motion, and this will be repeated in every succeeding revolution of the node. (649.) Now this is precisely the kind of motion which, as we have seen in art. 325, 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 phenomena of nutation and precession are intimately connected, or rather both of them essential constituent parts of one and the same phenomenon. 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. (650.) 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 a 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 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 :— 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 systems, and will give rise, in every member of it and in every part of each member, to periodic movements executed in equal period, with that to which they owe their origin, though not necessarily synchronous with them in their maxima and minima.1 1 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. 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. (651.) 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, the 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 revolution 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 earth 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 phenomenon. The explanation of the tides, however, belongs more properly to that part of the general subject of perturbations which treats of the action of the radial component of the disturbing force, and is therefore postponed to a subsequent chapter. 23 CHAPTER XIII. THEORY OF THE AXES, PERIHELIA, AND EXCENTRICITIES. VARIATION OF ELEMENTS IN GENERAL. -DISTINCTION BETWEEN PE- REPRESENTATION TIONS. ESTIMATION CIRCULAR ORBITS. EXPERI APPLICATION OF THE FOREGOING PRIN- CIPLES TO THE PLANETARY THEORY. VERY NEARLY CIRCULAR. EXCENTRICITIES. (652.) IN the foregoing chapter we have sufficiently explained the action of the orthogonal component of the disturbing force, and traced it to its results in a continual displacement of the plane of the disturbed orbit, in virtue of which the nodes of that plane alternately advance and recede upon the plane of the disturbing body's orbit, with a general preponderance on the side of advance, so as after the lapse of a long period to cause the nodes to make a complete revolution and come round to their former situation. At the same time the inclination of the plane of the disturbed motion continually changes, alternately increasing and diminishing; the increase and diminution, however, compensating cach other, nearly in single revolutions of the disturbed and disturbing bodies, more exactly in many, and with perfect accuracy in long periods, such as those of a complete revolution of the nodes and apsides. In the present and following chapters we shall endeavour to trace the effects of the other components of the disturbing force, those which act in the plane (for the time being) of the disturbed orbit, and which tend to derange the elliptic form of the orbit, and the laws of elliptic motion in that plane. The small inclination, generally speaking, of the orbits of the planets and satellites to each other, permits us to separate these effects in theory one from the other, and thereby greatly to simplify their consideration. Accordingly, in what follows, we shall throughout neglect the mutual inclination of the orbits of the disturbed and disturbing bodies, and regard all the forces as acting and all the motions as performed in one plane. (653.) In considering the changes induced by the mutual action of two bodies, in different aspects with respect to each other, on the magnitudes and forms of their orbits, and in their positions therein, it will be proper in the first instance to explain the conventions under which geometers and astronomers have alike agreed to use the language and laws of the elliptic system, and to continue to apply them to disturbed orbits, although those orbits so disturbed are no longer, in mathematical strictness, ellipses, or any known curves. This they do, partly on account of the convenience of conception and calculation which attaches to this system, but much more for this reason, that it is found, and may be demonstrated from the dynamical relations of the case, that the departure of each planet from its ellipse, as determined at any epoch, is capable of being truly represented, by supposing the ellipse itself to be slowly variable, to change its magnitude and excentricity, and to shift its position and the plane in which it lies according to certain laws, while the planet all the time continues to move in this ellipse, just as it would do if the ellipse remained invariable and the disturbing forces had no existence. By this way of considering the subject, the whole effect of the disturbing forces is regarded as thrown upon the orbit, while the relations of the planet to that orbit remain unchanged. This course of procedure, indeed, is the most natural, and is in some sort forced upon us by the extreme slowness with which the variation of the elements, at least where the planets only are concerned, develop themselves. For instance, the fraction expressing the excentricity of the earth's orbit changes no more than 0.00004 in its amount in a century; and the place of its perihelion, as referred to the sphere of the heavens, by only 19′ 39" in the same time. For several years, therefore, it would be next to impossible to distinguish between ar ellipse so varied and one that had not varied at all; and in a single revolution, the difference between the original ellipse and the curve really represented by the varying one, is so excessively minute, that, if accu rately drawn on a table, six feet in diameter, the nicest examination with microscopes, continued along the whole outlines of the two curves, would hardly detect any perceptible interval between them. Not to call a mo tion so minutely conforming itself to an elliptic curve, elliptic, would be affectation, even granting the existence of trivial departures alternately on one side or on the other; though on the other hand, to neglect a variation, which continues to accumulate from age to age, till it forces itself on our notice, would be wilful blindness. (654.) Geometers, then, have agreed, in each single revolution, or for any moderate interval of time, to regard the motion of each planet as elliptic, and performed according to Kepler's laws, with a reserve in favour of those very small and transient fluctuations which take place within that time, but at the same time to regard all the elements of each ellipse as in a continual, though extremely slow, state of change; and, in tracing the effects of perturbation on the system, they take account principally, or entirely, of this change of the elements, as that upon which any material change in the great features of the system will ultimately depend. (655.) And here we encounter the distinction between what are termed secular variations, and such as are rapidly periodic, and are compensated in short intervals. In our exposition of the variation of the inclination of a disturbed orbit (art. 636,) for instance, we showed that, in each single revolution of the disturbed body, the plane of its motion underwent fluctuations to and fro in its inclination to that of the disturbing body, which nearly compensated each other; leaving, however, a portion outstanding, which again is nearly compensated by the revolution of the disturbing body, yet still leaving outstanding and uncompensated a minute portion of the change which requires a whole revolution of the node to compensate and bring it back to an average or mean value. Now, the two first compensations which are operated by the planets going through the succession of configurations with each other, and therefore in comparatively short periods, are called periodic variations; and the deviations thus compensated are called inequalities depending on configurations; while the last, which is operated by a period of the node (one of the elements,) has nothing to do with the configurations of the individual planets, requires a very long period of time for its consummation, and is, therefore, distinguished from the former by the term secular variation. (656.) It is true, that, to afford an exact representation of the motions of a disturbed body, whether planet or satellite, both periodical and secular variations, with their corresponding inequalities, require to be expressed; and, indeed, the former even more than the latter; seeing that the secular inequalities are, in fact, nothing but what remains after the mutual destruction of a much larger amount (as it very often is) of periodical. But these are in their nature transient and temporary: they disappear in short periods, and leave no trace. The planet is temporarily |