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Let us take h — 1 spaces A and let us number them \, . . . j — 1, 

 i + l,..L 



These k — 1 spaces A determine with k — 1 spaces A{ suitably 

 chosen k — 1 spaces S r .+H-'- These spaces meet the corresponding 



spaces Bi in A' — 1 spaces «Sr+s.+p— n+i, (i=l, • •• ,i — 1,^+1, ...A). 



These spaces determine an Si=j— i i=n 



JS (ty+«i)+S (»ï+«d+(*-l) (/»-«+2)-l 

 i'=i «=ƒ+> 



This space has in common with Bj a space 



'YV t - + 'Fri + 2 .,- + (A - 1) (p + 2) - *n - 1 

 1=1 i=j+i i=i 



In its turn this space determines with J, a space 



'"£' ( n + ./) + (A - 1) (p + 2) - /■„ 

 t=l 

 On account of the equality (3) the latter meets C in a single 

 point, which determines with D a space A/. 



When /' varies from 1 to k, one obtains k series of spaces A 

 between which exists a (1, 1, . . . 1) correspondence. There are h 



coincidences. 



k 

 The variety described by the space S p is V( n — p )(p+\]—\ ■ 

 The locus of a space S p for which the «Sjyfp-fi which 



unite it to k fixed spaces S r . meet k S S( i n k S rf \- Si +p— n+l 

 Of a same S^( r . + , ( ) _f_ k(p—n + 2) — 1' # = lj ' • ' ^' is a va " 



riety F(„_ 79 )( p +i)_i . 



The spaces ^4, are evidently principal spaces of the locus of S p , 

 principal space having the same meaning as principal point or plane 

 of a complex of rays. 



In S t we find the following theorem: 



The locus of an S l for which the 5, which join it to 

 four S meet four S l in four S of a same space *S', is 

 a variety V\ (complex of order four). 



3. If we regard the ordinary space as if generated by right lines 

 we have a geometry of four dimensions. We shall now show two 

 generalizations of the theory of Grassmann in this geometry. 



Let us imagine k linear congruences G lt . . . Gk, and h plane pencils 

 (P «j),'. . '. (Pk, *fc). Let us imagine moreover to be given a linear 

 system C of linear complexes to the amount in number of oo 6- k . 



An arbitrary right line g determines k linear complexes with the 



