16 THE THEORY OF ROLLING CURVIX 



Therefore the curve ACH has a constant tangent equal to the diameter of 

 the circle, therefore ACH is the orthogonal tractory of the straight line, which 

 is the tractrix or equitangential curve. 



The operation of finding the fixed curve from the rolled curve is what 

 Sir John Leslie calls " divesting a curve of its radiated structure." 



The method of finding the curve which must be rolled on a circle to 

 trace a given curve is mentioned here because it generally leads to a double 

 result, for the normal to the traced curve cuts the circle in two points, either 

 of which may be a point in the rolled curve. 



Thus, if the traced curve be the involute of a circle concentric with the 

 given circle, the rolled curve is one of two similar logarithmic spirals. 



If the curve traced be the spiral of Archimedes, the rolled curve may be 

 either the hyperbolic spiral or the straight line. 



In the next case, one curve rolls on another and traces a circle. 



Since the curve traced is a circle, the distance between the poles of the 

 fixed curve and the rolled curve is always the same ; therefore, if we fix the 

 rolled curve and roll the fixed curve, the curve traced will still be a circle, 

 and, if we fix the poles of both the curves, we may roll them on each other 

 without friction. 



Let a be the radius of the traced circle, then the sum or difference of 

 the radii of the other curves is equal to a, and the angles which they make 

 with the radius at the point of contact are equal, 



dd 

 If we know the equation between 0, and r,, we may find -, ' in terms of 



substitute (a r t ) for r,, multiply by '^ t and integrate. 



*** 



Thus, if the equation between 0, and r, be 



r, = a sec 0,, 



