ox DEFLECTIVE FORCES. 35 



their weights, and fly off at the same instant. But, for the present purpose, one 

 of the stages only is required, and the time of revolution may be measured 

 by a half second pendulum. We may make the force, or the weight to be 

 raised, equal to the weight of the revolving body, and we shall find that this 

 body will fly off when its velocity becomes equal to that which would be ac- 

 quired by any heavy body in falling through a height equal to half the dis- 

 tance from the centre, and as much greater as is sufficient for overcoming the 

 friction of the machine. (Plate I, Pig. 13.) 



. From 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 gra- 

 vity. 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 

 he eight feet in a second, since eight times the square root of 1 is eight; and 

 this must be its velocity at the highest ])oint; with this velocity it would per- 

 form each revolution in about a second and a half, but its motion in other 

 parts of its orbit will be greatly accelerated by the gravitation of the stone. 



It may also be demonstrated, that when bodies revolve in equal circles, 

 their centrifugal forces are proportional to the squares of their velocities. 

 Thus, in the whirling table, the two stages being equally loaded, one of 

 them, which is made to revolve with twice the velocity of the other, will 

 lift four equal weights at the same instant that the other raises a single one; 

 But when two bodies revolve with equal velocities at different distances, the 

 forces are inversely as the distances ; consequently the forces are, in all cases, 

 directly as the squares of the velocities, and inversely as the distances. 



If two bodies revolve in equal times at different distances, the forces by 

 which they are retained in their orbits are simply as the distances. If one of 

 the stages of the whirling table be placed at twice the distance of the other, 

 it will raise twice as great a weight, when the revolutions are performed in 

 the sajne time. 



