138 PRACTICAL MATHEMATICS 



Then dx* a(l - cos a) 2 



8 . ,3. 



2 



o -\ ^ 



sm^a } 

 cos a) 2 -* 



/ 

 = I 



(1 



1 2 cos a + cos 2 a + sin 2 a"|^ 

 (1 cos a) 



[2(1 - cos g)\l 

 1(1 cos a) 2 



2\/2 



Then R = 



(1 cos a) 2 



*V} f 

 / 



da; 2 

 2V2 g(l - cos a) 2 



g 



(1 cos a)^ 



= 2a-\/2(l cos a) 



When a = 180, that is, at the highest point of the curve, 

 R = 4a, or twice the diameter of the rolling circle. 



Example. Find the radius of curvature and the co-ordinates 

 of the centre of curvature of the curve y z = 8x at the points where 

 x = 0, x = 2, and x = 8. 



y = 2 V% ar 



dy ,- -1 /2 

 = V 2 x = A/ - 

 oc > a; 



d 2 ?/ V2 3. 



dx* 



