ON DEFLECTIVE FORCES. 27 



linear motion, so as to diminish it very considerably, until the velocity is 

 so much reduced as to suffer it to describe a path evidently curved, and 

 becoming more and more so as the motion is slower. 



When a body is retained in a circular orbit by a force directed to its 

 centre, its velocity is every where equal to that which it would acquire in 

 falling, by means of the same force, if uniform, through half the radius, 

 that is, through one fourth of the diameter.* This proposition affords a 

 very convenient method of comparing the effects of central forces with 

 those of simple accelerating forces, and deserves to be retained in memory. 

 We may in some measure demonstrate its truth by means of the whirling 

 table : an apparatus which is arranged on purpose for exhibiting the pro- 

 perties of central forces, although it is more calculated for showing their 

 comparative than their absolute magnitude ; for accordingly as we place 

 the string on the pullies, the two horizontal arms may be made to revolve 

 either with equal velocities, or one twice as fast as the other. The sliding 

 stages, which may be placed at different distances from the centres, and 

 which are made to move along the arms with as little friction as possible, 

 are in a certain proportion to the weights, which are to be raised by means 

 of threads passing over pullies in the centres, as soon as the centrifugal 

 forces of the stages with their weights are sufficiently great; and the 

 experiment is to be so arranged, that when the velocity having been gradu- 

 ally increased, produces a sufficient centrifugal force, both stages may raise 

 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 mea- 

 sured 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 acquired by any heavy body in falling through a height equal to 

 half the distance from the centre, and as much greater as is sufficient for 

 overcoming the friction of the machine. (Plate I. Fig. 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 

 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 eight times the square root of 1 

 is eight ; and this must be its velocity at the highest point ; with this 

 velocity it would perform 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.t 



, 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. 



* Newton, Principia, I. Prop. 4, Cor. 9. f Ibid. I. Prop. 4, Cor. 1. 



