1450 
well with regard to #2 as to the absolute circle, and if both the 
cone of direction and the isotropic cone are considered for a point 
of this plane as vertex, they have the two isotropic rays in that 
plane and passing through that vertex in common, so that J, ,, / 
lie on 42, as well as on the absolute circle. 
1 eo? ~~ 200? 
Consequently the surface 2 may be imagined to have arisen’ 
more intuitively in the following way. 
Let the tangent ¢ be drawn in a point P of x’, and the point at 
infinity 7 of it be connected with Z,; the connecting line cuts 
fin two points A\, and A,,, and if these points are connected 
with P, the two generatrices 6,, 6,, passing through P have been 
found. 
From this construction the order of 2 ensues at once and that 
in two ways, if we suppose for the moment that the above mentioned 
numbers d, #, 4,7, €, 6 are all zero. For, in the first place, the com- 
plete intersection of {2 with the plane of /” is easy to indicate, it 
consists of £? itself, counted twice, as 4” is evidently a nodal curve 
of £2, and further only of such 45°-lines with regard to this plane 
as may be esteemed to lie at the same time in this plane, Le. 
isotropic straight lines. Through each of the two isotropic points 
ijk 
passing through such a tangent and Z, touches 4% (as Z, is the 
eds, Of B pass u(u—1) tangents of this curve, and the plane 
pole of 1, with regard to #5). and consequently contains of £2 two 
coinciding generatrices, or rather only one generatrix, which in this 
plane itself, however, counts for 2, in any other plane passing 
through that line, as for instance g, for one ; the order of 2 is therefore 
m = u + 2u(u—1) = 2u? or = 2u + 2». 
We may, however, also easily determine the intersection of {2 
with the plane at infinity of space. If we suppose an arbitrary point 
K, of & connected with Z, u(u—1) tangent planes of 4’ will 
then pass through the connecting line; the lines connecting the 
points of contact with A, are the generatrices of {2 passing through 
2 is therefore for 2 a u(u—1)- or v-fold curve. 
But £# possesses further u points at infinity, whose tangents meet 
/ in those points themselves; the lines connecting those points with 
Z,, are therefore edges of {2 and that nodal edges, because they 
cut £ in two points. £2 therefore contains at infinity w nodal edges, 
vic. the lines connecting Z,, with the points at infinity of k’, and 
from this it ensues that we again find for order m: 2 u(u—1) + 
+2u=2u*. At the same time we observe that Z, as intersection 
of u nodal edges is a 2u-fold point of 2. 
this point; & 
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