24 THE THEORY OF ROLLING CURVES. 



Ex. 3. When the curve A is the polar catenary 



the curve B is the tractory of the circle 



and the curve C is a circle of which the radius is - . 



Third. Examples of one curve rolling on another, and tracing a straight 

 line. 



Ex. 1. The curve whose equation is 



6 = Ar-" + &c. + Kr~* + Lr~ l + Mlog 

 when rolled on the curve whose equation is 



. . 



traces the axis of y. 



Ex. 2. The circle whose equation is r = acos# rolled on the circle whose 

 radius is a traces a diameter of the circle. 



Ex. 3. The curve whose equation is 



/2a . ... r 



6 . -- 1 versm ' - , 

 V r a 



rolled on the circle whose radius is a, traces the tangent to the circle. 



Ex. 4. If the fixed curve be a parabola whose parameter is 4a, and if we 

 roll on it the spiral of Archimedes r = a0, the pole will trace the axis of the 

 parabola. 



Ex. 5. If we roll an equal parabola on it, the focus will trace the directrix 

 of the first parabola. 



Ex. 6. If we roll on it the curve ^ = ja, the pole will trace the tangent 

 at the vertex of the parabola. 



