( 546 ) 
1 3 (V2) 
is successively 0, ove, 5, ove, 2/2 and as to Cos “ wegeta 
1 
point, a cube with edge 2 V2, a cube with edge V2, a polyhedron 
(32, 48, 18) limited by 6 squares and 12 hexagons with two axes 
of symmetry — which may be characterized as a rhombic dodecahe- 
dron truncated at the octahedral vertices by the faces of a cube — 
and this semiregular polyhedron itself (24, 36, 14) with one kind of 
face, which is also called granatohedron *). 
Sections normal to OK,, = (2, 1, 1, 0) GC, 
Here we have to distinguish two series of fractions, one related 
to the C,, itself, the other related to the box C,. The fractions 
0, a = KET placed below on the right hand correspond to the 
ke. 8 
1616’ B 
placed above at the left hand present themselves in the second case. 
In our second paper we have explained why the problem of the 
determination of the section of an eightcell loses one dimension in 
the case (2,1, 1,0) C, and all the sections are prisms with height 2, 
the bases of which are the sections of a cube with a series of parallel 
planes normal to the line connecting the origin with the point (2, 1, 1), 
seven sections in the first case, whilst the fractions 
1 
Le. of planes determining segments proportional to, 1,1 on the axis 
of coordinates. For the seven cases presenting themselves here fig. 4 
indicates the form of the bases; so it is not diffieult to draw the 
prisms represented in the second column of plate IL. As it is not 
quite so easy to deduce from the characteristics given on plate I 
the forms of the sections of C,,, the faces of these sections situated 
in the boundary of the prisms have been determined independently 
by means of the diagrams 54, 5%, 5e closely connected to fig. 4. If 
we suppose that ON, OX,, ON, OX, (fig. +) are the four edges of 
the eightcell concurring in © and that the intersecting space is brought 
Me YD) os eae oe. 
_ 1) Here too the vertices of the faces of the section of (24 ‘ visible in the limiting 
2) 
faces of the section of C8 have been brought to the fore; the shaded faces passing 
into one another by a parallel translation of the intersecting space are shaded in 
the same way. 
