I 267 ) 



Their equations are of two types; namely eight of the form 

 ± ,,» ± .,■ ± X,' ± .r 4 2 ± a.' ± «,' ± x. 3 ± *„' = . 

 where the combinations of signs must be derived from the sign 

 system ; and twenty-eight of the form : 



± as, x a ± x, .r 4 ± ,r 5 x t ± *, ,r s = . 



where the connection of the indices is given by the seven binary 

 substitutions of the regular G a , whilst the signs must be selected: 



+ + + + 



+ + - - 



+ - + - 



+ - - + 

 The sixty-three operations which transform an element into another 

 of the same sort are collineations \ so we obtain, analogous to the 

 Klein G ti in Sp t , a geometrical Abel group G liS , consisting of the 

 identity and sixty-three collmeations ; twenty-eigld focal systems in 

 involution and thirty-six polarities. 



§ 5. The hoenty-eight points in each Sj) t of K vn lie on a qua- 

 dratic Q;, and reciprocally. 



To prove this we regard the determinant of the terms of order 

 two, formed of seven of the eight homogeneous coordinates; so this 



2 



is of order 7 -f- I 7. ) = 28. The omission of a coordinate is geometri- 

 cally the projecting out of a vertex of the fundamental simplex on 

 the opposite ,?/>„ ; if the projections of 28 points lie in it quadrati- 

 cally, then the points themselves do so in their Sp 6 . 



Let us first restrict ourselves to Sp 6 : A x . 



The twenty-eight points are to be divided into seven quadruplets 

 of the same order of letters; the purely quadratic terms within such 

 a quadruplet are in each column alike, the mixed ones may differ 

 in sign. Let us call the four terms in a column p, q, r, s, then the 

 substitution 



Q=p + q — r — s 



R = p — <? + r — s 

 S = p — q — r -j- 8 



the A 



1111' 

 1 l-l—l I 

 1—1 1—1 

 1—1-1 1 ' 



of which is =|= 0, 



causes three of the four quadratic terms to disappear, the A 18 breaks 

 up into the product of a A, of quadratic and a A s , of mixed 

 terms. Here 



