26 JOURNAL OF THE 



^ is a variable for different latitudes and is 

 slightly over 32. Take it 32 for brevity. 

 s = 16 t^ . . . . {11). 

 In the time / -j- a^ the space described would 

 be jr -|- ^s (see fig. 3), and by the same law, 



V. 



/is 



'^^ (^ + A^) = 16 (/ + ^i)\ 



Subtracting the preceding equation and divid- 

 ^'r ^ ing by A/, 



A.9 



— = 16 (2/ + A/*). 

 A^ 



This o-ives the average rate or velocity with which the 

 small space a^ is described. 



As the rate or velocity is changing all the time, call 

 v^ and z'2 the least and greatest values of the velocity in de- 

 scribing the space a^-; then the spaces which would have 

 been described with uniform velocities z^,, v^ in time a/ are 

 z/, a/ and v^ a/, which are respectively less and greater than 

 the actual space /^s. 



Hence z'„ — and c>^ are in ascending order of magnitude. 



A/ 



As a/ (and a^ consequently) is diminished indefinitely, these 

 three quantities approach equality and the exact velocity 

 the body has at the beginning of the space as is given by 

 the constant to which they approach indefinitely but never 



AS 



attain. But lim. z\ = lini. v^ = limit — = velocity or 



At 



rate at the instant the space .? has been described (see Ed- 

 wards' Differential Calculus). 



Hence in the particular example above, the velocity the 

 falling body has, at the end of / seconds, when it has de- 

 scribed the space s^ is, 



ds AS 



— = lim. — ^ 32/ (12); 



dt At 



