Proceedings of the section of sciences - DWC - KNAW

( 759 )
Fig. 2
tetrahedl'on .. From a simple inspection of the figure appears th at
the three points, in which any of' the faces of the tetraheclron is cut
by the corresponcling three rays lying in the other races, are sitl1atecl
on the second asymptote of the hypel'bola passing through the three
vertices of that face anel having the fom-th corresponding ray lying
in that face as an asymptote. So this ensues inter alia for the face
AsAaA( ftom the three l'elations:
A 8e, ::::: C1A( , A4DJ::::: D1A, , A 2B(::::: B1As'
So already four lines rest on ll' l" la, l4' nameJy 011e in each face,
which proves that the lines l1' l" la, l4 have hyperboloidic position.
6. We leave Oul' ol'iginal problem for an other moment in order
to investigate first the series of ql1adratic surf aces furnished in the
last special case l1J1eler consic1eration by the quadl'uplets of C01'responding
rays. All these surfaces have eight points in common, the
foUl' vertices Ol' 0" 0 8 , 0 4 of the pencils and the foul' points 0["
0 0 , 0 7 , Os symmetric to these with respect to the common cenÜ'e
0; so they belong to the net lVp of the quadratic sUl'faces determined
by seven of those eight base-points Oj, fOl'ming in their turn
the vPl'lices of a eube. 'Ve can likewise point out eight ('ommon
53
Proceedings Royal Acad. Amsterdam. Vol. VUL
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