ANALYTIC THEORY. 67 
such transformations are inverse transformations. (Arts. 3 
and 27. ) 
78. The Correspondence of Lines is Also One-to-one. The 
transformations expressed by (1) and (3) always transform 
lines into lines. Let the equation of any chosen line be 
lx-+ my +n =0; substitute for « and y in this equation their 
values from (38) and we have 
Am + Aly +A! Bai+ Bly: +B! 
(eaeerera es cna tno! (4) 
Clearing of fractions and collecting, we get a linear equation 
in x, and y,, which represents a straight line. Hence the 
transformation (3) transforms straight lines into straight 
lines. In like manner the transformation (1) can be shown 
to transform lines into lines. 
79. A Collineation is Self-dualistic. The transformation 
expressed by equations (1) is capable of a double interpreta- 
tion according as the variables represent point or line coordi- 
nates. When (a,y) and (#,,y,) are point coordinates, 
equations (1) and (8) immediately show that. points are 
transformed into points; and we are able to prove as above 
that lines are also transformed into lines. 
On the other hand, if (a, y) and (~,, y,) are line coordinates, 
the equations show at once that lines are transformed into 
lines, and it can be shown by substituting in the equation of 
a point that points are also transformed into points. 
Although we shall have but little occasion for the explicit 
use of line coordinates, yet the dual interpretation should 
be held in mind and will often be of great use to us. 
THEOREM 1. A plane collineation transforms points into 
points and establishes a one-to-one correspondence between the 
points of the two configurations; it also transforms lines into lines 
and establishes a one-to-one correspondence between the lines of the 
twoconfigurations; a plane collineation is a self-dualistic transform- 
ation. 
