duced, so that X now shifts to the farther side of S. In all cases, the disturbing force is wholly effective in the plane MP S, in which the three bodies lie.

P

L

It is very important for the student to fix distinctly and bear constantly in his mind these relations of the disturbing agency considered as a single unresolved force, since their recollection will preserve him from many mistakes in conceiving the mutual actions of the planets, &c. on each other. For example, in the figures here referred to, that of Art. 611. corresponds to the case of a nearer disturbed by a more distant body, as the earth by Jupiter, or the moon by the Sun; and that of the present article to the converse case : as, for instance, of Mars disturbed by the earth. Now, in this latter class of cases, whenever M P is greater than MS, or SP greater than 2 S M, N lies on the same side of the plane of P's orbit with M, so that N S, the disturbing force, contrary to what might at first be supposed, always urges the disturbed planet out of the plane of its orbit towards the opposite side to that on which the disturbing planet lies. It will tend greatly to give clearness and definiteness to his ideas on the subject, if he will trace out on various suppositions as to the relative magnitude of the disturbing and disturbed orbits (supposed to lie in one plane) the form of the oval about M considered as a fixed point, in which the point N lies when P makes a complete revolution round S. (615.) Although it is necessary for obtaining in the first

instance a clear conception of the action of the disturbing force, to consider it in this way as a single force having a definite direction in space and a determinate intensity, yet as that direction is continually varying with the position of N S, both with respect to the radii SP, SM, the distance PM, and the direction of P's motion, it would be impossible, by so considering it, to attain clear views of its dynamical effect after any considerable lapse of time, and it therefore becomes necessary to resolve it into other equivalent forces acting in such directions as shall admit of distinct and separate consideration. Now this may be done in several different modes. First, we may resolve it into three forces acting in fixed directions in space rectangular to one another, and by estimating its effect in each of these three directions separately, conclude the total or joint effect. This is the mode of procedure which affords the readiest and most advantageous handle to the problem of perturbations when taken up in all its generality, and is accordingly that resorted to by geometers of the modern school in all their profound researches on the subject. Another mode consists in resolving it also into three rectangular components, not, however, in fixed directions, but in variable ones, viz. in the directions of the lines N Q, QL, and LS, of which LS is in the direction of the radius vector S P, QL in a direction perpendicular to it, and in the plane in which SP and a tangent to P's orbit at P both lie; and lastly, NQ in a direction perpendicular to the plane in which P is at the instant moving about S. The first of these resolved portions we may term the radial component of the disturbing force, or simply the radial disturbing force; the second the transversal; and the third the orthogonal. When the disturbed orbit is one of small excentricity, the transversal component acts nearly in the direction of the tangent to P's orbit at P, and is therefore confounded with that resolved component which we shall presently describe (art. 618.) under the name of the tangential

This is a term coined for the occasion. The want of some appellation for this component of the disturbing force is often felt.

force. This is the mode of resolving the disturbing force followed by Newton and his immediate successors.

(616.) The immediate actions of these components of the disturbing force are evidently independent of each other, being rectangular in their directions; and they affect the movement of the disturbed body in modes perfectly distinct and characteristic. Thus, the radial component, being directed to or from the central body, has no tendency to disturb either the plane of P's orbit, or the equable description of areas by P about S, since the law of areas proportional to the times is not a character of the force of gravity only, but holds good equally, whatever be the force which retains a body in an orbit, provided only its direction is always towards a fixed center. Inasmuch, however, as its law of variation is not conformable to the simple law of gravity, it alters the elliptic form of P's orbit, by directly affecting both its curvature and velocity at every point. In virtue, therefore, of the action of this disturbing force, the orbit deviates from the elliptic form by the approach or recess of P to or from S, so that the effect of the perturbations produced by this part of the disturbing force falls wholly on the radius vector of the disturbed orbit.

(617.) The transversal disturbing force represented by QL, on the other hand, has no direct action to draw P to or from S. Its whole efficiency is directed to accelerate or retard P's motion in a direction at right angles to SP. Now the area momentarily described by P about S, is, cæteris paribus, directly as the velocity of P in a direction perpendicular to SP. Whatever force, therefore, increases this transverse velocity of P, accelerates the description of areas, and vice versâ. With the area ASP is directly connected, by the nature of the ellipse, the angle ASP described or to be described by P from a fixed line in the plane of the orbit, so that any change in the rate of description of areas ultimately resolves itself into a change in the amount of angular motion about S, and gives rise to a departure from the elliptic laws. Hence arise what are called in the perturbational theory

Newton, i. 1.

equations (i. e. changes or fluctuations to and fro about an average quantity) of the mean motion of the disturbed body.

(618.) There is yet another mode of resolving the disturbing force into rectangular components, which, though not so well adapted to the computation of results, in reducing to numerical calculation the motions of the disturbed body, is fitted to afford a clearer insight into the nature of the modifications which the form, magnitude, and situation of its orbit undergo in virtue of its action, and which we shall therefore employ in preference. It consists in estimating the components of the disturbing force, which lie in the plane of the orbit, not in the direction we have termed radial and transversal, i. e. in that of the radius vector PS and perpendicular to it, but in the direction of a tangent to the orbit at P, and in that of a normal to the curve, and at right angles to the tangent, for which reason these components may be called the tangential and normal disturbing forces. When the orbit of the disturbed body is circular, or nearly so, this mode of resolution coincides with or differs but little from the former, but, when the ellipticity is considerable, these directions may deviate from the radial and transversal directions to any extent. As in the Newtonian mode of resolution, the effect of the one component falls wholly upon the approach and recess of the body P to the central body S, and of the other wholly on the rate of description of areas by P round S, so in this which we are now considering, the direct effect of the one component (the normal) falls wholly on the curvature of the orbit at the point of its action, increasing that curvature when the normal force acts inwards, or towards the concavity of the orbit, and diminishing it when in the opposite direction; while, on the other hand, the tangential component is directly effective on the velocity of the disturbed body, increasing or diminishing it according as its direction conspires with or opposes its motion. It is evident enough that where the object is to trace simply the changes produced by the disturbing force, in angle and distance from the central body, the former mode

of resolution must have the advantage in perspicuity of view and applicability to calculation. It is less obvious, but will abundantly appear in the sequel that the latter offers peculiar advantages in exhibiting to the eye and the reason the momentary influence of the disturbing force on the elements of the orbit itself.

(619.) Neither of the last mentioned pairs of resolved portions of the disturbing force tends to draw P out of the plane of its orbit PSA. But the remaining or orthogonal portion N Q acts directly and solely to produce that effect. In consequence, under the influence of this force, P must quit that plane, and (the same cause continuing in action) must describe a curve of double curvature as it is called, no two consecutive portions of which lie in the same plane passing through S. The effect of this is to produce a continual variation in those elements of the orbit of P on which the situation of its plane in space depends; i. e. on its inclination to a fixed plane, and the position in such a plane of the node or line of its intersection therewith. As this, among all the various effects of perturbation, is that which is at once the most simple in its conception, and the easiest to follow into its remoter consequences, we shall begin with its explanation.

(620.) Suppose that up to P (Art. 611. 614.) the body were describing an undisturbed orbit CP. Then at P it would be moving in the direction of a tangent PR to the ellipse P A, which prolonged will intersect the plane of M's orbit somewhere in the line of nodes, as at R. Now, at P, let the disturbing force parallel to N Q act momentarily on P; then P will be deflected in the direction of that force, and instead of the arc Pp, which it would have described in the next instant if undisturbed, will describe the arc Pq lying in the state of things represented in Art. 611. below, and in Art. 614. above, Pp with reference to the plane PSA. Thus, by this action of the disturbing force, the plane of P's orbit will have shifted its position in space from P Sp (an elementary portion of the old orbit) to P Sq, one of the new. Now the line of nodes SA B in the former is determined by prolonging