BY A SPACE E0TAT1NG ABOUT A PLANE. 13 



27, which <ire in reality the numbers we assign to the eightcells 

 of the middle layer; in the same manner the eightcells of the layer 

 above are designated by -f- 1 , -f- 2, -|-3,..., -f 27, those of 

 the layer below by — 1, — 2, — 3,..., — 27, so that above 

 and below 1 are -|- 1 and - — 1 , etc. 



It may be remembered that, while each of the 27 cubes of the 

 block is a central section of an eightcell parallel to two of its 

 opposite limiting cubes (the upper cube and the lower cube) , each 

 face is a section of a side-cube , each edge a section of a face and 

 each vertex a section of an edge. Also the cubes which are in 

 plane , line , or point contact — that means the cubes which have 

 a plane , line , or point in common — are sections of eightcells 

 respectively in space , plane , or line contact. 



11. In order to identify the limiting cubes of each eightcell as 

 their sections with the space S 3 (t) take different forms and posi- 

 tions we distinguish them by letters. We suppose the cubes a , b , 

 c , d, to meet at one of the vertices of the eightcell and represent 

 the parallel cubes meeting at the diagonally opposite vertex by 

 a , b' , c', d' ; of these d and d' are the upper cube and the lower 

 cube. The visible faces of the cube 1 (fig. 6) correspond in this 

 manner to the side cubes a , b', c of the eightcell 1 . 



The three pairs of parallel planes dividing the large cube of 

 three times the size into the 27 separate cubes are the intersections 

 of the initial position of the space S 3 (t) with three of the four 

 pairs of parrallel spaces dividing the large eightcell of three times 

 the size into the 81 separate eightcells. The fourth pair of dividing 

 parallel spaces is parallel to /S 3 (7r) and divides the fourdimensional 

 block into the three layers which we have called middle layer, 

 plus layer and minus layer. We may illustrate the positions 

 of the cubes limiting the eightcells, and the manner in which the 

 eightcells are numbered and their limiting cubes lettered , by giving 

 a small table of the space contact of the eightcell 14, the central 

 one of the block, with the surrounding ones. The eightcell 14 is: 



in ad contact with the eightcell 1 1 , 

 ,, bb „ ,, ,, ,, 23 , 



11 £-£. ,, ,, ,, ,, 1 o , 



,, Cl Cl ,, ,, ,, ,, 1 # , 



!l "" ,) 11 ') )> " > 



ii C C ,. ii ii ii *■ " > 



„ dd' „ „ „ „ -\~ 14 , 



,, (i (i ii 11 >> i» 14K 



