( 264 ) 



Mathematics. — ''The coiKiruence of the conies situated <>it l/ic cnhic 

 surfcices of ei pencil." Bv Prof. Jan de Tries. 



(Communicated in the meeting of September 24, 1904). 



1. On eac'Ii (Mil)ic' surface S^ of a pencil lie 27 systems of conies, 

 of which each system lias one of the 27 right lines for common 

 chord. Through any point F of space passes one S^ of the pencil; 

 so P bears 27 conies C^ of the congruence formed by I he C"' of 

 all S\ 



The pairs of lines of the congruence evidently forin the skew sur- 

 face of the trisecants of the tuisecurxe /i" ; for the points of inter- 

 section of a right line of one of the ^S'' with a second aS' belong to 

 all S^ of tlie pencil. Quite independent of this consideration we 

 can find the degree of the al)Ove mentioned skew surface in the 

 following wa^^ 



Let 'Uk be a homogeneous function of degree /■ in .'',//, c. If we 

 take a point of R" to l)o the origin of the system of coordinates, 

 the pencil has for equation 



?'. + ^'. + //, = (), 

 where the coefficients contain a [)aranieter X to the first degree. 



If a right line through O must lie on S^, for all values of ni 

 the equation 



?«'?«3 + m^ii^ -f niu^ = 



must be satisfied, so we ha^•e simultaneously 



tt, = , u, = , u, = 0. 



By eliminating .c, //, j out of these three equations a relation is 

 found containing the coefficients, so ). too, in degree 



The basecurve /i'' is thus an eJevenfokl curve on the skew surface 

 of the right lines lying oji the surfaces >S'' of the pencil. 



Each surface S'^ bearing 27 right lines, its section with the indicated 

 skew surface is of degree 27 -j- 9 X li = 126; the skew surface is 

 thus of degree 42 ^). 



2. Any right line / is a chord of 42 conies, whose planes are 

 determined by the 42 right lines of the skew surface (?'' resting on /. 



Let us now consider the surface (P) formed by the C\ the planes 



1) See a.o. Clebsch, Legons sur la geometrie, II, 13. 



2) See a.o. Kluyver, Kenmerkende getallen der algebraïsche ruimtekromme 

 {Characteristic numbers of the algrbrair /nvf<(<'d rurre.) Vorsl. K. A. v. W., 3''^ 

 series vol. Yll, page 152, 1889. 



