53G MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES, 



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



y = « log . „ + -J a- - x' 



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

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



the curve B is the tractory of the circle 



a N /^ 



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



'3d. Examples of one curve rolling on another, and tracing a straight line. 

 Ex. 1. The curve whose equation is 



^ = A *■-" + etc.' + K r-'- + L >-^' + M log r + N r + etc. + Z r'' 



when rolled on the curve whose equation is 



1/ = -^ A a?'-" + etc. + 2 K j:-' - L log a; + M x + h'S x^ + etc. + -^ Z 3■»^-' 



= cos 



a "^ r' 



»+ 1 



traces the axis of y. 



Ex. 2. The circle whose equation is /• = « cos Q rolled on the circle whose 

 radius is a traces a diameter of the circle. 



Ex. 3. The curve whose equation is 



7^" 



versin ' — 

 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 4 a, and if we 

 roll on it the spiral of Archimedes ;• = a 6, the pole will trace the axis of the para- 

 bola- 

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

 the first parabola. 



Ex. 6. If we roll on it the curve r = — the pole will trace the tangent at 



the vertex of the parabola. 



Ex. 7. If we roll the curve whose equation is 



= acos(^o) 



