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Mathematics. — "The theorem of Grassmann in a space of n 

 dimensions." By Lijcien Godevux, at Morlanwelz (Hainault). 

 (Communicated by Prof. P. H. Schoute). 



We shall designate by the letter *S a linear space and the number 

 of dimensions of this space shall be the index. 



The notation Vi represents a variety, the locus of oo< elements 

 and of order j. 



The order of a variety, locus of spaces S/ c occurring in an 

 [n — k) (k-\-l) — 1 times infinite number in a space S„, is the number 

 of Sic of an iSjfc-j-i through an Sk—i of this Sk+ t and belonging to the 

 variety . 



1. In an S 3 the theorem of Grassmann can be read thus: 



The locus of S for which the N, which unite it to 

 three f i x e d S meet three fixed S, in three S c of the 

 same S t is a variety V 3 . 



In an S s it has been given it the two following forms: 



The locus of an S„ for which the S t which unite it 

 to four fixed »S', meet four fixed <s', in four S of a 

 same *S, is a V*. (Le Paige, Sur la generation cle certaines 

 surfaces par des faisceaux quadrilinéaires, Bul. de Belgique, 1884, 

 3 e série, tome VII 1). 



The locus of an »S' for which the S l which unite it 

 to four fixed -S meet four fixed S, in four S of a 

 s a m e S t i s a V*. 



2. Let there be in an S„ k S r - which we shall designate by A t and 

 k S s - which we shall designate by B;, (i = l, . . , k). 



Let /; be a number satisfying the 2k inecpialities 



n + p + 1 ^n — 1 (1) 



n + v+p + l>n, (i = l,...k) . . (2) 



A space S P determines with the k spaces A{ k spaces S r .XjU-i. 

 These spaces meet the corresponding spaces Bi in k spaces 



S'i+'ji+P— » + !• 



If these k spaces belong to an Si= n , 



2{rt + sd+ Hp-n + 2)-l) 



the space S p describes a variety F( B — p )( P +i)_ i the order of which 

 is to be found. 



Let us suppose we have 



2 (n + a-) + i (p - n + 2) = n + 1 (3) 



Let C be an *S),+i and D an ^S^-; of C. 



Let us designate by A an S p passing through D and situated in C. 



