THE THEORY OF ROLLING CURVES. 9 



Thus we have found three independent equations, which, together with the 

 equations of the curves, make up six equations, of which each may be deduced 

 from the others. There is an equation connecting the radii of curvature of the 

 three curves which is sometimes of use. 



The angle through which the rolled curve revolves during the description of 

 the element ds 3 , is equal to the angle of contact of the fixed curve and the 

 rolling curve, or to the sum of their curvatures, 



ds 3 ds l ds, 



7~ = 7z; + 7^- 



But the radius of the rolled curve has revolved in the opposite direction 

 through an angle equal to d$ lt therefore the angle between two successive posi- 



ds 

 tions of r t is equal to ^ cZ# 2 . Now this angle is the angle between two 



**2 



successive positions of the normal to the traced curve, therefore, if O be the 

 centre of curvature of the traced curve, it is the angle which ds 3 or ds l subtends 

 at 0. Let OA = T, then 



ds, _ ^j _ 3 ,. _ t , ,* 

 ~~ ~~~' "'" " 



, _ ^ __ 

 'ds, T~R l + R a ds,' 



' r, \T 



As an example of the use of this equation, we may examine a property 

 of the logarithmic spiral. 



In this curve, p = mr, and R = , therefore if the rolled curve be the 



m 



logarithmic spiral 



1\ 1m 



m _ 



T~^' 



AO 



therefore AO in the figure = mR l , and -p-=m. 



*% 



Let the locus of O, or the evolute of the traced curve LYBH, be the 

 curve OZY, and let the evolute of the fixed curve KZAS be FEZ, and let 

 us consider FEZ as the fixed curve, and OZY as the traced curve. 



VOL. I. 2 



