( 705 ) 



/ • • • Q, P.r + S, r, + k (r, 7, + /?. ..,) ^ o - 



^/ • • • Q. /'v 4- '^^ '■-/ + /■(/', '/,, 4- /''. ^-/) = 'M 



///... Q, /'_- + S, r, + /• {/', 7, + A', x,) = ()'■■■ 



For wliicli \ allies of/: have these four ravs liyperlioloidie |i(isition ? 

 T(i Ihis eiul it is neccssai'j and siilüicieiil Ihal tlie points at infiiiitT of 

 I, II, til lie on a v\'^\\i line. So for / wc lind the cnbic eqnation : 



So in reality three reguli have separated from the surface ()". 



8. Finally we have still to indicate that all quadruplets of corre- 

 sponding' rays lie hyperI)oioidically if four (|nadra])lels do. This proof 

 we join on t(i the most general case of four arbitrary planes «,, «.^, 

 «,, «, and four arbitrary |)oints ()^. (>.., <>.,. <>_, in them. If .l,..],.!^ 

 is the face «^ no lontivr eipulateral, then tiie |)r()iective pencils (/J, (/j^ 

 (/,) describe on the sides .l^J,, J,,,l,, ,1.^J,, the projective series of 

 points ((',), (Z),), (/j^) possessing for /■ = 4 loui- tri|)lets of points on 

 the same right line. In that case the conies enveloped by the connect- 

 ing lines (\D, and I),B, have live common tangent.s, i.e.A^A^ and 

 the four lines liearing corresponding triplets of points ; then those 

 conies coincide. So the supposition of four siu-h triplets leads to the 

 case that there is an infinite number of such triplets. But then in 

 each of the four jtlanes «j, a.^, «,, «, lies a common trans\ersal of 

 each corresponding quadruplet, etc. 



We now conclude by showing that in order to deteriuiue four 

 projectively related pencils of rays \vitli merely hyperboloidic qua- 

 druplets the four planes ft,, tt^, k^, «, and the four vertices (>j, (J^, 

 0^,0^ in them can be taken arbitrarily by showing that to a ray /• 

 drawn arbitrarily in «j through O, only a single triplet of rays 

 l-i, /j> f, of the remaining pencils corresponds, forming with /, four 

 lines with hyperboloidic position crossing each other. 



If (/, I', Q represent successively the conditions that a quadratic 



surface contains a point, touches a line and touches a plane, then 



1 



_- F„ |2r^' - 3i-^ (ft + <)) + r(:i!i' + 2,,p + 3^)^) - 2(,t" + o")j 



indicates according to Hukwitz the number of surfaces through an 

 arbitrary line, which satisfy the sixfold condition F, {Mntlt. Ann., 

 vol, 10, page 354). So the nuntber of (puuiratic stu'faccs through /j 



