445 CHAPTER XIII. THEORY OF THE AXES, PERIHELIA, AND EXCEN 66 TRICITIES. Incipiunt magni procedere menses."—VIRG. Pollio. VARIATION OF ELEMENTS IN GENERAL. DISTINCTION BETWEEN - EXPE (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 each 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 variations of the elements, at least where the planets only are concerned, develope 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 an 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 accurately 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 motion 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 configu rations; 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 drawn from its orbit (its slowly varying orbit), but forthwith returns to it, to deviate presently as much the other way, while the varied orbit accommodates and adjusts itself to the average of these excursions on either side of it; and thus continues to present, for a succession of indefinite ages, a kind of medium picture of all that the planet has been doing in their lapse, in which the expression and character is preserved; but the individual features are merged and lost. These periodic inequalities, however, are, as we have observed, by no means neglected, but it is more convenient to take account of them by a separate process, independent of the secular variations of the elements. (657.) In order to avoid complication, while endeavouring to give the reader an insight into both kinds of variations, we shall for the present conceive all the orbits to lie in one plane, and confine our attention to the case of two only, that of the disturbed and disturbing body, a view of the subject which (as we have seen) comprehends the case of the moon disturbed by the sun, since any one of the bodies may be regarded as fixed at pleasure, provided we conceive all its motions transferred in a contrary direction to each of the others. Let therefore APB be the undisturbed elliptic orbit of a planet P; M a disturbing body, join MP, and supposing M K=M S take MN: MK:: M K2 : M P2. Then if SN be joined, NS will represent the disturbing force of M or P, on the same scale that S M represents M's attraction on S. Suppose ZPY a tangent at P, SY perpendicular to it, and N T, NL perpendicular respectively to SY and P S produced. Then will N T represent the tangential, TS the normal, N L the transversal, and LS the radial components of the disturbing force. In circular orbits or orbits only slightly elliptic, the directions PSL and SY are nearly coincident, and the former pair of forces will differ but slightly from the latter. We shall here, however, take the general case, and proceed to investigate in an elliptic orbit of any degree of excentricity the momentary changes produced by the action of the disturbing force in those elements G G |