( 554 ) 
: 46 
Case (= z) Of the two polyhedra (12, 21, 11), (18, 30, 14) 
Gd 
12 12 
orientations. The facial contact of two forms (12, 21,11) takes place 
by a triangle, that of two forms (18, 30,14) by a hexegon and that 
of two polyhedra of different shape by a trapezium. 
The projection on a plane normal to the axes is represented in 
fig. 9%. In the three shaded hexagons ABCDEF one recognizes 
4 6 
corresponding to (=) ; (a) only the first one occurs in two opposite 
; 6 
immediately the upper planes of three central sections =) and in 
LMN the base, in F,E,B‚A,D,C, the equatorial section of the body 
+ 
— | lying between, etc. 
(oa) lying 
Case (5: =). Here both forms (18, 27,11) and (80, 45, 17) 
occur in two different orientations. Two polyhedra (18, 27,11) of 
the same orientation have no face in common, two polyhedra 
(18, 27,11) of different orientation in contact have a hexagon in 
common, so as to make their axes one another's production. Contact 
of two polyhedra (30,45, 17) takes place either by a semiregular 
hexagon or by a hexagon with two axes of symmetry in a plane 
inclined with respect to the axes, according as the two polyhedra 
correspond or differ in orientation. Contact by a hexagon with two 
axes situated in a plane parallel to the axes MN takes place between 
two polyhedra of different shape. 
The projection of the space-filling normal to the axes MN is 
represented in fig. 9’. In the three shaded semiregular hexagons drawn 
; ONE 
in toto one recoguizes easily the three upper planes of the section 12m 
the other shaded parts of the figure portions of lower planes of 
0 : : : : 5 : 
the section i in Opposite orientation, in the two larger semiregular 
hexagons partially covering one another in the centre of the figure 
3 ; 
equatorial sections of the form (5 ‚ for which the thickness, 1. e. 
the segment on the axis lying within the polyhedron, is half as 
| 5 
large as that of ge , ete. 
12 
Space-jillings normal to OR,,. Finally we get here two known 
space-fillings, that of octahedra and combinations of cube and octa- 
hedron in equilibrium, and that of the body (24, 36, 14) of Lord 
KELVIN, 
