1207 
Of the a- (9-) planes through P two cut a straight line of R,; 
they intersect A, therefore in the a- and b-lines of a quadratic 
surface O?. On O' lies the projection & of the curve V,; & has the 
a- and 6-lines as p- and q-fold secants. The projections of % out of 
a point O of U? on a plane r of R* gives a curve k’, which has 
a p-fold and a q-fold point in the passages A, and B, of the a- and 
b-lines through O. The straight line A,B, has only the points A, 
and B, in common with 4’, hence &’ is of the order p+q. If r 
is the number of double points of 4, hence also of 4’, the class of 
Rs: 
(PN hl Ook) ag (G1) ar’ "2e 
so that out of A there can be drawn 
2 (pq—r) — 2g = 2g (pl) — 20 
and out of B 
2 (pq—r) — 2p = 2p (q—1) — 2r 
tangents to &’, touching this curve elsewhere. 
These numbers are at the same time the numbers of b- and a-lines 
touching & and also the numbers of the g- and the a-planes through 
P that have two coinciding points in common with V,. 
Now r is the number of bisecants of V, through P or the axis- 
degree (rank) of the congruence in consideration, hence: 
The focal surface of a congruence of the field-degree p, the sheaf- 
degree q and the axis-degree r is of the order 2p(q—1)—2r and of 
the class 2q(p—1)—2r. 
§ 5. I shall now consider the complete congruence of intersection 
of two complexes. 
If these are of the order m and n, they have as images the 
multiplicities which O*, has in common with two hypersurfaces 
Vi and V,". These two hypersurfaces cut each other in a multi- 
plicity V,m. The bisecants of V,”" passing through a point P and 
cutting a plane zr lie in the linear space Rh, =(P, 2); they are also 
the bisecants through P of a curve in PR, which is the complete 
intersection of a surface of the mt» order and a surface of the nth 
order. The number of these bisecants is 4 mn (m—1)(n—1), hence 
the bisecants of V,”" through P form a cone of 3 dimensions and 
of the order 5 mn (m—1)(n—1). By choosing P on O*, it appears 
that through this point there pass mn(m—1)(n—1) bisecants of Vy 
that lie on O?,, accordingly : 
The congruence of intersection of a complex of the m* order and 
a complex of the n™ order has the axis-degree mn (m—n)(n—1). 
