8 
ALGEBRAIC INVARIANTS 
If l and V are identical, we first project the range on V 
on to a new line (A 'B' in Fig. 2) and proceed as before. 
Any linear fractional transformation L is therefore a pro 
jective transformation of the points of a line or of the points 
of one line into those of another line. The cross-ratio of any 
four points is invariant. 
3. Homogeneous Coordinates of a Point in a Line. They 
are introduced partly for the sake of avoiding infinite coor 
dinates. In fact, if 7^0, the value —8/y of x makes x' 
infinite. We set x=xi/x2, thereby defining only the ratio of 
the homogeneous coordinates xi, X2 of a point. Letx' = xi / /x2 / . 
Then, if p is a factor of proportionality, L may be given the 
homogeneous form 
pX\ =aXi+/3X2, pX2 =7X1 + 5X2, «5—/37 5^0. 
The nature of homogeneous coordinates of points in a 
line is brought out more clearly by a more general definition. 
We employ two fixed points A and B of the line as points of 
reference. We define the homogeneous coordinates of a point 
P of the line to be any two numbers x, y such that 
x _ AP 
where c is a constant 9^0, the same for all points P, while 
AP is a directed segment, so that AP = —PA. We agree 
to take y = 0 if P=B. Given P, we have the ratio of x to y. 
Conversely, given the latter ratio, we have the ratio of AP 
to PB, as well as their sum AP-\-PB=AB, and hence can 
find AP and therefore locate the point P. 
Just as we obtained in plane analytics {cf. § 1) the relations 
between the coordinates of the same point referred to two 
pairs of axes, so here we desire the values of x and y expressed 
in terms of the coordinates £ and r] of the same point P referred 
to new fixed points of reference A', B'. By definition, there 
is a certain new constant k 9^ 0 such that