nearly perpendicular to the radius vector, and therefore the inclination of the path would be such as

distance from perihelion That is, when the planet

corresponds to a smaller than the planet really has. leaves e, the inclination of its path to the radius. vector is greater than it would have been if the planet had continued to move in the orbit cd b, but is the same as if its perihelion had been at some such situation as ƒ, supposing no disturbing force to act. Now let the disturbing force cease entirely to act; and the planet, which at e is moving as if it had come from the perihelion ƒ, will continue to move as if it had come from the perihelion f; it will proceed, therefore, to describe an elliptic orbit in which ƒSg is the line of apses: the line of apses has been twisted round in the same direction as before, or the line of apses has still progressed. The effect then of a disturbing force directed to the central body before and after passing the perihelion, is to make the line of apses progress.

*This result, and those which follow immediately, may be inferred from the construction in Newton's 'Principia,' book i. sect. 3. prop. xvii. If we assume (as we suppose in all these investigations) the excentricity to be small, the disturbing force directed to the sun will not sensibly alter the planet's velocity, but will change the direction of its path at P, the place of action (in Newton's figure); the length of P H, therefore, will not be altered (since that length depends only on the velocity), but its position will be altered, the position of P H being determined by making the angle R PH equal to the supplement of R P S. On trying the effects of this in different positions of P, and observing that the immediate effect of a disturbing force directed to the centre is to increase the rate of approach, or to diminish the rate of receding, and that the effect of a force directed from the centre is the opposite, all the cases in the text will be fully explained.

(51.) In the same manner it will be seen, that the effect of a disturbing force, directed from the central body before and after passing the perihelion, is to make the line of apses regress.

(52.) The motion of the planet, subject to such forces as we have mentioned, would be nearly the same as if it was revolving in an elliptic orbit, and this elliptic orbit was at the same time revolving round its focus, turning in the same direction as that in which the planet goes round, and always carrying it on its circumference. And this is the easiest way of representing to the mind the general effect of this motion; the physical cause is to be sought in such explanations as that above.

(53.) (VI.) Suppose a disturbing force directed to the centre, to act upon the planet when it is near aphelion. As the planet is going towards aphelion it is receding from the sun. The effect of the disturbing force is to diminish the rate of recess from the sun; and, therefore, to increase the inclination of the planet's path to the radius vector. The aphelion is the place where the planet's path is perpendicular to the radius vector. The effect of the disturbing force, then, which increases the inclination of the planet's path to the radius vector, will be to make that path perpendicular to the radius vector turbing force had not acted. be at aphelion sooner than it disturbing force had acted. The aphelion has as it were, gone backwards to meet the planet. If the

sooner than if the disThat is, the planet will would have been if not

disturbing force should entirely cease, the planet will move in an elliptic orbit, of which this new aphelion would be the permanent aphelion. The line passing through the aphelion has, therefore, twisted in a direction opposite to the planet's motion, or the line of apses has regressed. After passing aphelion, if the disturbing force still continues to act, the planet's approach to the sun will be quickened by the disturbing force, and, therefore, after some time, the planet's rate of approach will be greater than that corresponding in an undisturbed orbit to its actual distance from aphelion, and will be equal to that corresponding in an undisturbed orbit to a greater distance from aphelion. If, now, the disturbing force ceases, the planet, moving as if it came in an undisturbed orbit from an imaginary aphelion, will continue to move as if it came from that imaginary aphelion; and that imaginary aphelion having been at a greater distance behind the planet than the real aphelion, its place will be represented by saying that the line of apses has still regressed. The effect, then, of a disturbing force directed to the central body, before and after passing the aphelion, is to make the line of apses regress.

(54.) In the same manner it will be seen, that the effect of a disturbing force, directed from the central body, before and after passing the aphelion, is to make the line of apses progress.

(55.) (VII.) Since a disturbing force, directed to the

central body, or one directed from the central body, produces opposite effects with regard to the motion of the line of apses, according as it acts near perihelion, or near aphelion, it is easy to perceive that there must be some place between perihelion and aphelion, where the disturbing force, directed to the central body, will produce no effect on the position of the line of apses. It is found by accurate investigation, that this point is the place where the radius vector is perpendicular to the line of

(56.) (VIII.) The effects mentioned above are greatest when the excentricity is small. Thus, if we compare two orbits, as figures 8 and 9, in one of which the excentricity was great, and in the other small; and if (for instance) we supposed the disturbing force to act for a short time at the perihelion C, and supposed the forces in the two orbits to be such as to deflect the new paths from the old orbits by equal angles in the two cases: it is plain that in fig. 8, in consequence

*To the reader who is familiar with Newton's 'Principia,' sect. 3, the following demonstration will be sufficient :-The disturbing force, which is entirely in the direction of the radius vector, will not alter the area described in a given time, and, therefore, will not alter the latus rectum (to the square root of which the area is proportional). But half the latus rectum of the undisturbed orbit is the radius vector at the supposed place of action of the disturbing force (since that radius vector is supposed perpendicular to the major axis). Therefore, half the latus rectum of the new orbit is the radius vector at the point in question; and, consequently, the radius vector, at the point in question, is perpendicular to the major axis in the new orbit; but it was so in the undisturbed orbit; and, therefore, the major axes in the new orbit and the undisturbed orbit coincide.

of the curvature at C differing much from that of a circle whose centre is S, we should find the new perihelion c at a small distance from C; whereas in fig. 9, where the orbit does not differ much from the circle whose centre is S, c would be far removed from C. In fact, c would in both cases bisect the part of the orbit lying within that circle; and it is evident, that the angle at C being the same in both, the length of the part lying within the circle would be much less in fig. 8, where the orbit is almost a straight line, than in fig. 9, where the curvature of the orbit differs

little from that of the circle. Or we may state it thus: The alteration of the place of perihelion, or aphelion, depends on the proportion which the alteration in the approach or recess produced by the disturbing force bears to the whole approach or recess; and is, therefore, greatest when the whole approach or recess is least; that is, when the orbit is little excentric.

(57.) (IX.) To judge of the effect which a disturbing force, directed to the sun, will produce on the excentricity of a planet's orbit, let us suppose the planet