f 288 ) 



w or between w and // and an octagon if S lies between w and 

 w;', except when S coincides with the midpoint in which case the 

 section is a hexagon. In other words, with reference to the side 

 1\ l\ of the starpentagon of fig. 5 : the section is a hexagon if S 

 coincides with M lt an octagon if S lies elsewhere between Q t and 

 (2 4 , a pentagon if >S falls between I\ and Q t or Q 4 and P a . So 

 in the case k i we find two pentagons, since two points of intersec- 

 tion lie outside the pentagon with t he vertices Q t , a hexagon and 

 an octogon, etc. 



13. The method developed here has a slight drawback, revealing 

 itself to the utmost, in the determination of the exact position of the 

 trace A,. The difficulty consists in this that the method leaves us 

 in the dark as to the succession of the different values of X on the 

 trace /. If we have deduced that the different ratios of VI,, VII,, 

 IX,, X, present themselves and we have chosen for VII, the centre 

 J/, (fig. 5) we are obliged to investigate by a rotation of the rider 

 about J/„ on which side - and in which of the two different points 

 on this side - we must assume the point of division corresponding 

 to VI, in order to make the other points of intersection to corre- 

 spond to IX, and X,. We now indicate finally how this difficulty 

 can be overcome. 



To any chosen edge of C 8O0 projecting itself on plate I\ r/ ' in h on 

 the axis OF , there correspond tive adjacent points of C t00 . If now 

 it were possible : 



1. to select a determined edge projecting itself in h on OF , 



2. to point ont the tive adjacent vertices and to indicate in what 

 order these points are the vertices of a regular starpentagon, 



3. to find where these five points project themselves on the 

 same axis 0F , 



then it would be possible to make out, in what ratio the 

 successive sides of the starpentagon were divided in projection on 

 ()1\ by h, which would enable us to fix in lig. 5 on each of these 

 sides a quite definite point. Really in these suppositions the difficulty 

 would be quite dissolved. 



Now these suppositions are quite realisable, by means of the 

 tables published in my first memoir with the title " Regel massige 

 Schnitte u. s. w." (Regular sections and projections of G',, and 6' 600 , 

 Verhandelingen Amsterdam, first section, vol. II, No. 7, 1894) ; 

 we will explain this with the aid of l\^. i) for the case of the 

 trace It.,. 



