350 



EVOLUTE. POLAR FORMULA. 



Let be the centre of curvature corresponding to the 

 point P of a curve referred to polar co-ordinates. Let 

 the perpendicular on the tangent at P. 



Let SP = r, P0 = p, SY=p, 



S0 = r, p = perpendicular from S on PO. 

 From the triangle SOP we have 



Also 



p'* = r*- 



dr 



P ~ r ~dp 



(1). 

 ,(2). 



,(3). 



From the given equation to the curve we can find p in terms 

 of r, and then between (1) and (2) we can eliminate r, and 

 thus we have an equation between p' and / to determine the 

 locus of 0. Since PO is a tangent to the locus of 0, p' is 

 the perpendicular from the origin on the tangent to the 

 evolute at 0. 



In the figure the curve is drawn concave to the pole. 



f/* 



If the curve be convex to the pole -r- is negative (Ail. 29d), 



P 



dt* 



and we should take p = r -r- ; in this case we shall find in- 



dp 



stead of (1) the equation 



