( 266 ) 



The elements of two simplexes A and B in Mil can be arranged 

 only in one other way to two suchlike simplexes, namely as 

 first simplex P ■. A x , A„ A t , A t , B„ B„ B v B a , 

 second ,, Q-. B lt i> 3 , B l7 B„ A lt A e , A 7 , A e . 

 If we regard such a new simplex in connection with C, D, . . . H, 

 it then shows with each of these a new sort of position; for all 

 however of the same type, showing analogy to the pairs of tetrahedra 

 in STEiNER-position which can be separated in the same way from 

 K lUl ). We find for the c/(16 10 ) of two such simplexes a diagram 

 of the shape: 



«S x 

 x S, 

 where x again represents a system (8 8 ) which however does not dege- 

 nerate now, but is identical to the cyclic system which is obtained 

 out of the initial row : 1 2 . . 5 . . . 



Opposite elements of one simplex furnish, as in Sp s , no opposite 

 ones of the other. 



§ 4. The 28 operations determining in each e/'-space the c/'-points 

 incident to them and reciprocally, are focal-correlations ; thus e.g. 

 the Sp, ■■ A, 



(+ a, 4- b, + c, + d, + e, + ƒ, + g, + h) 

 is transformed into the point A, situated in it 



(-f b, -a, + d, - c, + ƒ, - e, + h, - g) 

 by operating with the skew-symmetrical determinant of transformation : 



These focalsystems are mutually in involution as the group of the 

 letter substitutions as well as that of the sign variations are Abel groups. 



The 36 remaining reciprocities are polarities with respect to some 

 36 quadratic Sp e , which serve for K vu as the 10 fundamental- 

 surfaces of order two for K 111 . 



x ) Martinetti, Bendic. Palermo 16 p. 196. 



