708 MECHANICS Now (1) is the general polar equation of a conic section, focus the pole ; and, as its nature depends on the value of the excentricity e. given by (4 ), we see that if V 2 >2/t/R , e>l, and the orbit is an hyperbola ; if V 2 = 2/i/R , e 1, and the orbit is a parabola ; if V 2 < 2/i/R, e< 1, and the orbit is an ellipse. But the square of the speed from rest at infinity to distance R, for the law of attraction we are considering, is 2/J./11 , and the above conditions may therefore be expressed more concisely by saying that the orbit will be an hyperbola, a parabola, or an ellipse, accord ing as the speed of projection is greater than, equal to, or less than, the speed from infinity. Illustrations of this proposition are found in the cases of comets and of meteor swarms. The speed of a particle moving in a circle is also often taken as the standard of comparison for estimating the velocities of bodies in their orbits. For the gravitation law of attraction the square of the speed in a circle of radius R is /J./R ; and the above con ditions may be expressed in another form by saying that the orbit will be an hyperbola, a parabola, or an ellipse, according as the speed of projection is greater than, equal to, or less than /2 times the speed in a circle at the same distance. Supposing the orbit to be an ellipse, we shall obtain its major axis and latus rectum most easily by a different process of integrat ing the differential equation. Multiplying it by 7i 2 ~ and inte- dd grating, we obtain du dd But when u = -rr v = V ; which gives n_JLv2 *L ~ 2 V R hence h 2 + u- (5). Now to determine the apsidal distances, we must put du dd and this gives us the condition = 0; which is a quadratic equation whose roots are the reciprocals of the tvo apsidal distances. But if a be the semi-axis major, and e the excentricity, these distances are a(l -e) and a(l +e). Hence, as the coefficient of the second term of (6) is the sum of the roots with their signs changed, we have __.. a(l-e) a(l+e) It? And the third term is the product of the roots, so that a"(l c") h"li It? _1 = A_L 2 a R p. or, by (7), Thus and therefore Equations (7) and (8) give the latus rectum and major axis of the orbit, and show that the major axis is indepsndent of the direc tion of projection. Equation (9) gives a useful expression for the speed at any point, and shows that the radius of the circle of zero speed is 2. The time of describing any given angle is to be obtained from the formula, c)} , by equation (7). From this, combined with the polar equation of the ellipse about the focus, we have dt _ r*__ / / ff*(l - r 2 ) 3 __ 1 dQ ~ V { (M(l~~c y ) } ~ V V /T ) l+ccosO) 2 " measuring the angle from the nearer apse. Integrating, we find the time of describing about the focus an angle 9 measured from the nearer apse, in the ellipse or hyperbola, expressed as 2/A of the sectorial area ASP (figure to 147), which might have been written down from the condition of uniform moment of momentum. In the parabola, if d be the apsidal distance, the integral becomes [ -VT From the result for the ellipse we see that the periodic time is 27rV^V. In the notation commonly employed for the further development of this most important question we write T=2jr/, where n, which is called the mean motion, " is /u/a 3 . 145. By laborious calculation from an immense series of observations of the planets, and of Mars in particular, Kepler was led to enunciate the following as the kine- matical laws of the planetary motions about the sun. I. The planets describe, relatively to the sun, ellipses of which the sun occupies a focus. II. The radius vector of each planet traces out equal areas in equal times. III. The squares of the periodic times of any two planets are as the cubes of the major axes of their orbits. 1 46. We proceed to the inverse problem of 8 (b), the determination of the force from the observed motions. From the second of the above laws we conclude that the planets are retained in their orbits by an attraction tending to the sun. If the radius-vector of a particle moving in a plane describe equal areas in equal times about a point in that plane, the resultant attraction on the particle tends to that point. For the datum is equivalent to the state ment that there is no change of moment of momentum about the sun, or that the accelerations all pass through the sun viewed as a point. From the first law it follows that the law of the intensity of the attraction is that of the inverse square of the dis tance. The polar equation of an ellipse referred to its focus is Kepler s laws (kinema tieal). Conse quences Kepler s laws. = -j-(l +ccos6), where Z is the latus rectum. Hence ^, and therefore the attraction to the focus requisite for the descrip tion of the ellipse is (g 47) Hence,, if the orbit be an ellipse described about a centre of attrac tion at the focus, the law of intensity is that of the inverse square of the distance. From the third law it follows that the attraction of the sun (supposed fixed) which acts on unit of mass of each of the planets is the same for each planet at the same distance. For, in the last formula in 144, T 2 will not vary as a 3 unless ^ be constant, i.e., unless the strength of attraction of the sun be the same for all the planets. We shall find afterwards that for more reasons than one Kepler s Law of laws are only approximate, but their enunciation was sufficient to gravita- enable Newton to propound the doctrine of universal gravitation, tion viz. , that every particle of matter in the universe attracts every other (physica with an attraction whose direction is that of the line joining them, and whose magnitude is as the product of the masses directly and as the square of the distance inversely ; or, according to Maxwell s formulation, between every pair of particles there is a stress of the nature of a tension, proportional to the product of the masses of the particles divided by the square of their distance. If we take into account that the sun is not absolutely fixed, then, neglecting the mutual attractions of the planets, Kepler s third law should be stated thus : The cubes of the major axes of the orbits are as the squares of the
periodic times and the sums of the masses of the sun and the planet.