of the equator, produce inequalities in the tides, which modern investigation has been able to trace to their corresponding causes, and thus to apply these irregularities themselves to the establishment of the general theory; and prove that the tides, as well as the inequalities to which they are subject, are the necessary effects of universal gravitation.

If the waters of the sea covered the whole surface of the Earth to an equal depth, and experienced the action of gravity only, without any disturbing influence from the motion of the Earth, they would take a spherical form: but, on account of the rotary motion of the Earth, they ought to assume the form of an ellipsoid of revolution, the axis of the Earth being its minor axis. If the force of gravity combined with the centrifugal force, arising from the diurnal rotation of the Earth, were the only forces which acted upon the waters of the ocean, the ellipsoid would finally attain a state of stability, and its form would, in consequence, undergo no farther alteration. But, as the sea is also subject to the action of the Sun, it gravitates towards that body, each of its molecules being attracted in proportion to its distance from that luminary supposing the Sun to be in the plane of the equator, this plane would then no longer retain its circular form, but become elliptic; the parts immediately beneath the Sun being raised above the other by his direct influence upon them, while those on the opposite side, being less attracted than the intermediate parts, would yield more freely to the influence of the centrifugal force resulting from the rotation of the Earth, and form the opposite part of the ellipse, which a section of the Earth through the equator would now exhibit. From this it follows, that there would result from the action of the Sun alone, two high and two low tides, the high tides being in the line of the Sun's action, or 180° from each other, and the low tides at right angles to this line, or at 90° from the first. If the Earth and the Sun had con

stantly the same angular velocity, or moved together, the solar action would always be exerted upon the same point; and the relative positions of the elevated and depressed parts of the ocean, with respect to the other parts, would then be constant; but, as the Sun continually changes his position with respect to the Earth, in his apparent progress from east to west, the elevations and depressions of the waters which arise from the influence of his attraction, combined with the rotary motion of the Earth, experience the same change, and follow his motions. The two high tides, therefore, happen at the time of the Sun's passage over the meridian of that place, and over the opposite meridian, and the true low tides at the meridians that are 90° distant from these; and as the Sun passes over every point of the equator in the course of 24 hours, the interval between the two high tides would therefore be 12 hours, and the time of low water 6 hours distance from these. If the Sun remained constantly in the plane of the equator, as has been supposed, the waters would always be highest at the equator, and diminish gradually from thence to the poles, where there would be no elevation; but, as the Sun is sometimes on one side of the equator and sometimes on the other, the most elevated points of the ocean will follow the same course, and consequently be affected by his declination.

The Sun, however, is not the only body which acts upon the waters of the ocean, and gives rise to the phenomena of the tides; the Moon also produces her effect; and by supposing her to act alone, as has been done with respect to the Sun, the same reasoning may be applied to her effects as has been employed relative to the action of that luminary. If the action of these two bodies were independent of cach other, there would be four high and four low tides every 24 hours, except when their actions were both exerted in the same direction, that is, when the luminaries were either in conjunction or opposition. But, Bb

as the Sun and Moon move with different angular velocities, the directions of their influence are constantly varying; and hence the intervals between these high and low tides are also subject to equal variation. But these two effects are not produced independently of each other; and, therefore, the high tide is the effect of the combined action of both bodies; and, consequently, the place where the waters rise the highest, as well as the time when the maximum takes place, will not only depend upon the relative actions of these bodies, but also upon the directions in which their influence is exerted. When they act in the same direction, either both the same way or in opposition to each other, their united effect will be the greatest, and will then be the sum of the two separate effects; for, when their actions are diametrically opposed to each other, one of them coincides with that of the centrifugal force, and by this means produces the same effect as if it had united with the other. Under these circumstances, it is equally evident that their effect will be the least when they are at right angles to each other, as their separate effects will then destroy most of each other's influence. In any other situation, the magnitude and direction of their combined influence may readily be calculated.

For, since the masses of the Sun and Moon, and their distances from the Earth, as well as their directions at any given time, are known, their separate forces may be reduced into one compound force acting in the plane of the three bodies. As the attractive power is always directly as the mass and inversely as the square of the distance of the attracting body, each of these forces may be found and represented by a line drawn in this plane; and then the force and direction of their combined influence calculated from these. In the annexed figure, let E represent the Earth, S the place of the Sun, and M that of the Moon; and P the point upon which the greatest

combined action of the Sun and Moon is exerted; also, PM the direction and influence of the Moon, and PS those of the Sun; then by completing the parallelogram, and drawing the diagonal PÃ, it will represent the compound force required.

M

E

Fig. 4.

From the above data we know the values of PM and PS, and the angle MPS, made by the direction of the two bodies; and as MA and PS are parallel to each other, the angle MAP is equal to the angle APS; and, consequently, the sum of the two angles MAP and APM is equal to the whole angle MPS, and as this sum is known, the angle AMP is consequently given. In the triangle AMP, then, we have the two sides PM and MA (PS), from which the side AP and the angle APM can readily be found: the former value will express the magnitude of the compound force, and the latter its direction, with respect to that of the Moon, which is previously known. Then, as the direction of this compound force with respect to one side of its plane has been found, and the inclination of that plane to the plane of the equator was previously known, this compound force may be reduced to one acting parallel to the plane of the terrestrial equator, as AQ; and its effect immediately combined with that resulting from the rotary motion of the Earth, in determining the position of the highest

point of the waters at the given instant, produced by the joint action of the Sun and Moon united with the effect of the centrifugal force.

The angular velocities of the two bodies being different, their effects will vary with their positions: they will be united in the syzygies when they produce the highest tides; but they must be subtracted from each other in the quadratures; and, consequently, the tides are then the least. The time of high water, according to the premises we have assumed, is, in the first case, the time of the passage of the two bodies over the meridian; and, in the quadratures, the time of flood is that of the passage of the body which exercises the greatest action. At all other periods, the time of high water is between the passages of the two bodies, but nearest to the passage of that which exercises the greatest force.

The mean interval between two consecutive passages of the Sun over any given meridian is a day; and the interval between two similar passages of the Moon is 1.03505 day. The interval of the conjunctions is 29-53059 days. At the moment of conjunction, the solar and lunar tides coincide with each other, and the compound tide is the greatest. At a period of 7.38264 days after this, the Sun and Moon being in quadrature, the compound tide is the least; because the full sea answering to the one, coincides with the low sea of the other. After an elapse of 14.76529 days from the conjunction, the bodies are in opposition, and the compound tide is again the highest, because it is the whole effect of their united action; and, at the end of 22-34795 days, the two bodies being in quadrature, their combined influence gives rise only to a small tide, because the effect of the one is again opposed to the other. At the expiration of 29-53059 days, the two bodies being again in conjunction, produce a high tide, as before. On this subject, the results of observation are in perfect coincidence with the deductions from theory, in