LAW OF THE UNIVERSE. 



229 



tangents to the curve at B and D respectively ; B C and D E 

 the arcs described in a given time ; C c and E e lines parallel 



Kg. 2 . 



V B 



to the radii vectores S B and S D respectively ; and C V, E d 

 parallel to the tangents. The centripetal forces at B and D 

 must be in the proportion of V B and d D (being the other 

 sides of the parallelograms of forces) if the arcs are evanescent, 

 so as to coincide with the diagonals of the parallelograms V c 

 and d e. Hence the centripetal forces in-B and D are as the 

 versed sines of the evanescent arcs ; and the same holds true 

 if instead of two arcs in the same curve, we take two arcs in 

 different but similar curves.* 



From these propositions another follows plainly, and its 

 consequences are most extensive and important. If two or 

 more bodies move in circular orbits (or trajectories) with an 

 equable motion, they are retained in those paths by forces 

 tending towards the centres of the circles ; and those forces 

 are in the direct proportion of the squares of the arcs described 

 in a given time, and in the inverse proportion of the radii of 

 the circles. 



First of all it is plain, by the fundamental proposition, that 

 the forces tend to the centres S, s, because the sectors A S B 

 and PBS being as the arcs A B, B P, and the sectors a s b, 



* If B C, D E, are bisected, the proportion is found with the halves of 

 V B, D d ; and that is the same proportion with the whole versed sines. 



