OF DEFLECTIVE FORCES. Ill 



reeled to its centre, the velocity is every 

 where equal to that which it would acquire in 

 falling, by the action of the same force, sup- 

 posed to be uniform, through the length of 

 half the radius : and the force is as the square 

 of the velocity directly, and as the radius 

 inversely. 



Scholium. By means of this proposition we may 

 easily calculate the velocity, with which a sling of a given 

 length must revolve, in order to retain a stone in its place 

 in all positions ; supposing the motion to be in a vertical 

 plane, it is obvious that the stone will have a tendency to 

 fall when it is at the uppermost point of the orbit, unless 

 the centrifugal force be at least equal to the force of 

 gravity. Thus if the length of the sling be two feet, we 

 must find the velocity acquired by a heavy body in falling 

 through a height of one foot, which will be eight feet in 

 a second, since 8v/l = 8; and this, at least, must be its 

 velocity at the highest point, in order that the string may 

 remain stretched throughout its revolution. With this 

 velocity it would perform each revolution in about a second 

 and a half; but its motion will be greatly accelerated 

 during its descent by the gravitation of the stone. 



260. Corollary 2. " 241.'' In equal 

 circles the forces are as the squares of the 

 times inversely. 



For the velocities are inversely as the times. 



Scholium. It may easily be shown, by the apparatus 

 called a whirling table, that when two sliding stages arc 



