68 THEORY OF COLLINEATIONS. 
80. Hight Conditions Determine a Collineation. Equa- 
tions (1) contain nine coefficients; but since we may divide 
the numerator and denominator of both fractions through by 
any one of the coefficients, it follows that there are only eight 
independent constants. Therefore eight independent condi- 
tions are sufficient to determine a collineation. Let the coor- 
dinates of four: points be (a, 4); (aga ey a (arena): 
and let their four corresponding points be respectively 
(2%;/; y;'); (x,", y.'); (x,'", Ue); (x,;", y:”) . Substituting in 
equations (1) successively the coordinates of each pair of cor- 
responding points we have eight equations from which to 
determine the eight independent constants in (1). These 
eight equations are linear and homogeneous in the nine coeffi- 
cients a, b, c, a’, ete., and therefore the eight independent 
constants are determined uniquely and completely. 
From the principle of duality we infer that a plane collinea- 
tion is also uniquely and completely determined by four lines 
and their four corresponding lines. 
It should be understood that the four points must be so 
chosen that no three of them lie on a line; if four lines are 
chosen, no three of them pass through a point. 
THEOREM 2. Any complete quadrangle or quadrilateral may be 
transformed into any other complete quadrangle or quadrilateral 
by a plane collineation in one and only one way. 
81. Cross-ratio Unaltered by a Collineation. It was es- 
tablished in Chapter I, article 9, that when two lines are 
projectively related, the cross-ratio of any four points of 
the one line is equal to the cross-ratio of the four correspond- 
ing points on the other line. This fact is independent of the 
position of the lines. They may be coincident, they may in- 
tersect and thus lie in the same plane, or they may be non- 
intersecting lines in space. The same theorem is true for 
two projectively related pencils, and is independent of the 
positions of the pencils. 
A plane collineation transforms a line g into g,; the range 
