Imagine that upon DR or fe, as a diameter, is described the 

 semicircle DER. We shall have OE = yhx' 0/2, and the arc 



caui >- rr." J^iSy^ 1 



we have therefore generally, 



OM = V -f OE + RE. 



To determine the constant C 7 , it must be observed that when 

 x f = 0, we have y f = 0. Therefore, since OE and RE then be- 

 come zero, we have C' = ; consequently OM OE + .K.E ; 

 and the curve sought is therefore the common semicycloid, of 

 ' which DER is the generating circle, and AD the semibase. 



The only thing which remains to be determined, is the quan- 

 tity h ; for the only things given are the two points A and B, 

 through which the body is to pass, h is determined in this man- 

 ner. 



Having drawn the vertical BK, which meets in K the hori- 

 zontal line AK passing through the point A, we describe upon 

 AK as a semibase, the semicycloid AVT, that is, a semicy- 

 cloid of which the generating circle has AK for the length of its 

 semicircumference. And having drawn AB cutting this cycloid 

 in F, we draw VK, and parallel to VK, through the point J5, we 

 draw BD, which determines JlD for the semibase of the cycloid 

 sought, that is, for the semicircumference of its generating circle. 

 This construction is founded upon the circumstance, that the cy- 

 cloids AVT, ABR, which have their bases upon AD, and the 

 point A common, are similar, as may be easily shown.* 



* Since the diameters of circles are as their circumferences, or 



as their semicircumferences ; 

 Geora. 



287. DR : KT :: DA : KA' t 



but 



DR : KT : : DB : tf F, 

 hence, - 



DB : KV : : DA : KA. 



