( 637 ) 



six series of spaces to deduce tlie sections from this tabiilarly; in 

 tiie second place we indicate tlie sections themselves in parallel 

 perspective in the eightcell. To each of those two closely allied 

 modes of transacting an extending plate is given. 



To promote the uniformity we indicate the axes OE, OK, OF, 

 OR by their ends (i, J,l,i), (1,1,1,0), (J, 1,0,0), (1,0,0,0). 

 Then we hax'e to deal successively with the six series 

 (1,1,1,1), (1,1,1,0), (1,1,0,0), (1,0,0,0), (3,1,1,1), (2,1,1,0) 

 and we have now to investigate for each of those six cases the two 

 parts into which the problem was above divided. 



2. Case (1,1,1,1). — This case was, as far as the tirst part of the 

 problem is concerned, completely solved in a foregoing study {Pro- 

 ceedincis, Jan. 1908, page 485). Hence the tirst part of the first plate 

 with the superscription (J, 1,1,1) OE^ is an extension of the tiist 

 diagram y^ = 4 of the plate given then. In order to be able to indicate 

 together with the projections of all bounding elements the projections 

 of the vertices of these elements, which considerably promotes the 

 insight into the spacial figure, the numbers of edges, faces, bounding 

 bodies are denoted here outside the scheme on the righthand side. More- 

 over the sections of the eightcell with the spaces of transition and 

 the intermediate spaces perpendicular to the diagonal of projection 

 are mentioned tabularly ; here use has been made of a method formerly 

 ( Verhaudclinc/cn, volume IX, n".4) developed in all details which acquaints 

 us not only with the characteristic numbers {e,k,f} of each section, 

 but also with the nature of the faces. Thus the central section is a 

 (6, 12, 8), because it contains 6 vertices and does not cut an edge, 

 intersects 12 faces and contains no edges, intersects 8 bounding cubes 

 and contains no faces ; this section is a regular octahedron in connec- 

 tion with which each cube of the two quadruples of bounding bodies 

 is cut according to an equilateral triangle of the same size. In this 

 way the adjacent intermediary section is a (12, 18, 8), because 12 

 edges, 18 faces and 8 bounding cubes are intersected, viz. a tetra- 

 hedron regularly truncated at the vertices, i. e. the first of the 

 semi-regular Archimedian |)olyliedra {Proccediiujs, page 488) because 

 four of the bounding cubes are intersected according to regular hexa- 

 gons, the four remaining ones according to equilateral triangles. Here 

 the number of edges is found back as half of the total number of 

 sides of the faces, thus 12 as half the product of eight and three, 

 18 as half the sum of four times six and four times three. Moreover, 

 when indicating the polygons lying in the faces, we have underlined 

 the figure of each group of regular polygons. 



