822 ANALYTIC GEOMETRY [On. XIII 



\\luoh rolls, without sliding upon a lixed ri^lit lint; (OX). 

 The point P is called the generating point ; the eirele Pll 

 the generating circle; tin* points O and A, tin- vertices; tin- 

 line EK) perpendicular to OA at its middle point, tin* axis; 

 and the line OA, the base of the cycloid. 



I <> derive the rectangular equation <>f tin- eyeloid let a be 

 the radius of the generating circle, and OX tin- lixed straight 

 line on which it rolls; also let P be the gnu-rating point, 

 and let PNS be any position of the generating circle. 

 Draw the radius CP, the ordinate MP, the line PL parallel 

 to OX* and the radius Off to the point of contact <>| the 

 generating circle and tin- lin- OX. Let OX and OY (the 

 perpendicular to it through 0) be chosen as axes, and let 

 6 be the angle PCff. 



Then, if />=(*, y). 



x= OM = OH -ME 

 = Off- PL 



= ad - a sin 6, [since OH arc Pff = aO] . 

 i.e., x = a(0-8in0). . . . (1) 



Similarly, y = a(l-cos0). ... (2) 



Solving equation (2) for 6 gives 





and substituting this value of in equation (1) gives 



x = a venr 1 - V2 ay - y*, ... (3) 

 which is the rectangular equation sought. 



