( 642 ) 



clearly evident from the second plate, ^ivin^ the sections for tiie cases 



OK and OF. As is shown in the three diagrams with the fractional 



3 2 1 

 symbols ~, , - belongin»; to OK here one of the dimensions of the 

 6 6 6 



section, viz. the dimension in the direction of the edge with K as 



centre, is of constant length, by which the sections become prisms 



with a height 2, namely an hexagonal prism //''-), a triangular 



prism pf2i 2) j^pjj .^ triangular prism 7^* -); with these symbols // 



and P the indices V^2 and 2 1 ^2 indicate the length of the sides of 



the bases. As a matter of fact we can now assert that with these 



prisms of which the endplanes are the determining variable elements, 



the problem of the intersection has lost a dimension; for, in 



order to determine the prism we have oidy to ask how the grpund- 



cube is intersected by a plane perpendicular to a diagonal of this 



bounding body of the eightcell, i. o. w. the i>roblem has become 



threedimensional. In the same way we tiiid in case OF rectangular 



prisms of which two dimensions remain constant, which has been 



• ■ 2 , , . 

 indicated for the section ot transition - and the intermediary sec- 



4 



\ 



tion -, whilst the section in case OR is an invariable cube, which 



4 

 is of course not designed. 



It is almost superlluous to stop for the two space-fillings of case OA', 

 that by H'^^ -^ and P^'-) together and that by P'-'» '^^ alone, as they 

 appear indeed as well-known plane-fillings. We sulTice by giving 

 the following relations : 

 P'>'^+i;r.^ (^.4.2)^7^^)+ (|._t_l)_^i/(i-2)4_ (^)^iy2) \ 



i/C^A+01 :>= 6(y[;+l), P'y^) 4- (3/'^+3/+J) //» 2)4_ ^,(kJ^i)JY~^ 



i/(H2) _ 3/(;2p(i/2) _L '6k^p{y^^) 



4. Case (3, 1, 1, J). — If the vertex A of the eightcell 6'f — see 

 first diagram of second i)hite — is point {[,[,{,[) then the point 

 li 1 is obtained bv dixidini»' tlic inner diagonal AB of 



V ' 3 ' 3 ' 3 ; 



the cube lying in the s|)ace ./■, = J into three equal parts and then 

 to take the first |»oint of (ii\i^ion C). The line (JO is for this case 



1) By mistake in the diagram I'or ^C' has been taken ^ AR instead of - AB. 



