ELISHA MITCHELL SCIENTIFIC SOCIETY. 21 



values of y) and since dy -=- dc represents the slope of the 

 locus at the point {:i\ j), we can represent dy and dx by the 

 length of certain lines. Thus in fig. 2, from the point of 

 tangency P, draw PR parallel to the axis of x a7iy distance 

 from P to R to represent dx and from the point R draw RT 

 parallel to the axis of y to intersection T with the tangent, 

 when RT will represent dy ; for then 



dv Ar 



— = tan TPR = lim. — , 

 dx AX 



as should be the case. 



If we choose to make dx = a r = PQ, then dy = QI, which 

 is less than aj' when S is above I, for a convex curve to X, 

 and greater than av when j is below, or for a curve concave 

 to X, just to the right of P. It is only in the case where 

 y =/(x) is the equation of a straight line that for dx = ax 

 we have dy = Ay, for here Ay -^ ax represents the slope of 

 the line. 



Other important formulas can be deduced from fig. 2. 



If we call the length of the curve from some point D to 

 P, J, then the increment of the arc PS corresponding to 

 the simultaneous increment ax of x will be called as. Call 

 the length of the chord PS = <r. 



Then we have, 



PQ AX 



c 



AS 



AS 



QS Ay 



c 



AS 



AS 



COS SPQ, 



sin SPQ. 



c 

 Now we have seen before that lim. — = i, so that the 



AS 

 3 



