18 



Proceeflings of the Royal Irish A caitemi/. 



The diagi-am rolled by a regular polygon of many sides,, as on fig. 13, can 

 be eonstructed with great acciuaey. For the half diagram the fii-st stroke 

 is upward at ti to the vertical, then down vertically, then up at plus 20, 

 down at miuus 6, and so on, like writing, plus 20 up, B down ; ever making 

 an equal advance on the base with the further check, point by point, that each 

 up stroke equals the last one down. 



CYCLOID /-•.- MODl.'^lED CYCLOID 



It is rolled hy any Regular Poljfgon 

 inscribed in the Generating Circle. 



Id die lim!t,\he Modined Cycloid has 

 the -same leugni as the Cycloid. The 

 slope and curratiue are doubled point 

 by point. The area oi^he .Generating 

 Circle is sqccezed ont. 



Its area is that of the Poly^n 



and twice the Circle. The Hodifipd 



Figjlre leaves ont the Polygon (shaded 



iangles) and is contiuuon-s. ItsPoly- 



^nal hail" Evniute equals a Jijrnctei. 



Fig. 1.3. 



"Wlien the black triangles are taken out (that is, the generating polygon) 

 and the white sectors closed up, we have a modified continuous curve, which 

 in the limit is equal in length to the cycloid, and the area inclosed between 

 it and its polygonal evolute is that of twice the rolling circle. 



By inspection of the diagram it .will be seen that this modified cycloid 

 has, from point to point, twice the slope and curvature of the cycloid. 



