( 547 ) 
parallel to ON, through the seven sections of the cube, it is clear 
that this space will be cut by the space O(CX, X, _X,) in a plane 
parallel to the endplanes of the prism, by the two spaces OX, X, X) 
and OY, X, X,) in planes parallel to the couples of parallel lateral 
faces of the middle section, by the space O(X, X, X,) in a plane 
not presenting itself in the middle section that according to 
its position in our figures — may be called the face behind. 
We now try to find in each of these four spaces of coordinates a 
plane normal to the indicated plane of intersection of that space 
with the intersecting space, on to which moreover the projection, of 
the limiting cube situated in that space of coordinates and the octa- 
hedron polarly inscribed in it is as simple as possible. So we get in 
OX, X, X,) the plane OLK, Y), in OK, X, X,) the plane OLX,_X,’, 
in OLX, X, X,) the plane OCX, X,), in OCX, NX, X,) the plane OLX, _X,). 
With omission of the case OCX, XN, Y,) equal to that of OY, X, XN) 
these projections are represented in the diagrams 5%, 52, 5e where 
the series of parallel intersecting planes are indicated by their 
parallel traces. For any position of the intersecting plane the required 
sections of the octahedra are easily found by means of these diagrams. 
So the section pgr of the octahedron in the lozenge os of fig. 5% is 
the hexagon of the endplanes and the section p’g’r’ of the octahe- 
dron in the rectangle o’r’ of fig. 5° is the deltoid of the lateral faces, 
of the middle section, whilst the section p'g'r's” of the octahedron 
in the rectangle of” furnishes to us the hexagon in the face behind of 
og With the aid of 
12 
the characteristics of the sections tabulated on plate I we then easily 
find how the visible faces of the section are to be completed to the 
total boundary of the polyhedron by means of faces situated within 
the prism. 
The form of the polyhedral sections of C,, obtained in this manner 
is rather complicated and therefore not easily described ; all the forms 
admit of two common characteristic features: they possess an axis 
with the period 3, in our figures the horizontal line MN, and four 
planes of symmetry, three through the axis and one normal to it. 
This axis MN is no axis for the prismal section of the circumscribed 
C, the middle section excepted; for this section it is an axis with 
the period 2 and in connection with this it becomes an axis with 
the period 6 for the middle section of C,, that admits of seven 
planes of symmetry. 
It is easily verified that the length of the axis MN within the 
seven different sections is successively 
i 
the section corresponding to the fractions To And 
ai 
