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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 MK=M S take M N:MK::M K? : M P. Then if S N 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 NT, NL perpendicular respectively to S Y and P S produced. Then will N T represent the tangential, TS the normal, N L the transversal, and L S the radial components of the disturbing force. In circular orbits or orbits only slightly elliptic, the directions PSL and S Y 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
on which the magnitude, situation, and form of the orbit depends (i.e. the length and position of the major axis and the excentricity), in the same way as in the last chapter we determined the momentary changes of the inclination and node similarly produced by the orthogonal force.
(658.) We shall begin with the momentary variation in the length of the axis, an element of the first importance, as on it depends (art. 487) the periodic time and mean angular motion of the planet, as well as the average supply of light and heat it receives in a given time from the sun, any permanent or constantly progressive change in which would alter most materially the conditions of existence of living beings on its surface. Now it is a property of elliptic motion performed under the influence of gravity, and in conformity with Kepler's laws, that if the velocity with which a planet moves at any point of its orbit be given, and also the distance of that point from the sun, the major axis of the orbit is thereby also given. It is no matter in what direction the planet may be moving at that moment. This will influence the excentricity and the position of its ellipse, but not its length. This property of elliptic motion has been demonstrated by Newton, and is one of the most obvious and elementary conclusions from his theory. Let us now consider a planet describing an indefinitely small arc of its orbit about the sun, under the joint influence of its attraction, and the disturbing power of another planet. This arc will have some certain curvature and direction, and, therefore, may be considered as an arc of a certain ellipse described about the sun as a focus, for this plain reason, — that whatever be the curvature and direction of the arc in question, an ellipse may always be assigned, whose focus shall be in the sun, and which shall coincide with it throughout the whole interval (supposed indefinitely small) between its extreme points. This is a matter of pure geometry. It does not follow, however, that the ellipse thus instantaneously determined will have the same elements as that similarly determined from the arc described in either the previous or the subsequent instant. If the disturbing force did not exist, this would be the case ; but, by its action, a
variation of the element from instant to instant is produced, and the ellipse so determined is in a continual state of change. Now when the planet has reached the end of the small arc under consideration, the question whether it will in the next instant describe an arc of an ellipse having the same or a varied axis will depend, not on the new direction impressed upon it by the acting forces, - for the axis, as we have seen, is independent of that direction, -not on its change of distance from the sun, while describing the former arc, — for the elements of that arc are accommodated to it, so that one and the same axis must belong to its beginning and its end. The question, in short, whether in the next arc it shall take up a new major axis or go on with the old one will depend solely on this — whether its velocity has or has not undergone a change by the action of the disturbing force. For the central force residing in the focus can impress on it no such change of velocity as to be incompatible with the permanence of its ellipse, seeing that it is by the action of that force that the velocity is maintained in that due proportion to the distance which elliptic motion, as such, requires.
(659.) Thus we see that the momentary variation of the major axis depends on nothing but the momentary deviation from the law of elliptic velocity produced by the disturbing force, without the least regard to the direction in which thạt extraneous velocity is impressed, or the distance from the sun at which the planet may be situated, at the moment of its impression. Nay, we may even go farther, for, as this holds good at every instant of its motion, it will follow, that after the lapse of any time, however great, the total amount of change which the axis may have undergone will be determined only by the total deviation produced by the action of the disturbing force in the velocity of the disturbed body from that which it would have had in its undisturbed ellipse, at the same distance from the center, and that therefore the total amount of change produced in the axis in any lapse of time may be estimated, if we know at every instant the efficacy of the disturbing force to alter the velocity of the body's motion, and that without any regard to the alterations which the action of that force may have produced in the other elements of the motion in the same time.
(660.) Now it is not the whole disturbing force which is effective in changing P's velocity, but only its tangential component. The normal component tends merely to alter the curvature of the orbit or to deflect it into conformity with a circle of curvature of greater or lesser radius, as the case may be, and in no way to alter the velocity. Hence it appears that the variation of the length of the axis is due entirely to the tangential force, and is quite independent on the normal. Now it is easily shown that as the velocity increases, the axis increases (the distance remaining unaltered *) though not in the same exact proportion. Hence it follows that if the tangential disturbing force conspires with the motion of P, its momentary action increases the axis of the disturbed orbit, whatever be the situation of P in its orbit, and vice versa.
(661.) Let ASB (fig. art. 657.) be the major axis of the ellipse A PB, and on the opposite side of A B take two points P' and M', similarly situated with respect to the axis with P and M on their side. Then if at P and M' bodies equal to P and M be placed, the forces exerted by M' on P and S will be equal to those exerted by M on P and S, and therefore the tangential disturbing force of M' on P' exerted in the direction P'Z' (suppose) will equal that exerted by M on P in the direction P Z. P' therefore (supposing it to revolve in the same direction round S as P) will be retarded (or accelerated, as the case may be) by precisely the same force by which P is accelerated (or retarded), so that the variation in the axis of the respective orbits of P and P' will be equal in amount, but contrary in character. Suppose now M's orbit to be circular. Then (if the periodic times of M and P be not commensurate, so that a moderate number of revolutions may bring them baek to the same precise relutive positions) it will necessarily happen, that in the course of a very great number of revolutions of both bodies, P will have been presented to M on one side of the axis, at some one moment, in the same manner as at some other moment on the other. Whatever variation may have been effected in its axis in the one situation will have been reversed in that symmetrically opposite, and the ultimate result, on a general average of an infinite number of revolutions, will be a complete and exact compensation of the variations in one direction by those in the direction opposite.
* If a be the semiaxis, » the radius vector, and v the velocity of P in any point of an ellipse, a is given by the relation v' = _, the units of velocity and force being properly assumed.
(662.) Suppose, next, P's orbit to be circular. If now M's orbit were so also, it is evident that in one complete synodic revolution, an exact restoration of the axis to its original length would take place, because the tangential forces would be symmetrically equal and opposite during each alternate quarter revolution. But let M, during a synodic revolution, have receded somewhat from S, then will its disturbing power have become gradually weaker, so that, in a synodic revolution the tangential force in each quadrant, though reversed in direction being inferior in power, an exact compensation will not have been effected, but there will be left an outstanding uncompensated portion, the excess of the stronger over the feebler effects. But now suppose M to approach by the same gradations as it before receded. It is clear that this result will be reversed; since the uncompensated stronger actions will all lie in the opposite direction. Now suppose M's orbit to be elliptic. Then during its recess from S or in the half revolution from its perihelion to its aphelion, a continual uncompensated variation will go on accumulating in one direction. But from what has been said, it is clear that this will be destroyed, during M's approach to S in the other half of its orbit, so that here again, on the average of a multitude of revolutions during which P has been presented to M in every situation for every distance of M from S, the restoration will be effected.
(663.) If neither P's nor M's orbit be circular, and if moreover the directions of their axes be different, this reasoning, drawn from the symmetry of their relations to each other, does not apply, and it becomes necessary to take a inore general