THE SLOPE OF A CURVE 81 



its effect is to make the perpendicular of the right-angled triangle 

 slightly less, and at the same time to make the base slightly 



perpendicular 



more. This makes the error in the fraction - *_ quite 



base 



pronounced. 



51. If we are given the law of a curve, then we are in a position 

 to find the slope of the curve without using the graphical method, 

 and the errors introduced by that method will not affect the result. 



Let the co-ordinates of the point P (Fig. 26) be x, y, and the 

 co-ordinates of the point Q (x+ Bx), (y + By), where Bx and By 

 are the increases in the values of x and y respectively. 



Bx and By are also the base and perpendicular of the right- 

 angled triangle PQR, whose hypotenuse is the chord PQ. 



Then the slope of the chord PQ = tan (0 + a) = ^ 



When the chord PQ becomes infinitely small, the slope of the 



r\ 



curve is the limiting value of the fraction -J^ when &c is made 



dii 

 infinitely small, and this limiting value is represented by -p 



du 



Then the slope of the curve = -p = tan 0. 



d'X 



As an example on the application of this method, let the law 

 of a curve be y = a + bx + cx z where a, b, and c are constants. 



Then at the point P, y = a + bx + cx z 



at the point Q, y +By = a + b(x + &e) + c(x + Bx) z 



= a +bx +b Bx +cx z +2cx Bx +c(Bx)* 

 Subtracting By = b Bx + 2cx Bx+ c (Bx) 2 



Slope of the chord PQ, ~ = b + 2cx + c Bx 



making 8# infinitely small. 



du 

 Slope of the curve at the point P, - = b + 2cx 



ml 



Referring again to Fig. 26, since -^- = tan (0 + a), it necessarily 



Bx 



Mows that 1 = cot (0 + a). 

 By 



In the limit when Bx becomes infinitely small, ~ becomes 



uX 

 fit* 



tan and -=- becomes cot 0. Because tan and cot are mutu- 

 dy 



dii di 



ally reciprocal, -p and -3- are mutually reciprocal. 



