12 THE THEORY OF ROLLING CURVES. 



Then 



' ' 1 7//J / = ^" 1,1*. \ r * ' 



then by integration =003"' I 1 J , 



2a 



7* ~~ 



l + cos0' 



This is the polar equation of the parabola, the focus being the pole ; there- 

 fore, if we roll a parabola on a straight line, its focus will trace a catenary. 



The rectangular equation of this parabola is a? = <iay, and we shall now 

 consider what curve must be rolled along the axis of y to trace the parabola. 



By the second equation (2), 



T + l, but Xt=p 



but q t is the perpendicular on the normal, therefore the normal to the curve 

 always touches a circle whose radius is 2a, therefore the curve is the involute 

 of this circle. 



Therefore we have the following method of describing a catenary by con- 

 tinued motion. 



Describe a circle whose radius is twice the parameter of the catenary; roll a 

 straight line on this circle, then any point in the line will describe an involute 



