080 



tv)o points A ; the correponding lines a fuiin the completing figure. 

 The conic {B,)" has in common with y' the two coincidences of 

 1' lying on it and the coincidence of the (P') lying on B. As it 

 cannot apparently contain a coincidence of an other P it must pass 

 through B^ and ^3, while it 'touches 7' in B^ . 



8. k conic is transformed by {P,P' ) into a figure of the tenth 

 order. For the conic {B,f it consists of twice (^1)' itself, the conies 

 {B^)\ {BsY and two lines a ; it bears consequently tivo points A, 

 which we shall indicate by A, and A^. As these points each form 

 a triangle of involution with B^ and another point of {B^Y, the 

 lines a, and «j* pass through B^ . 



Analogously we shall indicate the singular lines which meet in 

 B, and in ^3, by a„ a,* and a„ a,*; the points A, and ^/ are 

 then situated on {B^Y ; A^ and .^3* on {Bj-. 



On a^ two more points A are lying; one of them belongs to 

 {BJ\ the other to (B^Y; we may indicate them by .4,* and A*. 



If we act analogously with the remaining points A and lines a, 

 then the sides a„ a„ a, of the triangle Ji*^,*^3* will pass through 

 B^, B^, B^, and the same holds good concerning the sides a*, a/% a,* 

 of the triangle A^A^A^^. 



In connection with the symmetry, which is involved by tlie 

 quadratic correspondence {P,p), the lines />,, h^, b, contain respecti- 

 vely the pairs A„ J^*; A„ .4/; .4,, .43*. The triangle of the lines 

 b has C\, C„ C\ as vertices; analogously c„c,,c, are the sides of 

 B^,B^,B^. 



The j^Lc points A, and the ^Art^e points B form with the 6^2.r straight 

 lines a a conjiguration (9„ 63) B '), the points A with the straight 

 line a and the straight lines b the reciprocal configuration (63, 9^) B. 



9. That the involution (P') discussed above exists, may be proved 

 as follows. 



We consider the congruence formed by the twisted cubics (f\ 

 which pass through tioo given points G, G^ and has as bisecants 

 three given lines g„g,,g, ') By hkt and A% we indicate the trans- 

 versals 0Ï gic,gi, which may be drawn out of G and G*. 



Let us now consider the net of cubic surfaces ¥'', which pass 



1) A configuration (93,63) A consists of two triplets of lines Pi, p^, Ih ; Qi^ Q^^ Qs 

 and the 9 points {Pj^Qi). 



~) This congruence has been inquired into by analytic method by M. Stuyvaert 

 ("Etude de quelques surfaces algébriques ..." Dissertation inaugurale Gand, 

 Hoste, 1902). 



