536 MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 
the curve B is the tractory of the straight line, whose equation is 
x eth ee 
y = a log ————— + V@ 2 
4 nN ee ue ss 
and Cis a straight line at a distance a from the vertex of the catenary. 
Ex. 3. When the curve A is the polar catenary 
(ce NG eee 
r 
the curve B is the tractory of the circle 
ca aie Oe / a 
é = cos iz mR it 
and the curve C is a circle of which the radius iss. 
3d. Examples of one curve rolling on another, and tracing a straight line. 
Ex. 1. The curve whose equation is 
6=Ar”+ ete. + Kr? + Lr + Milogr+Nr+ete.+ Zr 
when rolled on the curve whose equation is 
pees Aaz'”+ ete. +2K24-—Llog2+M2xzi+iN 2’ + ete. + —_ Zgntt 
n—1 n+1 
traces the axis of y. 
Ex. 2. The circle whose equation is 7 = a cos @ rolled on the circle whose 
radius is a traces a diameter of the circle. 
Ex. 3. The curve whose equation is 
a= Rie — 1 — versin-? — 
ip a 
rolled on the circle whose radius is @ traces the tangent to the circle. 
_ Ex. 4. If the fixed curve be a parabola whose parameter is 4 a, and if we 
roll on it the spiral of Archimedes 7 = a @, the pole will trace the axis of the para- 
bola. 
Ex. 5. If we roll an equal parabola on it, the focus will trace the directrix of ; | 
the first parabola. 
Ex. 6. If we roll on it the curve r = = 
the vertex of the parabola. 
Ex. 7. If we roll the curve whose equation is 
ts Co) 
7 = a COS b 
the pole will trace the tangent at 

