Projective geometry, by Oswald Veblen and John Wesley Young.

~~ 107,.10o s

THE LINEAiR COMPLEX

319

EXERCISES
1. All lines of a congruence can be constructed froml four lines by means
of regtli all of which have two given lines in coimmon.
2. Given two involutions (both having or both not having double points)
on two skew lines. Through each point of space there are two and only two
lines which are axes of perspectivity projecting one involution into the other,
i.e. such that two planes through conjugate pairs of the first involution pass
tlhrough a conjugate pair of the second involution. These lines constitute
two congruences.
3. All lines of a congruence meeting a line not in the congruence form
a regulus.
4. A linear congruence is self-polar with regard to any regulus of the
congruence.
5. A degenerate linear congruence consists of all lines meeting two intersecting lines.
108. The linear complex. THEOREM 19. A linear complex consists of all lines linearly depenedent on. the edges of a simple skew
pentagon. *
Proof. By definition (~ 106) the complex consists of all lines
linearly dependent on five independent lines. Let a be one of these
which does not meet the other four, b', c', d', e. The complex consists of
all lines dependent on a and the congruence b'cd'e'. If this congruence is degenerate, it consists of all lines dependent on three sides
of a triangle cde and a line b not in the plane of the triangle
(Theorems 14, 15). As b may be any line of a bundle, it may be
chosen so as to meet a; c may be chosen so as to meet b, and e may
be so chosen as to meet a. Thus in this case the complex depends
on five lines a, b, c, d, e not all coplanar, forming the edges of a simple
pentagon.
If the congruence is not degenerate, the four lines b", c", d", e" upon
which it depends may (Theorem 15) be chosen so that no two of
them intersect, but so that two and only two of them, b" and e",
meet a. Thus the complex consists of all lines linearly dependent
on the two flat pencils ab" and aet and the two lines c" and d". Let
b and e be the lines of these pencils (necessarily distinct from each
other and from a) which meet c" and d" respectively. The complex
then consists of all lines dependent on the flat pencils ab, be", ae, ed".
* The edges of a simple skew pentagon are five lines in a given order, not all
coplanar, each line intersecting its predecessor and the last meeting the first.