1250 
curves A and B need at most be of degree (r—m) and (r—n). If 
this is not the case, however, the terms of the highest degree of 
AF, and BF, will cancel each other; as the terms of the highest 
degree in F, and F, have no common factor, those of AF, and 
BF, will be divisible by those of MM. 
Let us therefore suppose 
AF, =A'F,F, + A'F. 
BF, =BF UF, + BIE, 
in which we extend the division by /',/, only so far that the 
terms of the highest degree in Af, and ALI’, have disappeared, 
we have 
AZS B 
So we find 
P =A" F,. BF, 
in which A" and B" are of a lower degree than A and 5. So we 
may go on till we find 
F,=AOF, + BOF, 
in which A© and B are at most of degree “r—m) and (r—n). 
Mathematics. — “A bilinear congruence of rational twisted quintics’’. 
By Professor JAN pr Vrins. 
(Communicated in the meeting of March 27, 1915). 
1. The base-curves of the pencils belonging to a net | &*| of cubic 
surfaces form a bilinear congruence. For through an arbitrary point 
passes only one curve, and an arbitrary straight line is chord of 
one curve; for the involution /,*, which the net determines on that 
line, has one neutral pair of points. 
We shall consider the particular net, the base of which consists 
of the twisted cubic o*, the straight line s and the points F,, F,, F. *) 
The surfaces ®*, which connect this basis with a point P have 
moreover a twisted quintic g° in common. A bilinear congruence 
|9°| is therefore determined by | #*]. A plane passing through s 
intersects two arbitrary surfaces of the net in two conics; of their 
intersections three lie on o’, the fourth belongs to @° ; consequently 
this curve has four points in common with s, is therefore rational. 
The straight line s is apparently a singular quadrisecant. 
The figure consisting of s, 6° and o° is, as complete intersection 
') Two other particular nets | have considered in two communications placed 
in volume XVI (p. 733 and p. 1186) of these ‘ Proceedings”. They determine 
bilinear congruences of twisted quartics (1st and 2nd species). 
