6 
ALGEBRAIC INVARIANTS 
so that k is a finite constant =^0, if C is distinct from A and 
B, and hence C distinct from A' and B'. Solving for x', we 
obtain a relation 
L: 
In fact, 
a = b' — ka', @ = ka'b — ab', y — l—k, 8 = bk—a. 
If we multiply the elements of the first column of A by b and 
add the oroducts to the elements of the second column, we 
get 
_ b' — ka' b'{b—d) 
= {b — a) \ a ^ =k(b—a)(b'—a')9 i 0, 
— k 1 
l—k b—a 
if B and A are distinct, so that B' and A' are distinct. 
Hence a projectivity between two ranges defines a linear 
fractional transformation L between the coordinate x of a 
general point of one range and the coordinate x of the corre 
sponding point of the other range. The transformation is 
uniquely determined by the coordinates of three distinct points 
of one range and those of the corresponding points of the other 
range. If the ranges are on the same line and if A'=A, 
B'—B, C'=C, then k — \, a= 8, /3 = 7 = 0, and x'—x. Thus 
(.ABCD) = (ABCD') implies D' = D. 
Conversely, if L is any given linear fractional transfor 
mation (of determinant ^ 0) and if each value of x is inter 
preted as the coordinate of a point on any given straight line 
l and the value of x' determined by L as the coordinate of a 
corresponding point on any second given straight line l', the 
correspondence between the resulting two ranges is a pro 
jectivity. This is proved as follows: 
Let A, B, C, D be the four’ points of l whose respective 
coordinates are four distinct values xi, X2, xs, x± of x such 
that yXi+8^0. The corresponding values xi, X2 , x% , x± of