26 THE THEORY OF ROLLING CURVES. 



Ex. 12. The straight line whose equation is 



r = a sec 0, 



rolled on a catenary whose parameter is a, traces a line whose distance from 

 the vertex is a. 



Ex. 13. The part of the polar catenary whose equation is 



rolled on the catenary, traces the tangent at the vertex. 



Ex. 14. The other part of the polar catenary whose equation is 



rolled on the catenary, traces a line whose distance from the vertex is equal to 2a. 



Ex. 15. The tractory of the circle whose diameter is a, rolled on the 

 tractory of the straight line whose constant tangent is a, produces the straight 

 line. 



Ex. 16. The hyperbolic spiral whose equation is 



a 

 T= d> 



rolled on the logarithmic curve whose equation is 



x 



i 



a log - , 

 



a 



traces the axis of y or the asymptote. 



Ex. 17. The involute of the circle whose radius is a, rolled on an orthogonal 

 trajectory of the catenary whose equation is 



x 



traces the axis of y. 



Ex. 18. The curve whose equation is 



