The amplitudes of all the constituents depending on the inclination of the moon's orbit to the equator vary with the position of the node on the ecliptic. As the monthly mean level at any place also depends on the inclination of the moon's orbit to the equator, we have a small constit uent with a speed N• equal to that of revolution of the moon's nodes. The speeds of the constituents of solar origin may be similarly determined.
Imagine an equilibrium-surface corresponding to each separate har monic constituent. For a constituent of the semi-diurnal species there will be two maximum and two minimum elevations on the equator, occurring alternately and equally spaced. Along any meridian the eleva tion will steadily decrease away from the equator, reaching zero values at the poles. For a constituent of the diurnal species the elevation will be zero along the equator and at the two poles, but along any other parallel of latitude there will be one maximum and one minimum equally spaced. At any one time all the maxima and minima will lie on two meridians, a maximum changing into a minimum and vice versa on crossing the equator. For both species the motion of the surface relative to the earth will be one of uniform rotation round the earth's axis with the speed given in the table. For a constituent of the long period species, the correspond ing surface at any one time will be symmetrical about the earth's axis, and its motion will be a harmonic pulsation with the speed given in the table.
(7). For the dynamical theory we require a mathematical expression of the geographical distribution of the equilibrium-form in each species of constituent, and for the theory of earth-tides we require the distribution of the generating forces inside the earth as well as over its surface. For this purpose we use the potential of the forces, i.e., a quantity possessing magnitude only, whose space-rate of increase in any direction gives the component of force in that direction. For points on the earth's surface
the space-rate of change of potential is proportional to the corresponding inclination of the equilibrium-surface. Let 0 and C denote the centres of the earth and moon respectively and let P denote a point at which the potential is to be calculated. Let OC = D, OP = r, CP = R, angle COP =6 and let M denote the mass of the moon and I' the constant of gravitation. Then the moon's attraction at 0 has a magnitude FM/D2 and is directed along OC. This is the average force over the whole body of the earth, and a uniform distribution of this magnitude has a potential Here Ti= 31— E =1•8 ft', being the mean value of D, e denoting a constant for each constituent, and C the numerical coefficient whose value is given in the table of constituents (G. H. Darwin, Brit. Assoc. Report [1883]; A. T. Doodson, Proc. Roy. Soc., A [1921]). A similar analysis applies to the solar con stituents, and the value of H will be -460X 1.8 ft., but in the table of constituents the solar coefficients have been multiplied by -46o so as to make the coefficients themselves give the relative order of magnitude of both lunar and solar constituents.
In certain circumstances the equilibrium-form of tide provides an approximation to the actual tide. But, while retaining the slope of eleva tion as given by the above formulae, it is clearly necessary to arrange that the total volume of water may remain unchanged. This requires that a constant depending on the distribution of land and water be added to the functions of position in the above formulae, a process known as the "correction for continents." For the long period constituents the cor rection has been carried out and found to be of little consequence (H. H. Turner, Proc. Roy. Soc. 4o). But for the true equilibrium-form we re quire to take account of another circumstance not allowed for in the above. It is that the elevated water itself produces a disturbance in the earth's gravitational field, and the resulting modification may be im portant. So far it has only been numerically evaluated for an ideal ocean covering the whole earth, and in this case the effect is a uniform multipli cation by the factor I.12 (W. Thomson and P. G. Tait, Natural Philoso phy, Arts. 815, 817 [1879-83]). The complete determination of even the equilibrium-form for the actual oceans is as yet an unsolved problem.