( 544 ) 
: (2) 
intersecting space with respect to the axes of the including cells Cg’, 
whilst this position can be chosen in two different manners in the 
cases of the two other series. This gives rise to six different series 
of parallel sections of a Cer enclosed in a ce, which have to be 
considered in the following. 
2. We adopt here the method followed in the second and third 
papers and indicate the results of the determination of the section 
of a cell CS” in two ways. Once more the first plate gives us the 
projection of the limiting elements of the cell BA on the diameter 
normal to the intersecting space, and the characteristics of the sections 
deduced tabularly from these projections; in this only four series 
of sections present themselves. The second plate shows further the 
form of the sections in parallel perspective, enclosed in the circum- 
scribed eightcell; here we have to deal with six different series. 
Finally a third plate principally contains some diagrams with three- 
dimensional space-fillings generated by the intersection of the net, 
whilst the third of these diagrams numbered separately, which domi- 
nates the deduction of the projections of plate I, has been transferred 
thither, in order to facilitate comparison. We now proceed to the 
consideration of the diagrams 1 and 2 of plate III. 
The manner in which the cell peo is inscribed in the box Ge is 
characterized by this, that the vertices of are the centres of the 
faces of Ce ’ We indicate how these points combine themselves by 
twos to extrimeties of edges, by threes to vertices of faces and by 
sixes to vertices of limiting octahedra, by indicating these octahedra 
in the diagrams 1 and 2. It is immediately clear that eight of these 
94 octahedra are polarly inscribed in the eight limiting cubes of Ce 
fig. 1 exhibits two opposite faces ABC, A’B’C’ of one of ty 
octahedra, whilst fig. 2 shows two opposite faces ABC, A"B'C" o 
one of the sixteen remaining octahedra. Indeed the vertices of oe 
divide themselves with respect to the space of a limiting cube of 
Ce, the central space parallel to it, and the space of the opposite 
limiting cube into three groups of 6,12,6 points and the central 
section is evidently the combination (12, 24,14) of cube and octa- 
hedron in equilibrium; from this can be deduced that the second 
