298 THEORY OF COLLINEATIONS. 
Substituting the values of «,and y, from equations (15) we get 
[a(ax+by+e)+8(a'e+ b’y+c’)]? + 29 (av+ by+ce)+ 
2f(a'x + b’y+c¢’)+d=0; this may be written in the form 
[(aa + Ca')a+ (ab+ 2b’) y]?+2Ge+2Fy+C=0, (17) 
which is again the equation of a parabola. 
THEOREM 22. Parallel lines are transformed into parallel lines 
and parabolas into parabolas by all collineations of the group @;(/. ). 
354. All Areas are Altered by a Constant Ratio R. One 
of the most important properties of group G,(1~) is that any 
collineation T of the group changes all areas of the plane by 
a constant ratio R. For example, if A represents any area 
of the plane, it is transformed by T into a new area A,, such 
that A, = RA; where F is a function of the parameter of T 
only and independent of the position, shape or size of A. 
To prove* this consider any triangle (ABC) and its corre- 
sponding triangle (A,B,C,). Draw lines through the vertices 
of (ABC) parallel to the opposite sides; three new triangles 
will be thus constructed; draw new lines through these ver- 
tices and so continue until the whole plane is divided into a 
net of equal triangles; do the same for (A,B,C,). The tri- 
angles of the second net evidently correspond to those of the 
first since parallel lines are transformed into parallel lines. 
Consider any area A and the corresponding area A,._ Each 
area is made up of the same number of whole triangles and 
parts of triangles. Hence we have 
A nA+e 
: ee ee (18) 
where A is the area of the triangle ABC and A, that of the 
triangle A,B,C,, and where e is the sum of all the pieces of 
triangles within A and e, the sum of all the pieces of triangles 
within A,. Wecan make the quantities e and e, as small as 
we please by taking the triangle ABC and its corresponding 
*A.Emch, Annals of Mathematics, vol. 10, pp. 2-4. 
