OF ACCELERATING FORCES. 81 



substituting fluxions for increments, x^ vi, or(231) dx=:vdt 

 and vdtz=:atdt, and the fluent x is equal to ^at^ (49) or ^vt. 

 Consequently x varies as ^2^ and v being the velocity ac- 

 quired at the end of the time t, the space described by 

 it in that time would be vt, instead of ^vt, the space ac- 

 tually described with the accelerated motion. 



Scholium. 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, is oile 

 of the most useful and interesting propositions in the 

 whole science of mechanics. Its truth is easily shown in 

 a popular manner, by comparing the time with the base, 

 and the velocity with the perpendicular of a right angled 

 triangle gradually increasing in length and height, the area 

 of which will represent the space described. We may also 

 observe, by means of Atwood's machine, that a quadruple 

 space is always described in a double time, by the con- 

 tinued operation of any constant accelerating force. 



233, A. Theorem. The times are as the 



square roots of the spaces directly, and of the 



forces inversely ; they are also as the spaces 



directly*, and the final velocities inversely.^ 



2x 2x 



Sincexzz^at^, tzz V — ; but v=at, x—^vt, and t— — . 

 a V 



233, B. Theorem. The final velocities 

 are also as the spaces directly, and the times 

 inversely. 



That is, v=:at-— (233. A). 



