planets and satellites, or very eccentric, as in the comets. In both circumstances the series which determine the motions of the body may be made to converge rapidly, which would not be the case if the eccentricity bore a mean ratio to the greater axis.

Motion of Comets.

404. If the ratio of the eccentricity to the greater axis be made very nearly equal to unity, instead of a very small fraction, the preceding series will then give the place of a comet in a very eccentric orbit, with this difference, that the terms have the increasing powers of the difference between unity and the ratio of the eccentricity to the greater axis, as coefficients, instead of the powers of that ratio itself. This difference is zero in the parabola; then the value of the radius vector becomes

D being the perihelion distance: hence, in the parabola, the distance Sm is equal to the perihelion distance SP, divided by the square of the cosine of half the true anomaly PSm. If, then, the true anomaly were known, the distance of the comet from the sun would be determined from this equation. When the body moves in a parabola, the equation between the mean and true anomaly is reduced to a cubic equation between the time and the tangent of half the true anomaly PSm.

Arbitrary Constant Quantities of Elliptical Motion, or Elements of the Orbits.

405. There are six clements in the orbit of each celestial body: four of elliptical motion, namely, the mean distance of the planet from the sun; the eccentricity; the mean longitude of the planet at the epoch; and the longitude of the perihelion at the same epoch. The other two elements relate to the position of the orbit in space, namely, the longitude of the ascending node at the epoch, and the inclination of the orbit on the plane of the ecliptic. The mean values of all these must be determined by observation, before the

motion of the body can be ascertained, or tables computed. Hence there are forty-two elements to be determined for the seven principal planets, and twenty-four more for the four new planets, Ceres, Pallas, Juno, and Vesta, besides those of the moon and satellites. Tables have been computed for most of these bodies; some of the satellites, however, are but little known, and the theory of the four new planets is still imperfect.

The same series that determine the motions of the planets answer equally well for the elliptical motion of the moon and satellites, only the mass of the planet is to be employed in place of that of the sun, omitting the mass of the satellite.

Co-ordinates of a Planet.

406. The simplicity of analytical expressions very much depends on a skilful choice of co-ordinates, which are arbitrary and infinite in number, but so connected, that any one set may be expressed in values of any other. For example, the place of the planet m has been determined by the angles Sm, mSp, and Sm, fig. 77, but these have been changed into Sp, pSm, and Sp, which are the heliocentric longitude, latitude, and curtate distance of m. Again, from the latter, the geocentric longitude, latitude, and distance may be deduced, that is, the place of m as seen from the earth; and, lastly, the right ascension and declination of m, or its place referred to the equator, may be obtained from its geocentric longitude and latitude.

These quantities are given in terms of the mean longitude or time, since the first co-ordinates are given in series of the sines and cosines of that quantity. In the theory of the moon, the series are found to converge more rapidly, if the mean longitude, latitude, and distance are determined in functions of the true longitude. All these co-ordinates are connected by spherical triangles, so that they are easily deduced from one another.

Determination of the Elements of Elliptical Motion.

407. Were the primitive velocity with which the bodies of the solar system projected in space known, the values of the elements

of their orbits might be determined; for if the equation (90) be resumed, and if the first member, which is the square of the velocity, be represented by V, then

2

a

in which r is the radius vector, and a is half the greater axis of the conic section, being the masses of the sun and planet. Thus the velocity is independent of the eccentricity of the orbit.

If u be the angular velocity which the planet would have if it described a circle at the distance of unity round the sun, then r=a=1, and the preceding expression gives u2 = μ; hence

V being the primitive velocity with which the body moved in a conic section. This equation will give a value of a by means of the primi

that the direction of the primitive impulse has no influence on the nature of the conic section in which the planet moves; the intensity alone has that effect.

To determine the eccentricity of the orbit, let a be the angle TmS, that the direction of the relative motion of m makes with the radius vector r; then mn: mv :: ds: dr::1: cos a;

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

B

er

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

r sin 6 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 equation V2 = u2

2

{-}

하

a

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.

409. When an ellipse is infinitely flattened, it becomes a straight line; hence, in this case, I will express the velocity of m, if it were