to have left its perihelion, and to be moving towards aphelion, and, consequently, to be receding from the sun, and now let the disturbing force act for a short time. This will cause it to recede from the sun more slowly than it would have receded without the action of the disturbing force; and, consequently, the planet, without any material alteration in its velocity (and, therefore, without any material alteration in the major axis of its orbit (28.)), will be moving in a path more inclined to the radius vector than if the disturbing force had not acted. The planet may, therefore, be considered as projected from the point A, fig. 10, in

the direction A b instead of A B, in which it was moving; and, therefore, instead of describing the orbit A C G, in which it was moving before, it will describe an orbit Acg, more resembling a circle, or less excentric than before. The effect, therefore, of a disturbing force directed to the centre, while a planet is moving from perihelion to aphelion, is to diminish the excentricity of the orbit.

(58.) If we suppose the planet to be moving from

aphelion to perihelion, it is approaching to the sun; the disturbing force directed to the sun makes it approach more rapidly; its path is, therefore, less inclined to the radius vector than it would have been without the disturbing force; and this effect may be represented by supposing that at E, fig. 11, instead of moving in the

direction E F in which it was moving, the planet is projected in the direction E f. Instead, therefore, of describing the ellipse E G H, in which it was moving before. it will describe such an ellipse as Egh, which is more excentric than the former. The effect, therefore, of a disturbing force directed to the centre, while a planet is moving from aphelion to perihelion, is to increase the excentricity of the orbit.

(59.) In a similar manner it will appear, that the effect of a disturbing force, directed from the centre, is to increase the excentricity as the planet is moving from perihelion to aphelion, and to diminish it as the planet moves from aphelion to perihelion.

(60.) (X.) Let us now lay aside the consideration of a force acting in the direction of the radius vector, and consider the effect of a force acting perpendicularly to

the radius vector, in the direction in which the planet is moving. And first, its effect on the position of the line of apses.

(61.) If such a force act at one of the apses, either perihelion or aphelion, for a short time, it is clear that its effect will be represented by supposing that the velocity at that apse is suddenly increased, or that the velocity with which the planet is projected from perihelion is greater than the velocity with which it would have been projected if no disturbing force had acted. This will make no difference in the position of the line of apses; for with whatever velocity the planet is projected, if it is projected in a direction perpendicular to the radius vector (which is implied in our supposition, that the place where the force acts was an apse in the old orbit), the place of projection will infallibly be an apse in the new orbit; and the line of apses, which is the line drawn from that point through the centre, will

be the same as before.

(62.) But if the force act for a short time before the planet reaches the perihelion, its principal * effect will be to increase its velocity; the sun's attraction will, therefore, have less power to curve its path (25.); the new orbit will be, in that part, exterior to the old one. In fig. 12, we must, therefore, suppose that the planet, after leaving A, where the force has acted to accelerate

*It is supposed here, and in all our investigations, that the excentricity of the orbit is small, and, consequently, that a force perpendicular to the radius vector produces nearly the same effect as a force acting in the direction of a tangent to the ellipse.

its motion, instead of describing the orbit A CG, proceeds to describe the orbit Acd, which at A has the same direction (or has the same tangent A B) as the orbit A CG. It is plain now that c is the part nearest

FIG. 12.

G

B

to the sun, or c is the perihelion: and it is evident here, that the line of apses has altered its position from SC to Sc, or has twisted in a direction opposite to the angular motion of the planet, or has regressed.

(63.) If the force act for a short time after the planet has passed perihelion, as at D in fig. 13, the planet's velocity is increased there, and the path described by the planet is Df, instead of DF, having the same direction at D (or having the same tangent DE), but less curved, and, therefore, exterior to DF. If now we conceive the planet to have received the actual velocity with which it is moving in Df, from moving without disturbance in an elliptic orbit cDf (which is the orbit that it will now proceed to describe, if no disturbing force continues to act), it is evident that the part cD must be described with a greater velocity than CD, inasmuch as the velocity at D from moving in cD is

greater than the velocity from moving in CD; cD is, therefore, less curved than CD, and, therefore, exterior to it (since it has the same direction at D); and then

the perihelion is some point in the position of c, and the line of apses has changed its direction from SC to Sc, or has twisted round in the same direction in which the planet is moving, or has progressed.

(64.) If the force act for a short time before passing aphelion, it will be seen in the same manner that the line of apses is made to progress. It is only necessary to consider that (as before) the new orbit has the same direction at the point H, fig. 14, where the force has

acted as the old one, but is less curved, and, therefore, exterior to it; and the aphelion, or point most distant from the sun, is g instead of G, and the position of the