MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 535 
Before proceeding to give a list of examples of rolling curves, we shall state 
a theorem which is almost self-evident after what has been shewn previously. 
Let there be three curves, A, B,and C. Let the curve A, when rolled on itself, 
produce the curve B, and when rolled on a straight line let it produce the curve 
C, then, if the dimensions of C be, doubled, and B be rolled on it, it will trace 
a straight line. 
A Collection of Examples of Rolling Curves. 
lst. Examples of a curve rolling on a straight line. 
Ex. 1. When the rolling curve is a circle whose tracing-point is in the circum- 
ference, the curve traced is a cycloid, and when the point is not in the circum- 
ference, the cycloid becomes a trochoid. 
Ex. 2. When the rolling curve is the involute of the circle whose radius is 
2 a, the traced curve is a parabola whose parameter is 4 a. 
Ex. 3. When the rolled curve is the parabola whose parameter is 4 a, the 
traced curve is a catenary whose parameter is a, and whose vertex is distant a 
from the straight line. 
Ex. 4. When the rolled curve is a logarithmic spiral, the pole traces a straight 
line which cuts the fixed line at the same angle as the spiral cuts the radius vector. 
Ex. 5. When the rolled curve is the hyperbolic spiral, the traced curve is the 
tractory of the straight line. | 
Ex. 6. When the rolled curve is the polar catenary 
gat of p24 
r 
the traced curve is a circle whose radius is a, and which touches the straight line. 
Ex. 7. When the equation of the rolled curve is 
rt x2 4 2 
pang (VE-1+ 2) — or (YBa 2) 
the traced curve is the hyperbola whose equation is 
ye =a + @ 
2d. In the examples of a straight line rolling on a curve, we shall use the 
letters A, B, and C to denote the three curves treated of in page 555. 
Ex. 1. When the curve A is a circle whose radius is a, then the curve B is 
the involute of that circle, and the curve C is the spiral of Archimedes, 7 = a 0. 
Ex. 2. When the curve A is a catenary whose equation is 
VOL. XVI. PART V. 6¥ 
