32 REAL TRANSFORMATIONS. 
$5. Projective Transformations. 
38. The theory developed in the preceding $$ 1-4 is per- 
fect, complete, and perfectly general. The special case of 
greatest interest is that in which the variables « and «, and 
the constants a, b, c, d in the equation, 
az+b 
ty = copa? (1) 
are real numbers. Such a transformation transforms real 
points into real points; for if real values of x are substituted 
in equation (1), a, b, c, d being real, then ~«, is also real. 
The theory as developed in § 1 is modified in the case of 
real transformations in only one particular, viz. : in regard to 
the invariant points of the transformations. The invariant 
points of the transformation (1) are given by the roots of the 
quadratic equation 
cv +(d—-—a)x—b=0. 
With real coefficients the roots of this equation are real and 
unequal, real and equal, or conjugate imaginary, according as 
(a+d)— 4(ad — bc) : 0. There are thus three kinds of 
real projective transformations of the points ona line, dis- 
tinguished by the character of the invariant points. When 
the invariant points of the transformation are real and dis- 
tinct, it is called a hyperbolic transformation; when they are 
coincident, it is called parabolic; when they are conjugate 
imaginary, it is called elliptic. 
The character of the cross-ratio k is also different in the 
hyperbolic and elliptic cases. From equations (18) and (15) 
it follows that k is real when A and A’ are real; and com- 
plex, when A and A’ are conjugate imaginary. It follows 
also from equation (15) that in the elliptic case, k is a com- 
plex number, and |k| = 1, 2. e., k = e”, where 
a+d 
2 Nad —be 
1+k 
() = Aowyre GOS = 207TC COS, =, - 
