CURVATURE 



135 



If S.v be taken as a very small arc, it can be approximately 

 taken as the small arc of a circle, the radius of which is R and the 

 angle at the centre is 86, since the radii must be at right angles 

 to the tangents at P and Q respectively. 



FIG. 40. 



FIG. 41, 



Then 



or 



80 = 1_ 

 8s R 



The smaller the length of arc taken, the more nearly correct 

 does this approximation become, and, making 8s infinitely small, 



The circle whose radius is R, the value of which is given by 



ds 



-JE, is spoken of as the Circle of Curvature, and R is the Radius 



at) 



of Curvature. 



JA 



In order to find -=-, we want a relation giving us in terms of s, 

 ds 



but if we are given the law of the curve as y = f(x) (or y in terms 

 of x), it is a very difficult matter to obtain such a relation, and 

 therefore it is better to find the expression which is equivalent to 



d8. . 



-r- m terms of x and y. 



as 



If 8* is a very small arc. it may be taken as the hypotenuse 

 of a right-angled triangle whose base angle is 6, and whose per- 

 pendicular and base are 8y and 8# respectively. 



