where ρ is the mean density of the earth, δ that of the attracting mass, and A = ƒs−3xdv, in which dv is a volume element of the attracting mass within the distance s from the point of deflection, and x the projection of s on the horizontal plane through this point, the linear unit in expressing A being a mile. Suppose, for instance, a table-land whose form is a rectangle of 12 miles by 8 miles, having a height of 500 ft. and density half that of the earth; let the observer be 2 miles distant from the middle point of the longer side. The deflection then is 1″.472; but at 1 mile it increases to 2″.20.
At sixteen astronomical stations in the English survey the disturbance of latitude due to the form of the ground has been computed, and the following will give an idea of the results. At six stations the deflection is under 2″, at six others it is between 2″ and 4″, and at four stations it exceeds 4″. There is one very exceptional station on the north coast of Banffshire, near the village of Portsoy, at which the deflection amounts to 10″, so that if that village were placed on a map in a position to correspond with its astronomical latitude, it would be 1000 ft. out of position! There is the sea to the north and an undulating country to the south, which, however, to a spectator at the station does not suggest any great disturbance of gravity. A somewhat rough estimate of the local attraction from external causes gives a maximum limit of 5″, therefore we have 5″ which must arise from unequal density in the underlying strata in the surrounding country. In order to throw light on this remarkable phenomenon, the latitudes of a number of stations between Nairn on the west, Fraserburgh on the east, and the Grampians on the south, were observed, and the local deflections determined. It is somewhat singular that the deflections diminish in all directions, not very regularly certainly, and most slowly in a south-west direction, finally disappearing, and leaving the maximum at the original station at Portsoy.
The method employed by Dr C. Hutton for computing the attraction of masses of ground is so simple and effectual that it can hardly be improved on. Let a horizontal plane pass through the given station; let r, θ be the polar co-ordinates of any point in this plane, and r, θ, z, the co-ordinates of a particle of the attracting mass; and let it be required to find the attraction of a portion of the mass contained between the horizontal planes z = 0, z = h, the cylindrical surfaces r = r1, r = r2, and the vertical planes θ = θ1, θ = θ2. The component of the attraction at the station or origin along the line θ = 0 is
| k²δ ∫r2r1∫θ2θ1∫h0 | r²cosθ | dr dθ dz = k²δh (sinθ2 − sinθ1) log{r2 + (r2² + h²)1/2 / r1 + (r1² + h²)1/2}. |
| (r² + z²)3/2 |
By taking r2 − r1, sufficiently small, and supposing h also small compared with r1 + r2 (as it usually is), the attraction is
k²δ (r2 − r1) (sinθ2 − sinθ1) h/r,
where r = ½ (r1 + r2). This form suggests the following procedure. Draw on the contoured map a series of equidistant circles, concentric with the station, intersected by radial lines so disposed that the sines of their azimuths are in arithmetical progression. Then, having estimated from the map the mean heights of the various compartments, the calculation is obvious.
In mountainous countries, as near the Alps and in the Caucasus, deflections have been observed to the amount of as much as 30″, while in the Himalayas deflections amounting to 60″ were observed. On the other hand, deflections have been observed in flat countries, such as that noted by Professor K.G. Schweizer, who has shown that, at certain stations in the vicinity of Moscow, within a distance of 16 miles the plumb-line varies 16″ in such a manner as to indicate a vast deficiency of matter in the underlying strata; deflections of 10″ were observed in the level regions of north Germany.
Since the attraction of a mountain mass is expressed as a numerical multiple of δ : ρ the ratio of the density of the mountain to that of the earth, if we have any independent means of ascertaining the amount of the deflection, we have at once the ratio ρ : δ, and thus we obtain the mean density of the earth, as, for instance, at Schiehallion, and afterwards at Arthur’s Seat. Experiments of this kind for determining the mean density of the earth have been made in greater numbers; but they are not free from objection (see [Gravitation]).
Let us now consider the perturbation attending a spherical subterranean mass. A compact mass of great density at a small distance under the surface of the earth will produce an elevation of the mathematical surface which is expressed by the formula