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this surface should turn out, after precise measurements, to be exactly an ellipsoid of revolution is a priori improbable. Although it may be highly probable that originally the earth was a fluid mass, yet in the cooling whereby the present crust has resulted, the actual solid surface has been left most irregular in form. It is clear that these irregularities of the visible surface must be accompanied by irregularities in the mathematical figure of the earth, and when we consider the general surface of our globe, its irregular distribution of mountain masses, continents, with oceans and islands, we are prepared to admit that the earth may not be precisely any surface of revolution. Nevertheless, there must exist some spheroid which agrees very closely with the mathematical figure of the earth, and has the same axis of rotation. We must conceive this figure as exhibiting slight departures from the spheroid, the two surfaces cutting one another in various lines; thus a point of the surface is defined by its latitude, longitude, and its height above the “spheroid of reference.” Calling this height $\mathrm {N}$, then of the actual magnitude of this quantity we can generally have no information, it only obtrudes itself on our notice by its variations. In the vicinity of mountains it may change sign in the space of a few miles; $\mathrm {N}$ being regarded as a function of the latitude and longitude, if its differential coefficient with respect to the former be zero at a certain point, the normals to the two surfaces then will lie in the prime vertical; if the differential coefficient of $\mathrm {N}$ with respect to the longitude be zero, the two normals will lie in the meridian; if both coefficients are zero, the normals will coincide. The comparisons of terrestrial measurements with the corresponding astronomical observations have always been accompanied with discrepancies. Suppose $\mathrm {A}$ and $\mathrm {B}$ to be two trigonometrical stations, and that at $\mathrm {A}$ there is a disturbing force drawing the vertical through an angle $\delta$, then it is evident that the apparent zenith of $\mathrm {A}$ will be really that of some other place $\mathrm {A} '\!$, whose distance from $\mathrm {A}$ is $r\delta$, when $r$ is the earth’s radius; and similarly if there be a disturbance at $\mathrm {B}$ of the amount $\delta '\!$, the apparent zenith of $\mathrm {B}$ will be really that of some other place $\mathrm {B} '\!$, whose distance from $\mathrm {B}$ is $r\delta '\!$. Hence we have the discrepancy that, while the geodetic measurements deal with the points $\mathrm {A}$ and $\mathrm {B}$, the astronomical observations belong to the points $\mathrm {A} ',\mathrm {B} '\!$. Should $\delta ,\delta '\!$ be equal and parallel, the displacements $\mathrm {AA} ',\mathrm {BB} '\!$ will be equal and parallel, and no discrepancy will appear. The non-recognition of this circumstance often led to much perplexity in the early history of geodesy. Suppose that, through the unknown variations of $\mathrm {N}$, the probable error of an observed latitude (that is, the angle between the normal to the mathematical surface of the earth at the given point and that of the corresponding point on the spheroid of reference) be $\epsilon$, then if we compare two arcs of a degree each in mean latitudes, and near each other, say about five degrees of latitude apart, the probable error of the resulting value of the ellipticity will be approximately $\pm {\tfrac {1}{500}}\epsilon ,\epsilon$ being expressed in seconds, so that if $\epsilon$ be so great as 2″ the probable error of the resulting ellipticity will be greater than the ellipticity itself.

It is necessary at times to calculate the attraction of a mountain, and the consequent disturbance of the astronomical zenith, at any point within its influence. The deflection of the plumb-line, caused by a local attraction whose amount is $k^{2}\mathrm {A} \delta$, is measured by the ratio of $k^{2}\mathrm {A} \delta$ to the force of gravity at the station. Expressed in seconds, the deflection $\Lambda$ is

$\Lambda =12''\!\!\cdot \!447{\text{A}}\delta /\rho ,$

where $\rho$ is the mean density of the earth, $\delta$ that of the attracting mass, and ${\text{A}}=fs^{-3}xdv$, 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 $\mathrm {A}$ 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 $r,\theta$ be the polar co-ordinates of any point in this plane, and $r,\theta ,z,$ 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 $r=r_{1},r=r_{2}$, and the vertical planes $\theta =\theta _{1},\theta =\theta _{2}$. The component of the attraction at the station or origin along the line $\theta =0$ is

$k^{2}\delta \int _{r1}^{r2}\int _{\theta 1}^{\theta 2}\int _{0}^{h}{\frac {r^{2}\cos \theta }{(r^{2}+z^{2})^{\frac {3}{2}}}}drd\theta dz=k^{2}\delta h(\sin \theta _{2}-\sin \theta _{1})log{r_{2}+(r_{2}^{2}+h^{2})^{\frac {1}{2}}/r_{1}+(r_{1}^{2}+h^{2})^{\frac {3}{2}}}.$

By taking $r_{2}-r_{1}$, sufficiently small, and supposing $h$ also small compared with $r_{1}+r_{2}$ (as it usually is), the attraction is

$k^{2}\delta (r_{2}-r_{1})(\sin \theta _{2}-\sin \theta _{1})h/r,$

where $r={\tfrac {1}{2}}(r_{1}+r_{2})$. 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 $\delta :\rho$ 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 $\rho :\delta$, 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

$y=a\mu {(1-2u\cos \theta +u^{2})^{-{\frac {1}{2}}}-1},$

where $a$ is the radius of the (spherical) earth, $a(1-u)$ the distance