MEASUREMENT OF GRAVITY ON THE WATER SURFACE.

M. I. Polikarpov.

1. History of the Method.

The determination of the acceleration of gravity on the water surface, especially on the ocean, provides very valuable data for solving a number of questions relating to the figure of the earth and the structure of the earth’s crust. For this reason the problem was posed long ago of measuring gravity on the water surface with the greatest possible accuracy. The first very valuable observations were made by Hecker, who in 1901–1908 made voyages in the Atlantic, Indian, and Pacific Oceans and in the Black Sea. He employed the method proposed by the Norwegian Mohn, based on determining atmospheric pressure simultaneously in two different ways: by an ordinary mercury barometer and by observing the boiling point of a liquid (hypsothermometer). The first method gives a result dependent on the magnitude of gravity at the given place, while the second does not depend on it; from comparison of these data it was possible to obtain the desired value of gravity. All Hecker’s attempts to increase the accuracy of the method were not crowned with success; the accuracy of the measurement of gravity at sea remained considerably below that of the corresponding determinations on land. True, this circumstance did not prevent the well-known geodesist Helmert from drawing the conclusion, on the basis of Hecker’s observations, that the deviations of gravity on the ocean from the normal values measured on the continents were insignificant. Later attempts to improve Hecker’s method—the static method—did not increase the accuracy, but instead considerably complicated the performance of the observations themselves. This is fully confirmed by the recently published experiments of Duffield1 (Duffield).

There existed an opinion that the method of swinging pendulums would be entirely inapplicable for observations at sea. But measurements on land

pointed to the significant advantages of this dynamic method over the earlier static methods. It was therefore highly desirable to investigate in greater detail the question of the applicability of pendulums for measuring gravity at sea.

The fact that gave rise to this line of thought had its origin in the difficulties that V. Meinesz (Vening Meinesz) encountered in carrying out gravitational observations in Holland. The mobility of the ground throughout the western part of the country made it quite impossible to find sufficiently stable foundations for pendulums. Microseismic oscillations of the ground existed not only near carriage roads, but occurred everywhere. These movements depended on the wind and, moreover, increased on approaching the shores of the North Sea, so that it may be supposed that they were caused by the surf of the waves. Since the influence of these harmful movements could not be avoided, it was necessary to seek the solution of the problem in another direction: in eliminating the perturbations of the motion of the pendulums caused by oscillations of the ground. For this purpose Vening Meinesz proposed to carry out observations simultaneously with two pendulums, swinging with different phases on one and the same support. The method he developed gave quite satisfactory results, so that it became entirely obvious that the same idea should be used to eliminate the motions of a ship, although the motions of the latter, of course, considerably exceed the microseismic oscillations of the ground mentioned above.

The first observations were made in May 1922 on board a small steamer (of 1200 tons) in the North Sea, but the observations were unsuccessful owing to bad weather and the strong rolling of the vessel. During these observations it became clear that photographic recording of the motions of the pendulum had to be used, because the perturbations began rather suddenly and changed very rapidly. In addition, it turned out that this method can be applied provided that the rolling of the ship is insignificant; for example, the angular deviations must not exceed \(1^\circ\); otherwise a whole series of disturbing causes make the observations impossible. Here it is absolutely necessary that the agate knife-edges on which the pendulums hang should not slide over the support during the motion of the pendulums. A very successful suggestion was made by van Iterson at the congress in Maastricht—to achieve the necessary conditions even during bad weather by placing the instruments on a submarine, and to make observations when the submarine is submerged several tens of meters below the surface, which should considerably weaken the disturbing influence of the agitated surface of the sea.

After successful experiments on a submarine in the vicinity of Helder, the Geodetic Committee decided to undertake a more thorough experi-

…use of this method during the long voyage. A favorable opportunity soon presented itself: in September 1923 three submarines of the Dutch navy set out for the island of Java. The Minister of the Navy permitted Meinesz to carry out observations on submarine K. II, and thus it became possible to make a sufficiently large number of observations under the most varied conditions.

For the observations during the voyage, use was made of a Stückrath apparatus with four pendulums of brass. No special alterations were made to the instrument; it was only provided with an attachment for photographic recording and with the corresponding light source.

The squadron left Helder on September 18 and arrived in Batavia on December 24.

During the first six days of the passage from Helder to Gibraltar the sea was very rough. Even when the submarine was submerged to 30 meters, the oscillations of the vessel exceeded the permissible limits: the rolling reached several degrees. In the last days of this first passage the sea was calm, and several observations could be made; their preliminary reduction in Gibraltar gave quite satisfactory results.

In Gibraltar, with the assistance of the English authorities, a special suspension for the pendulums was constructed in the Admiralty dockyard, which considerably weakened the influence of the ship’s rolling, so that during the remainder of the voyage the rolling of the submerged submarine, in most cases, did not interfere with the observations.

Subsequently, measurements of the force of gravity were made at the following points: two before Gibraltar, south of the Spanish coast; two between Gibraltar and Tunis; three between Tunis and Alexandria; one between Alexandria and Port Said; four between Suez and Aden; seven between Socotra and Colombo; four between Colombo and Sabang; and three between Sabang and the Strait of Malacca.

In addition to these, special observations were also made: 1) measurements were made four times with the object of checking the “Eötvös effect”1 when the vessel was moving in exactly opposite directions (from east to west and back). The Eötvös effect was fully confirmed in the preliminary—

[[unclear: continuation of preceding sentence]] errors of the experiments; 2) observations were carried out four times in order to investigate the influence on the pendulums of the electric field on board the submarine: one observation was made with a metal casing enclosing the apparatus, another without the casing. In the latter experiments no appreciable difference was detected, so that the ordinary metallic cover of the apparatus is quite sufficient to eliminate the influence of the strong electric field present in the submarine.

To obtain control observations, measurements were made on shore in the following ports: Tunis, Alexandria, Suez, Aden, Colombo, and Sabang. In addition, ordinary observations of gravity with pendulums were made at three places in the Dutch Indies: at Sabang—in the north of the island of Sumatra, in Batavia, and in Bandoeng. In these observations four invar pendulums were used.

The reduction of the observations made in the submarine shows that the attempt to measure gravity at the water surface by means of pendulums was crowned with complete success and surpassed all expectations: the accuracy of the measurements proved to be no lower than on land. Some discrepancy among individual observations is due to an error in other reductions, but was not caused by oscillations of the vessel, since the influence of the pitching of the latter was almost completely eliminated. The period of oscillation of the pendulums, in calm weather, was determined with a mean error of \(2\) or \(3 \cdot 10^{-7}\) sec.; in a rough sea the mean error reached \(10 \cdot 10^{-7}\) sec. The mean error indicated here refers to the correction for the motion of the vessel, but other corrections—for example, for temperature, the rate of the chronometer, and so on—are not taken into account here.

In these observations it became clear that a whole series of perturbations which, at first sight, ought to have been taken into account, in practice proved to be very small, which considerably facilitated the computational work and reduced the error. The strongest perturbations of the motion of the pendulums were caused by horizontal accelerations of the points of suspension. But these perturbations can be completely eliminated by the Meinesz method. Perturbations caused by vertical acceleration and by changes in the inclination of the plane of swing of the pendulums can likewise be excluded with sufficient accuracy. All the remaining perturbations are very insignificant. For many of them theory shows that they should affect the amplitude of oscillation of the pendulums, but in reality the amplitude of the “hypothetical pendulum” is perfectly constant, which also indicates the insignificance of these perturbations. To eliminate all the perturbations mentioned above, two pendulums, oscillating simultaneously on one support and in one plane, would have been sufficient. In the Stückrath apparatus, with which the observations were made, there were two pairs of pendulums swinging in two mutually perpendicular

planes, and therefore two independent results were obtained simultaneously.

It may be said that the mean error of the result for \(g\) over the time of the entire voyage ranged from \(0.003\) to \(0.006\ \mathrm{cm/sec^{-2}}\).

This error is caused chiefly by the following three corrections:

1) Correction for temperature. When the submarine submerged, the temperature inside the vessel rose very rapidly and in some cases reached almost \(40^\circ\) C. It was risky to use invar pendulums because of their great sensitivity to the magnetic field, which is very strong on a submarine. Therefore pendulums of brass were used, whose temperature coefficient was equal to \(47\cdot 10^{-7}\ \mathrm{cm/sec^2}\) per \(1^\circ\) C. The temperature corrections reached a rather considerable magnitude.

To reduce temperature corrections in the future, B. Meinesz proposes to use quartz pendulums. But here it should be noted that the experience of using quartz pendulums in India (Survey of India), according to H. McColly Cowie, showed that pendulums of this material are extremely fragile and, despite all precautions during transport, break easily.

Moreover, in experiments with the same pendulums in Potsdam it was found that quartz very readily absorbs moisture, which must be borne in mind when working with quartz pendulums (made of fused quartz).

2) Correction for the speed of motion of the vessel. This correction is proportional to the component of the ship’s velocity in the direction of the parallel; at the equator the correction is \(0.0040\ \mathrm{cm/sec^2}\) for each \(\mathrm{km/hour}\), and therefore it is necessary to know the speed with an accuracy of at least \(0.5\ \mathrm{km/hour}\).

It is not very difficult to determine the speed of motion of the vessel relative to the water, but currents must also be taken into account. This requires a rather long time, since the mean value of the current velocity can be obtained only on the basis of two navigational determinations separated from one another by substantial intervals of time. These errors are inseparably connected with the circumstance that observations are made on a moving vessel.

Moreover, there is no certainty that the current at a depth of several tens of meters will not differ from surface currents.

Meinesz proposes to use a special gyroscopic apparatus for the precise determination of the vessel’s speed, but the implementation of this method will evidently encounter considerable practical difficulties.

In the voyage to the island of Java, the most advantageous speed of the boat’s motion, when observations were being made, proved to be a speed of 4½ miles per hour. Under these conditions the shocks were insignificant.

3) Correction for the rate of the chronometer. In observations with pendulums on land, the rate of chronometers is usually determined by receiving time signals by radio. Theoretically, it would have been desirable to distribute the pendulum observations evenly between two receptions of time signals. On a submarine it is very difficult to satisfy this condition for a whole series of reasons. Very often during the voyage the signals were received only once per day. Nor can observations be carried out on a submarine for a prolonged time, since it cannot remain submerged for very long.

During the voyage described, the observations usually continued for 15–20 minutes, and the error depended exclusively on the irregularity of the chronometer’s rate. Moreover, the expedition had at its disposal only one, though very good, Nordin chronometer, whose fluctuations in daily rate did not exceed 0.1 sec.

Such a duration of observation was caused by the fact that it was not considered possible to increase the observation time without increasing the amplitude (with an increase in the latter, the “bunny” went out of the field of view). On the basis of the experience obtained during the voyage, it proved possible and very desirable to bring the observation time up to 30 or 40 minutes.

Accurate reception of radio signals from distant stations on a submarine proved rather difficult because of the small size of the antenna, which, moreover, is usually wet.

It was necessary to increase the number of chronometers in order to reduce the error arising from their rate.

The absence in the East of stations giving rhythmic time signals did much harm to the observations.

The reduction of the observations made during the voyage, and subsequent experiments, show that it will probably be possible to carry out observations on board an ordinary steamship in a calm sea. It is necessary only that the vibrations of the ship’s hull arising from the operation of the engines be insignificant. On board a submerged submarine this condition is fulfilled, since in this case the boat is set in motion by an electric motor.

The following figures give an idea of the permissible rolling of the vessel. The mean error of an observation lasting about 20 minutes does not exceed \(10 \cdot 10^{-7}\) sec, if the vertical velocity of the vessel remains less than 15 cm/sec, and the amplitude of the keel rolling does not exceed \(0^\circ.5\) or \(1^\circ\).

Despite the fact that the possibility of observations with pendulums on board an ordinary ship is proven, nevertheless it is preferable—

it is more preferable to make measurements on a submarine, since here observations are possible even when the sea is not perfectly calm, and the accuracy of observation will be more substantial.

2. Theory of the Method

The principal perturbations caused by the motion of the ship may be divided into the following:

a) The effect due to the horizontal component of the acceleration of the point of suspension of the pendulums.

b) The effect due to the vertical component.

c) The effect due to the inclination of the plane of oscillation of the pendulums.

Below is given a brief exposition of the theory of the method which makes it possible, from observations of the simultaneous swinging of two pendulums on one and the same support and in one plane, with equality of the accelerations of the points of suspension of both pendulums, to eliminate the first two effects.

Let us recall that the greatest influence is exerted by the horizontal acceleration of the points of suspension.

The equation of motion of a pendulum whose point of suspension has horizontal acceleration $\ddot y$ will be:

\[ \frac{g}{l}\,\theta+\ddot{\theta}+\frac{\ddot y}{l}=0, \tag{1} \]

where $\theta$ is the angle of deflection of the pendulum, and $l$ is the length of the mathematical pendulum.

To lower the order of this equation we introduce a new complex variable $q$:

\[ q=\theta-\frac{i}{n}\dot{\theta}, \tag{2} \]

where $i\sqrt{-1}$, and $n=\sqrt{\frac{g}{l}}$.

The quantity $q$, as a complex number, may be represented by a vector referred to a rectangular system of coordinates, if along the axes one lays off $\theta$ and $-\dfrac{\dot{\theta}}{n}$.

In what follows we shall call $q$ the “pendulum vector,” its length $a$ the amplitude, and its argument $\varphi$ the angular phase of the pendulum.

The quantities $a$ and $\varphi$ are related to $q$ by the equation:

\[ q=a e^{i\varphi}. \]

Introducing \(q\) into the equation of motion (1) gives:

\[ \dot q-inq-i\frac{n\ddot y}{g}=0. \tag{3} \]

If \(\ddot y=0\), i.e., the pendulum is not subject to perturbations, then the solution of equation (3) will be:

\[ q=q_0 e^{int}, \]

i.e., \(q\) is a vector of constant length, rotating with constant angular velocity \(n\) about the point \(0\).

Fig. 1.

The projection of \(q\) onto the horizontal axis gives \(\theta\). In this case we obtain the well-known graphical representation of the motion of a pendulum by means of circular motion.

The period of oscillation of the pendulum will be equal to:

\[ \tau=\frac{\pi}{n}=\pi\sqrt{\frac{l}{g}}. \]

If the suspension points of both pendulums are subject to equal accelerations \(\ddot y\), then the equations of motion of these pendulums are written as follows:

\[ \dot q_1-in_1q_1-i\frac{n_1\ddot y}{g}=0. \]

\[ \dot q_2-in_2q_2-i\frac{n_2\ddot y}{g}=0. \]

Eliminating \(\ddot y\) and introducing the new notation \(n=\frac{1}{2}(n_1+n_2)\), we have:

\[ \left(\frac{n}{n_1}\dot q_1-\frac{n}{n_2}\dot q_2\right)-in(q_1-q_2)=0. \]

Introducing \(r=\dfrac{n}{n_1}q_1-\dfrac{n}{n_2}q_2\) and \(\Delta=\dfrac{n_1-n_2}{2}\), we obtain:

\[ \dot r-inr-i\Delta\left(\frac{n}{n_1}q_1+\frac{n}{n_2}q_2\right)=0. \tag{4} \]

Discarding the last term, which will be discussed below, we obtain the equation of motion of the unperturbed pendulum, which does not depend on \(\dot y\). We shall call this pendulum, to which the vector \(r\) corresponds, the “hypothetical pendulum,” replacing the two actual pendulums \(n_0 1\) and \(n_0 2\).

If the actual pendulums have almost identical periods of oscillation, then the vector \(r\) will be equal to the difference of the vectors corresponding to the actual pendulums, and then the last term will be very small. This term is equal to 0 for isochronous pendulums.

In order to clarify the influence of the last term, we divide equation (4) by \(r\) and separate the real part from the imaginary part; after some transformations we find:

\[ r=\frac{n}{n_1}q_1-\frac{n}{n_2}q_2, \tag{5a} \]

\[ \dot\varphi=n+\Delta\left(\frac{n^2}{n_1^2}\cdot\frac{a_1^2}{a^2}-\frac{n^2}{n_2^2}\cdot\frac{a_2^2}{a^2}\right), \tag{5b} \]

\[ \frac{\dot a}{a}=2\Delta\,\frac{n^2}{n_1n_2}\,a_1a_2\sin(\varphi_2-\varphi_1), \tag{5c} \]

where \(\varphi\) and \(a\) refer to the hypothetical pendulum. If the pendulums are almost isochronous, the multipliers \(\dfrac{n}{n_1}\) and \(\dfrac{n}{n_2}\) will be equal to 1, and then we obtain:

\[ r=q_1-q_2, \tag{6a} \]

\[ \dot\varphi=n+\Delta\,\frac{a_1^2-a_2^2}{a^2}, \tag{6b} \]

\[ \frac{\dot a}{a}=2\Delta\,a_1a_2\sin(\varphi_2-\varphi_1). \tag{6c} \]

Since the period of oscillation \(T\) is the time during which the angle \(\varphi\) increases to \(\pi\), formula (6b) may be replaced by the following:

\[ T=\tau+\delta\,\frac{a_1^2-a_2^2}{a^2}, \tag{6d} \]

where \(T\) is the period of the hypothetical pendulum, \(\tau\) is the mean value of the periods \(\tau_1\) and \(\tau_2\) of the actual pendulums, equal to the period of oscillation of the unperturbed pendulum, and \(\delta=\dfrac{1}{2}(\tau_1-\tau_2)\).

For absolutely synchronous pendulums, formulas (6) are simplified:

\[ r=q_1-q_2, \tag{7a} \]

\[ \dot{\varphi}=n, \tag{7b} \]

\[ \frac{\dot a}{a}=0, \tag{7c} \]

\[ T=\tau. \tag{7d} \]

With the aid of formulas (5a), (6a), and (7a), one can derive the magnitude of the vector of the hypothetical pendulum from the vectors of the actual pendulums, while formulas (5b), (6d), and (7d) give the period of the hypothetical pendulum.

The accuracy of the measurement will be the greater, the greater the length of the vector \(r\). Formulas (5a), (6a), and (7a) show that the greatest value for \(r\) is obtained when the phase difference is equal to \(\pi\). It is therefore advantageous to use pendulums performing oscillations with the same amplitude and opposite phases.

From the preceding it follows that there is no need necessarily to use two isochronous pendulums for the measurement; but the greater \(\delta\) is, the more accurately the factor

\[ \frac{a_1^{\,2}-a_2^{\,2}}{a^2} \]

in the second term of formula (6d) will have to be determined. Since the quantities \(a_1\) and \(a_2\) continuously change their values during the observations, finding the mean value of this factor over the entire time of measurement requires very lengthy computations in order to attain the required accuracy. A difference in the periods of the actual pendulums of \(200\cdot 10^{-7}\) sec does not cause special complications. In this case the use of formulas (5), which should be applied only to considerably larger differences, is not required.

It should be noted that the method becomes entirely unsuitable if the knives on which the pendulums are suspended slide along the support, and when the accelerations at the points of suspension are not equal to one another. Fortunately, no such sliding was observed during the voyage.

The theoretical study of other perturbations shows that they are considerably smaller than those considered by us above. This is because their expression includes as a factor the amplitude, which does not reach the value \(0.01\).

If the period of the ship differs from the period of the pendulum, as is usually the case, then for the most part these secondary perturbations have an irregular, random character, which affects both the amplitude and the period of oscillation of the pendulums. Between the latter quantities there is the following relation:

\[ [\delta\tau]=\frac{T}{na}[\delta\dot a], \tag{8} \]

where \([\dot{\delta \tau}]\) denotes the mean value of the perturbation in the period of oscillation, and \([\dot{\delta a}]\) the same change of the amplitude per unit time.

Observations during the voyage showed that the changes in amplitude were very insignificant, and consequently it may be asserted that the perturbations in the period of oscillation also belong to the category of very insignificant perturbations. A comparison of the results for both pairs of pendulums leads us to the same conclusion.

The following perturbations have a more systematic character:

1°. Perturbations caused by the vertical acceleration \( \ddot{x} \) of the point of suspension. This same cause produces other, very insignificant perturbations of an irregular character, which may be neglected and which will not be considered by us.

The perturbation mentioned here will affect the quantity \(n\), connected with \(g\) by the formula:

\[ n=\sqrt{\frac{g}{l}}. \]

owing to the change of \(g\) into \(g+\ddot{x}\).

Consequently, we shall observe not the quantity \(g\), but \(g\) increased by the mean change \(\ddot{x}\) over the time of observation. Fortunately, \(\ddot{x}\) is a quantity that changes rapidly both in magnitude and in sign, so that its mean value remains small for a fairly long time; it may be represented by the following formula:

\[ \frac{1}{t}\left(\dot{x}_{t}-\dot{x}_{0}\right), \tag{9} \]

where \(\dot{x}_{t}\) and \(\dot{x}_{0}\) are the vertical velocities at the end and at the beginning of the time \(t\). These vertical velocities can be determined, and consequently these perturbations can also be calculated.

In a calm sea the mean value of the change of \(\dot{x}\) reaches \(1\) cm per sec., and the mean error in determining the period of oscillation of the pendulum is \(2\) or \(3\cdot 10^{-7}\) sec. In a rough sea the mean value of the change of \(\dot{x}\) reaches up to \(10\) cm per sec., which gives a mean error in determining the period of oscillation of \(7—11\cdot 10^{-7}\) sec. These figures refer to an observation duration of 15—20 minutes; with an increase in the observation time, the mean error in the period of oscillation will be considerably smaller, since it is inversely proportional to the time.

2°. The effect depending on the change of the angle between the direction of the force of gravity and the plane of swing of the pendulums.

Discarding the insignificant irregular influences, which may be neglected, this effect of the inclination of the plane of oscillation can

be reduced to the fact that we shall measure not the quantity \(g\), but \(g\cdot\cos a\), where \(a\) is the angle of deviation of the plane of oscillation of the pendulums from the vertical plane.

The magnitude of the correction for \(g\) will be: \(-\dfrac{a^2}{2}g\), and for the period of oscillation:

\[ \delta t = + \frac{a^2}{4}\,T . \tag{10} \]

This correction has one and the same sign throughout the entire observation and reaches, for \(T\), a value of \(30\cdot 10^{-7}\) sec.

Consequently, it is necessary to record the angle of inclination \(a\), so that the mean value \(a^2\) for the entire time of observation can be calculated.

It must also be borne in mind that often the planes in which the pendulums of one and the same pair oscillate are not absolutely parallel, so that if \(a=0\) for one pendulum, then \(a\) for the other pendulum may also be nonzero. This deviation is very insignificant in ordinary relative determinations of gravity on land, since the error is constant for all observed stations; but for the case considered here, where the angle \(a\) is a variable quantity, this circumstance must be taken into account.

3°. Perturbations caused by acceleration arising from rotation about a horizontal axis.

It can be shown that, although these perturbations always have the same sign, they are small. The effect caused by rotation about a horizontal axis does not exceed a few percent of the preceding perturbations.

This effect will be insignificant even when the ship is not moving in a straight line.

All the remaining perturbations belong to the category of irregular ones and are insignificant in magnitude.

3. Instruments.

From the preceding section it is clear that, in order to eliminate perturbations, it is sufficient to have an apparatus with at least two pendulums, equipped with an arrangement for recording the motions of these pendulums. If one has at one’s disposal a larger number of pendulums capable of oscillating in one and the same plane, then it is possible to obtain a number of independent determinations smaller by one than the number of pendulums. In the apparatus of Stuckrath, which served Meinecke for measurement, there are four pendulums, oscillating in pairs in two mutually perpendicular planes. Thus, with the instrument described one can obtain in all only 2 independent results, but on the other hand

this instrument has the advantage that it does not require an auxiliary device for measuring the angle of inclination of the plane of oscillation of the pendulums, since it also records this angle.

Fig. 2.

Let us dwell on the consideration of the rotation of the instrument about an axis perpendicular to the plane of swing of the pendulums. This effect of the ship’s motion was not considered in the preceding section, since it does not influence the measured magnitude of the force of gravity, but it is nevertheless recorded rather accurately. As regards this effect, we may imagine that these motions occur in such a way that the pendulums, together with the recording apparatus, remain motionless, while the vertical line passing through the point of suspension performs oscillations in opposite directions.

Although the angle which the axis of the pendulum makes with the vertical is not recorded, the angle made by a certain fixed line with respect to the instrument is, on the other hand, determined. Thus, we measure not the angle of elongation \(\theta\), but \(\theta+a\), where \(a\) is the angle of deviation of the vertical in the plane of oscillation. This angle is at the same time the angle of inclination of the plane of oscillation of the other pair of pendulums. Thus \(\theta+a\) is found, and it is therefore necessary to separate from one another the angles \(\theta\) and \(a\). This is fairly simple to do, at least approximately, since \(a\) changes sufficiently slowly in comparison with the motions of the pendulums, so that during the time in which \(\theta+a\) changes between \(a+a\) and \(a-a\), \(a\) may be regarded as constant. The angle \(a\) is determined if one takes the mean of these two quantities, given by neighboring points of maxima and minima of the recording curve.

Fig. 3.

Therefore, when processing the observations one has to subtract the angle of inclination of the plane of oscillation of one pair of pendulums from the registration curve of the second pair.

A description of the Stückrath apparatus is available in a number of works, and we shall not dwell on it here, especially since Meinesz introduced no substantial changes into the construction of the instrument. The light source was a small device located at a distance of 1.05 meters from the pendulums. It contained a lamp (with an electric arc between tungsten poles), manufactured by the Philips factory according to Prof. Eindhoven. This lamp gave an intense and well-concentrated bundle of light rays and was very convenient for purposes of registration.

Fig. 4.

In front of the lamp there was a diaphragm of 0.2 mm, the aperture of which was closed every second for a very short interval of time by means of a small lever actuated by an electric current, which was closed and opened by a chronometer. Thus, on the registration curve there were marks at every second.

Fig. 5.

Reflecting from the four mirrors attached to the pendulums, the light rays, passing through the vertical slit of the recording device, fell on a drum with photosensitive paper. The drum rotated at a speed of 1.2 mm/sec. This apparatus was located at a distance of 1.15 m from the pendulums; under these conditions all four images of the diaphragm were obtained especially sharply.

The general arrangement of the instruments—the pendulum, the recording apparatus, the illuminator, and the chronometer—is clearly visible in Figs. 2 and 3.

On the photographic paper four perturbed sinusoids were obtained. The mirrors of the pendulums were arranged in such a way that the images from one pair of pendulums fell on the upper part of the paper, and those from the other pair on the lower part (see Figs. 4 and 5).

Experience showed that the above-mentioned rates of rotation of the drum should be somewhat changed. It is more convenient to use two different rates: about 1.8–2 mm/sec and 0.2–0.4 mm/sec; the first is better used at the beginning and at the end of the observation, when the “pendulum vectors” are determined, and the second when determining the amplitude and the angles of deflection \(a\).

The photographic paper was 12 cm wide and 75 m long and was specially manufactured by the firm “Schaeuffelen” in Heilbronn am Neckar.

As was mentioned in § 1, the apparatus with the pendulums was enclosed (during the stay in Gibraltar) in a special suspension, which made it possible to carry out registration even under fairly considerable rolling. This suspension could swing only along the axis of the boat; the direction of the recording light rays coincided with this same axis, so that oscillations of the pendulum apparatus relative to the apparatus with photographic registration did not interrupt the process of registration.

The pendulums were arranged so that the planes of their oscillation made angles of \(45^\circ\) with the major axis of the submarine. All the instruments were mounted near the metacenter of the boat. The commander of submarine K II required the crew to maintain complete immobility during the observations. Walking on the floor could cause the pendulums to slip.

Below are given reproductions of several photographic records obtained by Meinesz during the voyage to the island of Java.

Fig. 4 reproduces part of a record made in the port of Tunis. The submarine is on the surface; the sea is very calm. The amplitudes hardly change; there is a slight horizontal acceleration. The inclination of the plane of oscillation of the pendulums is very small.

Fig. 5 gives a record in the port of Suez. The submarine is on the surface. The amplitude changes greatly, probably from impacts of K II against neighboring boats. The inclination of the plane of oscillations is small. Despite the presence of horizontal accelerations, the mean error in the period of the hypothetical pendulum does not exceed \(4 \cdot 10^{-7}\) sec for an observation duration of 15 minutes.

4. Results of measurements in the Indian Ocean.

The measurements of V. Meinesz aroused great interest among a whole series of geodesists. The well-known American geodesist W. Bowie (William Bowie) asked Meinesz to communicate to him, if possible, more detailed information on the results of the measurements of gravity in the Indian Ocean. This information was obtained for 13 stations of the Indian Ocean. At the suggestion of W. Bowie, an employee of U. S. Coast

MEASUREMENT OF GRAVITY ON THE WATER SURFACE

and Geodetic Survey C. H. Swick introduced an isostatic reduction for some of Meinesz’s stations (2, 3, 4, 5, and 11). The results of this computation are given in the following table:

All five stations were selected in the open ocean over depths of 4,025–4,390 meters.

The calculation of the theoretical value of gravity $g_c$ was carried out by Helmert’s formula of 1901, with corrections introduced into this quantity for the submergence of the submarine, topographic and isostatic corrections over the entire surface of the globe. In the fourth column $(g)$ the observed values are given. In the column $(g - g_c)$ are given the topographic-isostatic anomalies, computed by Helmert’s formula of 1901, in which the first term is equal to 978.030. The greatest anomaly is 0.015. The mean value of the anomaly for the five stations, taking signs into account, will be $=-0.003$; without signs, $=0.012$.

In the last column are given the values of the anomalies computed by V. Bowie’s formula, which differs from Helmert’s formula only in the value of the first term, equal, according to Bowie, to 978.038. The mean values of the anomalies, according to Bowie, will respectively be: $-0.011$ and 0.014.

On the basis of these results Bowie comes to the conclusion that the insignificant magnitudes of the anomalies, both with signs and without them, indicate the circumstance that in the open deep parts of the Indian Ocean the earth’s crust is in a state of very perfect isostatic equilibrium.

In his article (Nature, No. 2878, Vol. 114, Dec. 1924) Bowie reports some additional information on the accuracy of V. Meinesz’s observations and points out that the following had a great influence on the accuracy: 1) the change in the temperature of the pendulums during the observations and 2) the rate of the chronometers.

In Bowie’s opinion, great difficulties in measuring gravity at sea are posed by obtaining accurate time intervals, since, for example, V. Meinesz checked the rate of the chronometers only once a day, receiving time signals by radio. For the po—

for obtaining more accurate results, Bauy proposes sending and receiving time signals during the observations every hour.

Discussing this question, he came to the idea of using a tuning fork for marking intervals of time when working with pendulums both on land and at sea. The tuning fork should be made of a suitable material with elastic properties as constant as possible, and changes in temperature should have only a very slight effect on the elastic properties of the tuning fork. If such an instrument can be made, then, according to Bauy, by comparing a small number of oscillations of the pendulum with the oscillations of the tuning fork, it will be possible to determine the period of oscillation of the pendulum with sufficient accuracy. Of course, the tuning fork must first be carefully verified and calibrated.

5. The Recent Experiments of 1925.

At the request of the Dutch Geodetic Committee, the Minister of the Navy once again permitted V. Meinesz to carry out observations on board a naval submarine bound for the island of Java. Meinesz sailed on submarine K XI from Den Helder on October 15 and arrived in Alexandria on November 12, having visited Seville and Tunis. On the return journey, on November 23, he boarded at Port Said the Dutch mail steamer “Koningin der Nederlanden” and arrived in Amsterdam on December 5.

The main purpose of the last voyage was to test the new pendulum apparatus designed by V. Meinesz and described in No. 5, January/March 1925, of the Bulletin Géodésique (the organ of the geodetic section of the Geodetic and Geophysical Union). The apparatus was built in the mechanical workshop of the Royal Meteorological Institute by the chief instrument maker Van Rest, with the cooperation of the assistant director Dr. C. Schoute.

A detailed description of the apparatus is to appear in the publications of the Geodetic Committee; for the time being only a brief description of this instrument is known.

The whole apparatus consisted of three main parts: a support with pendulums, the suspension, and the recording device.

On the support there were three pendulums (of length \( \tfrac{1}{4} \) m) of the usual Stückrath model, the same as had been used during the first voyage. The pendulums were made of brass.

All three pendulums could perform almost isochronous oscillations in one plane and could be set in motion simultaneously with precisely established amplitudes. The oscillations were recorded on a moving photographic tape by means of light reflected from mirrors fixed on the pendulums, but not as in the old appara-

MEASUREMENT OF THE FORCE OF GRAVITY ON THE WATER SURFACE

...on the apparatus—for each pendulum separately. The light beam first fell on the mirror of the first pendulum and, after being reflected, fell on the mirror of the second pendulum. With this method the difference of the angular deflections of the two pendulums was recorded. The record obtained on the tape immediately gave the curve of a “hypothetical pendulum,” in which the rotating vector was equal to the difference of the vectors of the actual pendulums. In this way perturbations due to the horizontal accelerations of the moving vessel were eliminated.

In exactly the same way the combined oscillations of the second and third pendulums were recorded. The two curves give two independent results, checking one another and thereby increasing the accuracy of the observation.

To introduce corrections for the change in temperature, the motion of the second pendulum was recorded separately; the beam performing this task was reflected on its path by the mirror of a small pendulum with strong damping, whose plane of oscillation was parallel to the plane of the second pendulum. Another pendulum with strong damping, of the same construction, whose plane of oscillation was perpendicular to the plane of oscillation of the main pendulums, served to record changes in the inclination of this plane over the course of the observation.

Changes in the temperature inside the apparatus during the observation were recorded by means of a metallic thermometer.

In the intervals between observations the pendulums were not removed and remained in the apparatus even during heavy rolling. Arresting of the pendulums was carried out as follows: by turning a handle, the blades from which the pendulums hang were raised from the agate plates on which they normally rest; by moving another handle it was possible to clamp the required ends of the pendulums between four clamps.

Parallel to the plane of oscillation of the pendulums there was a small level, the bubble of which, during the observation, was constantly in motion caused by the rolling of the vessel; the pendulums were set in motion at the moment when the bubble was at zero, so that the oscillations began with the desired and equal amplitude. The instrument also contained a dummy pendulum with a thermometer and a hair hygrometer.

The observations usually lasted 35 minutes.

It may be considered that the practical test of the new instrument exceeded all expectations. The records of the curves came out very clearly; their outward appearance differs substantially from the earlier records (Figs. 4 and 5). The amplitudes of the hypothetical pendulum change quite regularly, which indicates the complete elimination of the influence of horizontal accelerations. The remaining decreases in amplitude, caused by damping, did not exceed \(0.1\ \text{mm}\), i.e. \(2\text{–}3\%\) of the entire amplitude.

Slipping of the pendulums occurred only under especially unfavorable circumstances. Thanks to the special method of suspension of the apparatus, lateral rolling had almost no effect at all; if, however, keel rolling...

when the rolling exceeded certain limits, then slipping of the pendulums was observed, and the images could completely disappear from the photographic film. The pitching of the submarine, when submerged, could be greatly weakened by the use of the horizontal rudder.

Fig. 6.

Fig. 6 reproduces a record made with the new instrument on November 2 in the port of Tunis. \(A\) and \(B\) are the curves of the hypothetical pendulums (the curves overlap one another). There are four time curves in all: \(a\) and \(b\) give sidereal time (small period), \(c\) and \(d\) mean time (large period). \(C\) is the curve of the mean pendulum. Inclination of the plane of oscillation: \(D\) is the damped pendulum; \(D'\) is the undamped pendulum.

Fig. 7 gives a record in the Mediterranean Sea from November 10. The submarine is submerged. The designations are the same as in Fig. 6. The undamped pendulum \(D'\) is absent.

Advantages of the new apparatus over the old are as follows:

1) The observer becomes much less fatigued in making observations, since he does not have to keep his body in an uncomfortable position under the black cloth with which the old apparatus was covered.

Fig. 7.

2) Measurements of the records are much easier to make, owing to their distinctness and regularity.

3) The computations are considerably simplified, since the curves of the hypothetical pendulums are obtained directly on the film, and there is no need to derive them from the curves of the individual pendulums.

4) The accuracy of the observations is greater; it is no lower than that of ordinary measurements on land.

5) The new apparatus can be freely used in those cases in which the old apparatus failed to operate.

During the most recent voyage, observations were made at the following points: in the mouth of the Channel, not far from the shores of France; in the Bay of Biscay off the shores of Spain; in Seville on the Guadalquivir; in the Mediterranean Sea, between Gibraltar and Tunis; in the port of Tunis; in the Mediterranean Sea, between Malta and Alexandria; and in the port of Alexandria.

Some of the observations have already been processed; the results obtained in the ports of Tunis and Alexandria differ, respectively, by 0.001 and 0.007 cm/sec² from the observations of 1923.

The weather remained good throughout, and the sea was calm. The small number of observations in the Atlantic Ocean is explained by the fact that the time assigned for the passage from Helder to Seville did not permit frequent stops. The observations in the Mediterranean Sea constitute a valuable supplement to the observations of 1923.

On the return journey, Meinesz made a series of observations aboard a steamer between Crete and Sicily. The sea was very calm, and the ship’s rolling was slight. The apparatus was placed in a cabin, without suspension. The curves obtained show the existence of vibrations caused by the operation of the engines, but the course of the amplitude variations is fairly regular. Unfortunately, the weather during the rest of the voyage did not permit further measurements to be made. The results obtained show that, under favorable conditions, the new apparatus can also be used for measuring gravity on steamships.

In 1926 the Dutch Geodetic Committee proposes to organize a voyage to the island of Java through the Panama Canal, which will make it possible to carry out measurements of gravity at a whole series of points that are extremely interesting from the gravitational point of view1.

1. While the present article was in press, a report appeared in Nature 117, No. 2953, stating that this voyage had already taken place. ↩↩↩