715 
xr, with respect to ®,,®, envelops the reciprocal figure of a 
cubic curve, i.e. a developable of class 3, and the tangential planes 
of these two developables are conjugated through the points P one 
by one to each other; for, a tangential plane a, of the first devel- 
opable has only one pole P and this again only one polar plane 
with respect to ®,. Now tbe lines s are the lines of intersection of 
the conjugated tangential planes of the two developables; they form 
a scroll the order of which appears to be 6. Let us namely assume 
a line /; through a point P of this line pass 3 tangential planes ar, 
of the first developable, and to these three planes a, are conjugated ; 
if these intersect the line / in three points Q, then to one point P 
three points Q are conjugated, but of course inversely too; through 
each of the 6 coincidences passes one line s, so the line / interseets 
the demanded surface in 6 points. 
For each of the three points of intersection of 4° with one of 
the four tetrahedral faces the corresponding line s passes through the 
Opposite vertex; the four cone vertices are therefore threefold points 
of the surface. Moreover the surface possesses a nodal curve cut by 
each generatrix in 6—2=4 points and which proves to be of 
order 10; the four cone vertices are as points of intersection of 3 
generatrices of the surface also threefold points of the ncdal curve. 
The order of the nodal curve we determine again as in § 9 with 
the aid of ScnuBeRT's formula: 
2.63 — EO + 2.89, 
by conjugating each generatrix of the scroll as ray g to all others 
as rays h. The symbol eg, the number of coinciding pairs where g 
intersects an arbitrary line, is 6, viz. equal to the order of the sur- 
face; the question is now how great is eo, the number of pairs gh 
of which the components lie at infinitesimal distance and intersect 
each other; these are evidently the torsal lines of the surface. We 
shall show that their number is 8. 
The rays s conjugated to the points of a line / describe a regulus 
through the four cone vertices ($4) and so they cross each other all, 
then too when they lie at infinitesimal distance; they can intersect 
each other only when line / is itself a ray s (§10); however, they 
then intersect each other all and that in the same point, viz. the 
focus of s. If thus two rays s corresponding to two points of 4? 
are to intersect each other, then their connecting line must be a 
ray s; and if moreover these rays are to lie at infinitesimal distance 
then the line connecting the points must be a tangent of k°; so the 
question is simply this how many tangents of k° are rays of the 
