ELISHA MITCHELL SCIENTIFIC SOCIETY. 27 



i. e.^ the body at the end of i, 2, 3 . . , seconds is moving 

 with a rate of 32, 64, 96 . . , feet per second. The same 

 conclusion follows if we give a decrement to / in eq. (11). 

 Thus 



[s — A-y) = l6{t — i^tf 



. • . lim. — = lim. 16 (2/ — a/) ^ 32/. 

 a/ 

 The average velocity in describing the space A^y just above 

 the point considered is 16 (2/ — a/), that below, as found 

 above, is 16 (2/ — a/); the true velocity lies between them 

 and is equal to the limit 16 (2^) of either. 



The above general demonstration can be adapted to the 

 rate of increase of any function, n =/'(/'), which does not 

 change uniformly with the time, u representing a magni- 

 tude of any kind, as length, area, volume, etc. ; for if a// 

 is the actual change of the magnitude in time a/*, and 

 r, and r^ the least and greatest values of the rate of cJiange 

 of II in the interval a/, corresponding to the increments 

 r, a/, }\ c\t^ of the magnitude, if these rates were uniform 

 for the time a/, then r, a/, a// and i\ a/ are in the ascend- 



^u 

 ing order of magnitude; also ;-,, — and i\ are in the same 



a/ 

 order. Hence, as these quantities approach equality in- 

 definitely as a/ tends towards zero, the limit of any one of 

 them is equal to the actual rate of increase of the magni- 

 tude u which is thus represented by lim. r„ = lim. }\^ or, 



A?/ (ill 

 lim. — = — . 

 A/ dt 

 Thus the derivative of any function, which varies with 

 the time, with respect to /, gives the exact rate of increase 

 of the function at the instant considered. 



If It and X are both functions of /, connected by the 

 relation ?/ ^ F (,r), then 



