166 Dynamics. 



communicated at each unit of time, it is evident that the success- 

 ive velocities of the body will be i<, 2w, 3w, &c., so that after a 

 number of units of time, denoted by J, the velocity acquired will 

 be , taken as many times as there are units in t ; that is, it will 

 be t X u or t u. 



264. Hence, (1.) In the case of motion uniformly accelerated, 

 the number of degrees of velocity which the body acquires, in- 

 creases as the number of intervals or periods during which the 

 motion continues, which may be expressed by saying that for 

 different times the velocities acquired are as the times elapsed from 

 the commencement of the motion. 



Thus, if we call v the velocity that the body acquires at the 

 end of the time f, we shall have 



v = u t. 



(2.) The velocities which the body will be found to have 

 during the lapse of each successive interval, will form an arith- 

 metical progression or progression by differences, 



-f- u . 2 u . 3 u . &c. 



the last term of which is u t or T, and of which the number of 

 terms is f, that is, is denoted by the number of actions of the ac- 

 celerating force. 



(3.) Also, since the velocities w, 2 w, &c., are simply the spa- 



<>.> ces that the body would describe in the corresponding intervals 



of time respectively ; the whole space described during the time 



t will be the sum of the terms u + 2 u -f &c., of this progress- 



Alg.229. ion, that is, it will be expressed by (u -f- v) X | t. Therefore, 



if we call s this whole space passed over from the commence- 



of the motion, we shall have 



s = (u + v) X \ t. 



265. Let us suppose now, that the accelerating force acts 

 without interruption, or, which amounts to the same thing, that 

 the time is divided into an infinite number of infinitely small 

 parts, which we shall call instants ; and that at the beginning of 

 each instant, the accelerating force exerts a new impulse upon 

 the body. Let us suppose, also, that it acts by degrees infinitely 



