24 LECTURE III. 



with a very convenient mode of making experiments on accelerating forces. 

 The velocity produced by the undiminished force of gravity, is much too 

 great to be conveniently submitted to experimental examination ; but by 

 means of this apparatus we can diminish it in any degree -that is 

 required. Two boxes, which are attached to a thread passing over a 

 pulley, may be filled with different weights, which counterbalance each 

 other and constitute, together with the pulley, an inert mass, which is put 

 into motion by a small weight added to one of them. The time of descent 

 is measured by a second or half second pendulum, the space described 

 being ascertained by the place of a moveable stage, against which the bot- 

 tom of the descending box strikes ; and when we wish to determine im- 

 mediately the velocity acquired at any point, by measuring the space 

 uniformly described in a given time, the accelerating force is removed by 

 means of a ring which intercepts the preponderating weight, and the box 

 proceeds with a uniform velocity, except so far as the friction of the 

 machine retards it. By changing the proportion of the preponderating 

 weight to the whole weight of the boxes, it is obvious that we may change 

 the velocity of the descent, and thus exhibit the effects of forces of different 

 magnitudes. The most convenient mode of letting the weights go, with- 

 out danger of disturbance from their vibrations, is to hold the lowest 

 weight only, and to allow it to ascend at the instant of a beat of the pen- 

 dulum. 



That the velocity generated is proportional to the time of the action of 

 the force, or that the force of gravitation, thus modified, is properly called 

 a uniform accelerating force, may be shown by placing the moveable ring 

 so as to intercept the same bar successively at two different points : thus the 

 space uniformly described in a second, by the box alone, is twice as great, 

 when the force is withdrawn after a descent of ten half seconds, as it is after 

 a descent of five. And if we chose to vary the weight of the bar, we might 

 show, in a similar manner, that the velocity generated in a given time is 

 proportional to the force employed. 



We are next to determine the magnitude of the whole space described in 

 a given time with a velocity thus uniformly increasing. The law discovered 

 by Galileo,* that the space described is as the square of the time of descent, 

 and that it is also equal to half the space which would be described in the 

 same time with the final velocity,f is one of the most useful and interesting 

 propositions in the whole science of mechanics. Its truth is easily shown 

 from mathematical considerations, by comparing the time with the base, 

 and the velocity with the perpendicular of a triangle gradually increasing, 

 of which the area will represent the space described ; and we may observe, 

 by means of Atwood's machine, that a quadruple space is always described 

 in a double time, whatever may be the magnitude of the force. Of course, 

 if the forces vary, the spaces are as the forces and as the squares of the 

 times conjointly. (Plate I. Fig. 12.) 



It may also be demonstrated, that if a body falls through one foot in a 

 second by means of a certain force, it will require a quadruple force to 

 make it fall through the same space in half a second ; and in general, where 

 * Dial. III. Prop. 2. f Ibid. Prop. 1. 



