23 



APPENDIX A 

 GRAPHICAL EXTRAPOLATION METHOD FOR DETERMINING THE SLOPE OF A CURVE 



Any short segment of a curve which is continuous and fair may be approximated by a 

 quadratic equation of the form 



y = ax^ + bx + c [-20] , 



To obtain the slope of the curve, the differential quotient Ay/A x is found. At a neighboring 

 point X + Ax, y + Ay the approximate equation of the curve is 



y + Ay = a{x + Aa;)^ + b{x + Ax) + c [21] ■ 



The slope of the curve between these two points is 



Ay 



— =2aa;+6 + aAar rom 



A a; 1-22J . 



Since 2oaj + 6 is constant for any selected point, the slope is a linear function of A a;. 



To determine the slope at a point x^, y^, the x and y coordinates of four or five points 

 on one side of this point are recorded at convenient small intervals. Then the following dif- 

 ferential quotients are obtained 



Ax^ x^-Xq' Ax^ x^ -Xq 



If the values of these quotients are plotted against A a; on a large scale, the faired curve 

 Uirough them should be a straight line which, when extrapolated to A a; = 0, gives the slope at 

 the point aj„, y^. If the faired curve is not a straight line, the last increments chosen are too 

 large to satisfy the approximation made in Equation [21]. In practice it is preferable to obtain 

 the differential quotients on both sides of the selected point and use the best line through all 

 the points to find the intercept. This process makes it possible to obtain an extra significant 

 figure in the slope of the curve. 



