PHYSICAL ASTRONOMY. 239 



d. When v goes beyond this latter limit, or when 

 tr* r -^ 2 d^f, the value of a is negative, and the 

 trajectory becomes a hyperbola. 



244. From the fact that action is always ac- 

 companied by re-action, we conclude that gra- 

 vitation among terrestrial bodies is the mutual 

 tendency of the particles of matter to one ano- 

 ther. It is therefore reasonable from analogy 

 to suppose, that this is true in all cases, and 

 that the force of gravitation toward different 

 bodies, the distances being the same, is propor- 

 tional to the quantities of matter, or the masses 

 of the bodies. If the mass of a body be call- 

 ed m, the gravitation to it at any distance x 



will be . 



x* . 



Hence, in the formulas above, m may be inserted 

 instead of d*f. 



, The quantities of matter in any two prL 

 mary planets, are directly as the cubes of the 

 mean distances at which their satellites revolve, 

 and inversely as the squares of the periodic 

 times of those satellites. 



a. This is proved by substituting m a for df* 9 in the 

 formula t = ; we have thence 



