THE THEORY OF ROLLING CURVES. 



0, d6 t 1 r, 

 Here *, = tan-, ,.^ = __ = _ 



r, 



' a 2 + COS0,' 



therefore the first wheel is an ellipse, whose major axis is equal to $ of the 

 distance between the centres of the wheels, and in which the distance between 

 the foci is half the major axis. 



Now since 0, = 2 tan' 1 0, and r t = a-r v 



which is the equation to the wheel which revolves with constant angular velocity. 



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 pre- 

 viously. 



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 Rotting Curves. 



First. Examples of a curve rolling on a straight line. 



Ex. 1. When the rolling curve is a circle whose tracing- point is in the 

 circumference, the curve traced is a cycloid, and when the point is not in the 

 circumference, the cycloid becomes a trochoid. 



Ex. 2. When the rolling curve is the involute of the circle whose radius 

 is 2a, the traced curve is a parabola whose parameter is 4a. 



