CHAP. XI. NUTATION. 333 node, have arrived at e, having described not an exactly circular arc, but a single undulation of a wave-shaped or epicycloidal curve, a b c d e, with a velocity alternately greater and less than its mean motion, and this will be repeated in every succeeding revolution of the node. (525.) Now this is precisely the kind of motion which, as we have seen in art. 272., the pole of the earth's equator really has round the pole of the ecliptic, in consequence of the joint effects of precession and nutation, which are thus uranographically represented. If we superadd to the effect of lunar precession that of the solar, which alone would cause the pole to describe a circle uniformly about P, this will only affect the undulations of our waved curve, by extending them in length, but will produce no effect on the depth of the waves, or the excursions of the earth's axis to and from the pole of the ecliptic. Thus we see that the two phænomena of nutation and precession are intimately connected, or rather, both of them essential constituent parts of one and the same phænomenon. It is hardly necessary to state that a rigorous analysis of this great problem, by an exact estimation of all the acting forces and summation of their dynamical effects*, leads to the precise value of the co-efficients of precession and nutation, which observation assigns to them. The solar and lunar portions of the precession of the equinoxes, that is to say, those portions which are uniform, are to each other in the proportion of about 2 to 5. (526.) In the nutation of the earth's axis we have an example (the first of its kind which has occurred to us), of a periodical movement in one part of the system, giving rise to a motion having the same precise period in another. The motion of the moon's nodes is here, we see, represented, though under a very different form, yet in the same exact periodic time, by the movement of a peculiar oscillatory kind impressed on the solid mass of the earth. We must not let the opportunity pass of generalizing the principle involved * Vide Prof. Airy's Mathematical Tracts, 2d ed. p. 200, &c. in this result, as it is one which we shall find again and again exemplified in every part of physical astronomy, nay, in every department of natural science. It may be stated as the principle of forced oscillations, or of forced vibrations," and thus generally announced: 66 If one part of any system connected either by material ties, or by the mutual attractions of its members, be continually maintained by any cause, whether inherent in the constitution of the system or external to it, in a state of regular periodic motion, that motion will be propagated throughout the whole system, and will give rise, in every member of it, and in every part of each member, to periodic movements executed in equal periods with that to which they owe their origin, though not necessarily synchronous with them in their maxima and minima.* The system may be favourably or unfavourably constituted for such a transfer of periodic movements, or favourably in some of its parts and unfavourably in others; and, accordingly as it is the one or the other, the derivative oscillation (as it may be termed) will be imperceptible in one case, of appreciable magnitude in another, and even more perceptible in its visible effects than the original cause, in a third; of this last kind we have an instance in the moon's acceleration, to be hereafter noticed. (527.) It so happens that our situation on the earth, and the delicacy which our observations have attained, enable us to make it as it were an instrument to feel these forced vibrations,— these derivative motions, communicated from various quarters, especially from our near neighbour, the moon, much in the same way as we detect, by the trembling of a board beneath us, secret transfer of motion by which the sound of an organ pipe is dispersed through the air, and carried down into the earth. Accordingly, the monthly revo the See a demonstration of this theorem for the forced vibrations of systems connected by material ties of imperfect elasticity, in my treatise on Sound, Encyc. Metrop. art. 323. The demonstration is easily extended and generalized to take in other systems. Author. CHAP. XI. THE TIDES. 335 lution of the moon, and the annual motion of the sun, produce, each of them, small nutations in the earth's axis, whose periods are respectively half a month and half a year, each of which, in this view of the subject, is to be regarded as one portion of a period consisting of two equal and similar parts. But the most remarkable instance, by far, of this propagation of periods, and one of high importance to mankind, is that of the tides, which are forced oscillations, excited by the rotation of the carth in an ocean disturbed from its figure by the varying attractions of the sun and moon, each revolving in its own orbit, and propagating its own period into the joint phænomenon. (528.) The tides are a subject on which many persons find a strange difficulty of conception. That the moon, by her attraction, should heap up the waters of the ocean under her, seems to most persons very natural, that the same cause should, at the same time, heap them up on the opposite side, seems to many palpably absurd. Yet nothing is more true, nor indeed more evident, when we consider that it is not by her whole attraction, but by the differences of her attractions at the two surfaces and at the center that the waters are raised, that is to say, by forces directed precisely as the arrows in our figure, art. 510., in which we may suppose M the moon, and P a particle of water on the earth's surface. A drop of water existing alone would take a spherical form, by reason of the attraction of its parts; and if the same drop were to fall freely in a vacuum under the influence of an uniform gravity, since every part would be equally accelerated, the particles would retain their relative positions, and the spherical form be unchanged. But suppose it to fall under the influence of an attraction acting on each of its particles independently, and increasing in intensity at every step of the descent, then the parts nearer the center of attraction would be attracted more than the central, and the central than the more remote, and the whole would be drawn out in the direction of motion into an oblong form; the tendency to separation being, however, counteracted by the attraction of the particles on each other, and a form of equilibrium being thus established. Now, in fact, the earth is constantly falling to the moon, being continually drawn by it out of its path, the nearer parts more and the remoter less so than the central; and thus, at every instant, the moon's attraction acts to force down the water at the sides, at right angles to her direction, and raise it at the two ends of the diameter pointing towards her. Geometry corroborates this view of the subject, and demonstrates that the form of equilibrium assumed by a layer of water covering a sphere, under the influence of the moon's attraction, would be an oblong ellipsoid, having the semi-axis directed towards the moon longer by about 58 inches than that transverse to it. (529.) There is never time, however, for this spheroid to be fully formed. Before the waters can take their level, the moon has advanced in her orbit, both diurnal and monthly (for in this theory it will answer the purpose of clearness better if we suppose the earth's diurnal motion transferred to the sun and moon in the contrary direction), the vertex of the spheroid has shifted on the earth's surface, and the ocean has to seek a new bearing. The effect is to produce an immensely broad and excessively flat wave (not a circulating current), which follows, or endeavours to follow, the apparent motions of the moon, and must, in fact, if the principle of forced vibrations be true, imitate by equal, though not by synchronous, periods, all the periodical inequalities of that motion. When the higher or lower parts of this wave strike our coasts, they experience what we call high and low water. (530.) The sun also produces precisely such a wave, whose vertex tends to follow the apparent motion of the sun in the heavens, and also to imitate its periodic inequalities. This solar wave co-exists with the lunaris sometimes superposed on it, sometimes transverse to it, so as to partly neutralize it, according to the monthly CHAP. XI. synodical configuration of the two luminaries. This alternate mutual reinforcement and destruction of the solar and lunar tides cause what are called the spring and neap tides - the former being their sum, the latter their difference. Although the real amount of either tide is, at present, hardly within the reach of exact calculation, yet their proportion at any one place is probably not very remote from that of the ellipticities which would belong to their respective spheroids, could an equilibrium be attained. Now these ellipticities, for the solar and lunar spheroids, are respectively about two and five feet; so that the average spring tide will be to the neap as 7 to 3, or thereabouts. (531.) Another effect of the combination of the solar and lunar tides is what is called the priming and lagging of the tides. If the moon alone existed, and moved in the plane of the equator, the tide-day (i. e. the interval between two successive arrivals at the same place of the same vertex of the tide-wave) would be the lunar day (art. 115.), formed by the combination of the moon's sidereal period and that of the earth's diurnal motion. Similarly, did the sun alone exist, and move always on the equator, the tide-day would be the mean solar day. The actual tide-day, then, or the interval of the occurrence of two successive maxima of their superposed waves, will vary as the separate waves approach to or recede from coincidence; because, when the vertices of two waves do not coincide, their joint height has its maximum at a point intermediate between them. This variation from uniformity in the lengths of successive tide-days is particularly to be remarked about the time of the new and full moon. (532.) Quite different in its origin is that deviation of the time of high and low water at any port or harbour, from the culmination of the luminaries, or of the theoretical maximum of their superposed spheroids, which is called the "establishment" of that port. If the water were without inertia, and free from obstruction, either owing to the friction of the bed of the sea,- the narrow |