THE THEORY OF ROLLING CURVES. 27 



rolled on the witch, whose equation is 



traces the asymptote. 



Ex. 19. The curve whose equation is 



r = a tan 0, 

 rolled on the curve whose equation is 



traces the axis of y. 



Ex. 20. The curve whose equation is 



rolled on the curve whose equation is 



y? 



, or r = 



of 

 traces the axis of y. 



Ex. 21. The curve whose equation is 



r = a (sec 9 tan 6), 

 rolled on the curve whose equation is 



traces the axis of y. 



Fourth. Examples of pairs of rolling curves which have their poles at a fixed 

 distance = a. 



ie straight line whose equation is ^=sec~ 1 -. 

 Ex. 1. ' a 



vThe polar catenary whose equation is 6 = . / 1 



to 



r 



Ex. 2. Two equal ellipses or hyperbolas centered at the foci. 

 Ex. 3. Two equal logarithmic spirals. 



(Circle whose equation is r = 2a cos 6. 



IT r 



Curve whose equation is 6 = ^2 1+ versin J - . 



42 



