18 THE THEORY OF ROLLING CURVES. 



tangents is the same as in the catenary, namely, by describing a circle whose 

 radius is a on the production of the radius vector, and drawing a tangent to the 

 circle from the given point. 



In the next case the rolled curve is the same as the fixed curve. It is 

 evident that the traced curve will be similar to the locus of the intersection 

 of the tangent with the perpendicular from the pole ; the magnitude, however, 

 of the traced curve will be double that of the other curve ; therefore, if we 

 call r = < && the equation to the fixed curve, r 1 = <f> l l that of the traced curve, 

 we have 



?o ^ ' 



also, = '. 



Similarly, r t = 2 Pl = Zr, = 4 4r , 0, = 0, - 2 cos' 1 



r r t \ r o/ 



Similarly, r.= 2p._, - 2r n ., & &c. = 2V (|?V , 



r * \ r oi 



and ?= 



Let e n become 6J ; 0,, 0. 1 and &, . Let n l -0 n = a, 



a = $J - e n = t >-0 a - n cos' 1 ^i + cos' 1 



