PHYSICAL ASTRONOMY. 



Now v is the velocity of the falling body at L, as 

 well as of the revolving body at P ; therefore if 

 LN be taken equal to FP, the velocity of the fall- 



ing body atLis = c i/ But because SL is 



equaltoSP,andLNtoPF,SN=SP+PF=AB, 

 so that the point N is given. The centripetal 



force at L is therefore as ^Y~' $ 238., that is, as 



oiu 



^ , or inversely as the square of the distance. 

 Q. E. D. 



The forces, therefore, which make the 

 planets describe ellipses having the Sun in their 

 common focus, are inversely as the squares of 

 the distances from the centre of the Sun. 



a. As the same reasoning that is here applied to the 

 ellipse, might be applied^to the hyperbola or the 

 parabola, therefore the force tending to one of the 

 foci, which is requisite to make a body describe a 

 conic section, is inversely as the square of the dis- 

 tance. 



b. The converse of this is true, viz. that if the cen- 

 tripetal force be inversely as the square of the dis- 

 tance, the body will describe a conic section. 



c. The 



