ALONG THE CIRCUMFERENCE OF A CIRCLE. 



63 



But since 





NE : NZ : : NZ : 



NF : : Na : N/3 





NZ . V(NE 2 - Na 2 ) = NE . 



^(NZ 2 - N/3 2 ) = NE . Z/3 , 



and 



NE . ^(NF 2 - N/3 2 ) = NZ 



V(NZ 2 - Na 2 ) = NZ . Za 



wherefore 









v a = NE.Z/3; 



Vq — NZ . Za . 



Hence, under this supposition of two distinct intensities of gravitation, we 

 have 



v a : vp : : NE . Z/3 : NZ . Za , 



but we have shown in Article 6 that the minute distance aa is to the correspond- 

 ing distance /3/3' in the very same ratio, wherefore the time in which the body a 

 describes the distance aa is now equal to that in which j3 describes the distance 

 /3/3'. And consequently if a and /3 start at the same instant from N, they will 

 reach the points a and (3 simultaneously; and just when a has reached the 

 highest point Z, /3 will have reached its highest point F ; so that the periodic 

 times of the two conjugate motions have been made alike. 



11. In the figure 1 hitherto referred to, the points a and (3 have been placed 

 on opposite sides of the diameter NZ for the sake of perspicuity. We shall now, 

 in figure 2, suppose that they are both projected in the same direction from N 

 and at the same instant, so that when a has reached the point a, /3 has reached /3. 

 Proceeding onwards, when a comes to Z, (3 

 arrives at F, the velocity of a being then that 

 which is due to a fall from A to Z, and the 

 velocity of /3 being zero. Subsequently, while 

 a returns to N along the other semicircum- 

 ference, (3 returns to N by retracing its previous 

 path FN. In this way both bodies arrive at N 

 at the same instant, but moving in opposite 

 directions. While a, for the second time, de- 

 scribes the entire circumference of the circle, (3 

 ascends to L and thence returns to N at the 

 same instant that a reaches that point. The 

 two bodies are now moving in the same direc- 

 tion as at first, and these phases, all resistance 

 being set aside, are again and again reproduced. 



12. Let us now imagine that the arc a/3 is 

 continually bisected in 7, and let us trace the motion of this middle point. 



When a has reached Z and (3 has come to F, the point 7 must be at M the 

 middle of the arc FZ : when a has passed Z and /3 is descending from F towards 



VOL. XXIV. PART I. S 



