THE CENTRE OF CURVATURE 



137 



Then 

 and 



x - x l - CP 



= x l - R sin 



- y t + R cos 



; d y 



where is the angle whose tangent is -- when x - a? 



\i 



FIG. 42. 



Example. Find the radius of curvature of the cycloid 

 x = a(a sin a), y = a(l cos a) 



Then 



and 

 Then 

 To find 



dx 



-r- = a(l cos a) 

 dry. 



dy 



= a sin a 



da 



dw dw da sin a 



y z. x = = z 



da? dy. dx 1 cos a 



d 2 */ dz _ dz dy. 

 dx z dx dy. dx 



dz 



-j- log.,2 = log^ sin a log,. (1 cos a) 

 aa 



cos a 



sin a 



and 



; da sin a 1 - cos a 



_ cos a cos 2 a sin 2 a 

 sin a(l cos a) 

 cos a 1 1 



sin a(l cos a) sin a 



dz 1 sin a 



da sin a 1 cos a 



1 cos a 



