896 
And as regards finally the number of 4 > 102 = 408 points of 
intersection of the nodal curve with 9‘, in the four points 7; lie 
again 288 (comp. § 18), in the pinch points of the torsal lines of the 
second kind 54, in those of the six nodal edges 18 and in the four 
points S; which are sixfold for the nodal curve, 48, together 
288 + 54+ 18 + 48 = 408. 
20. The two particular pinch points on 7 which we have found in 
the preceding § were the two foci of the ray of the congruence / 
and the two torsal planes the two focal planes; for, in these points / 
was eut-by a ray of the congruence at infinitesimal distance. If 
henceforth with a slight modification in the notation the line / is 
called s,, the focus /,, then P, lies on @° and it is in general 
an ordinary point of this surface. Let us assume the tangential 
plane in this point and in it an arbitrary line ¢ through P,; then 
this has two conjugated lines crossing each other, and if therefore a 
point P describes the line ¢, the ray s of the complex conjugated 
to P will generate a regulus to which also belongs our ray s,, 
a ray of the congruence. As however t is a tangent of 2°, a second 
generatrix of the regulus lying at infinitesimal distance from s, will 
belong to the congruence, however without cutting s,. If however, 
we now imagine the complex cone at point P, and if we intersect 
it by the tangential plane, we find two lines ¢ which are at the 
same time lines s, viz. rays of the complex, and whose two conju- 
gated lines cut each other. Now the lines s conjugated to the points 
P of t will deseribe two cones containing also s,, and having their 
vertices on s, whilst we know out of our former considerations 
that these vertices are nothing but the foci of the two rays ¢; and 
now s, will be cut in each of these foci by a ray of the congru- 
ence at infinitesimal distance; the two cone vertices are thus the 
foci of s,. So: we find the foct of a ray s, of the congruence by 
determining the focus P, (lying on 2) of s,, by intersecting the com- 
plex cone of this point by the tangential plane in P, to 2°, and by 
taking the foci of the two lines of intersection t. And the two focal 
planes are the tangential planes through s, 10 the complex cones of 
the foci. 
If P, is a point of the nodal curve 4* of 2° then s, is a double 
ray of the congruence ($ 12); the complex cone of P, intersects the 
two tangential planes of P, in twice two rays ¢, so that we now 
have on s, two pairs of foci and through s, two pairs of focal planes ; 
and as the focal surface of the congruence is touched by each ray of 
the congruence in the two foci, so each double ray will touch the 
