( 2(55 ) 



The connection of Cf- elements can now be represented by a diagram 

 (pi. I) the rows of which indicate the Sp t , the columns the points, 

 whilst incidence of a Sp e with a point is indicated by hatching the 

 square common to the respective row and column. 



We see that the diagram can be brought to a more condensed 

 shape : 



ABCDEFGH 



Here S indicates a simplex-filling; each of the other letters a system 

 (8,) denoting the incidence connection between the elements of two 

 simplexes. These systems (8 8 ) have all degenerated into two (4 3 ), each 

 pair of our simplexes is thus connected in an equal way and forms 

 a C/{1Q 10 ) of the same type. 



§ 3. Analogous to the well-known decomposition of K m into 

 four tetrahedra lying in pairs in a MoBius-position, it is obvious 

 to call the position of two of the simplexes, e. g. A and B, by 

 that name. Each side-/S'^ n of one S contains three points, so a 

 face, of the other; each vertex of one lies in three side-/S)> 6 , so 

 in a side-S/^ of the other; the correspondence is such that opposite 

 elements of A x e.g. vertex^ and side-space A x also furnish opposite 

 elements of B, namely resp. the side-Sj), : B l B i B 6 B 7 B a and the 

 face B t B t B t , just as this is the case with the tetrahedra in Möbiüs- 

 position. 



There exists already however, provided with the same property, 

 an extension of this notion, that of Bekzolari l ) where each side-£p„ 

 of one S contains one vertex of the other, and is generated by 

 operation with a focal system on an arbitrary simplex ; let us call 

 this position MI, then it is evident that the discussed more specialized 

 Mil is to be regarded as a threefold MI. 



') Bendiconti del Circolo Matem. di Palermo 22. 



