902 
plane lie two rays of the complex cone and thus also two pairs gh, 
because each of the two rays can be either g or h; so op? =12. For 
ope finaly the point of contact must le in a given plane, the 
tangential plane must pass through a given point; so we can either 
apply the tangential planes in the points of a plane section of {2° 
and determine the class of the developable enveloped by it, or we 
can construct the cireumseribed cone and calculate the order of the 
curve of contact. The latter is the simplest; for the curve of contact 
is the intersection of 2° with the first polar surface of the vertex 
of the cone and therefore of order 6.5—2.3 == 24, because the 
first polar surface contains the nodal curve 4° and the latter counted 
twice separates itself from it. But the two complex rays through the point 
of contact and in the tangential plane count again for two pairs and 
so ope = 48, from which ensues sop = 48 + 48 + 12 — 48 = 60: 
so there lies on &° a certain curve k°° of order 60 having the property 
that the rays s conjugated to its points have coinciding foci and 
focal planes. 
We can ask how the curve &*° will bear itself with respect 
to the four cone vertices 7%, where the complex cone becomes 
indefinite. We now know however ont of § 21 that in the plane 
t; only one ray with coinciding foci lies, viz. the tangent in A to 
k?; so k°® will pass once through the four cone vertices. That for that 
tangent in A to 4’ the two focal planes do not coincide, is an 
accidental circumstance, which is further of no more inportance ; 
this result was based namely on the supposition that through an edge 
of the cone passes only one tangential plane of that cone; however, 
for the point A the complex cone breaks up into a pair of planes 
whose line of intersection is just the tangent in A to 4’, the tangential 
plane through that line to the cone is thus in first instance indefinite. 
The rays of the congruence with coinciding foci determine a scroll 
of which we will finally determine the order. To that end the scroll 
must be intersected by an arbitrary line and we now know that all 
rays of the congruence meeting a line / form a regulus 2*° and 
that the foci of those rays are situated on a curve £'* lying on £2° 
and passing singly through the 4 cone vertices. It is clear that to a 
point of intersection of 47° and °° a ray corresponds with coinciding 
foci and eutting /, with the exception of the cone vertices; for, to 7% 
is conjugated as regards /:"° the tangent in A to 4’, on the othér 
hand as regards #* the connecting line of the point of intersection 
of / and 1;- with A, as we now know. Now &” is, as we ‘know, 
the complete intersection of 2*° witha regulus; so the complete number of 
points of intersection of 4“ and £°° amounts to 120. If we set apart 
