( 383 ) 



In fig. 4 the diagonal D" D" of the cube, forming an axis OK a 

 of the eighteell, appears as axis of projection. Here I lie projection 

 of the eight vertices of the sixteencell on that line J J" /)" is found 

 in the easiest way by projecting these points first on to the space 

 of the cube with the diagonal D"D'" and by repeating this for the 

 eight projections obtained with respect to the line I)" I)'". For, the 

 projections of the eight vertices of the sixteencell on to the space of 

 the cube are the vertices of that cube, and these project themselves on 

 /)"/)'", if this line is divided by P and P' into three equal parts, 

 according to the arrangement (1,3,3,1) in the points D",P, P' , /)'". 

 From this arrangement (1,3,3,1) of the vertices can be deduced 

 immediately what appears on plate I under the beading (3, 1,1, J) OK s . 



For each of the live cases considered we repeat under the heading 

 "type" the manner in which the four couples of opposite vertices 

 of the eighteell project themselves on the different axes. 



3. We now proceed to the description of the sections, represented 



in parallel perspective on plate II, of the erect C\t l ~ ) and its envelope 



C$ on one side, and the inclined C id and 6 i? 2> and their 



envelopes C & on the other. The sections of the C 16 can be deduced 

 from the tables of projection of plate I, those of the circumscribed 



C' 8 and 6 8 " ; have already been given on plate II of the second 

 communication. 



By two thick vertical lines this plate II is divided into three parts, 

 respectively related to sections normal to OR s , normal to ( > l\ , 

 normal to OK 6 . Each of these three parts is divided by a thin 

 vertical line into two columns; of these two columns the lefthand 

 one always contains three sections of erect sixteencells, the right- 

 hand one live or more sections of inclined sixteencells. We now 

 consider separately each of the six columns so formed. 



Sections normal to OR i . 



a. Erect cells. This case is the simplest of all. If by a motion 

 parallel to itself of the intersecting space normal lo Ihe axis OE lt of 



6 ig " the point of intersection of thai space with thai axis moves 

 from one of the two vertices situated on thai axis to the other, the 



section with the circumscribed C'è' remains a cube with edge four 



and the section with the inscribed 6 ie itself, which is always 



