THE THEORY OF ROLLING CURVES. 



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-i r I<t~ 

 = cos l -- ./-= 1, 



a V f 



this is the equation to the orthogonal tractory of a circle whose diameter is 

 equal to the constant tangent of the fixed curve, and its constant tangent 

 equal to half that of the fixed curve. 



This property of the tractory of the circle may be proved geometrically, 

 thus Let P be the centre of a circle whose radius is PD, and let CD be 

 a line constantly equal to the radius. Let BCP be the curve described by 

 the point (7 when the point D is moved along the circumference of the circle, 

 then if tangents equal to CD be drawn to the curve, their extremities will 

 be in the circle. Let ACH be the curve on which BCP' rolls, and let OPE 

 be the straight line traced by the pole, let CDE be the common tangent, 

 let it cut the circle in D, and the straight line in E. 



Then CD = PD, .: LDCP=LDPC, and CP is perpendicular to OE, 

 .-. LCPE= LDCP+ LDEP. Take away LDCP= L DPC, and there remains 

 DPE=DEP, / 



