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corresponding rays luvs li_\ pcrboloidic position, the vertices not lying 

 in the same plane, the bearing planes not passing through the same 

 same point and those planes not being touched in tliose vertices by 

 all quadi-atic surfaces herewith generated. Analytically as well as 

 geometrically we can convince ourselves in a simple way of the 

 reverse. 



With respect to a rectangular system of coordinates {XYZ) the 

 four pairs of equations. 



y = p-v + 9 1 — y = p-e + q j // = — />•*■ -\-q) — n — — p-'' -f q | 



represent four lines 1^,1^,1^, It with hyperboloidic position. For the 

 conditions under Avhich the surface 



a,c' + Inf -\- cz^ = 1 



contains one of those lines are 



^ 4- ?,^;,^ + ,,^ ^ 0, hpq-^a'sz=zO, /y + ,>■'=.!, 



any of the four lines being taken. Now these lines /,, /,, /,, J^ hang 

 together in such a way, that by a rotation of 180° 



rounil the axis OX ihe liues l^ and /, and likewise tlie lines /j and l^ pass into each olher. 

 .. „ „ OY ., „ h ■■ h - ., ■■ V k .. /4 ■, ,. 



„ M :, OZ .,, ,. /, .. h .. ., ., : k .. h :■ ,. ., „ 



If now in a plane «, the line /, describes a pencil of rays with 

 0, as vertex, the lines 1^, 1^, 1^ will describe the pencils obtained by 

 making the pencil (/J undergo a rotation of 180° I'ound the axes 

 OX, OY, OZ, where the four vertices O^, 0„ 0„ 0, will not lie 

 in the same plane, and the bearing i)lanes will not pass through the 

 same point. And then is also excluded that the planes «j, «,,«,, «^ are 

 touched in 0^, 0^, 0,, 0^ by the generated quadratic surfaces. For two 

 quadratic surfaces touching each other in four points not situated in 

 one and the same plane coincide and the surfaces under consideration 

 do not. 



Let us consider geometrically a more special case connected with 

 a ]-egular tetrahedron. We start from a cube and take (fig. 2) one 

 of the two groups of four not adjacent vertices A^A^A^A^ as vertices 

 of this tetrahedron. Then the faces A^A^A,, A^A^At, A^A,A^,AiA,A, 

 of this tetrahedron are the bearing planes «j, «,, «3, «,, the centres 

 of those equilateral triangles are the vertices O^, O^, (),, O^ of those 

 pencils. And the rays I^, I,, /, corresponding to an arbitrary ray I^ 

 of the first pencil aic found again by a rotation of 180° round 

 the lines OX, OY, OZ through the centre of the cube parallel 

 to the edges of the cube, which are at the same time the connecting 

 lines EE' , FF', GO' of centres of pairs of opposite edges of the 



