897 
focal surface four times. The four tangential planes are the focal 
planes, however in such a way that if one pair of foci is called 
FF, the focal plane of #, is tangential plane in #,and reversely. 
Let P, be a point of £*, lying as a single curve on @' then s, is 
the tangent to £* in P, and it belongs to the congruence. The com- 
plex cone of /, intersects the tangential plane in this point to {2° 
according to s, itself and an other generatrix : so of the two foci 
of s, point P, is one whilst the other is the focus of the second 
generatrix of the complex cone of P, lying in the tangential plane; 
and of the two focal planes the osculation plane of k* in P, is one, 
because this really contains two rays of the congruence intersecting 
each other in P, and lying at infinitesimal distance (viz. two tangents 
of £©); so it touches the focal surface in the other focus, i.e. the 
surface of tangents of 4“ which is of order 8 envelops the focal 
surface, and the curve 4‘ itself lies on the focal surface. 
The question how the cone vertices 7; bear themselves with 
respect to the congruence, is already answered in § 11; @° 
intersects the plane rt; according to a plane 4° and the rays s con- 
iugated to these form a cone of order 9 with the vertex 7; and 
with three nodal edges and three fourfold edges, the latter of which 
coincide with the three tetrahedron edges through 7 
Let us assume an arbitrary point P of 4°, then-to this a ray s 
through 7; is conjugated; now the complex cone of P degenerates 
into a pair of planes, of which 1; is one component, whilst the other 
passes through 75, and this degenerated cone cuts the tangential plane 
in P to @° along the tangent ¢ in P to #* and according to an other 
line # through P. To that tangent the point 7; is conjugated as 
focus, so that for each ray of the congruence through 7; this point 
itself is one of the foci, the other being the focus of the line #. 
In order to find the foeal plane of the considered ray s in the 
point 7; we should have to know according to the preceding the 
complex cone of 7; which is in first instance entirely indefinite ; 
let us however bear in mind that in the general case that complex 
cone is at the same time the locus of the ray s conjugated to the 
points of the tangent ¢; then in this case also we can have a defi- 
nite cone, viz. the cone which replaces the regulus if the line / passes 
into a complex ray s, and which contains in general the four 
cone vertices and which will contain here, where 7; itself is the 
cone vertex, the three tetrahedron edges through this point. On this 
cone lie the two rays s conjugated to the two points of £° lying 
at infinitesimal distance from each other on ¢, and the plane through these 
is the focal plane of our ray s in 7; but those edges of the qua- 
