NEWTON S PRINCIPIA. 43 



O R 



central force in P is as Q v ^ n p2 . Lastly, if the revo- 



o i. X \^ -t 



lution be in a circle, or in a curve having at P the same 

 curvature with a circle whose chord passes from that point 

 through S to V, then the measure of the central force 



will be o Y2 pv* -^7 finding the value of those solids 



in any given curve, we can determine the centripetal force 

 in terms of the radius vector S P ; that is, we can find 

 the proportion which the force must bear to the distance, 

 in order to retain the body in the given orbit or trajectory ; 

 and conversely, the force being given, we can determine 

 the trajectory s form. 



This proposition, then, with its corollaries, is the foun 

 dation of all the doctrine of centripetal forces, whether 

 direct or inverse ; that is, whether we regard the method of 

 finding, from the given orbit, the force and its proportion 

 to the distance, or the method of finding the orbit from 

 the given force. We must, therefore, state it more in 

 detail, and in the analytical manner, Sir Isaac Newton 

 having delivered it synthetically, geometrically, and with 

 the utmost brevity. 

 . It may be reduced to five kinds of formula?. 



1. If the central force in two similar orbits be called 

 F and f, the times T and t, the versed sines of half the 



arcs S and s, then F : f :: : &quot;2 &amp;gt; anc ^ generally F is as 



JL t 



2. But draw SP to any given point of the orbit 

 in the middle of an infinitely small arc Q C. Let T P 

 touch the curve in P, draw the perpendicular S Y from 

 the centre of forces S to P T produced, draw S Q infi 

 nitely near S P, and Q R parallel to S P, Q o and R o 



