in a direction at right angles to the radius connecting the planet and the sun, then the planet would continue for ever to describe a circle around the sun.
§ 11. On the Elliptic Motion of the Planets.—The laws of the motions of the planets were discovered by Kepler by means of cal culations founded upon observations. They may be thus stated : 1. Each planet moves in an ellipse in one focus of which the sun is situated. 2. The radius vector drawn from the sun to the planet sweeps over equal areas in equal times. 3. The squares of the periodic times of the motions of the planets round the sun are in the same ratio as the cubes of their mean distances. These three laws form the foundations of that branch of astronomy which is called Physical Astronomy, and they are generally known as Kepler s Laws. The sun and the planets are all very nearly spherical, and for the present, whenever we speak of the motions of the sun or the planets, we are to understand the motions of the centres of the corresponding spheres. Indeed the diameters of these spheres are so small in comparison with the distances at which they are separated from each other, that we may generally regard them as mere physical points. Thus though the diameter of the sun is ten times greater than the diameter of the greatest planet, Jupiter, yet the sun s dia meter is only the forty-second part of its distance from the nearest planet, Mercury, and it is less than the three-thousandth part of its distance from the outermost planet, Neptune. The orbits of the planets are so little eccentric that at a first glance the majority of them appear to be circular. Among the larger planets the orbits of Mercury and of Mars have the greatest eccen tricity, being about and ^- respectively. Next to these comes the orbit of Saturn, which has an eccentricity of T V Next in order come Jupiter, Uranus, the Earth, Neptune, and Venus, with eccentricities f T?T) TT> T?T> TTS> TIB" respectively. The orbits of some of the minor planets are, however, much more eccentric.
Fig. 7.
§ 12. The Hodograph.—In discussing the actual motion of a planet around the sun, it is very convenient to introduce the elegant con ception of the hodograph. The use of this curve is originally due to Bradley, but for its practical development we are principally indebted to Sir "W. 11. Hamilton. Let AB (fig. 7) be a portion of the path of a particle P acted upon by any forces. From any point draw radii veetores OR 1; OR 2 , OR 3 , OR 4 (fig. 8), parallel to the tan gents to the curve AB drawn at the points, P u P 2 , P 3 , P 4 , and equal to the velocities at those points. Then the curve R 1 R 2 R 3 R 4 Is the hodograph of the orbit of P. To each position P of the Q^!____ particle in the path there is a corresponding point R in the hodograph ; and simultaneously with the motion of P in its orbit we have the motion of R in the hodograph.
The utility of the hodograph depends upon the theorem that the force which acts upon the particle is at any time equal and parallel to the velocity of the corresponding point in the hodograph. This theorem is thus proved. Let P a and P 3 , and therefore R 2 and R 3 , be very close together. The velocity imparted to the particle in passing from P 3 to P 3 in the small time St must, when compounded with the velocity OR. 2 , produce the velocity OR 3 . This increment of velocity must therefore bo equal and parallel to R 2 R 3 . If there fore v denote the velocity of R, we have v$t for the velocity ac quired in the time St. The force acting on P must therefore, by the second law of motion, be equal to v and parallel to the line R a R 3 , which ultimately coincides with the tangent. The theorem is there fore proved.
Fig. 9.
Kepler s second law states that in the movement of a planet around the sun the radius vector from the sun to the planet describes equal areas in equal times. It can be shown that when this is the casr the force which acts upon the planet must necessarily be directed towards the sun. Let the planet be supposed to move from P to P (fig. 9) in the time t with the velocity v. Then, since equal areas are described in equal times, we must have the area OPP equal to hSt, where h is a constant which represents the area described in the unit of time. Let fall OT perpendicular upon the tangent at P, then. irOT = 2A. "We therefore infer that the velocity of a planet is inversely proportional to the perpendicular 4 et fall from the centre of the sun upon the tangent to the orbit. Produce OT to Q, so that OT x OQ is constant ; the radius vector OQ will therefore be pro- portional to the velocity ; and as OQ is perpendicular to the tangent at P, it follows that the locus of Q must be simply the hodograph turned round through 90. Draw two consecutive tangents to tha orbit, and let Q x and Q 2 be the corresponding points (fig. 10). Then since OTj x OQ 1 = OT !1 x OQ 2 , the quadrilateral T^O^T., is inscrib- able in a circle, and therefore the angles OQjQa and OTjT x are equal ; whence it is easily seen that QjQ., is perpendicular to OP. It follows from the principle of the hodograph that the force on P must be directed along OP.
Fig. 10.
Let F (fig. 11) be the focus of the ellipse in which, according to Kepler s first law, the planet is moving, then, from a well-known property of the ellipse, the foot of the perpendicular FT let fall from F on the tangent drawn to the ellipse at the point P lies on the circle of which the diameter is the azis major of the ellipse. The line FT cuts the circle again at Q, and as FT x FQ is constant, the radius vector FQ must be con stantly proportional to the velocity, and there fore the hodograph will be the circle TAQ turned round through 90. "We have already shown that the tangent to the hodograph is parallel to the force. The line CQ must therefore be parallel to FP. If 6 be the angle PFC, then, in consequence of the law of equal description of areas in equal times, Wde + dt must be con stant, and therefore dQ + dt must vary inversely as FP 2 ; but the velocity of Q will be proportional to dfj + dt ; hence the velocity of Q, and therefore the force, will be inversely proportional to FP S . In this way it is shown that the planets are attracted by the sun with a force which varies inversely as the square of the distance.
