Steede — On Curvature of Pedal and Reciprocal Curves. 125 



3. Or, wliicli is the same thing, being given a curve and its 

 evolute, we have the following construction to 

 describe the evolute of the reciprocal curve : — 



PC being a normal to the curve touching the 

 evolute in C. 



From G draw CM perpendicular to the radius 

 vector OP. 



From M draw MJSf perpendicular to the 

 normal PC. 



Join NO, and produce this line so that 



NO : OC = PJY 



P 



(k being a constant, and P being perpendicular from origin on tangent 

 atP). 



Then the locus of C is the evolute of the reciprocal curve, the 

 fixed point being taken as the origin, and the constant k as the 

 radius, of reciprocation. 



4. Being given the centre of curvature C for any point P of a 

 curve, the centre of curvature for the corresponding point of the first 

 positive pedal is found by the following construction : — 



As before, draw CM perpendicular to OP, 

 and MN perpendicular to PC. 



Join T, the corresponding point on pedal 

 curve, with G, the middle point of OP. 



TG will intersect NO in C", the centre of 

 curvature of the pedal curve, for TG is evidently 

 the normal to the pedal curve ; and, since circles 

 of curvature at corresponding points of inverse 

 curves are inverse circles, the centre of curva- 

 ture of the pedal curve must lie in the line 

 NO, which we have seen passes through the centre of curvature of 

 the reciprocal curve. 



By reversing the construction, the centre of curvature of the first 

 negative pedal is determined, and hence — 



5, Being given the centre of curvature for any point on a curve, 

 the centre of curvature for the corresponding point on any positive or 

 negative pedal of either the curve or its inverse can be found by a 

 geometrical construction. 



