( 753 ) 



tlie eighteen points of intersection of tiic curve witli «, are easy to 

 indicate. 



The obtained surface '">" is intersected bv the given Hnc /" in eight 

 points. So in general tliere are eight lines resting on I" and on four 

 corresponding rays I^, /,, I^, I,. 



3. In the preceding we have assumed that four coi'responding 

 rays /;, 4, /„, /, always admit of two common transversals, not 

 taking into account the possibility that four corresponding rays 

 have an hyperboloidic position. In the general case this singularity 

 does not occur ; for the condition that four lines are situated hyper- 

 boloidically is a threefold one and the number of corresponding 

 quadruplets of rays is only slnyly infinite. However, this does not 

 prevent a proper selection of the data from leading to projectively related 

 pencils with a quadruplet of coi'responding rays lying hyperboloidi- 

 cally ; to this end we have but to assume the points 0^, 0^. (>,, 0^ 

 on four hyperboloidic lines /'j, /%, /'j, /', and the planes «,, «j, «,, «, 

 through these same lines, and to fix the projective correspondence in 

 such a wa}' that these four lines correspond. 



If the case of four hyperboloidic rays l\, l\, I\, l\ really occurs, 

 tiie scroll 0"^ of the lines intersecting these four rays belongs to the 

 locus under consideration; we have thus further to investigate whether 

 this 0'' joins the surface 0' of the general case or Avhether this 

 surface breaks up in this special case into the surface (P and a 

 completing surface 0\ At the outset only the first possibility occurred 

 to me and I contented myself with developing grounds why this 

 ele\ation of the order of the locus from eight to ten need not really 

 clash with the wellkwown principle of the conservation of the number '). 



Ajthougli at first sight it seems rather absurd that the infinitesimal 

 small difference between four nearly and four perfecthj hyperboloidic 

 rays should rule the locus obtained by means of the remaining 

 quadruplets so as to let us find in the first case an O" and in the second 

 an 0\ yet as will be i»roved directly the second of the two sup- 

 positions mentioned above is the right one, not the tii-st ; so in that 

 sense this paper has had to be modified. 



The surface Ö' of the common trans\'ersals of any hyperboloidic 

 quadruplet l\, l\, l\, l\ contains these lines and so it must admit 

 of a transversal through each of the vertices O^, O^, 0^, 0^ and in 

 each of the planes «„ a^, «,, «,. The deduction of the order of the 

 locus 0" has shown that through each of the four points three 



1) See for this a corresponding case of ajiparent contradiclion in my "Mehrdimen- 

 slonale Geometrie", vol. I, page 2U3. 



