So far, we have a general explanation of the occurrence of tides twice a day, and of their dependence on the moon. But we started with two assumptions which are not consistent with fact, viz., that the earth is spherical and uniformly covered with water, and that the moon is the only tide-producing body. The corrections to be made in consequence of the inaccuracy of these assumptions must now be explained. We com mence with the latter. The sun, although at an immense distance compared with that of the moon, has such an enormous mass, that his tide-producing influence is compara ble with that of the moon. In fact, it is easy to see that, as Newton showed, the tide producing power of an attracting mass is directly as the mass, and inversely as the cube of its distance. That it is directly as the mass, is obvious. To prove the other asser tion, let R be the earth's radius, D the distance of the attracting body from the earth's center, then the attraction per unit of mass on the earth is to that per unit of mass on the water nearest the attracting body as 1 1 D2 to (D — according to the law of gravitation. The difference between these quantities is propor tional to the tide-producing force. But 1 2R 1 2R = 1 = D2 (1 + + etc.) = + +, etc.
D2(1— D the remaining terms being omitted, since D is always much greater than R. The dif• ference is therefore approximately 2R Di' as stated above.
Now the mass of the sun is to that of the moon as 355,000 to 0.0125, and the sun's distance is about 400 times that of the moon. Hence the tide-producing power of the sun is to that of the moon as 355,000 to .0125 X or 355 to 800.
By calculations, which we cannot give here, it has been shown that the difference of length of the axes of the wave-spheroid produced by the moon alone is about 58 inches; so that in that due to the sun it will be about 25.7 inches.
In consequence of the extremely small amount of these effects on the sea-level, we are entitled to simply add or superpose the separate effects of the sun and moon, in order to obtain their joint effect. And now we have at once the explanation of what are called spring and neap tides. At new and at full moon, the wave-spheroids due to the sun and moon have their axes almost coincident, so that we have a tide which is to the lunar alone as 800+355 to 800, or as 13 to 9 nearly; while, when the moon is in her first or last quarter, the axes are nearly at right angles, and the compound tide is to the lunar tide alone as 800 — 355 to 800, or as 5 to 9 nearly. Thus the height of the spring-tide is to that of the neap-tide in the ratio of about 13:5. •Another curious phenomenon, which we can now easily account for, is the " priming" and " lagging" of the tides, or the acceleration and retardation of the time of high-water. If the tides were due to the sun or moon alone, they would recur at equal intervals of time; and, in fact, this is the case with the lunar and solar tides separately. But what we observe is the compound tide, and this will obviously have its maximum between two consecutive maxima of the lunar and solar tides; but nearer to the lunar tide as it is the greater. Thus, if about new moon the sun passes the meridian before the moon, the side is accelerated; if after, it is retarded. And the same is true about full moon, only that in this case our statement refers to passages of the sun and moon on opposite sides of the meridian. This retardation or acceleration has for its greatest value a period of
rather less than an hour; and the respective maxima occur about 4i days before and after the spring-tides.
But we meet with far more serious difficulties when we come to consider the actual distribution of water over the earth's surface; and it is here that future improvements must be looked for.
But even so inadequate an attempt at a solution as is the equilibrium theory, gives. us the moans of explaining a great many curious observed phenomena. It shows, for instance, how exceedingly small we should expect to find the tides in an inland sea such as the Mediterranean; for there, even when the moon is mbst favorably situated, the utmost difference of level would be (by calculations which we cannot give here) only about an inch or two; and of this part would be the rise in one portion of the sea, the rest the fall in others. The popular explanation of this phenomenon is very simple. We have but to notice that, according to the equilibrium theory, the form of the water is a spheroid of definite dimensions, its axes differing from each other by 58 inches. But a small portion of such a spheroid (of the dimensions of the Mediterranean, for instance) can hardly be distinguished from a sphere; so that the form of the surface of a limited mass of water will be but slightly altered by the attractions of the sun and moon.
It is obvious from what we have just said, that the rise of the water in tidal rivers, estuaries, and deep bays, where it sometimes amounts (even in calm weather) to more than 100 ft., cannot possibly be due to the moon's action upon the water of the mere river or bay, but must be almost entirely produced by the tidal wave in the ocean; and, in fact, this part of the problem presents comparatively little difficulty. Once grant the fact of the tidal disturbance of sea-level at the mouth of a river, and the calculation of the motion of the consequent wave in the river-channel is within the power of mathe matics. It is by means of investigations made from this point of view, and by others. concerning the effect of the moon on long canals, that Laplace's method has been im proved. For the details of the process, see Airy on "Tides and Waves," in the Encyc. Metrop. All we can do here is to point out a few of the immediate consequences of the periodic rise and fall of the sea-level as regards the motion of the water of a tidal river. Here the tide always runs up the river, even when, as in the case of the Severn, this is the opposite direction to that in which the moon appears to move. In the open sea at the mouth of the river, the interval from high to low-water is almost exactly equal to that from low to high-water, each being about 6-1.. hours nearly. But the further we go up the river, the greater becomes the disparity between these periods, high-water follow ing low-water at shorter and shorter intervals, while the intervals during which the tide falls are correspondingly increased. In some cases, as at certain points in the Seine and Severn, the interval from low to high-water is so short that the tide-wave rushes sud denly up, and spreading over the fiat sands at the side of the channel, forms a danger ous surf called a bore (q.v.).