The planets move in ellipses, having the sun in one of their foci; therefore let w represent the angle ySP, which the greater axis AP makes with the axes of the co-ordinates Sr, and let v be the angle ySm.

CS Then if the ratio of the eccentricity to the greater axis be e, and CP

the greater axis CP a, the polar equation of conic sections is

which becomes a parabola when e = 1, and a infinite; and a hyperbola when e is greater than unity and a negative. This equation gives a value of r in terms of the angle ySm or v, and thence it may be found that

planets and comets being conic sections, the force varies inversely as the square of the distance of these bodies from the sun.

Now as the force F varies inversely as the square of the distance,

expressing the intensity of the force. The equation of conic sections

h

will satisfy equation (84) when is put for F; whence as

forms an equation of condition between the constant quantities a and e, the three arbitrary quantities a, e, and w, are reduced to two; and as equation (83) is only of the second order, the finite equation of conic sections is its integral.

315. Thus, if the orbit be a conic section, the force is inversely as the square of the distance; and if the force varies inversely as the square of the distance, the orbit is a conic section. The planets and

comets therefore describe conic sections in virtue of a primitive impulse and an accelerating force directed to the centre of the sun, and varying according to the preceding law, the least deviation from which would cause them to move in curves of a totally different nature.

316. In every orbit the point P, fig. 63, which is nearest the sun, is the perihelion, and in the ellipse the point A farthest from the sun is the aphelion. SP is the perihelion distance of the body from the sun.

317. A body moves in a conic section with a different velocity in every point of its orbit, and with a perpetual tendency to fly off in the direction of the tangent, but this tendency is counteracted by the attraction of the sun. At the perihelion, the velocity of a planet is greatest; therefore its tendency to leave the sun exceeds the force of attraction but the continued action of the sun diminishes the velocity as the distance increases; at the aphelion the velocity of the planet is least: therefore its tendency to leave the sun is less than the force of attraction which increases the velocity as the distance diminishes, and brings the planet back towards the sun, accelerating its velocity so much as to overcome the force of attraction, and carry the planet again to the perihelion. This alternation is continually repeated.

318. When a planet is in the point B, or D, it is said to be in quadrature, or at its mean distance from the sun. In the ellipse, the mean distance, SB or SD, is equal to CP, half the greater axis; the eccentricity is CS.

319. The periodic time of a planet is the time in which it revolves round the sun, or the time of moving through 360°. The periodic time of a satellite is the time in which it revolves about its primary. 320. From the equation

it may be shown, that the force F varies, with regard to different planets, inversely as the square of their respective distances from the sun. The quantity 2a(1-e) is 2SV, the parameter of the orbit, which is invariable in any one curve, but is different in each conic section. The intensity of the force depends on

which may be found by Kepler's laws. Let T represent the time of the revolution of a planet; the area described by its radius vector in this time is the whole area of the ellipse, or Ta2. √ 1 — e2.

where 3.14159 the ratio of the circumference to the diameter. But the area described by the planet during the indefinitely small time dt, is cdt; hence the law of Kepler gives

but 2a (1

e) is 2SV, the parameter of the orbit.

Therefore, in different orbits compared together, the values of c are as the areas traced by the radii vectores in equal times; consequently these areas are proportional to the square roots of the parameters of the orbits, either of planets or comets. If this value of c be put in

in which 4 or h, is the same for all the planets and comets; the

k2

force, therefore, varies inversely as the square of the distance of each from the centre of the sun: consequently, if all these bodies were

placed at equal distances from the sun, and put in motion at the same instant from a state of rest, they would move through equal spaces in equal times; so that all would arrive at the sun at the same instant, properties first demonstrated geometrically by Newton from the laws of Kepler.

321. That the areas described by comets are proportional to the square roots of the parameters of their orbits, is a result of theory more sensibly verified by observation than any other of its consequences. Comets are only visible for a short time, at most a few months, when they are near their perihelia; but it is difficult to determine in what curve they move, because a very eccentric ellipse, a parabola, and hyperbola of the same perihelion distance coincide through a small space on each side of the perihelion. The periodic time of a comet cannot be known from one appearance. Of more than a hundred comets, whose orbits have been computed, the return of only three has been ascertained. A few have been calculated in very elliptical orbits; but in general it has been found, that the places of comets computed in parabolic orbits agree with observation on that account it is usual to assume, that comets move in parabolic curves.

322. In a parabola the parameter is equal to twice the perihelion distance, or

For, in this case, e 1 and a is infinite; therefore, in different parabolæ, the areas described in equal times are proportional to the square roots of their perihelion distances. This affords the means of ascertaining how near a comet approaches to the sun. Five or six comets seem to have hyperbolic orbits; consequently they could only be once visible, in their transit through the system to which we belong, wandering in the immensity of space, perhaps to visit other suns and other systems.

It is probable that such bodies do exist in the infinite variety of creation, though their appearance is rare. Most of the comets that we have seen, however, are thought to move in extremely

eccentric ellipses, returning to our system after very long intervals. Two hundred years have not elapsed since comets were observed with accuracy, a time which is probably greatly exceeded by the enormous periods of the revolutions of some of these bodies.

323. The three laws of Kepler, deduced from the observations of Tycho Brahe, and from his own observations of Mars, form an era of vast importance in the science of astronomy, being the bases on which Newton founded the universal principle of gravitation: they lead us to regard the centre of the sun as the focus of an attractive force, extending to an infinite distance in all directions, decreasing as the squares of the distance increase. Each law discloses a particular property of this force. The areas described by the radius vector of each planet or comet, being proportional to the time employed in describing them, shows that the principal force which urges these bodies, is always directed towards the centre of the sun. The ellipticity of the planetary orbits, and the nearly parabolic motion of the comets, prove that for each planet and comet this force is reciprocally as the square of the distance from the sun; and, lastly, the squares of the periodic times, being proportional to the cubes of the mean distances, proves that the areas described in equal times by the radius vector of each body in the different orbits, are proportional to the square roots of the parameters-a law which is equally applicable to planets and comets.

324. The satellites observe the laws of Kepler in moving round their primaries, and gravitate towards the planets inversely as the square of their distances from their centre; but they must also gravitate towards the sun, in order that their relative motions round their planets may be the same as if the planets were at rest. Hence the satellites must gravitate towards their planets and towards the sun inversely as the squares of the distances. The eccentricity of the orbits of the two first satellites of Jupiter is quite insensible; that of the third inconsiderable; that of the fourth is evident. The great distance of Saturn has hitherto prevented the eccentricity of the orbits of any of its satellites from being perceived, with the exception of the sixth. But the law of the gravitation of the satellites of Jupiter and Saturn is derived most clearly from this ratio,-that, for each system of satellites, the squares of their periodic times are as the cubes of their mean distances from the centres of their respective