it is the difference between the moon's attraction upon the given unit mass and the moon's attraction upon the entire earth divided by the mass of the earth. Because the depths of the oceans are small in comparison with the length of the earth's radius, and because of the smallness of the tidal forces, only the horizontal components of such forces are effective in the production of tides; and so these may, without impropriety, be spoken of as the tidal forces.
The vertical forces alter the intensity of the earth's gravity upon the waters of the ocean in a way similar to the alteration which would be occasioned were the density of the waters to undergo a fluctuation having a range of less than one ten-millionth part of itself. Now the greatest known ocean depth occurs near Guam Island and measures 5,269 fathoms or 31,614 feet; and it is obvious that the small density fluctuation just mentioned, and so the vertical forces, can create no sensible disturbance in the existing ocean.
For simplicity's sake, we shall here ignore that alternation in the forces which depends upon the declination of the moon, and is responsible for what is called the "diurnal inequality" in the tides. We shall also, as a rule, ignore that portion of the tidal forces resulting from the sun's attraction.
It is evident that at moonrise or moonset at a given point or locality upon the earth's equator, the horizontal forces vanish because the given point is then (very nearly) as remote from the moon as is the earth's center. On account of the moon's parallax, the moon at the times of rise or set does exert a downward disturbing force at the given point or locality; but, as already stated, this does not concern the tides. When the moon is on the meridian above or below the horizon, the disturbing force is all vertical and so the horizontal component does not exist. Therefore the tidal forces vanish four times during each lunar day. From moonrise to moon culmination, the force is directed eastward; and from culmination to moonset, westward. Also from moonset to lower culmination, the force is directed eastward, and from lower culmination to moonrise, westward.
For a point not upon the equator, there is also a meridional periodic force. In north latitude this force has its maximum southward value at the time of either culmination, and its maximum northward value at moonrise and moonset. The reverse of this is true for a point situated in the southern hemisphere.
In accordance with what has been said, a suspended plumb bob will, at the equator, make two complete oscillations daily in an east-and-west direction. The average amount of deviation either way from its undisturbed position will be about one thirteen-millionth part of the length of the line measured from the point of suspension to the center of inertia of the bob.