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termined 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 permanent effect of the disturbing forces is regarded as thrown upon the orbit, while the relations of the planet to that orbit remain unchanged, or only liable to brief and comparatively momentary fluctuation. 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 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.

(537.) 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 certain very small and transient fluctuations, 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, after all, any material change in the great features of the system will ultimately depend.

(538.) 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. 514.), 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 an immense period of time for its consummation, and is, therefore, distinguished from the former by the term secular variation.

(539.) 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, 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 to be neglected, but they are taken account of by a separate process, independent of the secular variations of the elements.

(540.) In order to avoid complication, while endeavouring to give the reader an insight into both kinds of variations, we shall henceforward 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

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in a contrary direction to each of the others. Suppose, therefore, S to be the central, M the disturbing, and P the disturbed body. Then the attraction of M acts on

P in the direction P M, and on S in the direction S M. And the disturbing part of M's attraction, being the difference only of these forces, will have no fixed direction, but will act on P very differently, according to the configurations of P and M. It will therefore be necessary, in analyzing its effect, to resolve it, according to mechanical principles, into forces acting according to some certain directions; viz., along the radius vector S P, and perpendicular to it. The simplest way to do this, is to resolve the attractions of M on both S and P in these directions, and take, in both cases, their difference, which is the disturbing part of M's effect. In this estimation, it will be found then that two distinct disturbing powers originate; one, which we shall call the tangential force, acting in the direction P Q, perpendicular to S P, and therefore in that of a tangent to the orbit of P, supposed nearly a circle-the other, which may be called the radial disturbing force, whose direction is always either to or from S.

(541.) It is the former alone (art. 419.) which disturbs the equable description of areas of P about S, and is therefore the chief cause of its angular deviations from the elliptic place. For the equable description of areas depends on no particular law of central force, but only requires that the acting force, whatever it be, should be directed to the center; whatever force does not conform to this condition, must disturb the areas.

(542.) On the other hand, the radial portion of the disturbing force, though, being always directed to or from the center, it does not affect the equable description of areas, yet, as it does not conform in its law of variation to that simple law of gravity by which the elliptic figure of the orbits is produced and maintained, has a tendency to disturb this form; and, causing the disturbed body P, now to approach the center nearer, now to recede farther from it, than the laws of elliptic motion would warrant, and to have its points of nearest approach and farthest recess otherwise situated than they would be in the undisturbed orbit, tends to

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derange the magnitude, excentricity, and position of the axis of P's ellipse.

(543.) If we consider the variation of the tangential force in the different relative positions of M and P, we shall find that, generally speaking, it vanishes when P is at A or C, see fig. to art. 540. i. e. in conjunction with M, and also at two points, B and D, where M is equidistant from S and P (or very nearly in the quadratures of P with M); and that, between A and B, or D, it tends to urge P towards A, while, in the rest of the orbit, its tendency is to urge it towards C. Consequently, the general effect will be, that in P's progress through a complete synodical revolution round its orbit from A, it will first be accelerated from A up to B-thence retarded till it arrives at C- thence again accelerated up to D, and again retarded till its re-arrival at the conjunction A.

(544.) If P's orbit were an exact circle, as well as M's, it is evident that the retardation which takes place during the description of the arc A B would be exactly compensated by the acceleration in the arc D A, these arcs being just equal, and similarly disposed with respect to the disturbing forces; and similarly, that the acceleration through the arc B C would be exactly compensated by the retardation along C D. Consequently, on the average of each revolution of P, a compensation would take place; the period would remain unaltered, and all the errors in longitude would destroy each other.

(545.) This exact compensation, however, depends evidently on the exact symmetry of disposal of the parts of the orbits on either side of the line CS M. If that symmetry be broken, it will no longer take place, and inequalities in P's motion will be produced, which extend beyond the limit of a single revolution, and must await their compensation, if it ever take place at all, in a reversal of the relations of configuration which produced them. Suppose, for example, that, the orbit of P being circular, that of M were elliptic, and

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