80 OF SIMPLE ACCELERATING FORCES. 



when thus modified, is properly called a uniform accelerat- 

 ing 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 with- 

 drawn 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. 



231. Theorem. The increment of space 

 described is as the increment of the time, and 

 as the velocity, conjointly. 



This is evident from the definition of velocity (45) ; and 

 calling the space described x, and its increment x, we have 

 x:=.vt'y or AJczivAf ; if we make the unities of time and 

 space equivalent. This proposition is true of all incre- 

 ments, when the motion is uniform, but when variable, of 

 evanescent increments only, 



232. Theoeem. The space described, 

 by means of a uniformly accelerating force, is 

 as the square of the time of its action ; it is 

 also equal to half the space which would be 

 described in the same time with the final velo- 

 city ; and if the forces vary, the spaces are as 

 the forces, and the squares of the times, con- 

 jointly : or a:=iaf. 



Since the velocity v is expressed by at, the product of 

 the force and the time (280), and since xzzvt' (231), or 



