EARTH-TIDES (2o). The main body of the earth yields to tidal forces, either with per fect or imperfect elasticity, and these minute tides of the solid earth have been measured. The measurement of earth-tides consists in determining the tilt of the earth's surface about its mean position in the earth's figure, and this has been carried out with apparatus of two kinds. One instru ment, viz., the horizontal pendulum, is of a type also used in Seismology (0. Hecker, Veroff. K. Preuss. Geod. Inst. [1907]). Its action is analogous to that of a door whose hinges are not exactly one above the other, so that it lies in a definite vertical plane depending on the position of its line of hinges. The other instrument consists of a horizontal tube, of length from ioo to 500 ft., acting as a water-level and read by optical interferom eter devices (A. A. Michelson, Astrophys. Journ. [1914D. But it should also be possible to detect earth-tides by means of observations of ordinary water-tides. The general form of the dynamical equations, when allow ance is made for the yielding of the earth, is— But in the absence of full observational knowledge of water-tides over a region we must have recourse to a certain amount of theory. Up to the present the method which has been adopted has utilized the ocean tides of long period and in particular the lunar fortnightly constituent Mf. It has been assumed that this constituent followed the equilibrium law, and this is not certain, while in addition there is the fact that the complete equilibrium-form has not yet been numerically calculated. There is a further serious difficulty in isolating the purely astronomical long period constituents in analysis, and it may well be doubted whether reliable constants for the Mf constituent have ever yet been obtained from the records of observation.
When we come to consider the dynamical theory of earth-tides we see that there are various factors involved. There is the direct response of
the solid earth to the tide-generating forces of the moon and sun; there is the yielding produced by the varying pressure of the tidal load on the ocean-floor; there is also the yielding of the earth's varying gravitational field itself, as produced by the moving water and solid earth. The con tributions from these factors have not yet been completely disentangled from the results of observation. The period of the slowest free tidal oscillation of a homogeneous sphere of the same radius and mass as the earth, and as rigid as steel, is about one hour (H. Lamb, Proc. Loud.
Math. Soc. [1882]). The corresponding period for a homogeneous fluid sphere, of the same size and mass, is about one and a half hours (W. Thomson, Phil. Trans. [1863]). It is therefore probably correct to as sume that the period of the free oscillations of the actual earth are all small compared with those of the chief constituents of the tides. This means that the earth-tides will approximate closely to their equilibrium forms, or in other words, that they may be calculated on the principles of Statics.
For an earth of uniform density and elasticity and with an ocean of uniform depth over the whole of it, a complete solution of the tidal problem, both ocean and earth, has been obtained, allowing for all the factors enumerated above (R. 0. Street, Mon. Not. R. A. S. Geophys. Stipp. [1925]). For the simpler case in which there is no ocean, if the assumed rigidity be that of steel, the earth-tide would be everywhere equal to 1/3 of the equilibrium ocean-tide, while if the rigidity be that of glass the fraction would be 3/5. Such determinations as have yet been made indicate that the effective rigidity of the earth as a whole is somewhat greater than that of steel (W. D. Lambert, Bull. U. S. Nat. Research Council [19 2 2 ] .