THE THEORY OF ROLLING CURVES. 23 



Ex. 3. When the rolled curve is the parabola whose parameter is 4a, the 

 traced curve is a catenary whose parameter is a, and whose vertex is distant 

 a from the straight line. 



Ex. 4. When the rolled curve is a logarithmic spiral, the pole traces a 

 straight line which cuts the fixed line at the same angle as the spiral cuts 

 the radius vector. 



Ex. 5. When the rolled curve is the hyperbolic spiral, the traced curve 

 is the tractory of the straight line. 



Ex. 6. When the rolled curve is the polar catenary 



the traced curve is a circle whose radius is a, and which touches the straight 

 line. 



Ex. 7. When the equation of the rolled curve is 



a 

 the traced curve is the hyperbola whose equation is 



Second. In the examples of a straight line rolling on a curve, We shall 

 use the letters A, B, and C to denote the three curves treated of in page 22. 



Ex. 1. When the curve A is a circle whose radius is a, then the curve B 

 is the involute of that circle, and the curve is the spiral of Archimedes, r = a0. 



Ex. 2. When the curve A is a catenary whose equation is 



a , 



I aa 

 ' 3TI C 



the curve B is the tractory of the straight line, whose equation is 



a* y. 



y = a log - / + /a a X 2 , 

 ^a + N/a'-x 2 



and C is a straight line at a distance a from the vertex of the catenary. 



