( 386 ) 



b. Inclined cell*. The beven eases corresponding to the fractions 



12 6 n , , t , 



. -...., are all represented here. In the case corre- 



' 12 12 12 ' 



sponding to nought the section with 6» is a line, here a vertical one, 



the section with Ci6 a point, here the upper extremity of that line. 



12 3 4 



In the cases — , — , — > - we tind an irregular octahedron, inscribed 

 12 12 12 12 



in a triangular prism, bounded by two equilateral triangles of diffe- 

 rent size and two sets of three isosceles triangles of different form ; 



the smaller of the two equilateral triangles is always inscribed in 

 the upperplane of the prism, whilst the larger forms a normal section 



3 2 1 



of the prism, successively at the heighl — , — , — -, 0. Finally in the 



4 4 4 



5 6 

 cases — , - we tind a semiregular hexagonal prism regular as to 

 12 12 



the angles and a regular one in which polyhedra (12, 24, J 4) are 



inscribed, once more bounded by two equilateral triangles, six isosceles 



triangles and six isosceles trapezia. But here, in opposition to the form 



(12, 24, 14) found above, the two equilateral triangles instead of 



being nomothetic, have an opposite orientation. ') 



4. Before we pass to the generation of more or less regular space- 

 fillings by intersecting the net (6' 16 ) we wish to say a single word 

 about the diagonal planes appearing in the sections of the cell C\ e 

 represented on plate II. In my communication "On groups of poly- 

 hedra with diagonal planes, derived from polytopes" published in 

 these Proceedings of October (p. 277 — 290) it has been explained 

 that any space intersecting 6' 1S and not passing through one of the 

 edges intersects this cell in a polyhedron with the property that 

 through any edge of it passes one and only one diagonal plane, and 

 that we only can obtain sections, through one or more edges of which 

 pass two diagonal planes, it' we choose an intersecting space passing 

 through one or more edges of C le . We have especially to show here 



l ) In the paper "Regelmassige Schnitte und Projeolionen des Achtzelles u.s.w." 

 (Regular sections and projections of the eightcell. the sixteencell, etc.", Verhande- 

 lingen of Amsterdam, first section, vol. II, N '. 2, 1894), I restricted myself princi- 

 pally to central sections; I only added incidentally a remark about the sections by 

 spaces not passiug through the centre. The figures 11 and 13 of that former paper, 

 being not quite correct, should be replaced by the second figures of the third and 

 fifth columns of plate II of this study. 



