ORBITS RESULTING FROM ASSUMED LAWS OF MOTIOX. 207 



The possibility of determining the total areas is discussed above in 

 section VIII. 



When ic = 1, ?i + 3 = 1, and the ratio of the areas in a closed orbit 

 is that of half the products of the major and rninor semi-axes. The 

 value of P in this case is B^/A, the proof of which need not here be 

 repeated. The conclusion follows that the times of revolution are 

 proportional to the square roots of the cubes of the major serai-axes. 



When w = 2, n + 3 = 4, and \/(i^"^) = P^'- The area described 

 by the radius vector from periastron to apastron is one quarter of 

 the area of the ellipse, and apastron always occurs. Hence the ratio 

 of the two areas Ui and a2 is that of Ai Bi and A2 B2, in which Ai and 

 A2, Bi and Bo, respectively denote the major and minor semi-axes. 

 We have found the value of P^ in this case to be AB. Hence 

 aiV(P2'^)/a2ViPi'^) becomes AiBiA2Bo/A-2B2AiB^ = 1; that is, 

 when tv = 2, the same time is required for all orbits of that system. 

 This result is ob\dous in the case of circular orbits, as Newton states 

 in the first book of the Principia. 



