THE THEORY OF ROLLING CURVES. 7 



Let the angle DCA = 6 l and CA=r^ and let 



Let this curve remain fixed to the paper. 



Let there be another curve BAT, whose pole is B. 



Let the angle MBA = d i} and BA = r 3 , and let 



Let this curve roll along the curve KAS without slipping. 

 Then the pole B will describe a third curve, whose pole is C. 

 Let the angle DCB=6 a , and CB = r 3> and let 



We have here six unknown quantities ^iff^fff, ; but we have only three 

 equations given to connect them, therefore the other three" must be sought for 

 in the enunciation. 



But before proceeding to the investigation of these three equations, we must 

 premise that the three curves will be denominated as follows : 



The Fixed Curve, Equation, 6 1 = fa (r^. 

 The Rolled Curve, Equation, 1 = fa(r^. 

 The Traced Curve, Equation, 6, = fa (r 3 ). 



, 



When it is more convenient to make use of equations between rectangular 

 co-ordinates, we shall use the letters xy lf x^, x 3 y 3 . We shall always employ the 

 letters $,&,$ to denote the length of the curve from the pole, p^p 3 for the per- 

 pendiculars from the pole on the tangent, and q t q. 2 q 3 for the intercepted part of 

 the tangent. 



Between these quantities, we have the following equations : 



x 

 = rsin 0, 



s = 



ydx xdy 



= 



