( 549 ) 
— the five sections of C,, already obtained, but enclosed now in 
parts of rhombohedra, regularly truncated at one of the axial vertices’), 
and 
as far as the middle section is concerned — enclosed in an 
unmutilated rhombohedron. For these sections of the circumscribed 
eightcell one may compare plate IL of our second paper. 
Here too the sections of C,, and C, correspond principally with 
one another with respect to axis and planes of symmetry. Only the 
plane bisecting the axis MN perpendicularly is not a plane of 
symmetry for the truncated rhombohedra and the axis maintains its 
period 5 for the unmutilated one. 
Sections normal to OR,, = OR,. 
In this simplest case partially mentioned above we find — see the upper 
region of the fourth part of plate I! — corresponding successively 
to the fractions O, DE 
(24, 36, 14) with one kind of vertex, and the combination (12, 24, 14) 
of cube and octahedron in equilibrium, always enclosed in an 
invariable cube. 
the octahedron, the semiregular polyhedron 
Sections normal to OR,, = OK, 
Here the three sections once more must be characterized by pairs 
| 2 ; 
of fractions, below to the right O, ener. related to C,,, above to 
tee 
the left —, —, — related to C,. 
Sue Ee 
The three sections of C,, found above reappear here — see the 
lower region of the fourth part of plate Il — successively inscribed 
in a tetrahedron, a semiregular polyhedron (12, 18, 8) with one kind 
of vertex, and an octahedron, represented in their turn inscribed in 
cubes for the more simple deduction of the true measures. 
1) These rhombohedra bounded by lozenges with acute angles of 84°15'39" are 
represented in toto, the section with the truncating plane being indicated by 
heavy lines, 
In arranging the sections on plate Il I mistook the sections of the Cy occurring 
here for cubes, though I myself had indicated their true nature in the second 
paper. After having read the manuscript Mrs. A. Boore Srorr had the goodness 
to set me right as to the text; but the diagrams could no more be corrected. 
Happily the difference between these cubes and the slightly elongated cubes that 
should replace them is hardly perceptible. 
