280 MOTION UNDER A VARIABLE FORCE 



LAW II. The areas described by the radii drawn from the planet 

 to the sun are, in the same orbit, proportional to the times of 

 describing them. 



LAW III. The squares of the periodic times of the various orbits 

 are proportional to the cubes of their major axes. 



From the first of these laws Newton proved that the law of 

 force between the planets and the sun must be the law of the 

 inverse square. The third law is seen to express the same fact as 

 equation (114). 



Motion of Two Particles about One Another 



228. A pair of objects known as a double star is of common 

 occurrence in the sky. This consists of two stars describing orbits 

 about one another, neither star being fixed. 



By the theorems proved in Chapter IX, the center of gravity of 

 the two stars must either remain at rest, or else must move with 

 uniform velocity in a straight line, in which case it may, as we 

 have seen, be treated as fixed, provided all motion is measured 

 relative to a frame of reference moving with it. 



Let A, B be the positions of the two stars at any instant, and let 

 G be their center of gravity. Let the masses of the stars be m, m', 

 and let a, b denote their distances from G. Then 



m m f m + m' 





The complete law of gravitation is expressed by the law 



where m, m f are the two masses, r the distance between them, 7 a 

 constant whose value can be found by experiment, and F is the 

 force of attraction between the two masses. Thus the force acting 



on the star B is f 



//Lin 



F ry - , 



7 



