MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 535 



Before proceeding to give a list of examples of rolling ciu-ves, 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 roUed 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. 



1st. Examples of a curve rolling on a straight line. 



Ex. 1. When the rolhng 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 ciuTre 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 Avhose 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 



I 



2 a 



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 



the traced curve is the hyperbola whose equation is 



If' = a^ + x'- 



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 invohite of that circle, and the curve C is the spiral of Archimedes, r = a 6. 



Ex. 2. When the curve A is a catenary whose equation is 



VOL. XVI. PART V. 6 Y 



