difference in rate between them correspond to from 0.001 to 0.002 sec. For short-time measurements this is a good result. However, for long periods this type of quartz clock is apparently not so suitable. Loomis, Brown, and Brouwer^29,30 investigated, with the aid of Marrison clocks in the laboratory of the Bell Company (for which the clocks had been built by Marrison), the rate of three Shortt clocks. They came to the conclusion that the Marrison clocks give greater accuracy over small intervals. For large intervals, however, the advantage proves to be on the side of the Shortt clocks.

Mac Illraith^40 attempted to improve crystal clocks by placing them in a second thermostat, and also by regulating the room temperature. As a result of these measures, no substantial improvements could be obtained. At the present time the constancy of rate of crystal clocks, as was shown by Adelsberger^41 by comparing specimens of both clocks, proves to be, in order of magnitude, lower than the constancy of PTR quartz clocks.

In view of this, in the following sections, when discussing the rate of quartz clocks, and also when comparing with results obtained by means of pendulum clocks, we shall not use data obtained with crystal clocks. Comparison with them would be easily feasible only in the case that, concerning their rate, there existed material as extensive as that available for PTR quartz clocks.

d. Factors Affecting the Rate of Quartz Clocks

1. The rate of quartz clocks \(g\) is in the following relation to the frequency of the controlling quartz:

\[ g = \frac{N}{t} - 86\,400. \]

where \(N\) is the total number of oscillations of the controlling quartz. The dependence of the rate \(g\) on the frequency \(f\) is obtained from this formula in the following form:

\[ \frac{\delta g}{86400}=-\frac{\delta f}{f}. \]

Thus, in order to find the change in the rate \(\delta g\), it is only necessary to investigate the factors determining the natural frequency \(f\) of the controlling quartz.

The frequency of an unloaded quartz rod undergoing longitudinal oscillations is expressed as

\[ f=\frac{1}{2l}\sqrt{\frac{E}{s}}, \]

where \(l\) is the length of the rod, \(E\) is the modulus of elasticity, and \(s\) is the density.

1. Since \(l\), \(E\), and \(s\) change with temperature, there must undoubtedly exist a dependence of the rate \(g\) of quartz clocks on temperature.

In Section 44 we already indicated that, by cutting a quartz rod in a certain way, it is possible to ensure that the influence of temperature fluctuations at a mean temperature of \(36^\circ\mathrm{C}\) is eliminated. Thus, only factors other than temperature, acting on the length of the rod \(l\), the modulus of elasticity \(E\), and the density \(s\), can change the rate of the clock. Mechanical changes in length due to the breaking off of pieces of quartz, observed under severe overload of piezoquartz devices, may be disregarded under the operating conditions of quartz clocks, where the amplitude of oscillation of the rod is negligible. Less clear is the question of whether, during prolonged operation, \(E\) and \(s\) may undergo gradual or instantaneous changes.

Fig. 30. Change in the differences of rate of quartz clocks I and II

Both Marrison’s crystal clocks and the quartz clocks of Scheibe and Adelsberger show a rapid decrease in the daily rate at first and, with the passage of time, a decreasing—gradual or sudden—decrease in the daily rate. The curve in Fig. 30 shows the change in the daily rate of quartz clock II relative to clock I after they were set in operation. Owing to the decrease in the daily rate of clock II, the difference I—II becomes larger; this change in the differences of rate ended long ago. At the time when this phenomenon was discovered, because of insufficient data it was assumed that it was caused by aging of the contact thermometer. But since subsequently quartz clocks III and IV, insensitive to temperature fluctuations, showed the same thing, it must be considered that this explanation was incorrect.

Scheibe and Adelsberger[^7] at the present time, on the basis of their experiments, cannot subscribe to the opinion that this phenomenon should be ascribed to aging of the quartz rod, i.e., to a change in its modulus of elasticity or density with time. These authors incline to the view that this decrease in the daily rate is a consequence of a gradual change in the electrical data of the quartz clocks.

1. It should be remembered that, owing to its connection with the generator, the quartz rod performs oscillations whose frequency, though only slightly, nevertheless measurably depends on the generator frequency. Fortunately, it turns out that those circuit data which, in the course of prolonged operation of the clocks, are most readily subject to changes—namely, the anode voltage and the filament voltage—have an exceptionally small influence on the rate of the clocks. A change of the anode voltage by 1 V (with 50 V at the anode) causes a change in the rate of only 0.001–0.002 sec. The constancy of the anode voltage can be maintained without any difficulty and without constant observation to an accuracy of up to 0.01 V. The influence of changes in the filament voltage proves to be still considerably smaller.

Fig. 31. Change in the differences of the rates of quartz clocks II and III (above) and I and III (below)

The situation is different with changes in the properties of the tubes, and also with changes in the capacitances connected between the anode and the grid and between the anode and the cathode. These quantities affect the rate of the clocks. When they are increased—for example, when additional capacitances are connected in parallel—the rate slows down. It has been established with complete certainty that the phenomena described above, i.e., the gradual change of the daily—

...of the rate of clocks are due to the influence of capacitances connected between the electrodes of the tube. As a result of the study of these phenomena, it proved possible to change the magnitude of the variations in the daily rate of newly switched-on quartz clocks in such a way that the variations in the daily rate of two clocks relative to one another are small from the very beginning. These measures therefore lead to a reduction in the time required for stabilization of the rate of quartz clocks.

1. In practical time service, such uniform changes in rate, which moreover are very small (in quartz clocks III, for example, they amount to \(0.00002\) sec. per day), are in no way an obstacle to the use of quartz clocks, since they are taken into account by the rate formula. Moreover, by rapidly determining the instantaneous values of the rate variation, it is easy to establish at what moment these gradual changes in rate occur and what their magnitude is. Fig. 31 shows a control sheet for quartz clocks I and II, which are sensitive to temperature influences; for these clocks the instantaneous values of the rate variation were measured daily relative to quartz clocks III, which are not subject to temperature influences. In both clocks

Fig. 32. Change in the rate differences of quartz clocks III and IV

jumps are observed in the daily change of rate, caused by changes in the room temperature1. The jump-like changes in rate correspond to vertical lines, whereas the daily changes in rate are represented in the figure by sloping straight lines drawn through the points corresponding to the individual measurements. Since the absolute value of the daily rate of quartz clock III, equal to \(0.00002\) sec/day, is known with great accuracy, by means of this control sheet it is possible to give the absolute values of the rates of clocks I and II for any moment.

Fig. 32 shows the results of checking the difference in rate of the two quartz clocks not subject to temperature influences (clocks III and IV) from the moment clock IV was switched on for service as a time standard. As can be seen, a slow decrease continuously occurs

difference in rate, amounting over 200 days to 0.003 sec., i.e. 0.000015 sec. per day. The scatter of the points about the mean straight line does not represent actual fluctuations of the rate, but reflects the mean error of the measurements, which at the end of March was 0.0006 sec. and from that time decreased to 0.0003 sec. The figure clearly shows the reduction in the scatter of the points that occurred.

The facts considered in this section show that changes of rate occur in quartz clocks, caused by the phenomena taking place in them. On the other hand, we see that, by measuring the instantaneous values of the change in rate, it is easy to determine the magnitude of these measurements. Since the PTR has at its disposal four quartz clocks of two different types, this provides confidence that factors, perhaps existing, which affect the rate of all the clocks simultaneously, can be detected. This confidence is based on the fact that, as has been established, different types of clocks react to influences with unequal strength.

1. Quartz clocks have one more significant advantage over pendulum clocks. The frequency of oscillation of the controlling quartz, and with it the rate of quartz clocks, is completely independent of the acceleration of gravity. Changes in gravity have no effect whatever on their rate, and therefore only with the aid of such clocks can the daily rate be determined at any moment with great accuracy (to several thousandths or ten-thousandths of a second). Consequently, for precise measurements of frequencies under conditions of short measurement durations, only quartz clocks are applicable.

The influence of shocks on the rate of quartz clocks is also absent, since quartz rods have small mass, and the mounting of the electrodes that determine the position of the rod between the exciting electrodes is sufficiently rigid. From this, of course, it does not follow that quartz clocks need not be installed on damped pedestals eliminating all shocks and oscillations.

e. Rate of Quartz Clocks

1. The determination of the absolute daily rate of quartz clocks is carried out by connection to the Nauen time signal at \(13^{h}01^{m}\). The absolute daily rate \(g_{\nu}\) found in this way contains the error inherent in the time signal. This error is eliminated by introducing corrections \(S_z\), published by the institutes of the time service. As a result one obtains the corrected daily rate \(G_{\nu}\), computed from \(g_{\nu}\) by the following formula:

\[ G_{\nu}=g_{\nu}+\left(S_{z,\nu}-S_{z,\nu-1}\right), \]

where \(\nu\) denotes the number of the day.

In Section 18 we indicated that the correction \(S_z\) itself contains considerable errors, so that the determination of the daily rate

of quartz clocks by time signals leads to erroneous values of the rate. To reduce the influence of the error associated with \(S_z\), it is advisable to extend the measurements over a considerable number \(n\) of days, with \(n\) not less than 30. Then the mean daily rate is obtained

\[ \overline{G}=g_y+\frac{1}{30}(S_{z,30}-S_{z,0}) \]

for a 30-day interval, correct to \(1—2^{0}/_{0}\), if the average from the data of several institutes is taken as \(S_z\).

1. Quartz clocks I and II were put into operation in the spring of 1932, and clocks III and IV in 1933. Since then, continuous measurements of the mean daily rate \(G\) have been made for all these clocks. The principal result of these measurements is, as was already indicated in Section 22, the establishment of the linear formula for the rate \(g\):

\[ g_t=g_0+\Delta g t. \]

To determine the initial rate \(g_0\) and the daily changes of rate, in accordance with what was set forth above it is advisable to use not the values of the daily rate, but its mean value, determined from 30-day measurements. In this case the rate formula takes the form

\[ G=G_0+\Delta g n, \]

where \(G_0\) denotes the mean daily rate corresponding to the beginning of the measurements, and \(n\) is the number of series of measurements; \(\Delta g\) represents the change of rate from one sequence to another. The high constancy of the rate of quartz clocks, checked by measuring instantaneous values of the changes in rate, makes it possible, by dividing by the number of days belonging to one series of measurements, to find the daily changes of rate \(\Delta g\) for each day.

1. In Fig. 33 are given curves representing the daily rate of quartz clocks I, II, and III, beginning with 1933. For clocks I and II, from October 1933 to May 1934, data are lacking, since during this interval the clocks were being rebuilt. This gap is covered by clock III, so that data for the whole period are available for determining the time. Nevertheless, we shall confine ourselves to considering the rate of the clocks not for the whole period, but specifically: for clock III, beginning from the moment it was put into operation, and for clocks I and II—from May 1934. For the daily changes of rate \(\Delta g\) the following values are obtained:

Quartz clock I — \(-0.00002\) sec/day from 1/V 1934 to 2/IX 1934.
\(+0.00002\) sec/day from 20/IX 1934 to 10/III 1935.
\(-0.00001\) sec/day from 10/III 1935.

Quartz clock II — \(-0.00004\) sec/day from 1/V 1934 to 20/VII 1934.
\(0.0000\) sec/day from 20/VII 1934 to 1/XI 1934.
\(+0.00008\) sec/day from 1/XI 1934 to 8/IV 1935.
\(-0.00001\) sec/day from 8/IV 1935.

Quartz clock III — \(-0.00009\) sec/day from 1/VI 1933 to 31/XII 1933.
\(-0.00002\) sec/day from 31/XII 1933.

According to this table, quartz clocks I and II more often exhibit changes in the magnitude of the daily change in rate than clock III. This is a consequence of their greater susceptibility to temperature effects, so that in this respect quartz clock III is superior to both clock II and clock I. If, however, changes in the rate of the latter two are taken into account by determining instantaneous values of the rate, then they prove to be quite applicable for the determination of time.

Fig. 33. Measured values of the mean rate \(\overline{G}\) of quartz clocks I (...), II (+++) and III (∘∘∘)

The decrease in the magnitude of the change in the daily rate of clock II from \(9\cdot10^{-5}\) to \(2\cdot10^{-5}\) sec. must be regarded as a phenomenon caused by a change in the properties of the clock. This phenomenon is invariably observed in the first period after new clocks are put into operation. The subsequently constant change in the rate of quartz clock III, equal to \(2\cdot10^{-5}\), is very small. It gives an annual change in rate of only \(0.0073\) sec. If this decrease had not been taken into account in the precomputation of the state of the quartz clocks, then over an annual interval it would have differed from astronomical time by \(1.3\) sec. The available experimental material undoubtedly indicates that the cause of these deviations should be sought in the quartz clocks themselves. Only if all quartz clocks, after a sufficiently long time following their start, showed identical magnitudes of change in rate could one consider that the cause of the discrepancy was not the clocks themselves.

As regards quartz clocks IV, which for almost two years served at the PTR for the investigation of certain questions connected with the change in rate during the first period after switching on, and which since October 1935 have also been used for maintaining the time service, there are as yet insufficient data on changes in the absolute rate. All that can now be said about them is that they are at least no worse than clock III.

1. Knowledge of the quantity \(\Delta g\) makes it possible, by means of the rate formula, to compute for each quartz clock the daily changes in rate, or else the mean changes in rate over 30 days. According to Section 22, the differences between these computed values and the observed changes in rate are random fluctuations of the rate. Calculations of this kind were carried out by Scheibe and Adelsberger for quartz clocks I, II, and III. The course of the rate fluctuations thus found, as a function of time, is shown in Fig. 34. The rate fluctuations for all three clocks are of the order of several thousandths of a second. From these quantities the mean random fluctuation is determined; for quartz clock III (for a period of 30 days) it is \(\pm 0.0013\) sec. For clocks I and II this quantity proves to be larger by several ten-thousandths of a second.

Fig. 34. Change in the difference between the measured and computed values of \(G\) according to quartz clocks I (···), II (+++) and III (ooo)

In this calculation of errors it is assumed that the fluctuations in rate which occur are caused only by the quartz clocks themselves. This assumption, however, is not confirmed by Fig. 34, from which it is clearly seen that the rate fluctuations of all clocks, in certain intervals of time, turn out to be the same to within several ten-thousandths of a second. If the rate fluctuations are regarded as a random phenomenon, then the points belonging to the individual clocks at identical moments of time should have been scattered within the limits of 0.0013 sec, which, as we see, is not the case.

In view of this, one may confidently put forward the supposition that the saw-toothed course of the curve in Fig. 34 is connected with the fact that the readings of the quartz clocks are compared with astronomical days. This view becomes still more convincing when Fig. 35 is examined, where the curve is given of the decrease in the change between the computed and measured values of the rate for an earlier period, beginning in January 1933.

This curve also reveals a saw-toothed course with just as small a scatter of the points belonging to the individual clocks. Together

...thereby the above-indicated value of the random variation of the rate, equal to 0.0013 sec., loses its reality. The mean random variation must be smaller. In accordance with this, Scheibe and Adelsberger take the mean value of the random variation of the rate of quartz clocks, for a measurement interval of 30 days, to be \(\pm 0.00020\) sec.

This value proves to be 4.5 times smaller than the value of the mean random variation of the rate of mean astronomical clocks,

Fig. 35. Change in the difference between the measured and computed values of \(\overline{G}\) according to quartz clocks I \((\ldots)\), II \((++++)\), III \((ooo)\), and IV \((\cdot\cdot)\)

compiled from data of three time institutes and equal to 0.00090 sec. From this it undoubtedly follows that quartz clocks, with respect to constancy of rate, surpass mean astronomical clocks. And since the mean astronomical clocks of the institutes of the time service have as their primary source a large number of pendulum clocks, it is clear that comparison of quartz clocks with single pendulum clocks alone must be still more favorable for the former.

The correctness of these arguments is disputed by Stoyko in his work \(^{32}\). Objections on this question, however, would be out of place here.

1. Less simple than finding the random variations of the rate over a period of 30 days is finding the random daily variations of the rate of quartz clocks. If this operation is carried out using time signals, then, owing to large fluctuations in the corrections, the variations of the rate of the quartz clocks prove to be excessively large. Admissible values are obtained when measurements of the instantaneous values of changes in the rate are used. From these data, and from consideration of the possibilities of the action of extraneous factors on quartz clocks, it follows unambiguously that the upper lim-

the limit of the daily deviations of the rate lies at \(\pm 0.001\) sec. Thus the mean daily fluctuation of the rate proves to be below 0.001 sec.

This upper limit of the random daily fluctuations of the rate determines not only the internal precision of the rate of quartz clocks, but also characterizes the true fluctuations of their rate relative to absolutely constantly running clocks. The possible internal random fluctuations of the rate, found from measurements of instantaneous values of rate differences, according to measurements carried out at the PTR, are equal to 0.0003 sec.; this value includes the error of measurement, which is of the same order of magnitude.

Pavel and Unk\(^{43}\), by checking against time signals, found for two quartz clocks of the Geodetic Institute in Potsdam (belonging to type III) an internal daily random fluctuation of the rate equal to from 0.00076 to 0.00085 sec.; moreover, they indicate that this value is a maximum value. Therefore the data cited may be regarded as evidence of the exceptionally small magnitude of the internal fluctuations of the rate of quartz clocks.

IV. Constancy of the Rate of Quartz Clocks and Constancy of the Duration of the Astronomical Day

60. According to Scheibe and Adelsberger\(^{7}\), the deviation of the computed rate of quartz clocks from the observed rate, represented by the sawtooth curves of the rate differences in Figs. 34 and 35, has its origin not in the properties of the quartz clocks themselves, but in the determination of time made by the institutes of the time service. It has not yet been clarified to what these fluctuations should be attributed—to systematic errors admitted in the determination of time, or to changes in the duration of the astronomical day. Noteworthy is the increase in rate that took place in June 1934 (Fig. 35), equal to 0.004 sec. and, to an accuracy of several ten-thousandths of a second, the same for all clocks (I, II, and III). Critical consideration of this fact leads to the conclusion that, with a high degree of probability, this change of rate must be

Fig. 36. Change in the difference between the computed and measured values of the rate of the quartz clocks of the Geodetic Institute in Potsdam

PRECISE MEASUREMENT OF TIME

is to be attributed not to the quartz clocks, but to the astronomical time scale, i.e., to the length of the astronomical day.

The correctness of this view is indicated not only by the fact that all the PTR quartz clocks show the same thing. Its correctness is also confirmed by the results of Pavlov and Unk43, obtained with both quartz clocks of the Potsdam Geodetic Institute, built according to the PTR type and with the assistance of the PTR. The rate of this pair of clocks was determined by finding sidereal time by the Geodetic Institute itself. If, from the indications of this pair of clocks, one forms mean quartz clocks and, by the method indicated above, computes the differences between the observed and computed rates, then the curve of the rate differences proves to have the form shown in Fig. 36. When Fig. 36 is compared with Fig. 34, a great similarity is found in the course of both curves. In particular, on the curve for the Potsdam clocks there is also clearly seen the sharp increase in June 1934 and the trough in October—November—December. Beginning in March 1935, there is, between the two curves, noticeably no further similarity. The reason for this should be sought in the difficulties of finding the magnitude of the acceleration of the rate that enters into the rate formula, which would be suitable for long intervals of time. As a consequence of small changes in this magnitude, the computed differences in the rate of the PTR quartz clocks, while fully preserving the general form of the curve, could, for example, after a sufficiently long time, turn out to be negative. In this case it would appear that the end portions of the curves in Figs. 34 and 35 have a greater similarity.

Thus, there can in reality be no doubt about the existence of the phenomenon discovered. The most recent measurements by Scheibe and Adelsberger12 show, moreover, that changes in the length of the astronomical day apparently occur periodically. In June 1935 an increase in the rate was again detected, amounting to 0.004 sec. The curve of rate differences revealed the same sawtooth course that was observed in 1934. Fig. 37 gives a curve depicting the rate differences of quartz clock III for 1934 and 1935. Quartz clocks I and II reveal the same change in differences, which is also observed, so far as can be judged, for the recently put into operation quartz clocks IV. (In computing the acceleration \(\Delta g\) from the data of both years, its value proves to be somewhat different than when computed only from the data of 1934. As a consequence of this,

Fig. 37. Course of the changes between the measured and computed values of \(G\) for quartz clock III.

(as was indicated above, the rate differences for the end of 1934 turn out to be negative.) Thus the results of the measurements for 1935 confirm the results of 1934.

1. The measurement data that we have concerning the rate of various quartz clocks undoubtedly indicate that the uniformity of the duration of the days determined by them exceeds that of the astronomical days determined by the time-service institutes. One may think that the small oscillations which we see in Figs. 34, 35, and 37 are the result of systematic errors in determining time from the stars, or of other errors introduced by the time-service institutes. Such an assumption, however, cannot be made with respect to the large oscillations observed every June. The cause of these oscillations should be sought in the astronomical time scale itself.

If we dwell on the latter assumption, admitting the existence of an error in the astronomical time scale itself (i.e., errors connected with the Earth–time-star system), then we must suppose either that there exist systematic errors in determining the positions of the time stars, which manifest themselves especially strongly in June, or that oscillations occur in the velocity of the Earth’s rotation. In favor of the first assumption is the fact that all time-service institutes, from whose data the rate of the quartz clocks is determined, make use in exactly the same way in April, May, and June of time stars different from those which they use in July, August, and September. But since up to the very writing of the present article no indications of a purely astronomical character on this matter have appeared, the correctness of this assumption does not seem especially plausible.

The situation is otherwise with the second of the assumptions given above—with the assumption that oscillations occur in the velocity of the Earth’s rotation about its axis, since the deviations of the actually observed duration of the lunar month from that calculated on the basis of the theory of gravitation indicate that oscillations exist in the velocity of the Earth’s rotation about its axis. Meyermann^44 studied the indicated data in detail with respect to the Moon and the planets and came to the conclusion that “it may be considered undoubted that the velocity of the Earth’s rotation about its axis is subject to irregular oscillations. In this case the annual rate of the Earth (the terms ‘rate’ and ‘state’ have here the same meaning as in application to clocks) may reach ±1 sec., and the state of the Earth may have a magnitude of ±30 sec. (and perhaps much more).”

At the present time, of course, it is still impossible to attempt to draw, on the basis of the oscillations in the duration of the astronomical days found with the aid of quartz clocks, any definite conclusions, except that, thanks to the exceptional constancy of the rate of these clocks, it becomes possible to observe the constancy of the velocity of the Earth’s rotation about its axis.

Conclusion

62. The most precise measurement of time is the basis and a necessary prerequisite for many precise physical measurements, an example of which may be the measurement of high frequencies, necessary in radio engineering. Therefore the purpose of the present article was to give a survey of the experimental work of many investigators devoted to the transfer of time from time stars, to the “preservation” of time by means of terrestrial chronometers, and also to the development of new instruments and the improvement of old instruments that measure time—pendulum clocks and quartz clocks.

The cosmic measure of time is the astronomical day, which is an interval of time determined by one revolution of the terrestrial globe about its axis. Establishing the instant of time by means of this scale is at present not possible with an accuracy better than \(\pm 0.02\) sec. This accuracy proves to be lower than that with which the instant of time can be determined by the best astronomical clocks existing at present. Therefore an increase in the accuracy of time transfer, necessary for refining measurements of time, is highly desirable. Until this increase in accuracy has been achieved, reducing the influence of the accuracy of time measurements on the daily rate of clocks, and at the same time on the physical measurements made with the aid of these clocks, is possible only by increasing the duration of the time intervals during which the clocks are observed.

The investigations of Riefler, Shortt, and Schuler therefore had as their aim the creation of such pendulum clocks which, owing to the high constancy of their rate, would make it possible to cover large intervals of time. We have considered the various factors influencing the rate of clocks, and the methods of eliminating their influence, applied in various designs. At present the best pendulum clocks are the Shortt clocks.

Alongside pendulum clocks, in view of the need for an even more perfect time-measuring instrument, especially for short intervals, quartz clocks were developed. A detailed examination of the construction and properties of the quartz clocks built by Scheibe and Adelsberger shows that, with respect to the uniformity of their rate, they surpass both astronomical pendulum clocks and the crystal clocks of Marrison (which represent another type of quartz clock), for small as well as for large and very large intervals of time. The observed fluctuations of the mean rate of these clocks, determined from measurements over intervals of 30 days, which prove to be almost the same for four different specimens, must with great probability be attributed to systematic errors allowed in the determination of time by the instruments of the time service, and also to the nonconstancy of the speed of the Earth’s rotation about its axis.

References

1. Berliner Astronomisches Jahrbuch für 1936, herausgeg. vom Astronomischen Rechen-Institut Berlin.
2. Arch. dtsch. Seewarte 54, H. 5, 1936.
3. A. Scheibe u. U. Adelsberger, Z. Hochfrequenztechn. u. El. Ak. 43, 37, 1934.
4. K. Schütte, Astronom. Nachr. 254, 251, 1934.
5. H. C. Freiesleben u. F. Lange, Ann. Hydrogr. Berl. 64, 60, 1936.
6. N. Stoyko, Sur la mesure du temps et les problèmes, qui s’y rattachent, p. 19, Paris 1931.
7. A. Scheibe u. U. Adelsberger, Physik. Z. 37, 185, 1936.
8. A. Repsold, Astronom. Nachr. 246, 231, 1932.
9. K. Schütte, Astronom. Nachr. 247, 173, 1932.
10. Bull. Horaire 5, 268, 1935.
11. N. Stoyko, Ann. franç. Chronom. 4, 221, 1934.
12. A. Scheibe u. U. Adelsberger, Physik. Z. 37, 415, 1936.
13. A. Scheibe u. U. Adelsberger, Ann. Physik 18, 1, 1933.
14. F. Göpel u. W. Keil, ATM J. 151, 1, 2, 1932.
15. M. Schuler, Physik. Z. 30, 884, 1929; Z. techn. Physik 10, 392, 1929; 12, 678, 1931.
16. J. Jackson a. W. Bowyer, Monthly not. 88, 465, 1928.
17. J. Jackson a. W. Bowyer, Monthly not. 90, 268, 1930.
18. M. Schuler, Schriftreihe Ges. Zeitmesskde u. Uhrentechn. 6, 95, 1935.
19. G. Gengler, Astronom. Nachr. 247, 185, 1932.
20. A. Repsold, Astronom. Nachr. 247, 301, 1932.
21. G. Schmerwitz, Z. Instrumentenkde 52, 1, 1932.
22. H. Gebelein, Z. Geophysik 8, 272, 1932.
23. H. Gräff, Z. Geophysik 11, 144, 1935.
24. Ferrié et Jouaust, C. R. 184, 56, 1927.
25. A. L. Loomis a. W. A. Marrison, J. Am. Inst. electr. Eng. 51, 542, 1932.
26. H. Gebelein, Astronom. Nachr. 248, 33, 1933.
27. E. Rieckmann, Physik. Z. 34, 831, 1933.
28. R. Tomaschek, Erg. exakt. Naturwiss. 12, 36, 1933.
29. A. L. Loomis, Monthly not. 91, 569, 1931.
30. E. W. Brown a. D. Brouwer, Monthly not. 91, 575, 1931.
31. E. Trousein u. H. C. Freiesleben, Astronom. Nachr. 242, 233, 1931.
32. Bull. Horaire 6, 25, 1936.
33. W. G. Cady, Proc. Inst. Radio Eng. 10, 88, 1922.
34. J. Curie et P. Curie, C. R. 91, 294, 1880.
35. E. Giebe u. A. Scheibe, Z. Physik 46, 607, 1928; Z. Hochfrequenztechn. u. El. Ak. 41, 83, 1933.
36. G. W. Pierce, Proc. Am. Acad. Arts. a. Sci. 60, 271, 1925.
37. W. A. Marrison, Proc. Inst. Radio Eng. 17, 1103, 1929.
38. F. R. Lack, Proc. Inst. Radio Eng. 17, 1123, 1929.
39. E. Giebe u. A. Scheibe, Elektr. Nachr.-Techn. 5, 65, 1928.
40. McIlwraith, Nat. Res. Counc. 29, 1931.
41. U. Adelsberger, Elektr. Nachr.-Techn. 12, 83, 1935.
42. A. Scheibe u. U. Adelsberger, Physik. Z. 33, 835, 1932.
43. F. Pavel u. W. Uhink, Astronom. Nachr. 257, 365, 1935.
44. B. Meyermann, Erg. exakt. Naturwiss. 7, 92, 1928.
45. G. Simon and R. Zuurman, Photocells and Their Applications, ONTI, 1936, p. 246 ff.

1. These clocks are in a room whose temperature, according to the seasons of the year, varies from 12 to 28° C. ↩↩