| ga = G · 4⁄3 | πR³Δ | + G · 4πhδ |
| (R + h)² |
(since the attraction of a shell h thick on a point just outside it is G · 4π(R + h)²hδ/(R + h)² = G · 4πhδ).
Therefore
| ga = G · 4⁄3πRΔ ( 1 − | 2h | + | 3h | δ | ) nearly, | |
| R | R | Δ |
whence
| ga | = 1 − | 2h | + | 3h | δ | , | |
| gb | R | R | Δ |
and
| Δ | = | 3h | / ( − 1 + | 2h | + | ga | ). |
| δ | R | R | gb |
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 gb/ga= 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.