# 1911 Encyclopædia Britannica/Gravitation

GRAVITATION (from Lat. gravis, heavy), in physical science, that mutual action between masses of matter by virtue of which every such mass tends toward every other with a force varying directly as the product of the masses and inversely as the square of their distances apart. Although the law was first clearly and rigorously formulated by Sir Isaac Newton, the fact of the action indicated by it was more or less clearly seen by others. Even Ptolemy had a vague conception of a force tending toward the centre of the earth which not only kept bodies upon its surface, but in some way upheld the order of the universe. John Kepler inferred that the planets move in their orbits under some influence or force exerted by the sun; but the laws of motion were not then sufficiently developed, nor were Kepler's ideas of force sufficiently clear, to admit of a precise statement of the nature of the force. C. Huygens and R. Hooke, contemporaries of Newton, saw that Kepler's third law implied a force tending toward the sun which, acting on the several planets, varied inversely as the square of the distance. But two requirements necessary to generalize the theory were still wanting. One was to show that the law of the inverse square not only represented Kepler's third law, but his first two laws also. The other was to show that the gravitation of the earth, following one and the same law with that of the sun, extended to the moon. Newton's researches showed that the attraction of the earth on the moon was the same as that for bodies at the earth's surface, only reduced in the inverse square of the moon's distance from the earth's centre. He also showed that the total gravitation of the earth, assumed as spherical, on external bodies, would be the same as if the earth's mass were concentrated in the centre. This led at once to the statement of the law in its most general form.

The law of gravitation is unique among the laws of nature, not only in its wide generality, taking the whole universe in its scope, but in the fact that, so far as yet known, it is absolutely unmodified by any condition or cause whatever. All other forms of action between masses of matter, vary with circumstances. The mutual action of electrified bodies, for example, is affected by their relative or absolute motion. But no conditions to which matter has ever been subjected, or under which it has ever been observed, have been found to influence its gravitation in the slightest degree. We might conceive the rapid motions of the heavenly bodies to result in some change either in the direction or amount of their gravitation towards each other at each moment; but such is not the case, even in the most rapidly moving bodies of the solar system. The question has also been raised whether the action of gravitation is absolutely instantaneous. If not, the action would not be exactly in the line adjoining the two bodies at the instant, but would be affected by the motion of the line joining them during the time required by the force to pass from one body to the other. The result of this would be seen in the motions of the planets around the sun; but the most refined observations show no such effect. It is also conceivable that bodies might gravitate differently at different temperatures. But the most careful researches have failed to show any apparent modification produced in this way except what might be attributed to the surrounding conditions. The most recent and exhaustive experiment was that of J. H. Poynting and P. Phillips (Proc. Roy. Soc., 76a, p. 445). The result was that the change, if any, was less than 110 of the force for one degree change of temperature, a result too minute to be established by any measures.

Another cause which might be supposed to modify the action of gravitation between two bodies would be the interposition of masses of matter between them, a cause which materially modifies the action of electrified bodies. The question whether this cause modifies gravitation admits of an easy test from observation. If it did, then a portion of the earth's mass or of that of any other planet turned away from the sun would not be subjected to the same action of the sun as if directly exposed to that action. Great masses, as those of the great planets, would not be attracted with a force proportional to the mass because of the hindrance or other effect of the interposed portions. But not the slightest modification due to this cause is shown. The general conclusion from everything we see is that a mass of matter in Australia attracts a mass in London precisely as it would if the earth were not interposed between the two masses.

We must therefore regard the law in question as the broadest and most fundamental one which nature makes known to us.

It is not yet experimentally proved that variation as the inverse square is absolutely true at all distances. Astronomical observations extend over too brief a period of time to show any attraction between different stars except those in each other's neighbourhood. But this proves nothing because, in the case of distances so great, centuries or even thousands of years of accurate observation will be required to show any action. On the other hand the enigmatical motion of the perihelion of Mercury has not yet found any plausible explanation except on the hypothesis that the gravitation of the sun diminishes at a rate slightly greater than that of the inverse square—the most simple modification being to suppose that instead of the exponent of the distance being exactly – 2, it is – 2.000 000 161 2.

The argument is extremely simple in form. It is certain that, in the general average, year after year, the force with which Mercury is drawn toward the sun does vary from the exact inverse square of its distance from the sun. The most plausible explanation of this is that one or more masses of matter move around the sun, whose action, whether they are inside or outside the orbit of Mercury, would produce the required modification in the force. From an investigation of all the observations upon Mercury and the other three interior planets, Simon Newcomb found it almost out of the question that any such mass of matter could exist without changing either the figure of the sun itself or the motion of the planes of the orbits of either Mercury or Venus. The qualification “almost” is necessary because so complex a system of actions comes into play, and accurate observations have extended through so short a period, that the proof cannot be regarded as absolute. But the fact that careful and repeated search for a mass of matter sufficient to produce the desired effect has been in vain, affords additional evidence of its non-existence. The most obvious test of the reality of the required modifications would be afforded by two other bodies, the motions of whose pericentres should be similarly affected. These are Mars and the moon. Newcomb found an excess of motions in the perihelion of Mars amounting to about 5′′ per century. But the combination of observations and theory on which this is based is not sufficient fully to establish so slight a motion. In the case of the motion of the moon around the earth, assuming the gravitation of the latter to be subject to the modification in question. the annual motion of the moon's perigee should be greater by 1.5′′ than the theoretical motion. E. W. Brown is the first investigator to determine the theoretical motions with this degree of precision; and he finds that there is no such divergence between the actual and the computed motion. There is therefore as yet no ground for regarding any deviation from the law of inverse square as more than a possibility. (S. N.)

Gravitation Constant and Mean Density of the Earth

The law of gravitation states that two masses M1 and M2, distant d from each other, are pulled together each with a force ${\displaystyle \textstyle {\frac {G.M_{1}M_{2}}{d^{2}}}}$, where G is a constant for all kinds of matter—the gravitation constant.

The acceleration of M2 towards M1 or the force exerted on it by M1 per unit of its mass is therefore ${\displaystyle \textstyle {\frac {G.M_{1}}{d^{2}}}}$. Astronomical observations of the accelerations of different planets towards the sun, or of different satellites towards the same primary, give us the most accurate confirmation of the distance part of the law. By comparing accelerations towards different bodies we obtain the ratios of the masses of those different bodies and, in so far as the ratios are consistent, we obtain confirmation of the mass part. But we only obtain the ratios of the masses to the mass of some one member of the system, say the earth. We do not find the mass in terms of grammes or pounds. In fact, astronomy gives us the product GM, but neither G nor M. For example, the acceleration of the earth towards the sun is about 0.6 cm/sec2 at a distance from it about 15 x 1012 cm. The acceleration of the moon towards the earth is about 0.27 cm/sec2 at a distance from it about 4 x 1010 cm. If S is the mass of the sun and E the mass of the earth we have ${\displaystyle \textstyle 0.6={\frac {GS}{{(15\times 10^{12})}^{2}}}}$ and ${\displaystyle \textstyle 0.27={\frac {GE}{{(4\times 10^{10})}^{2}}}}$ giving us GS and GE, and the ratio SE = 300,000 roughly; but we do not obtain either S or E in grammes, and we do not find G.

The aim of the experiments to be described here may be regarded either as the determination of the mass of the earth in grammes, most conveniently expressed by its mass ÷ its volume, that is by its “mean density” Δ, or the determination of the “gravitation constant” G. Corresponding to these two aspects of the problem there are two modes of attack. Suppose that a body of mass m is suspended at the earth's surface where it is pulled with a force w vertically downwards by the earth—its weight. At the same time let it be pulled with a force p by a measurable mass M which may be a mountain, or some measurable part of the earth's surface layers, or an artificially prepared mass brought near m, and let the pull of M be the same as if it were concentrated at a distance d. The earth pull may be regarded as the same as if the earth were all concentrated at its centre, distant R.

Then

${\displaystyle w={\frac {{\text{G}}\cdot {\frac {4}{3}}\pi {\text{R}}^{3}\Delta m}{{\text{R}}^{2}}}={\text{G}}\cdot {\frac {4}{3}}\pi {\text{R}}\Delta m,}$

(1)

and

${\displaystyle p={\frac {{\text{GM}}m}{d^{2}}}}$

(2)

By division

${\displaystyle \Delta ={\frac {3{\text{M}}}{4\pi {\text{R}}d^{2}}}\cdot {\frac {w}{p}}.}$

If then we can arrange to observe wp we obtain Δ, the mean density of the earth.

But the same observations give us G also. For, putting m = wg in (2), we get

${\displaystyle {\text{G}}={\frac {d^{2}}{\text{M}}}\cdot {\frac {p}{w}}\cdot g.}$

In the second mode of attack the pull p between two artificially prepared measured masses M1, M2 is determined when they are a distance d apart, and since ${\displaystyle \textstyle p={\frac {{\text{G}}.{\text{M}}_{1}{\text{M}}_{2}}{d^{2}}}}$ we get at once ${\displaystyle \textstyle {\text{G}}={\frac {pd^{2}}{{\text{M}}_{1}{\text{M}}_{2}}}}$. But we can also deduce Δ. For putting w = mg in (1) we get

${\displaystyle \Delta ={\frac {3}{4}}{\frac {g}{\text{G}}}\cdot {\frac {1}{\pi {\text{R}}}}.}$

Experiments of the first class in which the pull of a known mass is compared with the pull of the earth may be termed experiments on the mean density of the earth, while experiments of the second class in which the pull between two known masses is directly measured may be termed experiments on the gravitation constant.

We shall, however, adopt a slightly different classification for the purpose of describing methods of experiment, viz:—

1. Comparison of the earth pull on a body with the pull of a natural mass as in the Schiehallion experiment.
2. Determination of the attraction between two artificial masses as in Cavendish's experiment.
3. Comparison of the earth pull on a body with the pull of an artificial mass as in experiments with the common balance.

It is interesting to note that the possibility of gravitation experiments of this kind was first considered by Newton, and in both of the forms (1) and (2). In the System of the World (3rd ed., 1737, p. 40) he calculates that the deviation by a hemispherical mountain, of the earth's density and with radius 3 m., on a plumb-line at its side will be less than 2 minutes. He also calculates (though with an error in his arithmetic) the acceleration towards each other of two spheres each a foot in diameter and of the earth's density, and comes to the conclusion that in either case the effect is too small for measurement. In the Principia, bk. iii., prop. x., he makes a celebrated estimate that the earth's mean density is five or six times that of water. Adopting this estimate, the deviation by an actual mountain or the attraction of two terrestrial spheres would be of the orders calculated, and regarded by Newton as immeasurably small.

Whatever method is adopted the force to be measured is very minute. This may be realized if we here anticipate the results of the experiments, which show that in round numbers Δ = 5.5 and G= 115,000,000 when the masses are in grammes and the distances in centimetres.

Newton's mountain, which would probably have density about Δ2 would deviate the plumb-line not much more than half a minute. Two spheres 30 cm. in diameter (about 1 ft.) and of density 11 (about that of lead) just not touching would pull each other with a force rather less than 2 dynes, and their acceleration would be such that they would move into contact if starting 1 cm. apart in rather over 400 seconds.

From these examples it will be realized that in gravitation experiments extraordinary precautions must be adopted to eliminate disturbing forces which may easily rise to be comparable with the forces to be measured. We shall not attempt to give an account of these precautions, but only seek to set forth the general principles of the different experiments which have been made.

I. Comparison of the Earth Pull with that of a Natural Mass.

Bouguer's Experiments.—The earliest experiments were made by Pierre Bouguer about 1740, and they are recorded in his Figure de la terre (1749). They were of two kinds. In the first he determined the length of the seconds pendulum, and thence g at different levels. Thus at Quito, which may be regarded as on a table-land 1466 toises (a toise is about 6.4 ft.) above sea-level, the seconds pendulum was less by 11331 than on the Isle of Inca at sea-level. But if there were no matter above the sea-level, the inverse square law would make the pendulum less by 11118 at the higher level. The value of g then at the higher level was greater than could be accounted for by the attraction of an earth ending at sea-level by the difference 1111811331 = 16983, and this was put down to the attraction of the plateau 1466 toises high; or the attraction of the whole earth was 6983 times the attraction of the plateau. Using the rule, now known as “Young's rule,” for the attraction of the plateau, Bouguer found that the density of the earth was 4.7 times that of the plateau, a result certainly much too large.

In the second kind of experiment he attempted to measure the horizontal pull of Chimborazo, a mountain about 20,000 ft. high, by the deflection of a plumb-line at a station on its south side. Fig. 1 shows the principle of the method. Suppose that two stations are fixed, one on the side of the mountain due south of the summit, and the other on the same latitude but some distance westward, away from the influence of the mountain. Suppose that at the second station a star is observed to pass the meridian, for simplicity we will say directly overhead, then a plumb-line will hang down exactly parallel to the observing telescope. If the mountain were away it would also hang parallel to the telescope at the first station when directed to the same star. But the mountain pulls the plumb-line towards it and the star appears to the north of the zenith and evidently mountain pull/earth pull=tangent of angle of displacement of zenith.

Fig. 1.—Bouguer's Plumb-line Experiment on the attraction of Chimborazo.

Bouguer observed the meridian altitude of several stars at the two stations. There was still some deflection at the second station, a deflection which he estimated as 114 that at the first station, and he found on allowing for this that his observation gave a deflection of 8 seconds at the first station. From the form and size of the mountain he found that if its density were that of the earth the deflection should be 103 seconds, or the earth was nearly 13 times as dense as the mountain, a result several times too large. But the work was carried on under enormous difficulties owing to the severity of the weather, and no exactness could be expected. The importance of the experiment lay in its proof that the method was possible.

Maskelyne's Experiment.—In 1774 Nevil Maskelyne (Phil. Trans., 1775, p. 495) made an experiment on the deflection of the plumb-line by Schiehallion, a mountain in Perthshire, which has a short ridge nearly east and west, and sides sloping steeply on the north and south. He selected two stations on the same meridian, one on the north, the other on the south slope, and by means of a zenith sector, a telescope provided with a plumb-bob, he determined at each station the meridian zenith distances of a number of stars. From a survey of the district made in the years 1774–1776 the geographical difference of latitude between the two stations was found to be 42.94 seconds, and this would have been the difference in the meridian zenith difference of the same star at the two stations had the mountain been away. But at the north station the plumb-bob was pulled south and the zenith was deflected northwards, while at the south station the effect was reversed. Hence the angle between the zeniths, or the angle between the zenith distances of the same star at the two stations was greater than the geographical 42.94 seconds. The mean of the observations gave a difference of 54.2 seconds, or the double deflection of the plumb-line was 54.2–42.94, say 11.26 seconds.

The computation of the attraction of the mountain on the supposition that its density was that of the earth was made by Charles Hutton from the results of the survey (Phil. Trans., 1778, p. 689), a computation carried out by ingenious and important methods. He found that the deflection should have been greater in the ratio 17804 : 9933 say 9 : 5, whence the density of the earth comes out at 95 that of the mountain. Hutton took the density of the mountain at 2.5, giving the mean density of the earth 4.5. A revision of the density of the mountain from a careful survey of the rocks composing it was made by John Playfair many years later (Phil. Trans., 1811, p. 347), and the density of the earth was given as lying between 4.5588 and 4.867.

Other experiments have been made on the attraction of mountains by Francesco Carlini (Milano Effem. Ast., 1824, p. 28) on Mt. Blanc in 1821, using the pendulum method after the manner of Bouguer, by Colonel Sir Henry James and Captain A. R. Clarke (Phil. Trans., 1856, p. 591), using the plumb-line deflection at Arthur's Seat, by T. C. Mendenhall (Amer. Jour. of Sci. xxi. p. 99), using the pendulum method on Fujiyama in ]apan, and by E. D. Preston (U.S. Coast and Geod. Survey Rep., 1893, p; 513) in Hawaii, using both methods. FIG. 1.-Bouguer's Plumbline

Airy's Experiment.—In 1854 Sir G. B. Airy (Phil. Trans. 1856, p. 297) carried out at Harton pit near South Shields an experiment which he had attempted many years before in conjunction with W. Whewell and R. Sheepshanks at Dolcoath. This consisted in comparing gravity at the top and at the bottom of a mine by the swings of the same pendulum, and thence finding the ratio of the pull of the intervening strata to the pull of the whole earth. The principle of the method may be understood by assuming that the earth consists of concentric spherical shells each homogeneous, the last of thickness h equal to the depth of the mine. Let the radius of the earth to the bottom of the mine be R, and the mean density up to that point be Δ. This will not differ appreciably from the mean density of the whole. Let the density of the strata of depth h be δ. Denoting the values of gravity above and below by ga and gb we have

${\displaystyle g_{b}={\text{G}}{\frac {4}{3}}{\frac {\pi {\text{R}}^{3}\Delta }{{\text{R}}^{2}}}={\text{G}}\cdot {\frac {4}{3}}\pi {\text{R}}\Delta ,}$

and

${\displaystyle g_{a}={\text{G}}{\frac {4}{3}}{\frac {\pi {\text{R}}^{3}\Delta }{({\text{R}}+h)^{2}}}+{\text{G}}\cdot 4\pi h\delta .}$

(since the attraction of a shell h thick on a point just outside it is ${\displaystyle \textstyle {\frac {{\text{G}}.4\pi ({\text{R}}+h)^{2}h\delta }{({\text{R}}+h)^{2}}}={\text{G}}.4\pi h\delta }$).

Therefore

${\displaystyle g_{a}={\text{G}}.{\frac {4}{3}}\pi {\text{R}}\Delta \left(1-{\frac {2h}{\text{R}}}+{\frac {3h}{\text{R}}}{\frac {\delta }{\Delta }}\right)}$ nearly,

whence

${\displaystyle {\frac {g_{a}}{g_{b}}}=1-{\frac {2h}{\text{R}}}+{\frac {3h}{\text{R}}}{\frac {\delta }{\Delta }},}$

and

${\displaystyle {\frac {\Delta }{\delta }}={\frac {\frac {3h}{\text{R}}}{\left(-1+{\frac {2h}{\text{R}}}+{\frac {g_{a}}{g_{b}}}\right)}}}$

Stations were chosen in the same vertical, one near the pit bank, another 1250 ft. below in a disused working, and a “comparison” clock was fixed at each station. A third clock was placed at the upper station connected by an electric circuit to the lower station. It gave an electric signal every 15 seconds by which the rates of the two comparison clocks could be accurately compared. Two “invariable” seconds pendulums were swung, one in front of the upper and the other in front of the lower comparison clock after the manner of Kater, and these invariables were interchanged at intervals. From continuous observations extending over three weeks and after applying various corrections Airy obtained gagb = 1.00005185. Making corrections for the irregularity of the neighbouring strata he found Δδ = 2.6266. W. H. Miller made a careful determination of δ from specimens of the strata, finding it 2.5. The final result taking into account the ellipticity and rotation of the earth is Δ = 6.565.

Von Sterneck's Experiments.—(Mitth. des K.U.K. Mil. Geog. Inst. zu Wien, ii., 1882, p. 77; 1883, p. 59; vi., 1886, p. 97). R. von Sterneck repeated the mine experiment in 1882–1883 at the Adalbert shaft at Pribram in Bohemia and in 1885 at the Abraham shaft near Freiberg. He used two invariable half seconds pendulums, one swung at the surface, the other below at the same time. The two were at intervals interchanged. Von Sterneck introduced a most important improvement by comparing the swings of the two invariables with the same clock which by an electric circuit gave a signal at each station each second. This eliminated clock rates. His method, of which it is not necessary to give the details here, began a new era in the determinations of local variations of gravity. The values which von Sterneck obtained for Δ were not consistent, but increased with the depth of the second station. This was probably due to local irregularities in the strata which could not be directly detected.

All the experiments to determine Δ by the attraction of natural masses are open to the serious objection that we cannot determine the distribution of density in the neighbourhood with any approach to accuracy. The experiments with artificial masses next to be described give much more consistent results, and the experiments with natural masses are now only of use in showing the existence of irregularities in the earth's superficial strata when they give results deviating largely from the accepted value.

II. Determination of the Attraction between two Artificial Masses.

Fig. 2.—Cavendish's Apparatus.
h h, torsion rod hung by wire l g,; x, x, attracted balls hung from its ends; WW, attracting masses.

Cavendish's Experiment (Phil. Trans., 1798, p. 469).—This celebrated experiment was planned by the Rev. John Michell. He completed an apparatus for it but did not live to begin work with it. After Michell's death the apparatus came into the possession of Henry Cavendish, who largely reconstructed it, but still adhered to Michell's plan, and in 1797–1798 he carried out the experiment. The essential feature of it consisted in the determination of the attraction of a lead sphere 12 in. in diameter on another lead sphere 2 in. in diameter, the distance between the centres being about 9 in., by means of a torsion balance. Fig. 2 shows how the experiment was carried out. A torsion rod hh 6 ft. long, tied from its ends to a vertical piece mg, was hung by a wire lg. From its ends depended two lead balls xx each 2 in. in diameter. The position of the rod was determined by a scale fixed near the end of the arm, the arm itself carrying a vernier moving along the scale. This was lighted by a lamp and viewed by a telescope T from the outside of the room containing the apparatus. The torsion balance was enclosed in a case and outside this two lead spheres WW each 12 in. in diameter hung from an arm which could turn round an axis Pp in the line of gl. Suppose that first the spheres are placed so that one is just in front of the right-hand ball x and the other is just behind the left-hand ball x. The two will conspire to pull the balls so that the right end of the rod moves forward. Now let the big spheres be moved round so that one is in front of the left ball and the other behind the right ball. The pulls are reversed and the right end moves backward. The angle between its two positions is (if we neglect cross attractions of right sphere on left ball and left sphere on right ball) four times as great as the deflection of the rod due to approach of one sphere to one ball.

The principle of the experiment may be set forth thus. Let 2a be the length of the torsion rod, m the mass of a ball, M the mass of a large sphere, d the distance between the centres, supposed the same on each side. Let θ be the angle through which the rod moves round when the spheres WW are moved from the first to the second of the positions described above. Let μ be the couple required to twist the rod through 1 radian. Then ${\displaystyle \textstyle \mu \theta ={\frac {4{\text{GM}}ma}{d^{2}}}}$. But μ can be found from the time of vibration of the torsion system when we know its moment of inertia I, and this can be determined. If T is the period ${\displaystyle \textstyle \mu ={\frac {4\pi ^{2}{\text{I}}}{{\text{T}}^{2}}}}$, whence ${\displaystyle \textstyle {\text{G}}={\frac {\pi ^{2}d^{2}{\text{I}}\theta }{{\text{T}}^{2}{\text{M}}ma}}}$, or putting the result in terms of the mean density of the earth Δ it is easy to show that, if L, the length of the seconds pendulum, is put for ${\displaystyle \textstyle {\frac {g}{\pi ^{2}}}}$, and C for 2πR, the earth's circumference, then

${\displaystyle \Delta ={\frac {3}{2}}{\frac {\text{L}}{\text{C}}}{\frac {{\text{M}}ma}{d^{2}{\text{I}}}}{\frac {{\text{T}}^{2}}{\theta }}.}$

The original account by Cavendish is still well worth studying on account of the excellence of his methods. His work was undoubtedly very accurate for a pioneer experiment and has only really been improved upon within the last generation. Making various corrections of which it is not necessary to give a description, the result obtained (after correcting a mistake first pointed out by F. Baily) is Δ = 5.448. In seeking the origin of the disturbed motion of the torsion rod Cavendish made a very important observation. He found that when the masses were left in one position for a time the attracted balls crept now in one direction, now in another, as if the attraction were varying. Ultimately he found that this was due to convection currents in the case containing the torsion rod, currents produced by temperature inequalities. When a large sphere was heated the ball near it tended to approach and when it was cooled the ball tended to recede. Convection currents constitute the chief disturbance and the chief source of error in all attempts to measure small forces in air at ordinary pressure.

Reich's Experiments (Versuche über die mittlere Dichtigkeit der Erde mittelst der Drehwage, Freiberg, 1838; “Neue Versuche mit der Drehwage,” Leipzig Abh. Math. Phys. i., 1852, p. 383).—In 1838 F. Reich published an account of a repetition of the Cavendish experiment carried out on the same general lines, though with somewhat smaller apparatus. The chief differences consisted in the methods of measuring the times of vibration and the deflection, and the changes were hardly improvements. His result after revision was Δ = 5.49. In 1852 he published an account of further work giving as result Δ=5.58. It is noteworthy that in his second paper he gives an account of experiments suggested by J. D. Forbes in which the deflection was not observed directly, but was deduced from observations of the time of vibration when the attracting masses were in different positions.

Let T1 be the time of vibration when the masses are in one of the usual attracting positions. Let d be the distance between the centres of attracting mass and attracted ball, and δ the distance through which the ball is pulled. If a is the half length of the torsion rod and θ the reflection, δ = aθ. Now let the attracting masses be put one at each end of the torsion rod with their centres in the line through the centres of the balls and d from them, and let T2 be the time of vibration. Then it is easy to show that

${\displaystyle {\frac {\delta }{d}}={\frac {a\theta }{d}}={\frac {{\text{T}}_{1}-{\text{T}}_{2}}{{\text{T}}_{1}+{\text{T}}_{2}}}.}$

This gives a value of θ which may be used in the formula. The experiments by this method were not consistent, and the mean result was Δ = 6.25.

Baily's Experiment (Memoirs of the Royal Astron. Soc. xiv.).—In 1841–1842 Francis Baily made a long series of determinations by Cavendish's method and with apparatus nearly of the same dimensions. The attracting masses were 12-in. lead spheres and as attracted balls he used various masses, lead, zinc, glass, ivory, platinum, hollow brass, and finally the torsion rod alone without balls. The suspension was also varied, sometimes consisting of a single wire, sometimes being bifilar. There were systematic errors running through Baily's work, which it is impossible now wholly to explain. These made the resulting value of Δ show a variation with the nature of the attracted masses and a variation with the temperature. His final result Δ = 5.6747 is not of value compared with later results.

Cornu and Baille's Experiment (Comptes rendus, lxxvi., 1873, p. 954; lxxxvi., 1878, pp. 571, 699, 1001; xcvi., 1883, p. 1493).—In 1870 MM. A. Cornu and J. Baille commenced an experiment by the Cavendish method which was never definitely completed, though valuable studies of the behaviour of the torsion apparatus were made. They purposely departed from the dimensions previously used. The torsion balls were of copper about 100 gm. each, the rod was 50 cm. long, and the suspending wire was 4 metres long. On each side of each ball was a hollow iron sphere. Two of these were filled with mercury weighing 12 kgm., the two spheres of mercury constituting the attracting masses. When the position of a mass was to be changed the mercury was pumped from the sphere on one side to that on the other side of a ball. To avoid counting time a method of electric registration on a chronograph was adopted. A provisional result was Δ = 5.56.

Fig. 3.—Diagram of a Section of Professor Boys's Apparatus.

Braun's Experiment (Denkschr. Akad. Wiss. Wien, math.-naturw. Cl. 64, p. 187, 1896).—In 1896 Dr K. Braun, S.J., gave an account of a very careful and excellent repetition of the Cavendish experiment with apparatus much smaller than was used in the older experiments, yet much larger than that used by Boys. A notable feature of the work consisted in the suspension of the torsion apparatus in a receiver exhausted to about 4 mm. of mercury, a pressure at which convection currents almost disappear while “radiometer” forces have hardly begun. For other ingenious arrangements the original paper or a short abstract in Nature, lvi., 1897, p. 127, may be consulted. The attracted balls weighed 54 gm. each and were 25 cm. apart. The attracting masses were spheres of mercury each weighing 9 kgm. and brought into position outside the receiver. Braun used both the deflection method and the time of vibration method suggested to Reich by Forbes. The methods gave almost identical results and his final values are to three decimal places the same as those obtained by Boys.

G. K. Burgess's Experiment (Thèses présentées à la faculté des sciences de Paris pour obtenir le titre de docteur de l'université de Paris, 1901).—This was a Cavendish experiment in which the torsion system was buoyed up by a float in a mercury bath. The attracted masses could thus be made large, and yet the suspending wire could be kept fine. The torsion beam was 12 cm. long, and the attracted balls were lead spheres each 2 kgm. From the centre of the beam depended a vertical steel rod with a varnished copper hollow float at its end, entirely immersed in mercury. The surface of the mercury was covered with dilute sulphuric acid to remove irregularities due to varying surface tension acting on the steel rod. The size of the float was adjusted so that the torsion fibre of quartz 35 cm. long had only to carry a weight of 5 to 10 gm. The time of vibration was over one hour. The torsion couple per radian was determined by preliminary experiments. The attracting masses were each 10 kgm. turning in a circle 18 cm. in diameter. The results gave Δ = 5.55 and G = 6.64 × 10–8.

Eötvos's Experiment (Ann. der Physik und Chemie, 1896, 59, p. 354).–In the course of investigations on local variations of gravity by means of the torsion balance, R. Eötvos devised a method for determining G somewhat like the vibration method used by Reich and Braun. Two pillars were built up of lead blocks 30 cm. square in cross section, 60 cm. high and 30 cm. apart. A torsion rod somewhat less than 30 cm. long with small weights at the ends was enclosed in a double-walled brass case of as little depth as possible, a device which secured great steadiness through freedom from convection currents. The suspension was a platinum wire about 150 cm. long. The torsion rod was first set in the line joining the centres of the pillars and its time of vibration was taken. Then it was set with its length perpendicular to the line joining the centres and the time again taken. From these times Eötvos was able to deduce G = 6.65 × 10–8 whence Δ = 5.53. This is only a provisional value. The experiment was only as it were a by-product in the course of exceedingly ingenious work on the local variation in gravity for which the original paper should be consulted.

Wilsing's Experiment (Publ. des astrophysikalischen Observ. zu Potsdam, 1887, No. 22, vol. vi. pt. ii.; pt. iii. p. 133).—We may perhaps class with the Cavendish type an experiment made by J. Wilsing, in which a vertical “double pendulum” was used in place of a horizontal torsion system. Two weights each 540 gm. were fixed at the ends of a rod 1 metre long, A knife edge was fixed on the rod just above its centre of gravity, and this was supported so that the rod could vibrate about a vertical position. Two attracting masses, cast-iron cylinders each 325 kgm., were placed, say, one in front of the top weight on the pendulum and the other behind the bottom weight, and the position of the rod was observed in the usual mirror and scale way. Then the front attracting mass was dropped to the level of the lower weight and the back mass was raised to that of the upper weight, and the consequent deflection of the rod was observed. By taking the time of vibration of the pendulum first as used in the deflection experiment and then when a small weight was removed from the upper end a known distance from the knife edge, the restoring couple per radian defection could be found. The final result gave Δ = 5.579.

J. Joly's suggested Experiment (Nature xli., 1890, p. 256).—Joly has suggested that G might be determined by hanging a simple pendulum in a vacuum, and vibrating outside the case two massive pendulums each with the same time of swing as the simple pendulum. The simple pendulum would be set swinging by the varying attraction and from its amplitude after a known number of swings of the outside pendulums G could be found.

III. Comparison of the Earth Pull on a body with the Pull of an Artificial Mass by Means of the Common Balance.

The principle of the method is as follows:—Suppose a sphere of mass m and weight w to be hung by a wire from one arm of a balance. Let the mass of the earth be E and its radius be R. Then ${\displaystyle \textstyle w={\frac {{\text{GE}}m}{{\text{R}}^{2}}}}$. Now introduce beneath m a sphere of mass M and let d be the distance of its centre from that of m. Its pull increases the apparent weight of m say by δw. Then ${\displaystyle \textstyle \delta w={\frac {{\text{GM}}m}{d^{2}}}}$. Dividing we obtain ${\displaystyle \textstyle {\frac {\delta w}{w}}-{\frac {{\text{MR}}^{2}}{{\text{E}}d^{2}}}}$, whence ${\displaystyle \textstyle {\text{E}}={\frac {{\text{MR}}^{2}w}{d^{2}\delta w}}}$ and since ${\displaystyle \textstyle g={\frac {\text{GE}}{{\text{R}}^{2}}}}$, G can be found when E is known.

Von Jolly's Experiment (Abhand. der k. bayer. Akad. der Wiss. 2 Cl. xiii. Bd. 1 Abt. p. 157, and xiv. Bd. 2 Abt. p. 3).—In the first of these papers Ph. von Jolly described an experiment in which he sought to determine the decrease in weight with increase of height from the earth's surface, an experiment suggested by Bacon (Nov. Org. Bk. 2, § 36), in the form of comparison of rates of two clocks at different levels, one driven by a spring, the other by weights. The experiment in the form carried out by von Jolly was attempted by H. Power, R. Hooke, and others in the early days of the Royal Society (Mackenzie, The Laws of Gravitation). Von Jolly fixed a balance at the top of his laboratory and from each pan depended a wire supporting another pan 5 metres below. Two 1-kgm. weights were first balanced in the upper pans and then one was moved from an upper to the lower pan on the same side. A gain of 1.5 mgm. was observed after correction for greater weight of air displaced at the lower level. The inverse square law would give a slightly greater gain and the deficiency was ascribed to the configuration of the land near the laboratory. In the second paper a second experiment was described in which a balance was fixed at the top of a tower and provided as before with one pair of pans just below the arms and a second pair hung from these by wires 21 metres below. Four glass globes were prepared equal in weight and volume. Two of these were filled each with 5 kgm. of mercury and then all were sealed up. The two heavy globes were then placed in the upper pans and the two light ones in the lower. The two on one side were now interchanged and a gain in weight of about 31.7 mgm. was observed. Air corrections were eliminated by the use of the globes of equal volume. Then a lead sphere about 1 metre radius was built up of blocks under one of the lower pans and the experiment was repeated. Through the attraction of the lead sphere on the mass of mercury when below the gain was greater by 0.589 mgm. This result gave Δ = 5.692.

Experiment of Richarz and Krigar-Menzel (Anhang zu den Abhand. der k. preuss. Akad. der Wiss. zu Berlin, 1898).—In 1884 A König and F. Richarz proposed a similar experiment which was ultimately carried out by Richarz and O. Krigar-Menzel. In this experiment a balance was supported somewhat more than 2 metres above the floor and with scale pans above and below as in von Jolly's experiment. Weights each 1 kgm. were placed, say, in the top right pan and the bottom left pan. Then they were shifted to the bottom right and the top left, the result being, after corrections for change in density of air displaced through pressure and temperature changes, a gain in weight of 1.2453 mgm. on the right due to change in level of 2.2628 metres. Then a rectangular column of lead 210 cm. cross section and 200 cm. high was built up under the balance between the pairs of pans. The column was perforated with two vertical tunnels for the passage of the wires supporting the lower pans. On repeating the weightings there was now a decrease on the right when a kgm. was moved on that side from top to bottom while another was moved on the left from bottom to top. This decrease was 0.1211 mgm. showing a total change due to the lead mass of 1.2453 + 0.1211 = 1.3664 mgm. and this is obviously four times the attraction of the lead mass on one kgm. The changes in the positions of the weights were made automatically. The results gave Δ = 5.05 and G = 6.685 × 10–8.