If the moon moved with constant angular speed in the plane of the earth's equator and at a constant distance from the earth, we should have, at any place, equilibrium high water occurring regularly at in tervals of time equal to 12 hr. 25 min., with a maximum range of eleva tion at the equator and a zero range at the two poles. The rise and fall of the water would be approximately harmonic with the speed of 2 ( -y = cr) , where denotes the angular speed of the earth's rotation and a the mean motion of the moon, and the equilibrium- tide would be repre sented very approximately by a single harmonic constituent. But the fact that the moon does not move as here supposed causes many modifica tions. Let us still suppose the moon to move in the equator but take into account the elliptic inequality in her distance. This distance D may be expressed approximately in the form magnitudes of the forces are given by the numbers on the figure, Al being in the direction of the moon. The separate attractions of the moon at the earth's centre and at a point on the earth's surface are each inversely proportional to the square of the moon's distance, so that the difference between the two, which gives the tide-generating force, is approximately inversely proportional to the cube of the moon's distance.
The vertical component of the tide-generating force coincides in direc tion with the gravitational force of the earth itself, and thus acts as a very slight modification of weight. This component does not tend to alter the position of equilibrium which the water would take up in the absence of any disturbance from an extra terrestrial body. The effective tide generating forces, therefore, are the horizontal components of those in dicated in fig. I.
Equilibrium-form of Tide.—(6). For many purposes it is con venient to specify the distribution of tide-generating forces at any instant by reference to a fictitious tide in an ocean covering the whole earth. If the earth carried such an ocean and were to remain at rest, and if the tide-generating forces were to remain constant, there would be an invariable elevation of water at each point of the ocean-surface and no tidal currents. If the distribution of forces were the same as that of the actual tide-generating forces at any definite instant, then we might use the consequential elevation of water as a specification of these forces. This distribution of fictitious tidal elevation is known as the equilibrium form of the tides. It is a real and accurate specification of the actual tide generating forces at any instant, and when the equilibrium-form is stated for every instant we have a complete specification of these forces. In this equilibrium-form the inclination, to the horizontal, of the surface of the water would be always such that the consequential differences of pressure produced horizontal forces balancing the tide-generating forces.
It is therefore clear that the surface of the water would slope upwards from DD towards the points V and I, the water being raised by a maxi mum amount at V and I and depressed by a maximum amount along the great circle through DD. The surface of the water would be that of a nearly spherical surface of revolution. The volume of this surface of revolution would be the same as that enclosed by the surface of the un disturbed ocean. Owing to the motion of the moon relative to the earth, this surface of revolution moves over the earth so that V is always directly under the moon and I always directly opposite, while owing to the varying distance of the moon the surface of revolution changes slightly in shape.
The tide-generating forces due to the sun's gravitation may be sim ilarly specified. The sun's mass is nearly 27,00000o times the moon's mass and the sun's distance is about 390 times the moon's distance. Con sequently the sun's tide-generating forces are to those of the moon in the so that we should have two new harmonic constituents. The moon's motion, however, also varies in her elliptic orbit, increasing and decreas ing with the reciprocal of her distance and this causes the period of the equilibrium-tides to increase and decrease with their range. On this account there is reinforcement of the constituent of speed 27-3o-+w as compared with that of speed 27 — a —w.
If the moon moved with constant angular speed in a parallel of latitude other than the equator, consecutive high waters would be unequal except at the equator, and we should require the introduction of a new constitu ent of speed (7— a) with an amplitude vanishing at the equator. Also, the amplitude of the original constituent of speed 2 (-y — a), would be less than when the moon was in the equator. Since the declination of the moon changes, this new constituent also requires modification. If its amplitude could be regarded as changing harmonically with speed a it would be replaced by two harmonic constituents of equal amplitude and speeds Owing to the fact that this is not quite so, the ampli tude of the constituent of speed• 2o- is a little greater than that of speed -y. Also, the changing declination of the moon causes the ampli tudes of the semi-diurnal constituents to vary, but it is sufficiently ac curate to take mean values in all cases except that of speed 2 cr) . As the effect is to make the speed and range of tide increase or decrease to gether, we get a new constituent of speed 2(7-e) + 2a. The daily mean level in the equilibrium-form depends upon the particular curve of the spheroid which passes over the place. On this account the changing dec lination of the moon introduces a constituent of speed 2u.