PHYSICAL ASTRONOMY. 



tained in the whole elliptic area, of so many seconds 

 does the time of the revolution consist. Now the 

 area of the ellipsis of which a is the semi- transverse, 

 and b the semi-conjugate, is a b ; therefore the 



time of revolution =r * a b X ~ _ = ^ 

 b df* </* " 



If, therefore, t and ^ be the times of the revolu- 

 tion of two planets, 



The third and last of the laws of KEPLER, is thus 

 derived from the action of a centripetal force, o- 

 beying the law already shewn to be conformable 

 to the other two, that is, being inversely as the 

 squares of the distances. 



Suppose a body to gravitate to a centre, 

 as in the preceding proposition, and to be pro- 

 jected from a given point, with a given veloci- 

 ty, in a direction perpendicular to the line drawn 

 to the centre j required to find the Conic Sec- 

 tion which it will describe. 



a. Let tfce semi-transverse or the mean distance = 

 a ; the semi-conjugate = b ; the velocity at the 

 mean distance = c ; and at the apsis A, (fig. 24.) 

 from which the body is projected, = v ; let the 

 line SA = r. 



From 



