LAW OF INVERSE SQUARE 279 



225. For an elliptic orbit, the periodic time is that required to 

 sweep out an area Trab, where a, b are the semi-axes of the ellipse. 

 Since the area is swept out at a rate J- h per unit time, the periodic 

 time T will be 



b* 

 The semi latus-rectum / is equal to > and has also been seen 



h 2 a 



to be equal to > so that 



2 Trab 2 TTO /IIA\ 



whence T = - = -=- - (114) 



h 



Since this does not depend on the eccentricity, it is clear that 

 the periodic time of any orbit is the same as that in a circle of 

 radius equal to the semi major-axis. 



226. The law of force of the inverse square is that of gravitation : 

 the law which we have been investigating is therefore that which 

 governs the motions of the planets in their orbits round the sun, 

 as well as the motions of comets and meteorites. For reasons 

 which cannot be explained here, the conies described by the planets 

 are all of them ellipses of small eccentricity. A wider range is 

 found in the orbits of comets. These bodies generally come from 

 far outside the solar system. To a close approximation many of 

 them may be treated as coming from infinity, and as starting x with 

 relatively small velocity. In this case the orbit is approximately 

 parabolic. 



Kepler's Laws 



227. Long before the theory of the planetary orbits had been 

 worked out mathematically by Newton, three of the principal 

 laws governing the motion of the planets had been discovered 

 empirically by Kepler. Kepler's three laws are as follows : 



LAW I. Every planet describes an ellipse having the sun in one 

 of its foci. 



