Gravity

( Originally Published 1915 )

CERTAINLY one of the mechanical phenomena, one of the manifestations of movement, which most strikes the least experienced observer is the fall of bodies. When a material body, which has been raised to a certain height, is deprived of its support, it falls to the ground, following the line which joins the original position of the body to the centre of the Earth, at least as nearly as our senses and instruments can show.

This line is called the vertical of the place. All bodies obey this law of falling, which is the law of gravity. Consequently, liquids by reason of their fluidity dispose themselves so that their free surfaces are at each point normal to the vertical at the point in question. Such surfaces are therefore curvilinear, and their formation is due to the combined action of the laws of gravity and centrifugal force which give to the Earth its ellipsoidal form. The surface of the oceans imaginarily prolonged under the continents is, as we have seen, called the geoid. If it be of very small extent, the curvilinear surface of a liquid mass coincides with its tangent plane and in this case only does it form a horizontal plane perpendicular to the vertical of the place.

Gravity thus furnishes us with the data as to our fundamental directions of horizontality and verticality. If we suspend a heavy body on a string attached to a fixed point, the string, being flexible, takes the direction along which the body is drawn towards the Earth. It forms a plumb line which indicates the vertical of the place.

The weight of a body, which makes it fall to-wards the Earth, is a particular case of the universal attraction between portions of matter; it is not a distinct kind of force. The attractive force between two bodies is proportional to the product of masses and in inverse ratio to the square of their distances. In the case we are considering, the Earth is one of the bodies, viz., that with the preponderating mass. Since it is nearly spherical, its effect, on anything outside it, is the same as if its entire mass was accumulated at its geometrical centre. The other body, that which is attracted, is the one which falls to the ground. We have thus, under our very eyes, an illustration of the law which Newton enunciated, and which governs the movements of the bodies in infinite space.

Father Ximénès, as early as 1757, had pointed out that the balance could demonstrate the identity of weight and gravitation, but lack of precision of the instruments of his period prevented him from realising his idea. The experiment devised by the Spanish scientist was carried out by Jolly towards the end of the nineteenth century. The essentials are as follow: A balance is placed on an elevated support, such as, for example, the flooring of a higher storey of a house, and a long fine metallic wire is suspended from below one of its scale-pans. On the latter, say the right pan, a weight of one kilogram [2 lb. 3 oz. 4 dr.] is placed, while on the other, the left pan, a similar weight is put to give equilibrium. If the weight be now hung on the end of the wire, after removal from the pan where it formerly was, the other weight being untouched, the equilibrium is destroyed because the first weight is now nearer to the centre of the Earth, and so is attracted with greater force.

It is easy to calculate the difference; for a height of 300 metres [990 ft.], such as that of the Eiffel Tower, the variation is 1/10,000 of the weight. In other words if the above experiment were made from the height of the Tower with the weight of one kilogram suspended at the end of the wire near the ground, it would be necessary to add one decigram [1.54 gr.] to the other pan to restore equilibrium. For a height of 30 metres [99 ft.], one centigram [.154 gr.] would be required, and for one of 3 metres [9.9 ft.] the difference would be one milligram [015 gr.] Now, the sensitive balances to be found in modern physical laboratories will easily weigh a kilogram to within 1/12 of a milligram. The experiment may thus be easily carried out between the ceiling and floor of a room; it is extremely instructive, and should be done in schools and colleges at the beginning of every course of physics.

Once it is realised that weight is identical with universal gravitation, it will be seen that it is not strictly correct to say, as is done in courses of elementary physics, that "gravity is a force constant in magnitude and direction." It is not constant in magnitude, for it varies with the least vertical displacement, and we shall see that it also varies with the least horizontal displacement; neither is it constant in direction, since it is directed along the vertical, and two neighbouring verticals meet about the centre of the Earth. And, if within the limits even of a room it is possible to demonstrate a variation in its intensity, it would be equally possible to prove astronomically, by means of the meridian circle and variation in the direction of the vertical with respect to the celestial sphere, in the same space.

The laws of the fall of bodies given in similar courses of physics are similarly not exact. The law of velocities and the law of distances are verified with an apparatus called Atwood's machine, generally 2 metres [6.5 ft.] in height. Now, Jolly's experiment succeeds with a difference of height of two metres; it indicates a difference of two-thirds of a milligram [.o10 gr.] for the weight of one kilogram [2 lb. 3 oz. 4 dr.] transferred from the top of the apparatus to the bottom. Therefore, if the experiment with Atwood's machine appears to succeed, it is thanks to the systematic lack of precision of the apparatus, to the unconscious assistance of the experimenter, and the inexperience of his pupils.

On the other hand, we may demonstrate for purposes of instruction the existence of gravitation by that of weight; if the story be true, Newton in watching the fall of an apple had the first inspiration of his discovery of the law which governs the movements of the celestial bodies in space.

The variation of the force and the direction of gravity demands methods and instruments of exceptional precision for its measurement.

The determination of the direction of gravity resolves itself into the determination of the true vertical at each point on the Earth's surface. This vertical is normal to the ideal surface which is called the geoid, formed, as before stated, by the prolongation of the oceanic surface underneath the emergent land masses. The determination of the direction of gravity therefore necessitates the study of the exact form of the geoid, that is to say, the form of the Earth itself. A special science, geodesy, deals with this matter and we shall shortly have to return to its methods more fully.

The measurement of the intensity of gravity at a given place is a problem in mechanics. Since gravity is a force, we may study that force by the effects which it produces; there are two distinct ways of doing this, the dynamic method and the static method. The first consists in studying the movement impressed on a given system by the action of the force in question; in the second we maintain a state of equilibrium of the body, which is submitted to the action of the force which we require to measure by counterbalancing it with another force, the value of which is known.

The dynamic method has been, up to the present, the one almost wholly employed, and the only form which this method takes is that of oscillations. When a body capable of oscillation, such as a magnetised needle, for example, be displaced from its position of equilibrium, it tends to revert to it, and executes a series of oscillations, the amplitude of which decreases logarithmically. If we place the pole of an electromagnet near this needle, the latter will execute a certain succession of oscillations, which can be determined by observation. Now, if the intensity of the current which circulates around the iron nucleus of the electromagnet be increased and the intensity of the magnetic force due to it consequently also augmented, the needle will, under the influence of the greater force, oscillate more and more quickly in proportion as the force increases.

This is exactly analogous to the method used in studying gravity. A heavy body is taken, suspended at the extremity of a fine, but inextensible, thread. This constitutes a plumb line. The body is displaced from its equilibrium position and left to itself ; under the action of gravity, which tends to bring it to a position as near as possible to the Earth's centre, it executes a series of oscillations of gradually decreasing amplitude, thus constituting a pendulum. Should the intensity of gravity increase, the oscillations will be more rapid and, on the other hand, in a place where the intensity is feebler, they will be slower.

This apparatus would be what mathematicians call a simple pendulum. If the thread were free from friction at its point of suspension, if it were without mass, while still remaining rigid and in-extensible, and if the suspended body had no dimensions but were merely a heavy material point, the law governing its movement would be also simple; the oscillations, while subjected to a logarithmic decrement, would continue indefinitely. When they became of extremely small amplitude, they would be isochronous, and their period would be proportional to the square root of the length of the thread, and inversely proportional to the square root of the intensity of gravity at the place where the pendulum is.

Unfortunately, this ideal pendulum is absolutely unrealisable in practice; however fine the thread may be, it has some mass; the body which is suspended on it, and which is usually in the form of a ball, has dimensions and so cannot be considered as a mathematical point however great its density may be. Furthermore, whether the thread be suspended from a knife-edge or held in a vice, friction is bound to come into play; also the whole oscillates in a resisting medium. For these reasons the experiment so beautifully simple in principle is extremely difficult to carry out actually.

Nevertheless, Bouguer and La Condamine attempted to measure the intensity of gravity with a pendulum approaching as nearly as possible to the simple pendulum, but it is Borda to whom the credit is due of making the first really precise experiment, from which the law of the oscillations and the value of gravity at a given place could be deduced. The famous sailor tried to realise as far as possible the conditions of the simple pendulum. For the heavy body he used a platinum ball, the high density of which enabled it to be of relatively small size; the suspending wire was also of platinum and hung from the knife-edge of a balance. The wire carried at its lower end a hollow greased metallic cap to which the ball of platinum was fixed by simple adherence. The whole was therefore composed of two parts: the ball, and the knife-edge wire-cap system. Borda measured the duration of oscillation of the complete pendulum; then, removing the ball, he displaced, by means of a screw with a heavy head, the centre of gravity of the knife and increased this displacement until the knife-edge wire-cap system, oscillating as a pendulum, had the same period of oscillation as the original pendulum furnished with the ball. In these circumstances, the suspension system did not enter into the calculation at all, and the result was as if Borda was using a pendulum constituted of a wire without mass supporting a heavy sphere. Now the formula of mechanics enable one to calculate the moment of inertia of a sphere with respect to an exterior axis around which it oscillates; the problem was thus solved, save for the corrections due to the perturbing action of air-currents. Borda had previously devised a formula which allowed for the amplitude of the oscillations when these were not extremely small.

This method has been abandoned, though it is difficult to assign a reason for this; it only requires the measurement of the duration of oscillation, a measurement which is equally necessary in all cases, whatever form of pendulum method be employed, and also the measurement of the distance between the knife edge and the centre of the ball. It allows of moving the heavy ball with reference to the cap in which it fits, by twisting it round, and so eliminating any error due to the non-homogeneity of the ball, the mean of experiments with different positions being taken. The whole experiment may be carried out in vacuo, and, consequently, it follows that it is susceptible of the maximum degree of precision that we can attain at the present time.

In spite of this fact, modern geodesists have given up the simple pendulum method and only use that of the compound pendulum.

The compound pendulum consists merely of any body whatsoever, which is caused to oscillate about any axis not passing through its centre of gravity. Under the action of gravity the body originally takes up a position of equilibrium, and, when it is displaced from this position, it executes a series of oscillations according to the pendulum law. But, even if the oscillating body had some definite geometrical form, the experiment would hardly be suitable for the precise determination of the intensity of gravity if De Prony, the inventor of the dynamometer which bears his name, had not discovered a curious and unexpected property characterising the compound pendulum.

Let us take a body of any form whatever. Near one of its extremities we fix a transverse axis and make the body oscillate about this axis, which is called the axis of suspension. These oscillations would have a definite period which is measured and noted with care. We next make use of a second axis parallel to the first one and placed in such a way that if the body be caused to oscillate round this new axis, to which the name axis of oscillation is given, the new period of oscillation is exactly the same as the former one. The science of mechanics shows that it is always possible to find the point where this second axis must be placed, in any body; there are actually several such places. The two axes, those of suspension and oscillation, are reciprocal to one another.

This being done, it may be proved that when the axes have been adjusted as above described, the distance between them is equal to the length of the simple pendulum that would have the same period of oscillation. This is the salient feature of Prony's discovery.

The importance of this will now be seen. The simple pendulum, unrealisable as such, is indirectly realised by means of the compound pendulum. All that is necessary is to find by repeated experimental trials the position of the two axes, to measure the distance between them with all possible precision, and to determine the period and amplitude of the oscillations of the body. The value of the intensity of gravity at the place where the experiment is made may be deduced from the formula for the simple pendulum.

Although the principle of this method is not complicated, its practice is a very delicate matter on account of the precision which is necessary, and which exceeds 1/300,000 part.

In the first place, it is necessary to correct the pendulum for the variations in length due to changes in the temperature of its surroundings. Then there is the influence of the nature of the medium in which the body oscillates, that is to say, the disturbing effects of the air.

One effect which the air has on the pendulum is to exert a thrust upon it, lessening its apparent weight, according to the principle of Archimedes, with the result that the pendulum oscillates a little more easily under the action of a given force than if the same experiment were conducted in vacuo. Again, it offers resistance to the movement of the apparatus, a resistance which affects all moving bodies. This is easily recognised by artillerymen and by cyclists, and it is on account of such resistance that birds and aeroplanes can move through the atmosphere. The degree of resistance increases very rapidly in proportion as the velocity of the body in question is augmented. In the particular case of the pendulum, which oscillates slowly, it is very feeble, but nevertheless not negligible when the degree of precision we require is taken into account. Furthermore, there is a third effect ; the air is to a certain extent carried along with the moving pendulum, and from this it follows that the loss of weight due to the atmospheric thrust above mentioned is doubled. Finally, as the air is far from being a perfect fluid, it possesses a certain viscosity and this viscosity helps to retard the oscillatory movement of the pendulum.

These complex, and by no means negligible, effects of the air may be eliminated by making the pendulum oscillate in a vacuum, a procedure which has only recently been carried out in practice.

The determination of the length of the pendulum, that is to say the distance separating the two parallel axes, is a delicate operation. These axes are represented by two knife-edges, the edges being turned towards one another. It is possible to obtain the value of this distance to within a micron.

The measurement of the duration of the oscillations is an even more delicate matter; it necessitates the determination of a period of about one second with a precision of the same order of magnitude as that which we wish to attain in the resulting value of gravity.

An idea which readily occurs to one is to measure by means of an astronomical clock, regulated to keep sidereal time, the duration of say 1000 oscillations and to divide this quantity by 1000. This is the method of passages; it is long and tedious and tends to tire the observer and cause him to make large errors. It would be possible to re-introduce this method at the present time, registering the oscillations by photography on cinematograph films and perhaps a very good result could be thus obtained.

The other artifices are preferred: the method of coincidences devised by Mairan, and the method of phases applied by the Austrian general, von Sterneck. The latter is the one now most employed.

The pendulum may be used to give information of two different kinds, either to furnish the absolute value of the intensity of gravitation at a given place, or to give the relative value of the intensities at different stations on the Earth's surface.

The absolute measurement is difficult as we have already seen, since it implies the determination of a length and an interval of time with the greatest precision. The relative measurement is easier and thanks to the method instituted by General von Sterneck it is now in current use.

It consists in taking a pendulum, which of course remains of invariable dimensions, and causing it to oscillate successively at two distinct stations, in identical conditions, measuring in each case the duration of the oscillation. The formula for the simple pendulum shows that the intensities of gravity at the two stations are in the inverse ratio of the squares of the oscillation periods.

This being so, it suffices, in order to determine the absolute value of gravity at various places on the Earth's surface, to measure it absolutely at any one place, Paris for example; then the same pendulum is taken to the required places and the values relative to that at Paris obtained. In this way a map showing the values at different places may be made.

Although the method used for this practical application of the pendulum to the determination of gravity is a very good one, nevertheless it requires a series of complex operations and long and delicate manipulations. Also, the measurements are difficult, and the apparatus fragile and bulky. An astronomical clock is, in fact, necessary, as is, also, an instrument to observe the stars and so regulate the clock to keep sidereal time. Then there is the pendulum, or rather pendulums (von Sterneck used four), a firm support to hold them and an air-pump and receiver to create a vacuum in which to place them. Finally, an apparatus is necessary to measure the coincidences, and the whole constitutes, as will be seen, a complicated arrangement, exacting as regards the personnel and necessitating an expenditure of much time.

Physicists have therefore endeavoured to find if it would not be possible to measure the intensity of gravity directly by a static method, attaining equilibrium between gravity which tends to draw a heavy body to the ground and an opposing force, which is known or measurable, equal and of contrary sign. The principle of such an arrangement is attractive because if it could be realised with the necessary precision, the intensity of gravity at a place could be deduced by a simple reading of the graduation.

The most simple kind of such an apparatus is the spring balance. The elasticity of stretching of a spiral spring depends only on the nature of the metal and not on the value of gravity. If, therefore, a very sensitive balance be made use of and taken to different places, the same body being always suspended from it, the body will appear to weigh more or less according as the intensity of gravity at the place in question is stronger or weaker. Consequently the spring is stretched to various extents in the different parts of the globe to which it is taken and the variations in length give us the relative values of gravity in these places.

But it is a far cry from theory to practice and the good spring gravity measuring machine has yet to be constructed. However, Threifall has made an instrument which has a high degree of sensitiveness; he does not employ the property of stretching but that of the torsion of a very fine thread of quartz, and geophysicists anticipate that they will be able to do good work with it.

An elasticity to which scientists have for a long time given their attention is that of a gas forced to occupy a constant volume by the pressure of a column of mercury. The mercury, the weight of which is responsible for the pressure to which the gas is submitted, weighs more or less according to the value of the intensity of gravity at the place of the experiment. The result is, therefore, that we balance a pressure, which always remains the same, with mercury columns of different densities of which the heights will thus be in inverse ratio to the densities, that is to say to the corresponding intensities of gravity. The method is simple and ingenious but unfortunately the great sensitiveness of the pressure of the gas to variations of temperature (1/273 part per degree Centigrade) renders the method very difficult in actual practice.

Count Wüllersdorff-Urbair, of Vienna, has attempted to avoid this difficulty by only making use of a gaseous mass as an intermediary between two manometric arrangements, one of which depends on gravity while the other, serving as a standard, does not vary with anything. The mass of gas chosen for this purpose is that of the atmosphere itself. Let us imagine that the pressure of the atmosphere is measured simultaneously by means of two barometers, one a mercury barometer and the other a spring barometer, such as an aneroid. At the first station, the two instruments are made to agree, but this agreement will not hold good at a second station, for which gravity has a different value. In fact, the mercury weighs more in the place where the force of gravity is stronger. It is then denser, and in this denser condition a column of mercury of less height will balance a pressure which would have necessitated a longer column of mercury of normal density. The aneroid barometer should always indicate, on the contrary, the true pressure. The greater or less extent of disagreement between the two instruments enables the variations of gravity to be deduced, the atmospheric pressure acting only as an intermediary agent.

The method appears good but it is vitiated by a weak point, viz., the aneroid barometer. Depending upon the elasticity of a spring it is subject to vicissitudes ; the elasticity of steel changes greatly with temperature and also slowly varies with time. The method, therefore, would not have been susceptible of the necessary precision if the Swiss physicist Guillaume had not conceived the idea of replacing the aneroid barometer by an instrument called the hypsometer, which gives the value of the atmospheric pressure by deter-mining the value of the boiling point of water, which depends on that pressure. The tables of boiling points and the respective atmospheric pressures they correspond to are drawn up for the normal values of the pressures, measured by a column of mercury at zero Centigrade, at sea level, in a definite latitude on the Earth's surface, viz., that of 45°. So that if the indications of the barometer and hypsometer do not agree, the difference gives the variation of gravity. The hypsometer has been brought to a high state of perfection, due to the progress of thermometry. Professor Mohn of Christiania has made use of the method with great success on land, and the German geodesist, Dr. Hecker, has attempted to utilise it in the course of two voyages, one in the Pacific and one in the Atlantic, in order to obtain the value of gravity on the open sea, where the employment of a pendulum is quite impossible. The maximum degree of precision which may be obtained by this method appears to be one part in 50,000, and this is amply sufficient to detect certain anomalies in the normal value of gravity.

In a general way all physical phenomena, in the analytical formula for which occurs the value of the intensity of gravitation, the symbol for which is g may be utilised to determine this quantity. Thus, the fall of bodies, the velocity of sound, and the pitch of the musical note emitted by a vibrating wire stretched by means of a weight on a sounding box will give data from which the value of g may be deduced. The value of g also enters into many phenomena in connection with wave systems. The velocity of propagation of a seismic wave of translation over the surface of a large ocean is a function of the depth of the ocean and the mean value of the intensity of gravitation at its surface. Furthermore, all ex periments which have to do with a pressure may be made to give the required value of g.

Nevertheless, the pendulum method remains by far the most precise; the value of g which it will give in the hands of a good experimenter may attain a precision of one part in 300,000. It is true that many geodesists give values for g as if they were precise to the extent of one part, or even less, in 1,000,000. But this precision is only apparent and has its origin in the figures resulting from the application of the method known as that of least squares, made use of in the discussion of experimental errors. This method, though excellent in certain cases, often masks errors of experiment. If we have twelve numbers each of seven figures, which express twelve different measurements of some one quantity, and if the first five figures are common to all the twelve values, we may affirm that the experimental precision of the measure is expressed by the decimal order of the last of these figures; in the case cited, this will be a precision of the order of one part in 100,000. If the results be discussed by means of the mathematical theory of probabilities, in particular by the method of least squares, a probable error will be found less than a certain number in taking a value which the calculation determines. But this is a theoretical precision and not an experimental one. If the twelve numbers above taken as an example express, let us say, twelve determinations of the coefficient of expansion, a physicist who will have to use the value of this coefficient in subsequent work should only take the first five figures as exact ones; they are the only ones which he can be quite sure about, since they are common to all the twelve determinations. The use of one of the subsequent figures may lead to a greater or less degree of probability, but not to certainty.

We shall now briefly consider the results that the methods above summarised have given, with reference to the point of view of the variation of the intensity of gravitation over the surface of the globe.

First of all, what is the absolute value of gravity? This quantity has been determined by General Wefforges at the laboratory of weights and measures at Sèvres, in 1890-2, and at Vienna by General von Sterneck. The absolute value of gravity at Paris is 980.97 centimetres [386.208 in.] and at Vienna 979.98 centimetres [385.818 in.]. Some explanation must be given as to why the values should be expressed in centimetres. The reason is the application of a law relating to uniformly accelerated movement. It may be shown by mechanics, that when a constant force acts on a body originally at rest, it communicates a movement of uniform acceleration to the body, or in other words a movement the velocity of which increases, during each unit of time, by the same constant quantity, which is called the acceleration. Acceleration is therefore a length, and is naturally expressed in centimetres. The contradiction between the expression of gravity by a length and what has been shown above as to its variability will be noted. The expression of the intensity of a force by the degree of acceleration which it imparts to a material mass implies that the force in question remains constant. Now gravity is only a constant force on condition that the body on which it acts remains absolutely immobile; if it move, either up or down or parallel to the surface of the globe, the value of gravity changes. In order to obtain an acceleration to measure the force producing it, the body must fall and hence there will be a variation in the value of the force acting on it.

To avoid this difficulty we conventionally define the acceleration of gravity, and imagine that gravity, which has a fixed value when it acts on a body placed at a certain point, will maintain the same value during the fall of the body on which it acts. Then, and then only, will the movement of fall be uniformly accelerated and such uniform acceleration may serve as the measure of the intensity of gravity at the initial position of the body. It will be seen that these imaginary conditions are not practically realisable.

With this understanding, viz., the imaginary hypothesis as to the constancy of gravity throughout the extent of fall of the body, we shall now consider how these ideas may be applied to the measurement of the intensity of gravity by means of the pendulum.

A pendulum is a falling body; at its position of equilibrium it merely represents a plumb line, but when pulled aside from 0 this position it tends to return to it, since it is acted on by gravity and caused to fall down again towards its original position, which is that nearest the Earth's centre. The height of the fall is thus the distance mM. This distance de fines the degree of precision of pendulum measures. For a fall of 300 metres [990 ft.], the variation in the value of gravity is about 1/10,000 part of its original value. For a fall of 30 metres [99 ft.] it will be 1/100,000 part; for one of 3 metres [9.9 ft.], 1/1,000,000 part, and for a fall of 3 centimetres [1.17 in.] it will be one hundred millionth, while for one of three-tenths of a millimetre [.012 in.] it will be one ten thousand millionth part. Now for an angle a of oscillation equal to a degree, the height mM is equal to three-twentieths of a millimetre [.005 in.]. Consequently, the corresponding variation in the intensity of gravity is about one twenty thousand millionth part of its value, and this fraction therefore expresses the limiting precision of pendulum measurements. If it be required in the future to attain a greater precision, it will be necessary to take account of this little variation in gravity during the distance mM and to devise a new mathematical analysis dealing with the phenomena in these conditions.

But the experimental precision of pendulum measures does not actually exceed the 1/500,000 part and perhaps does not attain even this, so that in practice, in the operation of measuring the acceleration of gravity by the aid of pendulum observations, we may neglect the almost imperceptible variation in the force along the path of the fall, the resulting error being far smaller than the errors of experiment. Consequently the actual methods used for the determination of g are legitimate and accurate.

There is a force which partially opposes that of gravity, namely, the centrifugal force due to the Earth's rotation and we have to compound the two forces to find the resultant effect. A point A (Fig. 13) on the surface of the Earth (supposed spherical) is attracted towards the centre by a force AF but at the same time the Earth's rotation tends to drive it in the direction AQ, perpendicular to the polar axis. The point A is consequently subjected to two forces, the attraction AF and the centrifugal force AQ; this last increases in proportion as the point lies nearer the equator and diminishes with decreasing distance from the pole. The resultant of these two forces, that is to say the real force to which the point A is subjected, is thus a force AN which is not directed toward the centre of the sphere. Since the Earth was originally fluid it attained an equilibrium surface such that it was perpendicular at every point to the resultant force at that point. It may be shown mathematically that such a surface would be an ellipsoid of revolution rotating about its minor axis and this is why the Earth is not spherical but has an ellipsoidal figure. As a result, the pole P is nearer to the centre O than a point E on the equator, the former distance being less and the latter one greater than it would be if the Earth were spherical. In this case the equator would occupy the position ee. It follows that the Earth bulges at the equator, having the added portion Ee formerly mentioned.

It may be calculated that at the equator the centrifugal force is 2/286 part of that of gravity, and 289 being the square of 17 we have noted the consequences that would result if the Earth turned seventeen times quicker. Gravity would then be, at the equator, exactly balanced by the centrifugal force, and no body would have any apparent weight. As it is, this force is sufficient to cause the variation of gravity to the extent of 1/289 part between the equator and the pole.

Another cause also produces a variation of gravity between the equator and the poles. Because of the Earth's ellipticity, points on the equator, farther from the centre, are attracted less than points close to the poles which are nearer the centre. This second cause of a continuous variation of gravity with position on the Earth's surface acts in the same way as the first, viz., a decrease of gravity towards the equator. There is also a third cause, which on the contrary, acts in the opposite way from the first two, viz., the influence of the equatorial protuberance. This mass, Ee in section, exercises an additional attraction beyond what would occur in the case of a perfect sphere, and so increases the force of gravity at the equator.

Geodesists have found a simple formula to express the resultant variation of gravity between the Earth's poles and equator, which allows for all three causes. If we know the value go of gravity at the equator, its value in a place whose latitude is is given by adding to go a fraction of go obtained by multiplying it by the 1/193 part of the square of the sine of the latitude.1 Since 1/193 is nearly 1/200, we can state that in round numbers a body moved from the equator to the pole apparently gains in weight 1/200 part of its original weight. If the body weighs a kilogram [2 lb. 3 oz. 4 dr.] at the equator, on a dynamometer (not on a balance), it will weigh 5 grams [77 gr.] more at the pole.

The value go of gravity at the equator is expressed by 978.07 centimetres [385.057 in.].

Pendulum observations can therefore serve for the determination of the flattening of the Earth. Clairaut was the first to give a complete analysis of this matter and it was again undertaken by the eminent German geodesist Helmert, who in 1901 found the value e from a discussion of all the determinations of gravity. Geodetic measurements by direct measures of meridian arcs gave Bessel the result 1/299, and astronomical calculations based on precession, nutation, and inequalities in the Moon's movements have led to the value 1/297 deduced by the mathematicians Radau, Poincaré, and Hill. The agreement between the three figures obtained by such different methods is truly wonderful.

To the continuous regular variations of gravity must be added that caused by the elevation of the point of observation above the geoid, the cause of which is illustrated by the experiment of Jolly, previously described; its calculation is easy. But there are also accidental variations of gravity, anomalies, which we shall now deal with.

We have seen, in the course of the preceding pages, that the measurements of the intensity of gravity enable a precise value of the flattening of the terrestrial globe to be obtained. We have seen that the laws of the central attraction and centrifugal force combined would impose on the originally fluid Earth the shape of an ellipsoid of revolution, turning about its minor axis, which produces the flattening experimentally found.

The problem of finding the exact form of the Earth is one of a very great complexity. If it were required to determine point by point a figure similar to that of the Earth with its contour projections and representation of the various altitudes, the problem would surpass human power. Fortunately several simplifications are possible. In order to resolve the question, sufficiently, the thing to do is to consider three surfaces, each defined in a different way. These surfaces are as follows: First, the geographical surface, that is to say, the exterior surface of the terrestrial globe, comprising the surface of the continents and the free surface of the seas and on which rests the atmosphere that envelops us; secondly, the geoid, of which we have already spoken, and which is the oceanic surface supposed to be prolonged below the continents; thirdly, the geodetic surface, which is defined geometrically and which will be a surface of reference for the purpose of our study; it is an ellipsoid of revolution calculated to agree exactly with the most precise and extended measurements of meridian arcs.

The geographical or real surface is that on which we live and on which consequently all our observations and experiments are made, including the astronomical determination of fixed reference points in the celestial sphere. On this surface of the crust we have measured arcs of the meridian which have enabled us to determine the dimensions of the ellipsoid of reference. But what we wish to know is the exact form of the second surface above mentioned, viz., the geoid, the liquid surface which by its fluidity obeys the combined laws of attraction and centrifugal force. More-over we know that, on account of the relatively very slight mean continental altitude, as compared with the Earth's dimensions, the real surface does not differ much from the surface of the geoid.

We are, thus, lead to study a surface, the geoid, that is intimately related to the real surface, and which does not differ greatly from the surface of reference, that is to say, the theoretical ellipsoid.

If gravity were always the resultant of the centrifugal force and of the attractive force, the geoid would theoretically coincide with the theoretical ellipsoid. But gravity, the direction of which is found at every point by that of a plumb line, that is to say by the true vertical, is not simply this resultant at every point; it is subject to anomalies arising from the local irregularities of the crust, such as high continental plateaux, mountains, beds of minerals of different densities from the surrounding rocks, the discontinuities between land and sea, etc., which introduce local attractions acting on and deviating the heavy mass suspended at the end of the plumb line, thus deflecting the vertical.

And, as the geoid is normal to the direction of the vertical at each point of its surface, it follows that every anomaly of gravity, every local deviation of the vertical, introduces a deformation into the surface of the actual geoid.

It is, however, to be noted that an analysis of the matter will afford us much useful information with which to begin our study.

In the first place, it may be shown that it is always possible to place a given ellipsoid of revolution, chosen as surface of reference, in such a way that its surface shall contain a given point of the geoid and that at this point the astronomical longitude, latitude, and azimuth shall be respectively equal to the geodetic longitude, latitude, and azimuth defined by means of the surface of reference. Also, when this is done the axis of revolution of the ellipsoid is parallel to the polar axis of the Earth. Then in every point where the real vertical of a place on the geographical surface intersects the ellipsoid of reference, we may obtain the astronomical elements, above mentioned, by observation and the geodetic elements by calculation.

We have said that the local attractions modify, in places, the actual surface of the geoid, and in such places the geoid does not coincide with the surface of reference. Consequently as the latter is known by definition, in order to find the true surface of the actual geoid, it is necessary to know the vertical distance separating the points where the normal to the ellipsoid meets, in the first place, the ellipsoid, and in the second place, the geoid; this is astronomical levelling. It is thus necessary to have the largest possible number of observations of precise astronomical observations made on the real geographical surface.

The lack of homogeneity in the constitution of the terrestrial crust, and the separation of lands and seas with the consequent discontinuity of density, are the chief causes of local anomalies and hence of the corresponding deviations of the direction of the vertical. The pendulum, with its precision of about I part in 500,000, may be used to discover local gravitational anomalies. If gravity at a place is not affected by such causes it has the value which the formula previously given assigns, depending on the square of the sine of the latitude. If the pendulum observation gives a different value, there is an anomaly, positive or negative according as the observed value of gravity is greater or less than the calculated value. The numerical value of the difference between observation and calculation gives the anomaly, qualitatively and quantitatively after the value measured has been reduced to sea-level by correction, and after the altitude of the place of observation, which implies a diminution of the attractive force that is easily worked out, has been allowed for.

The anomalies of gravity are of two kinds : sometimes they are purely local and due to the attraction of neighbouring masses, whether the higher parts of the relief, such as mountains or whether underneath the ground in the form of mineral deposits of abnormal density. Sometimes also they have a systematic character and follow a veritable law with regard to their variations.

Local anomalies have a very great significance for geologists; they give valuable indications as to the density and position of mineral masses which it is impossible to see or sometimes even to reach at all. An earnest and profound study of the anomalies has led German geodesists and geologists to discover a long subterranean mass, directed from west to east, passing under the Elbe and the Oder. Swiss geodesists, from anomalies observed in their country have arrived at some very remarkable conclusions as to the constitution of the subsoil, in particular in the region of the Engadine.

Heim of Zurich has emphasised the importance of these studies, in relation to the conditions of formation of the terrestrial crust. The earth is constituted by a lithosphere (mineral crust) and a barysphere (internal nucleus of great density). On the site of the great original masses, such as Mt. Blanc, heavier internal masses were nearer the surface, and were raised up on the spot. On the contrary, where the density of the lithosphere is less, it has been accumulated by a succession of layers and has sunk down into the fluid part below in proportion to the added burden, the upper regions of the barysphere being driven back laterally. We should thus expect to find the vertical attraction, that of gravity, stronger near the great primary masses than on the regions formed by the accumulation of layers, and this is what observation confirms. So, reciprocally, the study of the values of gravity will enable us to distinguish between one or other of these two categories of country.

The study of the local anomalies exhibited by gravity has thus an enormous importance from the point of view of the constitution of the subsoil. This subject has been carried to a quite unexpected degree of precision by the work and methods of the famous Hungarian physicist, Baron Roland Eötvös, in the course of the last ten years. He endeavoured to find what was the form of a surface normal in every point to the real directions of the vertical, the minutest deviations of which he has managed to disclose. His apparatus consists of a torsion balance with a unifilar suspension enclosed in a copper cage having double walls; the system has a very considerable period of oscillation, which extends to about twenty minutes. The oscillations of the horizontal bar were observed in two successive perpendicular planes and from this Baron Eötvös showed that it is possible to calculate mathematically the principal radii of curvature of the unknown surface whose form is required. That being done, he took a second instrument in which the equal weights, suspended from the extremities of the oscillating lever, are no longer in the same horizontal plane, but are at different levels, one of the weights hanging below the lever, on a thread. This arrangement enables the variations of the force of gravity along the unknown surface to be measured.

By the aid of these torsion balances, Baron Eötvös succeeded in showing that the mountainous slopes of Buda were prolonged under the soil of the town of Pest, for which purpose the pendulum lacked sufficient sensitiveness and precision. He has also been able to prove variations of level in the Hungarian rivers and lakes; in the case of the Danube he was able to demonstrate such variations about 10o metres from the bank.

As to the systematic anomalies, they seem to follow laws which may be elucidated by dividing the observation stations into three categories. First, continental stations, situated in the midst of continents ; secondly, insular stations, situated upon islands, in the middle of oceans, and, thirdly, coast stations, lying near the line of separation of a continent and a sea.

Generally speaking, continental stations, for example those situated on the Thibetan plateau, in the centre of Asia, show a deficit of gravity as compared with the theoretical value. Conversely, insular stations surrounded by a mass of water of less density exhibit an excess of gravity over the calculated value. At the Sandwich Islands, to take a particular example, the excess difference reaches 1/2000 part of the theoretical value of gravity, and the extraordinary magnitude of this irregularity has given rise to numerous hypotheses. One of these supposes that, taking into consideration the immense extent of the Pacific Ocean, its waters are banked up on its two shores by the attraction exercised upon them by the Asiatic and American continents, so that in the central region there would be a compensating lowering of the sea-level. This would explain the anomaly, because the sea-level at the Sandwich Islands would be nearer to the Earth's centre than would be the case if no such lowering existed. This hypothesis is ingenious, but unfortunately if the experimental data be worked out it is found that in order to produce the above-mentioned difference of 1/2000 part of the entire value, by approaching nearer the Earth's centre, a lowering of the level of the Pacific by more than 1200 metres [3/4 of a mile] would be required. We are not able by any measurement to prove this lowering of the level of the open ocean.

As to the shore stations, there is usually an excess, though on some shores a deficit. It is a remarkable fact that along any one coast the difference between observed and calculated values has always the same sign.

It has for a long time been believed that the excessive attracting force experienced on oceanic islands is only a particular case of a general excess of gravity existing at the surface of the oceans. Dr. Hecker of Potsdam has attempted gravity determinations at sea by the hypsometer method. We have seen that the precision of this hypsobarometric method does not exceed 1/40,000 or 1/50,000 part. Dr. Hecker has shown that, to this degree of approximation, the force of gravity is sensibly normal in the Atlantic between Lisbon and Bahia. On the other hand, he has proved large local anomalies on the Pacific, reaching as much as 1/5000 and even 340 part, some positive (south of Australia and in the Honolulu roads) and others negative (above the depths round the Tonga Isles, which surpass 9000 metres [5.6 miles]). Over the rest of the Pacific Ocean, gravity seems to be normal. Nevertheless these results indicate serious anomalies in the distribution of the intensity of gravity in the Pacific.

The real form of the geoid will not be fully known until we can measure gravity at sea with as great a precision as can now be attained on land, since the land surface covers scarcely a quarter of the surface of the globe and the form of the geoid is, therefore, unknown for three-fourths of its extent.

If continuous and precise experiments should enable rigorous measurements of gravity to be made on the oceans, and if these showed that the excess values exhibited on the oceanic isles were a general fact and applied to the entire oceans, and furthermore if a similar certainty could be arrived at with regard to regions in the midst of the great continents, such as the Asiatic plateau, viz., that the value of gravity is always in deficit there, the beautiful hypothesis of Lippmann, relating to the constitution of the terrestrial crust, would be of great importance.

We have already said a few words about this hypothesis; at the commencement of the solidification of the crust, the first solid scoriae floated on the surface of the still liquid spheroidal bath, each one being sustained by the Archimedean thrust exerted on it by the liquid. In the case of those pieces surmounted by mountains or heavy continental masses the weight, being relatively more considerable, would cause them to sink lower and they would accordingly plunge more deeply into the mass of liquid material on which they floated. On the other hand, those which carried an ocean, of relatively slight weight, would not sink in so deeply. In other words, the terrestrial crust ought to be thicker under the continents than under the oceans.

The general anomalies of gravity would thus be explained; under the continents the greater thickness of the crust, increased by the height of the superincumbent mass, would render the distance from the surface of the soil to that of the barysphere greater, and it is the barysphere, the nucleus of very great density, which gives rise to the greater part of the attracting force. This, being inversely as the square of the distance, would consequently be reduced for points of the continental surface. Also this diminution is not compensated by the additional attraction exerted by the continental mass itself, for that has only the density of its constituent rocks, 2.5 times that of water, while, as we have seen, the density of the central nucleus, the barysphere, is much higher.

As regards the oceans, on the contrary, the relative thinness of the crust results in the barysphere being much nearer to the geographical surface and consequently exerting, at that surface, an attractive force of greater intensity. In this way, the excess of oceanic gravity, always assuming it is proved, would be readily explained. Thus, the future of the study of gravity lies on the sea as also does that of other sciences, meteorology in particular. This should not be surprising, considering the extent of the water superficies and its uniform, homogeneous and regular surface. It is natural that the general laws to which the Earth is subject apply fundamentally to these great liquid areas.

In any case, the real geoid, after all we have just said, is not an exact ellipsoid, but one modified in places by local anomalies which produce protuberances or hollows as the case may be; the differences between the two surfaces are not very considerable, since the German geodesist Helmert has shown that the real geoid never deviates more than 200 metres [656 ft.] from the theoretical ellipsoid.

We have seen, in the course of the pages of this chapter, how the Earth's attraction produces at the surface the continual phenomenon of the fall of bodies towards the centre. Each such fall has an effect on the intensity of gravity at the surface of the globe. When I raise a weight a small height in my laboratory in Paris and let it fall, I change at the same moment the intensity of gravity at Honolulu, and, in general, over all the Earth's surface, for in allowing the body to drop to the ground I add the attracting mass of this body to that of the terrestrial spheroid.

Another cause of the variability of gravity is thus adduced. It is true that the variation from this cause is infinitesimal, and so cannot be experimentally shown, but nevertheless it is real. The variability of gravity from other reasons is sufficient to confirm the idea of the law of in-stability, change, and general evolution which governs the existence of the planet on which we live.

There is still another cause of variation of gravity, which produces a secular variation. At the present time this variation is insensible, though it must have been large on taking into consideration the change in our globe since the time of its formation. The cause here referred to is the contraction of the globe due to its cooling process.

The force of gravity acting on a body placed on the surface of the Earth is in inverse proportion to the square of its distance from the Earth's centre, that is to say, in general, to the square of the terrestrial radius. If, therefore, the radius decrease owing to cooling, the force of gravity will increase accordingly.

A contraction of one-fifth part of the radius would produce an increase in gravity of 9/16 of the value of the latter, in other words a little more than 50% of the value.

The hypothesis of a shortening of the radius of our globe agrees well with the fact of the foldings of its crust. Gravity at the Earth's surface has consequently increased. The importance of this fact is considerable in connection with the pressure of the atmosphere which, formerly, must have been much less than at present, in the same conditions of temperature. Even the composition of the atmosphere must have changed from this cause and would vary in proportion to the contraction of the Earth as cooling unceasingly continued.

Since this cooling, though now quite insensible, yet goes on, the slow contraction is still occurring and the force of gravity at the surface of the globe consequently increasing.

To conclude, this force, which was formerly stated to be constant in magnitude and direction, is, on the contrary, subject to continuous variation, both as regards time and space.