CONTACT. CURVATURE. EXAMPLES. 351 



Thus in both cases we have 



333. Involute of a circle. 



Let S be the centre of a circle, APQ a portion of the 

 involute, OP= OA the portion of the string unwound. Let 

 SO = a, 08A = <f>, and let x, y be the co-ordinates of P, 

 the origin being at S, and SA the direction of the axis of x. 



Then 



OP = 0$, 



x = a cos <f> + a</> sin <, 

 y = a sin < ci(f> cos <. 

 Let AP = s, then 



Hence, as we shall see in the Integral Calculus, 



5 = 



2 ' 

 EXAMPLES. 



1. In the curve 



the ordinate at any point is a mean proportional between 

 the radius of curvature there and at the lowest point. 



