134 



PRACTICAL MATHEMATICS 



At the point A. Inclination of the tangent = tan- 1 ( 4) 



= 104 2' 



Required angle = 104 2' - 63 26' = 40 36' 

 At the point B. Inclination of the tangent = tan -1 ( 1) 



= 135 

 Required angle = 135 - 63 26' = 71 34' 



82. The Change in Direction along a Curve between two Given 

 Points. This is the angle between the tangents to the curve at 

 the given points. Let P and Q (Fig. 39) be the two points whose 

 co-ordinates are x v y l and x z , y z respectively. 



FIG. 39. 



Let 0j be the inclination of the tangent at P and 2 the inclina- 

 tion of the tangent at Q. 



But tan X = -j- when x = x lt thus giving the value of X 

 dx 



and tan 2 = -p then x = x 2 , thus giving the value of 2 

 The change in direction = 2 0j 



83. Curvature. Curvature is the rate at which the direction 

 changes with the arc. Let PQ (Fig. 40) represent the small arc 

 of a curve, and let and + 80 be the inclinations of the tangents 

 to the curve at P and Q respectively. Then the change in direc- 

 tion is 80, and this occurs over a length of arc 8s. 



The average curvature of the arc PQ = TT-, and, making 8* in- 

 finitely small, this becomes the actual curvature at the point P. 



Actual curvature at P = -3- 



ds ' 



