MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 

 A 



529 



b 



Then CD = PD . • . ^ DCP = ^ DPC, and CP is pei-pendicular to OE, 

 . • . ^ CPE = ^ DCP + ^ DEP. Take away ^ DCP = ^ DPC, and there remains 

 DPE = DEP . ■ . PD = DE . • CE = 2 PD. 



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 roUed 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 line traced be a tangent to the circle, the rolled curve is either of the 

 parts of the polar catenary. 



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 roUs 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 roU 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 



