( 259 ) 



of r form a system oo'' of quadrics; the corresponding cones liave 

 their vertices on the surface of Jacobi of the system. This surface 

 contains ten lines, which are double lines of as many pairs of planes 

 belonging to the system. From this ensues that T contains ^6?z figures 

 consisting each of two conies cutting each other twice. 



Mathematics. — "A quartlc surface with twelve strnhjht Unes''' 

 By Prof. Jan de Vries. 



1. We regard as given the three pairs of straight Imas n, a' \b,b' ; 

 c, c'. Let ta denote a transversal of a and cd ; and let ti and tc have 

 an analogous meaning. The points P sending out three transversals 

 ta, t(j, tr lying in a plane, form a surface (P) of which we intend to 

 determine the order. 



First we notice that the six given lines belong to [P). For, if P 

 is a point of c and Q the point of intersection of c' with the plane 

 through the transversals ta,th, the transversal tc^PQ lies with 4, /^ 

 in a plane. 



We can designate six other lines lying on (P), viz: the two trans- 

 versals tab,t'ni> of the pairs a, a' ; b, b' and the analogous lines /f/,,. , /'i,^^ ; 

 tact'ac- For, tij coincides with ta for a point P on t,,ij, so that ta, 

 th and tc are complanar. 



Let tc be an arbitrarj^ transversal of c, c, in a plane t through tc. 

 The lines ta and //, lying in t determine on tc two points A and B 

 which describe [irojective series of points when x revolves; the two 

 coincidences A and B are evidently points of {P). The points of 

 intersection of 4- with c and c' also belonging to [P) the locus to 

 be found is a quartic surface. 



If we allow tc to describe a pencil, whose vertex C lies on c, 

 then the above mentioned coincidences describe a curve of order 

 three; for, if C' is the point of intersection of c' with the plane of 

 the lines ta, tb through C, then one of the coincidences A^ B or 

 tc=CC' lies in C. 



2. The surface is entirely determined by the ten lines a,a' ; b,b' ; c,c'; 

 tah,t'ab\ tact'ac- For, if ou each one of tiie first six lines we take 

 arbitrarily five points and on each of the remaining four lines one 

 point then the quartic surface determined by those thirt3-four points 

 will contain the ten lines mentioned. 



Being moreover as locus of the point P entirely determined a. 

 quartic surface tlirouyh tlie above-mentioned ten lines must contain 

 tivo other lines {viz: tbc, t' be)- 



