( '59 ) 



Fig. 2 



tetrahedron. From a simple inspection of the figure appears that 

 the three points, in which any of the faces of the tetraiiedron is cut 

 by the corresponding three rays lying in the other faces, are situated 

 on the second asymptote of the hyperbola passing through the three 

 vertices of that face and having the fourth corresponding ray lying 

 in that face as an asymptote. So this ensues inter alia for the face 

 ^'1,^4, J^ from the three relations : 



A,C, = i\A, , A,D,^I),A, , A,B, = B,A,. 

 So already four lines rest on /j, 4, /,, I,, namely one in each face, 

 which pro\es that the lines h, /,, l„ /^ have hyperboloidic position. 



6. We leave our oi'iginal problem for an other moment in order 

 to investigate first the series of quadratic surfaces furnished in the 

 last special case under consideration by the quadruplets of corre- 

 sponding rays. All these surfaces have eight points in common, the 

 four vertices 0^, 0^, 0,, 0, of the pencils and the four points 0^, 

 0^, 0., O3 symmetric to these with respect to the common centre 

 0; so they belong to the net jV^ of the quadratic surfaces deter- 

 mined by seven of those eight base-points O,-, forming in their turn 

 the vertices of a cube. We can likewise point out eight common 



53 



Proceedings Royal Acad. Amsterdam. Vol. Vlll. 



