526 MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 



(d /-.A ^ _ r^^ r* 



\d 6,1 ' 



d 6 

 dr 



/ *■ _ 1 *^^° ^y integration 



n't ~ 



(2 a 



. _ 2a 



I + cos 6 



This is the polar equation of the parabola, the focus being the pole, therefore, 

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



The rectangular equation of this parabola is x- = 4 a y, and we shall now 

 consider what ciu-ve must be rolled along the axis of y to trace the parabola. 



By the second equation (2.), 



^2 = ^3 J ^' + 1 l"!* •^a = P-2 



■ ■ ■ r.-, = sf 4 a^ + p^^ 

 r.^ = jfj^- = 4 c? 



but q, is the perpendicular on the normal, therefore the normal to the curve al- 

 ways touches a circle whose radius is Q a, therefore the ciu-ve is the involute of 

 this circle. 



Therefore we have the following method of describing a catenary by continued 

 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 

 of the circle ; roll this curve on a straight line, and the centre of the circle will 

 describe a parabola ; roll this parabola on a straight line, and its focus will trace 

 the catenary required. 



We come now to the case in which a straight Mne rolls on a curve. 



When the tracing-point is in the straight line, the problem becomes that of in- 

 volutes and evolutes, which we need not enter upon, and when the tracing-point is 



