form of an oblong elliptic spheroid, differing but little from a sphere.

Mr. Laplace has taken compassion on the apprehensions of some timid persons, who seem to have been afraid that the attraction of a great elevation of the ocean should by some accident overcome the natural equilibrium of the earth and sea, so as to overwhelm the whole of our continents; and he has been at the pains to calculate that this can never happen, while the mean density of the earth remains greater than that of the sea. We cannot help being persuaded that the security is still greater than Mr. Laplace supposes; for it seems to us to be demonstrable, that the density of the sea must be to that of the earth as five to three, in order that any elevation, resembling that of a tide, may produce an attraction tending to increase its own magnitude: although we will not assert, that the same conditions of stability are applicable to every possible disturbance of the form of the sea which can be imagined.

But to return to our author; we have only to recollect, with respect to the first objection, that we are by no means required to imagine that the moon repels the remoter parts of the earth and sca; but merely to understand, that these parts are left a little behind, while the central parts fall more within the tangent towards the moon, and the nearer parts still more than the central parts; nor is this a fact of which our belief must rest on any observed phenomena of the tides, since it is completely demonstrable from the general laws of gravitation and of central forces: so that if no tides were under any circumstances observable, their non-existence would afford an unanswerable argument against the universality and accuracy of these laws, as they are inferred from other pheno

mena.

The second objection is already answered in the statement of the mode of operation of the disturbing force. The action of this force is only supposed to be sufficient to retain a particle of water in equilibrium on a surface of which the inclination to the horizon is scarcely perceptible, or to cause the whole gravitation of a column four thousand miles in height immediately under the luminary, to be equal only to the gravitation of a column shorter by a few inches or feet in another part of the spheroid. The objector has confounded this very slight modification of the force of gravitation with its complete annihilation by a greater force; and with respect to the force of cohesion, it is so little concerned in counteracting any elevation of this kind, that to attempt to calculate the magnitude of any resistance derived from it would be perfectly ridiculous.

The third objection is only so far more valid, as we may admit the truth of the imperfect and superficial notions, which our author,

in common with some other writers, entertains, of the supposed operation of the forces concerned. In fact, however, it is just as likely to happen, in the open ocean, that the transit of the luminary may coincide with the time of low water as with that of high water; and in more limited seas and lakes, there is no hour in the twentyfour at which high water may not naturally be expected to take place, according to the different breadth and depth of the waters concerned; while, under other circumstances, it may happen to be high water only once a day, or ouce a fortnight, or there may be no tide at all, without any deviation from the strictest regularity in the operation of the causes concerned. As an answer to Mr. Cuthbert, it is sufficient to have asserted this, and to refer to Laplace's Mécanique Céleste, or to Dr. Young's Natural Philosophy, for a more detailed demonstration and illustration of the assertion. But since the subject has hitherto been considered as extremely intricate, and is not indeed yet freed from all its embarassments, we shall here enter into it at some length, and endeavour to explain the principles on which the investigation has been conducted.

The attempts that were made by Newton, to determine the effects of the solar and lunar attraction on the sea, went no farther to the investigation of the magnitude of the elevation which would at any given time afford a temporary equilibrium: and even Maclaurin was satisfied with having ascertained the precise nature of the form which the waters must assume in such a case. But it is obvious that these determinations are by no means sufficient for ascertaining the motions, which arise from the change of relative situation induced by the earth's rotation; since the form, thus ascertained, ouly affords us a measure of the force by which the waters are urged when they do not accord with it, and by no means enables us to say, without farther calculation, how nearly they will at any time approach to it. In fact, the change of the conditions of equilibrium determines only the magnitude of this force, such as it would be if the sea remained at rest, while it is in reality materially modified at any given time by the effect of the motions which have previously taken place: and supposing its true magnitude to be ascertained, its immediate operation will at all times be complicated with the conditions under which an impulse of any kind is capable of being communicated to the neighbouring parts of the sea, which depend on the depth of sea, as well as on the form of the earth.

Mr. Laplace has undertaken the investigation of the theory of the tides with all these additional complications; and he considers it as constituting, without exception, the most difficult department of the whole science of astronomy; and yet this consummate mathematician has omitted to include in his calculation the effects to

be

be attributed to resistances of various kinds, and to the irregularities of the form of the sea, which appear to us to constitute by far the most material difficulties in the inquiry. The general problem, relating to the oscillations of a fluid completely covering a sphere, and moving with little or no resistance, which Mr. Laplace has solved by a very intricate analysis, is capable of being exhibited in a much less embarrassed, and we apprehend, even in a more accurate manner, by a mode of investigation which is equally applicable to the tides of narrow seas and of lakes, and which may easily be made to afford a correct determination of the effects of resistance, as well as a ready method of discovering the laws of motions governed by periodical forces of any kind; at least so far, as those forces are capable of being represented by any combinations of the sines of arcs, which increase uniformly with the time. The essential character of this method consists in comparing the body actuated by the given force to a pendulum, of which the point of suspension is caused to vibrate regularly to a certain small extent: the length of the pendulum being supposed to be such, as to afford vibrations of equal frequency with the spontaneous vibrations of the moveable body, and the point of suspension to be carried by a rod of such a length, as to afford vibrations of equal frequency with the periodical alternations of the force. It is then shown, that such a pendulum may perform regular vibrations, contemporary with the alternations of the periodical force, and inversely proportional in their extent to the difference between the length of the two rods and that, whatever may have been the initial state of the pendulum, the motion thus determined may be considered as affording a mean place, about which it will at first perform simple and regular oscillations; but that a very small resistance will ultimately cause these to disappear.

Now the sea, or any of its portions, may be considered as bodies susceptible of spontaneous vibrations, precisely similar to the small vibrations of a pendulum; and the variation of the form which would afford an equilibrium, in consequence of the solar and lunar attractions, is perfectly analogous to the regular vibration attributed to the point of suspension of the pendulum. The frequency of the simple oscillations of the sea, or of any of its parts, supposing their depth and extent known, may easily be deduced from the important theorem of Lagrange, by which the velocity of a wave of any kind is shown to be equal to the velocity of a heavy body, which has fallen through half the height of the fluid concerned: but in the case of a tide, extending to any considerable portion of the surface of the globe, this velocity must be somewhat modified according to the comparative density of the central and superficial parts.

The

The most remarkable consequence of this analogy is the law, that if the simple oscillations, of which the moving body is susceptible, be more frequent than the periods of the recurring force, the pendulum will follow its point of suspension with a direct motion; but if the spontaneous vibrations be slower, the motions will be inverted with respect to each other: and, with respect to the tides, we may infer from this mode of calculation, that supposing the earth to be between five and six times as dense as the sea, the oscillations of an open ocean can only be direct, if its depth in the neighbourhood of the equator be greater than 15 or 16 miles and that if the depth be smaller than this, the tides must be inverted, the time of low water corresponding, in this case, to the transit of the luminary over the meridian.

This distinction has not been explicitly made by Mr. Laplace, although he has calculated, that for a certain depth, of a few miles only, the tides of the open ocean must be inverted, and that for greater depths they will be direct: but the intricacy of his formulæ seems to render their use laborious, and perhaps liable to some inaccuracy and in the application of his theory, he seems to have lost sight even of the possibility of an inverted tide. In narrower seas, which Mr. Laplace has not considered, a smaller depth will constitute the limit between those two species of tides; and in either case the approach of the depth to this limit will be favourable to the magnitude of the tide.

However the primitive oscillation may be constituted, it is easy to understand, that it will be propagated through a limited channel connected with the main ocean, in a longer or shorter time according to the length and depth of the channel; and that if the channel be open at both ends, the tide will arrive at any part within it by two different paths; and the effects of two successive tides may in this manner be so combined, as to alter very materially the usual course of the phenomena: for instance, if there were about six hours difference in the times occupied in the passage of the two tides over their respective paths, the time of the high water belonging to one tide would coincide with that of the low water belonging to the other, and the whole variation of height might in this man ner be destroyed, as Newton has long ago observed with respect to the port of Batsha: and it may be either for a similar reason, or from some other local peculiarity of situation, that no considerable tides are observed in the West Indies; if indeed it is true, that the tides are so much smaller there than might be expected from calculation for in fact the original tides of an open sea, not exceeding a mile or two in depth, would amount to a few inches only, even without allowing for the effects of resistance. In the middle of a lake, or of a narrow sea, there can be little or no primitive ele vation

VOL. VI. NO. XI.

F

vation or depression; and the time of high water on its shores must always be six hours before or after the passage of the luminary over the middle: so that from this source we may derive an infinite diversity in the times, at which these vicissitudes occur in different ports.

The effects of resistances of various kinds, in modifying the time of high water, cannot easily be determined in a direct and positive manner from immediate observation. Mr. Laplace appears to be of opinion that these resistances are wholly inconsiderable; but if any dependence can be placed on the calculations of Du Buat, we ought to expect a very different result, since according to Du Buat's formula, the resistance, in the case of a tide of any moderate magnitude, must far exceed the moving power. From this result, however, nothing can be concluded with certainty, except that the formula is extremely defective with respect to great depths and slow motions; yet we may infer, as a probable conjecture, that the resistance must be great enough to produce some perceptible effects, and even that it must be greater than would be expected from another mode of calculation founded on the same experiments, (Ph. Tr. 1808,) which would give the proportion of the greatest resistance to the greatest moving force only as of the height of the tide, increased by about 10 feet, to the whole depth of the ocean concerned, at least on the supposition of a uniform depth and a smooth bottom, which indeed must be far from the truth; since the inequalities of the bottom of the sea must tend very greatly to increase the resistance, especially that part of it which varies as the square of the velocity.

Now it may be demonstrated, that a resistance, simply proportional to the velocity, would not disturb the perfect regularity of the oscillations concerned, and that it would only retard them when direct, and accelerate them when inverted, by the time corre sponding to an arc of which the sine is to the radius, as the greatest resistance to the greatest periodical force which would act on the body or surface if it remained at rest; the extent of the true oscillations being reduced by the resistance in the ratio of the cosine of the same arc to the radius. Nor will the displacement produced by an equal mean resistance, varying as the square of the velocity, be materially different; the body or surface merely oscillating a little about its mean place, in consequence of the different distribution of the resistance.

Here then we have another source of very great diversities in the times of the tides, according to the dimensions of the seas concerned, even in those parts in which the tides may be supposed to be rather original than derivative, not excepting the most widely extended oceans. There are however other considerations, which