MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 539 



4th. Examples of pairs of rolling curves which have their poles at a fixed 

 distance = a 



Ex. 1. 



The straight line whose equation is 6 = sec"' - 



The polar catenary whose equation is «=±vl± — 



Ex. 2. Two equal ellipses or hyperbolas centred at the foci. 

 Ex. 3. Two equal logarithmic spu-als. 

 [ Circle whose equation is 



r = 2 a cos 



Ex. 4. 



Curve whose equation is «=v2- — i+ versin-' - 



I. r a 



( Cardioid whose equation is r = 2 a (1 + cos e.) 



I Curve whose equation is s = sin-' - + log 



C Conchoid, r = a (sec 6 - 1.) 



Ex. 6. 



[ Curve, «=v 1 - ^ + 



[ Spiral of Archimedes, r = a6 



^''•''- Curve, 6= -+log- 



a 



a a 



Ex. 



Ex. 9. 



[ Hyperbolic spiral, 



a 

 r = — 



1 



Curve, r = 



a 



e" + 1 



Ellipse whose equation is, r = a 



Crxrve, '■ = "(1 + 2(2-0^)) 



Ex. 10. 



(Involute of circle, « = v ^ - 1 sec ' - 



a- a 



Curve, 6=\/'^±2-±\og (-±i +^^±2-) 



bih. Examples of curves rolling on themselves. 



Ex. 1. When the curve which rolls on itself is a circle, equation 



r = a COS 6 



the traced curve is a cardioid, equation r = a (1 + cos d). 

 Ex. 2. When it is the curve whose equation is 



r = 2" al cos — I 



the equation of the traced curve is 



cos ) 



« + 1/ 



VOL. XVI. PART V. 6 Z 



