ANALYIK //;) [Cii. Xlll. 



line A7* parallel to OA, where A is the point with which /' 

 coincided when in its initial position. Let OAX and OY, 

 the perpendicular to it through 0, be chosen aa -.., nlinair 

 axes; also let the angles AOQ.PO'Q and O'PK be desig- 

 nate.!, respectively, by 0, & and <f>. 



Then OM= OH+HM= OH+ KP 



COS 0+P 0^08(0' -0), 



[since <f> = 0' - 0] 

 .., z=(a-)cos0 + &cos(0'-0). . . . (1) 



Mut since arc AQ = arc PQ, therefore aB = bQ\ whence 



0' =?0, and equation (1) becomes 

 6 



x = (a - 6) cos0 + b cosfr* ~ ^ *. ... (2) 



Similarly, y = (a 6) sin 06 sin ^ ^ ^- . . . . ( 



6 



Equations (2) and (3) are together the equations of the 

 hypocycloid. A single equation representing the same 

 curve may be found, as in the case of the cycloid (Art. 191), 

 by eliminating between equations (2) and (3). 



NOTE. If the radii of the circles be commensurable, i.e., if b equals a 

 fractional part of a, then the hypocycloid will be a closed curve ; but if 

 these radii are incommensurable, then the curve will not again pass 

 through the initial point A. 



In particular, if a:b = 4 : 1, then the circumference of the fixed circle 

 is 4 times that of the rolling circle, and the hypocycloid becomes a closed 

 curve of four arches, as shown in Fig. 134. In this case, equations (2) 

 and (3) become, respectively, 



