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 μθ = 4GMma/d². 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 μ = 4π²I/T², whence G = π²d²Iθ/T²Mma, 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 g/π², and C for 2πR, the earth’s circumference, then
| Δ = 3⁄2 | L | Mma | T² | . | ||
| C | d²I | θ |
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 deflection, 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
δ/d = aθ/d = (T1 − T2) / (T1 + T2).