Page:EB1911 - Volume 12.djvu/403

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 1/14 that at the first station, and he found on allowing for this that his observations 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 9/5 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 Japan, and by E. D. Preston (U.S. Coast and Geod. Survey Rep., 1893, p. 513) in Hawaii, using both methods.

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 g_a and g_b we have

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

and

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

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

Therefore

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

whence

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

and

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

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 g_a/g_b＝1·00005185. Making corrections for the irregularity of the neighbouring strata he found Δ/δ＝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