722 
likewise a fourfold “branchplane”) Q. E. D. In $ 17 we shall see 
a confirmation of the considerations given here. 
$ 15. In order to be able to point out the eventual existence of 
the torsal lines of the third kind, we must include a new auxiliary 
surface in our consideration, which we deduce from the tetrahedral 
complex. All complex rays lying in one and the same plane envelop 
a conic which also touches the four tetrahedral planes, and indeed 
in $ 13 we have already drawn attention to the fact that the 
fourteen generatrices of °° lying in a plane À through / are the 
tangents of a conic; the auxiliary surface which we must introduce 
to find the torsal lines of the third kind is the locus of these conics, 
thus the locus of the complex conics lying in the planes À through 2. 
In each plane 4 lies one and through each point of 2 pass two of these 
conics, as is easy to prove. For, let us imagine an arbitrary plane 2 and an 
arbitrary point P on /, then 2 intersects the complex cone of P in 
two rays s, and these are the tangents out of P to k* lying in 2; 
therefore if k° is to pass through P then the two tangents out of P 
must coincide, and this takes place in the two tangential planes 
through / to the complex cone. The locus to be faund is therefore an 
2* with double line 1. 
If a surface possesses a double line it is an ordinary pheno- 
menon that only a part is efficient, the rest parasitical; so applied 
to our case that through certain points of / two real conics go, 
through others two conjugated imaginary ones, and through the limit- 
ing points between both groups two coinciding ones; for the surface 
we have here under discussion those limiting points are the points of 
intersection of / with the four tetrahedral planes. Let us namely assume 
the point of intersection s, of / with r,. The complex cone of S, 
breaks up into two planes, viz. t, and a plane through S, and 7, 
cutting t, along a line s, through S,; s, is nothing else but 
the generatrix which 2% has in common with t,. Now the tangen- 
tial planes through / to this degenerated cone coincide in the plane Js,, 
which bears a complex conic touching rt, in S, (with tangent s,) ; 
this conic is the only one passing through S,. 
Of great importance for our surface 2* are furthermore the planes 
through / and the four cone vertices. We know ia. that of the 
fourteen generatrices of @° in the plane /7, nine pass through 7, 
and the other five through a point 7,* lying in r,, and really the 
complex conic in this plane breaks up into the pair of points 77, 
T,*, which means for the surface @* that it is intersected by the 
plane /7, (except in the nodal line / of course) in the line 
