hence

a (1 − e2)2 = r2 sin2 a {-},

which gives the eccentricity of the orbit. The equation of conic

Thus the angle v, that the radius vector makes with the perihelion distance, is found, and, consequently, the position of the perihelion. The equations (96) will then give the angle u, or eccentric anomaly, and, by means of it, the instant of the passage at the perihelion.

In order to have the position of the orbit, with regard to a fixed plane passing through the centre of S, fig. 77, supposed immoveable, letbe the inclination of the two planes, and 6 mSN; also let

mp=z be the primitive elevation of the planet above the fixed plane, which s supposed to be known; then

B

r sin sinz.

So that, the inclination of the orbit, will be known when 6 shall be determined. For that purpose, let

n

fig. 79.

which is given, because A is supposed to be known; therefore

The elements of the orbit of the planet being determined by these formulæ in terms of r, z, the velocity of the planet, and the direction of its motion, the variations of these elements, corresponding to the supposed variations in the velocity and its direction, may be obtained; and it will be easy, by means of methods that will be hereafter given, to have the differential variations of these elements, arising from the action of the disturbing forces.

Velocity of Bodies moving in Conic Sections.

408. As the actual motions of the bodies of the solar system afford no information with regard to their primitive motions, the elements of their orbits can only be known by observation; but when these are determined, the velocities with which the bodies of the solar system were first projected in space, may be ascertained. If the equa√2= u2

tion

2

be resumed, then in the circle ra, since the eccentricity is zero;

hence

v=u

therefore Vu :: 1; √T.

thus the velocities of planets in different circles are as the square roots of their radii.

In the parabola, a is infinite; hence

Thus the velocities in different points of a parabolic orbit are reciprocally as the square roots of the radii vectores, and the velocity in each point is to the velocity the planet would have if it moved in a circle with a radius equal to r, as √2 to 1.

fig. 80.

409. When an ellipse is infinitely flattened, it becomes a straight line; hence, in this case, V will express the velocity of m, if it were to descend in a straight line towards the sun; for then Sm, fig. 80, would coincide with SA. If m were to begin to fall from a state of rest at A, its velocity would be zero at that point;

P

m

LA

n

Now, suppose that, in falling from A to n, the body had acquired the velocity V, then the equation would be

and eliminating a, which is common to the two last equations,

in which r' Sn. This is the relative velocity the body m has acquired in falling from A through r— r' An. Imagine the body m to have acquired, by its fall through An, the same velocity with a body moving in a conic section; the velocity of the latter body is

This expression gives the height through which a body moving in a conic section must fall, from the extremity A of the radius vector, in order to acquire the relative velocity which it had at A.

In the circle ar, hence An = r; in the ellipse, An is less thanr: in the parabola, a is infinite, which gives An = r; and in the hyperbola a is negative, and therefore An is greater than r.

CHAPTER V.

THEORY OF THE PERTURBATIONS OF THE PLANETS.

410. THE tables computed on the theory of perfectly elliptical motion, are soon found inadequate to give the true place of a planet, on account of the reciprocal disturbances of the system. It is therefore necessary to investigate what these disturbances are, and to determine their effects.

n

fig. 81.

In the first approximation to the celestial motions, the mutual action of the sun and of one planet was considered: it then appeared that a planet, m, moves round the sun in an ellipse NmPn, fig. 81, inclined to the ecliptic NBn, at a very small angle Pnp. Now, if m be attracted by another planet m', which is much smaller than the sun, it will no longer go on in its elliptical orbit Nmn, but will be drawn out of that orbit, and will move in some curved line, caD, which may either be nearer to, or farther from,

a

T

B

EP

the plane of the ecliptic, according to the position of the disturbing body. In the first infinitesimal of time, the troubled orbit coincides with the ellipse through an indefinitely small space ca; in the second infinitely small interval of time, am will be the path of the planet in the ellipse, and aD will be its path in its troubled orbit: am is described in consequence of the action of the sun alone; aD by the combined action of the sun and of the disturbing body: am is the second increment of the space; aD is the second increment of the space, together with some very small space, FD, introduced by the action of the disturbing force. In consequence of the addition of FD, the longitude of m is increased by Bb; its latitude is changed by the angle DSE, and the radius vector is increased by the difference between SD and Sm,-these three quantities are the perturbations of the planet in longitude, latitude, and distances.

411. It is evident that the perturbations are true variations; and as the longitude, latitude, and radius vector of a planet moving in

an elliptical orbit, have been represented by v, s, and r, the arcs Bb dv, ED = ds and SD - Smdr, are the variations of these co-ordinates.

=

412. The perturbations in longitude, latitude, and distance, depend on the configuration of the bodies; that is, on the position of the bodies with regard to each other, to their perihelia and to their nodes. These inequalities, after going through a certain course of increase and decrease, are renewed as often as the bodies return to the same relative positions, and are therefore called Periodic Inequalities.

413. Thus the place of a planet, m, moving in its troubled orbit caD, will be determined by the co-ordinates v + dv, s + ds, r + dr. These, however, are modified by a variation in the elements of the ellipse; for it is evident that, the path of the planet being changed from aE to aD, the elements of the ellipse NmE must vary. The variations of the elements are independent of the configuration or relative position of the bodies, and are only sensible in many revolutions; whereas those depending on the configuration, accomplish their changes in short periods. Thus v + dv, s +ds, r + dr, may be regarded as the co-ordinates of the planet in its true orbit, provided the elements contained in these functions be considered to vary by very slow degrees. This perfectly accords with observation, whence it appears that the perihelia of the orbits of the planets and satellites have a very slow direct motion in space; that the nodes have a slow retrograde motion; and that the eccentricities and inclinations are perpetually varying by very slow degrees. These very slow changes are really periodic, but many ages elapse before they accomplish their revolutions; on that account they are called Secular Inequalities, to distinguish them from the Periodic Inequalities, which pass rapidly from their maxima to their minima. Thus the Periodic Inequalities only depend on the configuration of the bodies, whereas the Secular Inequalities depend on the configuration of the perihelia and nodes alone.

414. La Grange took a new and very elegant view of the subject:-he considered the changes Sv, ds, Sr, to arise entirely from periodic and secular variations in the elements of elliptical motion, thus referring all the inequalities, to which a planet is liable, to changes in the elements of its orbit alone. In fact, as the curve aD