12 TYPES AND NORMAL FORMS. 
which shows that the difference of the reciprocals of the dis- 
tances of a pair of corresponding points from the invariant 
point is constant for all pairs of corresponding points. Let x 
be the point at infinity on the line; ¢ is thus seen to be the 
reciprocal of the segment Ax, where x, is the point into which 
the point at infinity is transformed. 
THEOREM 9. Ina transformation of type I, &, the cross-ratio of 
the invariant points and a pair of corresponding points, is constant 
for all pairs of corresponding points; in a transformation of type II, 
t, the difference of the reciprocals of the distances of a pair of corres- 
ponding points, is constant for all pairs of corresponding points. 
16. The Natural Parameters. When the transformation 
is written in the form of equation (1), we see that there are 
three independent parameters viz., —, oe -*, when it is of 
type I; in the case of a transformation of type II, the relation 
(a+d)’=4(ad—bc), is satisfied, and there are but two inde- 
pendent parameters. The coefficients, a, b, c, d, have no 
simple geometric meanings; but in the normal forms A, A’, k 
and A, t have definite important geometric meanings. The 
parameters A, A’, k and A, t are called the natural parame- 
ters of the transformation. 
17. Explicit Normal Forms. Equations (16) and (19) may 
be put into the forms: 
ze 1) 0. ra wie (0 
Ay ie A A Olina 
! AyW || 
i = = ang: —_ : —s (22) 
A 1 1 A 1 1 
A’ 1 k 1 0 t 
These are called the explicit normal forms of types I and 
II, respectively. 
