THE THEORY OP ROLLING CURVES. 17 



which is the polar equation of a straight line touching the traced circle whose 

 equation is r = a, then 







r* ~ TI Jrf-a? 



gfl, _r t a o_ 



<& r * (r, a) Jr*2r,a 



a 



2a 2a 



T s = = . 



Now, since the rolling curve is a straight line, and the tracing point is 

 not in its direction, we may apply to this example the observations which 

 have been made upon tractories. 



Let, therefore, the curve r = -~ - be denoted by A, its involute by B, and 



v I 



the circle traced by C, then B is the tractory of C ; therefore the involute 



2ct 

 of the curve r = ^r is the tractory of the circle, the equation of which is 



_.r la- __ . . 2a 



= cos ] / I. The curve whose equation is r= 7f . - seems to be among 



. a \J r" 6 f I 



spirals what the catenary is among curves whose equations are between rec- 

 tangular co-ordinates ; for, if we represent the vertical direction by the radius 

 vector, the tangent of the angle which the curve makes with this line is 

 proportional to the length of the curve reckoned from the origin ; the point 

 at the distance a from a straight line rolled on this curve generates a circle, 

 and when rolled on the catenary produces a straight line ; the involute of this 

 curve is the tractory of the circle, and that of the catenary is the tractory 

 of the straight line, and the tractory of the circle rolled on that of the straight 

 line traces the straight line ; if this curve is rolled on the catenary, it produces 

 the straight line touching the catenary at its vertex; the method of drawing 



VOL. I. 3 



