Mathematics. — "TVte Complex of the Conies ivhich cut Five 

 Given Straiqht Lines." By Dr. G. Schaakk. (Communicated 

 by Prof. Hendrik de Vries). 



(Communicated at the meeling of June 30, 1923). 



^ 1. We can represent the conies X' cutting five given straight 

 lines rt,, a,, at,, a^, </, on the points of space by associating to each 

 of these conies the pole K of its plane z relative to a given quadratic 

 surface 0. To any point K there corresponds tlie conic P in the 

 polar plane x of A' passing through the points of intersection 

 A^, A,, . . , A^ of ttj, (/,, . . , (/j with X. 



For this representation the points of the straight lines a'l, fl'',, .., a', 

 which are associated to a^, a,, . . , a^ relative to 0, are singular. 

 If we take for instance K on (/',, x passes through a, and A^ 

 becomes accordingly indefinite. To K there corresftonds the pencil 

 of conies I' passing through the points of intersection A^, ■ . , A^ 

 of X with (7,, . . , <?(. These are double conies of the system «S, under 

 consideration of go' individuals. 



There are accordingly jive straight lines a'u of singular points of 

 the second order. To a point of any of these straight lines there 

 corresponds a pencil of double conies of »S,. Each of these straight 

 lines is the representation of a system of oo' conies the planes of 

 ivhich pass throtigh one of the straight lines a^ and ivhich cut the 

 other four of these lines. 



If we choose K on one of the two straight lines t\^ and t\^ 

 cutting the lines a\, a\, a\, a\, e.g. on i',,, x passes through the 

 associated straight line <,, intersecting a„..,a^, and this plane 

 contains oo' degenerate conies of S, associated to K consisting 

 of <i, and a straight line through the point of intersection ^4, of 

 X and (7,. 



There are aecordingly ten straight lines t\^, t\, . . . ., /',,, t\, of 

 singular points of the frst order. To a point of any of these 

 lines there corresponds a pencil of degenerate conies and each of 

 these straight lines is the representation of a system of oo' degenerate 

 conies of which one straight line is fi.ved and the other straight lines 

 form a bilinear congruence. 



§ 2. If K describes a straight line /', x revolves round the asso- 



34 



Proceedings Royal Acad. Amsterdam. Vol. XXVI. 



