### Conjugate Line Congruences of the Third Order Defined by a Family of
Quadrics

Cornell University, 1910

Abstract

The first systematic study of algebraic congruences was made by Kummer, who
derived most of the forms of those of order 2. His work is both synthetic
and analytic, and is a model of elegance and directness. All of these
congruences can be arranged on a system of quadric surfaces in such a way
that only one system of generators comes in.

No other congruences have been studied from this point of view; the
present paper aims to do for congruences of order 3 what Kummer did for
those of order 2.

The totality of all lines of space satisfying two conditions is called a
congruence. The order of a congruence is the number of lines of the
congruence which pass through any point of space. The class of a congruence
is the number of lines of the congruence which lie in any plane in space.
A singular point is a point through which pass an infinite number of lines
of the congruence. A singular plane is a plane in which lie an infinite
number of lines of congruence. A basis point of a system of quadrics is a
point common to every quadric of the system.

Published in the American Journal of Mathematics, Vol. XXXV, No. 3 (1913),
323-356.