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Mr. Archibald Sharp on 



OX the axis of the wheel W turns uniformly in a horizontal 

 plane, and the distance rolled through by the wheel during 

 any short interval is equal to the corresponding displacement 

 of the tracer P in the direction OY. If the axis of the rolling 

 wheel W makes one complete turn while the tracing- point P 

 moves over one complete period of the curve (fig. 1), the 

 point of contact of the rolling wheel will describe a curve 

 Off/B (fig. 2). 



Let OY' (fig. 2) be the initial direction of the plane of the 



, •*) 

 --if 



p\ 



' o 



I 



_ . ir 



F/C 2 



rolling wheel, t. e. corresponding to zero abscissa of the tracer 

 (fig. 1) . Let P be any point on the curve iig. 1, p the cor- 

 responding point on the curve fig. 2. Let P' and p be two 

 corresponding points infinitely close to P and p respectively. 

 In fig. 1 draw P'Px and PP X parallel to OY and OX respec- 

 tively, and in fig. 2 draw p'q and pq respectively at right 

 angles and parallel to OY'. 

 Then 



p'g=pp' sin #=sin xdy, 

 pq =pp' cos # = cos xdy. 



Draw RR' perpendicular to OY' (fig. 2), R being the 

 position of p corresponding to # = 27r. Then 



C 2n . 

 RR' = 2p'q = sin x dy, 



OR' = — %pq= — 1 cos x dy. 



