OUTLINES OF NATURAL PHILOSOPHY. 



But FH X SG = CD Z , therefore 

 SP : PF : : SG 2 : CD*. 



7 CD* PF 

 1. Hence 



. If a body urged by a centripetal force, 

 directed to a fixed point, describe an ellipsis of 

 which that point is a focus, the centripetal 

 force must be inversely as the square of the dis- 

 tance. 



Suppose a body falling in a straight line to the focus 

 S, (fig. 24.) to have at A the same velocity that 

 the body revolving in the ellipsis has at A. Let 

 P be any other position of the revolving body, and 

 with the radius SP let an arch of a circle be de- 

 scribed, meeting AB in L ; the velocity of the fall- 

 ing body at L, and of the revolving body at P, will 

 be equal. If, then, the velocity of the revolving 

 body at the mean distance, or at the point D, be 

 called c, and its velocity at P be v, v : c : : CD : SG, 



( 236, .), or * = , and * = 



PF PTT 



X ^p, (j 239, a.) Therefore also =c JEL . 



Now 



