526 MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 
dr, PEM Se er aie 
d 6, a. on)" 2 
2 

7 
d é, 
dr, 5 A ee 
Maa ae. = 
dr, 2 34 
= 4. —a 
(aH 2 (r—a) 
do 1 
r ik then by integration 
TN a 
@ = cos* (=* _1) 
F: 
ee i sal 
~ 1 + 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 «* = 4 a y, and we shall now 
consider what curve must be rolled along the axis of y to trace the parabola. 
By the second equation (2.), 
eee 
= wile aad but 27, = Pp, 
3 
r= NN 4a +p? 
De ao ae 2 
Yr, =p, =4a 
2a= Nt." — py? = % 
but g, is the perpendicular on the normal, therefore the normal to the curve al- 
ways touches a circle whose radius is Q a, therefore the curve 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 line 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 

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