( 545 ) 
limiting octahedron of which ABC is a face has for opposite face 
one of the two faces of this combination parallel to ABC and then 
— it goes without saying — the triangular face differing in orientation 
from ABC. So we get indeed sixteen new octahedra, each of the 
eight triangular faces A"5"C'" of the combination (12, 24, 14) procuring 
two of them. 
The projections given under the headings OL, OK, OF ,,, OR,, 
on plate I can be easily deduced from plate I of the second paper 
by means of the projections of the faces and limiting bodies of the 
including eighteell given there. By tracing the centres of all these 
faces and the octahedra polarly inscribed in all these limiting cubes we 
obtain the results tabulated in the four diagrams 3%, 3°, 3¢, 3d of 
plate I, i.e. in the cases OF,,, OK,, only eight, in the cases OF,,, 
OR,, only sixteen of the 24 limiting octahedra. As the laws of reci- 
procity require that the arrangement of the 24 limiting octahedra 
into groups for the cases OZ, OK,,, OF, OR,, corresponds to 
that of the 24 vertices for the cases OR,,, OF,,, OK,,, OE,, 
respectively, the obtained numbers (2, 4, 2), etc. of the octahedra 
can be completed to the really occurring numbers (6, 12, 6) added 
between brackets. Then from the projections of vertices and octahedra 
those of edges and faces are easily deduced. 
3. We now proceed to the consideration of the sections represented 
on plate II in parallel perspective; of these the sections of the CS 
have been derived from the tables of the plate I, whilst those of the 
enveloping CS have been taken from the second paper. 
This plate is divided by three heavy vertical lines into four parts 
successively concerned with sections normal to OF,,, to OK,,, to 
OF,,, to OR. Of these parts the third and the fourth are subdi- 
vided into two parts, in relation to the two possible positions of the 
circumscribed eightcell. 
Sections normal to OF,, = OF. 
If we restrict ourselves here to the sections of transition and the 
intermediate sections bisecting the distances between these, we have 
; 23 4 
to deal with five cases corresponding to the fractions 0, B'B'E'E 
As to the circumscribed Cs” we then find a rectangular parallelopi- 
pedon, the base of which is a square with side 2, whilst the height 
37 
Proceedings Royal Acad. Amsterdam. Vol. XI. 
