( 75fi ) 



transversals pass and likewise tliere are three in eaci) of the four 

 planes «. Hence the question, whetlier besides the scroll 0' a surface 

 0' or a surface 0° presents itself, can be decided by the fact whether 

 the four generators through the points and the four generators 

 in the planes « are common to the two parts of the locus or not. 

 Now, as a matter of fact, those two parts can have but two gene- 

 rators in common, viz. those two common transversals of /'j, /',,/',, /', 

 joining the preceding pairs of transversals and the following of tlie 

 adjacent quadruplets. So the eight indicated transversals oï I\, l\, l\, l\ 

 are not situated on the other part of the locus and consequently the 

 latter is cut by each of the planes «,- according to a curve c^ with 

 a node in Oi and two right lines ; so the remaining part is a surface 

 0' with a nodal curve of order nine. For the scroll 0^ of genus 

 three appears instead a combination of a regulus 0' and a scroll 0' 

 of genus one cutting each other in two right lines and a twisted 

 curve of order ten. 



From the preceding follows immediately what will happen when 

 the singularity of the hyperboloidic quadruplet presents itself more 

 than once. If two of those particular quadruplets are at hand 0^ 

 breaks up into three parts, two quadratic reguli and a scroll 0* 

 with a twisted cubic as nodal curve ; so the latter principal com- 

 ponent part of the locus is of genus zero and has Avith each of the 

 two quadratic surfaces two generating transversals and a twisted 

 curve of order six in common. If the projectively related pencils 

 contain three hyperboloidic quadruplets 0" breaks up into four 

 quadratic reguli, three of which answer to these quadruplets whilst 

 the fourth, really the locus, is supplied by all the remaining qua- 

 druplets ; tlie latter surface is intersected by each of the others 

 according to the edges of a skew quadi-ilateral, whilst these three 

 intersect each other in general according to twisted curves of order 

 four. And if there are four hyperboloidic quadruplets, as will 

 appear later on, all quadruplets are situated hyperboloidically; then 

 the case presents itself where the order of the locus, so far always 

 eight, becomes infinite. 



4. The following simple example will show that it is not difficult 

 to choose the data so as to allow each quadruplet of corresponding 

 rays to lie hyperboloidically. 



We imagine the four pencils (/,), (/,), (/j), (/J situated in the four 

 sides of a cube (tig. 1), we assume the vertices O^, 0^, 0,, C\ of 

 the pencils in the centres of these sides and we allow those rays 

 A' ^i' 4. ^i tx) correspond which form the same angle (f with their 



