14 THE THEORY OF ROLLING CURVES. 



When a = 0, T=0, or the centre of curvature of the curve B is at the 

 point of contact. Now, the normal to the curve C passes through this point, 

 therefore 



"The normal to any curve passes through the centre of curvature of its 

 tractory." 



In the next case, one curve, by rolling on another, produces a straight 

 line. Let this straight line be the axis of y t then, since the radius of the 

 rolled curve is perpendicular to it, and terminates in the fixed curve, and 

 since these curves have a common tangent, we have this equation, 



rf<A_ .^0, 

 ^c&T ' dr t " 



10 



If the equation of the rolled curve be given, find -,- ? in terms of r s , sub- 



ar, 



stitute x t for r,, and multiply by x lt equate the result to 3?-, and integrate. 



Thus, if the equation of the rolled curve be 



8 = Ar-" + &c. + Kr~* + Lr~ l + M\og r + Nr + &c. 



-f- = - nAx~* - &c. - 2Kx~* - Lx~ l + M+ NX + &c. + nZx n , 





. . 



' ^~ I lv "j" 1 



which is the equation of the fixed curve. 



If the equation of the fixed curve be given, find -r- in terms of x, sub- 



stitute r for z, and divide by r, equate the result to -.-, and integrate. 



Thus, if the fixed curve be the orthogonal tractory of the straight line, 

 whose equation is 



alos 



- 

 a + V a* x 1 



dy _ N/a* x* 

 dx~ x 



