883 
The complex cone of point P has PB for double edge; for PB 
cuts two generatrices a and at the same time the corresponding 
lines 6. So the congruence (3,3) of the general case must break up 
here into a (1,0), a (0,1) and a (2,2). 
In order to check this we consider the correspondence between 
the points A a,a, and the corresponding planes 8 = 6,b,. If A 
describes a line, a, and a, generate an involution; as 6, and 6,” do 
then likewise, 8 will rotate about a fixed axis. So the correspondence 
(A, 8) is a correlation. Therefore plane « contains a conic «,?, each 
point A, of which is incident with the trace 6, of the homologous 
plane @,. So each point A, is the vertex of a pencil belonging to 
the complex and lying in plane 3,. These pencils generate a con- 
gruence (2,2). For their planes envelop a quadratic cone with vertex 
B, two tangential planes 8, of which pass through the arbitrarily 
chosen point P; so the lines connecting P with the homologous 
points A, belong to the congruence in question, which evidently is 
dual in itself. 
$ 6. We will now suppose that the tangents a of the conic a? 
in plane @ and the tangents 5 of the conic 3° in plane @ are in 
(1, 1)-correspondence. Then the congruence with any pair of corre- 
sponding tangents a, 4 as director lines generates once more a 
complex of order four, evidently not dual in itself. jj 
By the correspondence (a, 4) the points of the line -c common to 
« and g are arranged in a (2,2)-correspondence. The four coinci- 
dencies are principal points of the complex and the lines a, 4 con- 
curring in any of these points determine a principal plane. So we 
have indicated four sheaves of rays and four fields of rays belonging 
to the complex. 3 
The planes «a and @ are also fields of rays of the complex; for 
any line s of a is cut on c by two lines 5 but also by the eorre- 
sponding lines a; so s belongs twice to the complex. 
We account for this by saying that « and 3 are double principal 
planes. 
The complex cone of any point ? meets c in four points, i.e. in 
the four principal points; so we deal with a biguadratie complex. 
The complex cone is rational, its edges corresponding one to one 
to the tangents of a’; therefore it has to admit three double edges. 
Likewise the complex curve of any plane has to admit three double 
tangents. 
§ 7. In order to investigate this more closely we consider ‚the 
