6 ONE-DIMENSIONAL PROJECTIVE TRANSFORMATIONS. 
cua,’ +dx/ —ax’ —b=0, 
ex!" /’ +dax/’ — ax’ —b=0, (9) 
Cee + dg!" = aa!” i b = 0. 
These equations are linear and homogeneous in a, 8, ¢, d, 
and determine the ratios of these quantities uniquely and 
completely, provided no two of these equations are identical 
or have their coefficients proportional. 
THEOREM 3. There is one and only one projective transforma- 
tion that transforms three given points on a line into three other 
given points. 
8. The Identical Transformation.—Suppose that the trans- 
formation (1) leaves three points of the line invariant. If 
we put o/—a)) o> 2 and /"— a" in equations (9) these 
reduce to the following: 
cu? +(d—a)a’ —b=0, 
cu’? +(d—a)u” —b=0, (10) 
ce’? + (d—a)a’’—b=0. 
The determinant of these equations, 
ei2, oo! Th 
wa lt 1) = (a! — oe") (a — 2!") (a' — 2"), Gals) 
apftl2 rogtlt S14) 
does not vanish so long as the three points are distinct ; con- 
sequently, the coefficients of the above equations must vanish 
identically. Thus, ¢c=0,b=0, d=a. Putting these values 
in (1) we get #,=«, which is the identical transformation. 
The identical transformation we know transforms every point 
of the line into itself. 
THEOREM 4. A projective transformation which leaves three 
points of a line invariant is the identical transformation and leaves 
all points of the line invariant. 
9. Invariance of Cross-ratio. Let x, x’, v7”, x’, be the 
coordinates of any four points on the line. The function 
oa hl — ar 5 . : 
=: ~? 8 called the cross-ratio (Doppelverhaltniss, ra- 
gl—a! * all — 
tio anharmonique) of the four points. Let & be the value of 
