This form will be defined with reference to a spheroid of revolu-
tion, and it is not a matter of much importance what spheroid
is chosen as the figure of reference so long as it fits the actual
geoid reasonably well. Whatever spheroid is selected, careful
measurements will reveal the fact that the geoid is slightly
irregular with regard to it. The spheroid may be assumed
tangential to the geoid at any one point; the two surfaces will
not usually be coincident elsewhere. The inclination between
the normal to the geoid and the normal to the spheroid is called
the deflection of the plumb-line.
The relation of the geoid to the spheroid and the necessity for making the assumption that they are tangential to one another at some arbitrarily chosen point have not always been kept clearly in mind. The idea has been held that the deflection of the plumb-line was everywhere something real and measurable, if the means of measuring it could be found; whereas in reality the spheroid of reference has no existence in nature, and in order to fix this imaginary surface with reference to the real geoidal surface it is necessary to assume that at some particular point the two surfaces touch each other or are parallel. It is not neces- sary to assume that the origin of the survey is the point at which the surfaces are parallel; we may assume that there is a deflec- tion of the plumb-line therefor not, just as we please. Once the deflection at the origin, the height of the latter above the spheroid, and the axes of this spheroid have been selected, deflections at other points may be derived. If this is done at sufficient points the separation of the geoid from the spheroid can be deduced, and, as the form of the spheroid is known by assumption, that of the geoid also becomes known. The de- termination of the plumb-line deflections is made by means of a combination of results derived from observations to terres- trial and astronomical objects. If the deflections are found to vary smoothly, so that it appears justifiable to derive interme- diate values by interpolation, it becomes possible to integrate the separation of geoid and spheroid due to them, and so arrive at the separation at any point in the area dealt with.
In regions where the deflection is large it is generally also irregular and then interpolation between ordinary triangulation 'stations may prove inadequate, and it may be necessary to fix additional intermediate stations by triangulation and to make the necessary astronomical observations at these.
An alternative method of measuring the separation of geoid and spheroid may be based on the measurement of vertical angles between triangulation stations, combined with spirit levelling. To do this it is necessary that the plumb-line deflec- tions at the stations shall have been determined. The ray from ' one station to another is curved in a vertical plane by atmos- pheric refraction. If this (5) refraction is known it is a simple matter to compute the height of the observed station (whose distance is known) above the horizontal plane through the observing station. It is, however, necessary to find the height with respect to some general datum surface. As the form of the geoid, that is to say the mean sea-level surface, is still unknown, and is jn fact one of the objects of the measurement, it is im- possible to make a formula applicable to it, and it is necessary to have recourse to some assumed reference figure, and the obvious figure is the spheroid which has already been used in connexion with the deflections of the plumb-line. The vertical angles corrected for refraction can be reduced to the spheroidal vertical by applying the component of the plumb-line deflection, and then it is quite straightforward to compute the height of the observed point above the spheroid. By means of spirit levelling it is possible to find the geoidal height of the station observed, for spirit levelling with its short rays intimately follows the geoidal level surface. Both spheroidal and geoidal heights of the observed points are thus obtained and the difference between them is the separation of geoid and spheroid.
It has not generally been pointed out that the triangulated heights and spirit-levelled heights are not strictly the same thing. Triangulated heights have very rarely been properly reduced, taking account of plumb-line deflections and refraction, and so have not meant anything very precise, but they are
certainly not geoidal heights. Geoidal heights have many practical uses and are what would generally be required by engineers, but from the geodesist's point of view they do not mean much until the form of the geoid, to which they refer, has been determined.
Isoslasy. The theory of isostasy postulates that the apparent excesses of matter in the earth's crust, consisting of continents and mountains, and the apparent deficiencies, corresponding to oceans, are compensated by underlying variations of density, mountains being compensated by a low density and oceans by a high density in the material below them. These variations of density constitute the isostatic compensation of the topo- graphical features.
In 1909 Mr. Hayford of the U.S. Coast and Geodetic Sur- vey published his work on the figure of the earth and isostasy (8); he suggested the idea of isostatic compensation being complete at a depth small in comparison with the earth's radius; that is to say, he supposed that all those arrangements of crustal density required to make good the deficiency of height in a column under the sea, or to balance the excess of height in one under a mountain, would be found in a crust of moderate thick- ness, and that all matter at a greater depth was either homo- geneous or arranged in homogeneous layers. He also assumed, partly from considerations of convenience, that the excess or defect of matter was distributed uniformly from sea level to that depth, which he named the " depth of compensation." Computing the attraction of the visible topographical features upon those stations of the U.S. Triangulation at which the deflection of the plumb-line had been determined, and trying the effect of isostatic compensation complete at various depths, he arrived at the conclusion that the most probable depth of compensation is 122 kilometres. Using this depth and calculating the deflections that the visible topography and its compensation would produce, he finds that the average residual, that is the difference between the observed and the calculated deflection, is only one-tenth of what it was before the correction for com- pensation was applied. From this he concludes that in the U.S.A. the existence of a close approximation to isostatic equilibrium is proved, and that this equilibrium is complete at a depth which does not differ very greatly from 122 kilometres.
Hayford's hypothesis was subsequently applied to the pen- dulum stations of the U.S. C. and G.S. (9) and the results obtained were in good accord with those deduced from the deflections of the plumb-line. The hypothesis was also applied to the deflections and pendulum observations in India but the results did not appear to be so favourable. In a further discus- sion of the American results Mr. W. Bowie (10) endeavours to trace a connexion between the gravity residuals and the geology of the regions surrounding the stations, and has a certain measure of success. He points out, however, that if there is, for instance, a surface sheet of dense rock of wide extent, compen- sated by lightness in the deeper crust, the pendulum will not be able to reveal the fact, for the attraction of an extensive disc on a point above its centre is independent of the height of the point so long as the height is small in comparison with the radius of the disc, so that the effect of the excess of matter in the dense sheet of rock immediately under the station will be exactly counterbalanced by the negative effect of the corresponding deficient density in the lower strata of the crust. If the dense sheet were of small extent the pendulum would reveal its pres- ence, for its closeness to the pendulum would make its effect more potent than that of the more deeply situated deficiency which compensates it, and gravity would therefore be greater than if the whole were homogeneous. Sir Sidney Burrard (n) applies the idea of allowing for geological peculiarities to India, where the Gangetic plain is an example of an area covered by alluvium of low density which probably extends downward to a considerable depth. He also shows that the distribution of crustal density required to account for the low values of gravity found at stations of the Gangetic plain will go some way, at least, towards explaining the high values of gravity found at stations along the margin of the alluvium. The
