no or DEFLECTIVE FORCES. 



curvature, which is in the direction of the 

 force; and the velocity in the curve is equal 

 to that which would be generated by the same 

 force, during the description of one fourth of 

 the chord by its uniform action. 



For the force is as the space described by 



its action, beginning from a state of rest, or 



as the evanescent sagitta through which the 



body is drawn from the tangent of the curve 



in a given instant of time : but the portion 



AB of the tangent spontaneously described 



in a given instant is as the velocity, and BC the sagitta iszz 



ABq , . , ABq , . , „ ,, 

 , or ultimately >, that is, as the square oi the 



velocity directly, and inversely as the chord of curvature 

 of the arc AC. 



Now the velocity generated during the description of 

 BC is expressed by 2BC, since the force may be consi- 

 dered for an instant as constant, and the final velocity is 

 measured by twice the space actually described (232) : the 

 velocity generated is therefore to the orbital velocity as 

 2BC to AB, or as2AB to BD, or as AB to half BD : and 

 if the time were increased in the ratio of AB to half BD, 

 the velocity generated by the force would be equal to the 

 orbital velocity, but in this time half BD would be de- 

 scribed by the velocity in the orbit, and half as much, 

 or one fourth of BD, by a uniformly accelerated velocity 

 (232). 



259. Corollary 1. "240/' AVhenabody 

 describes a circle by means of a force di- 



