ON THE LAW OF UNIVERSAL GRAVITATION, DEDUCED CHAPTER II. 309. THE three laws of Kepler furnish the data from which the principle of gravitation is established, namely: i. That the radii vectores of the planets and comets describe areas proportional to the time. ii. That the orbits of the planets and comets are conic sections, having the sun in one of their foci. iii. That the squares of the periodic times of the planets are proportional to the cubes of their mean distances from the sun. A 310. It has been shown, that if the law of the force which acts on a moving body be known, the curve in which it moves may be found; or, if the curve in which the body moves be given, the law of the force may be ascertained. In the general equation of the motion of a body in article 144, both the force and the path of the body are indeterminate; therefore in applying that equation to the motion of the planets and comets, it is necessary to know the orbits in which they move, in order to ascertain the nature of the force that acts on them. B 311. In the general equation of the motion of a body, the forces acting on it are resolved into three component forces, in the direction of three rectangular axes; but as the paths of the planets, satellites, and comets, are proved by the observations of Kepler to be conic sections, they always move in the same plane: therefore the component force in the direction perpendicular to that plane is zero, and the other two component forces are in the plane of the orbit. 312. Let AmP, fig. 62, be the elliptical orbit of a planet m, hav component forces to be in the direction of the axes Sx, Sy, then the component force Z is zero; and as the body is free to move in every direction, the virtual velocities dr, dy are zero, which divides the general equation of motion in article 144 into =Y; d2x dt2 = X; giving a relation between each component force, the space that it causes the body to describe on or, or oy, and the time. If the first of these two equations be multiplied by y, and added to the second multiplied by x, their sum will be d(xdy—ydx) = Yx d'y But xdy - ydx is double the area that the radius vector of the planet describes round the sun in the instant dt. According to the first law of Kepler, this area is proportional to the time, so that d(xdy - ydx) Yx be multiplied by dr, and Xy = 0, d2x dt2 day dt = 0, therefore whence so that the forces X and Y are in the ratio of x to y, that is as Sp to pm, and thus their resulting force mS passes through S, the centre of the sun. Besides, the curve described by the planet is concave towards the sun, whence the force that causes the planet to describe that curve, tends towards the sun. And thus the law of the areas being proportional to the time, leads to this important result, that the force which retains the planets and comets in their orbits, is directed towards the centre of the sun. Xy. 313. The next step is to ascertain the law by which the force varies at different distances from the sun, which is accomplished by the consideration, that these bodies alternately approach and recede from him at each revolution; the nature of elliptical motion, then, ought to give that law. If the equation = X = Y, by dy, their sum is drder + dydy and its integral is d.x2 + dy2 2f(Xdx + Ydy), the constant quantity being indicated by the integral sign. Now the law of areas gives dt= 0 = which changes the preceding equation to c2(dx2+dy2) = (dy—ydr) = dv= so that the equation (82) becomes = Xdr + Ydy, xdy - ydr " с (82) In order to transform this into a polar equation, let r represent the radius vector Sm, fig. 62, and v the angle mSy, + then Sp = x = r cos v; pm = y = r sin v, and r = √x2 + y2 whence dx dy2 = r2dv2 + dr3, xdy ydx = r2dv; and if the resulting force of X and Y be represented by F, then F:X:: Sm: Sp :: 1: cos v; hence X = Fcos v the sign is negative, because the force F in the direction mS, tends to diminish the co-ordinates; in the same manner it is easy to see that Y = – F sin v; F= √X+Y2; and Xdx + Ydy = Fdr ; F = 2f(Xdx + Ydy). c2{ r2dv2 + dr3} + 2fFdr. r4dv2 cdr whence r √ — c2-2r2f Fdr 314. If the force F be known in terms of the distance r, this equation will give the nature of the curve described by the body. But the differential of equation (83) gives dr2 2 :} d \r+dv2 (84) 7.3 dr Thus a value of the resulting force F is obtained in terms of the variable radius vector Sm, and of the corresponding variable angle mSy; but in order to have a value of the force F in terms of mS alone, it is necessary to know the angle ySm in terms of Sm. (83) The planets move in ellipses, having the sun in one of their foci; 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 Sm or v, and thence it may be found that T= dr2 The coefficient = which substituted in equation (84) gives C2 F= a(1 h = 2 a(1 e2) 1 a(1 versely as the square of r or Sm. Wherefore the orbits of the 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, it may be represented by, in which h is a constant coefficient, expressing the intensity of the force. The equation of conic sections will satisfy equation (84) when is put for F; whence as h J is constant, therefore F varies in e2) e) forms an equation of condition between the constant quantities a and e, the three arbitrary quantities a, e, and ☎, 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 с 1 e2) p2 F= it may be shown, that the force F varies, with regard to different c9 c9 SV' a(1-e) or |
