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 that 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 P B, 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 back to the same precise relative positions) it will necessarily happen, that in the course of a very great

* If a be the semiaxis, r the radius vector, and v the velocity of P in any point of an ellipse, a is given by the relation

force being properly assumed,

=

2

T

a

, the units of velocity and

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.

(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 more general

view of the matter. Among the fundamental relations of dynamics, relations which presuppose no particular law of force like that of gravitation, but which express in general terms the results of the action of force on matter during time, to produce or change velocity, is one usually cited as the "Principle of the conservation of the vis viva," which applies directly to the case before us. This principle (or rather this theorem) declares that if a body subjected at every instant of its motion to the action of forces directed to fixed centers (no matter how numerous), and having their intensity dependent only on the distances from their respective centers of action, travel from one point of space to another, the velocity which it has on its arrival at the latter point will differ from that which it had on setting out from the former, by a quantity depending only on the different relative situations of these two points in space, without the least reference to the form of the curve in which it may have moved in passing from one point to the other, whether that curve have been described freely under the simple influence of the central forces, or the body have been compelled to glide upon it, as a bead upon a smooth wire. Among the forces thus acting may be included any constant forces, acting in parallel directions, which may be regarded as directed to fixed centers infinitely distant. It follows from this theorem, that, if the body return to the point P from which it set out, its velocity of arrival will be the same with that of its departure; a conclusion which (for the purpose we have in view) sets us free from the necessity of entering into any consideration of the laws of the disturbing force, the change which its action may have induced in the form of the orbit of P, or the successive steps by which velocity generated at one point of its intermediate path is destroyed at another, by the reversed action of the tangential force. Now to apply this theorem to the case in question, let M be supposed to retain a fixed position during one whole revolution of P. P then is acted on, during that revolution, by three forces 1st. by the central attraction of S directed always to S; 2nd. by that to M, always directed to M; 3rd. by a force equal to M's attraction on S; but in the direction M S, which