TYPES AND NORMAL FORMS. 11 
14. Implicit Normal Form of Type II.—A transformation 
of type II, whose single invariant point is A, is reducible to 
the form 
1 1 
ay |e ee == be (18) 
To verify this, solve for x,; thus, 
an 2 GareQeao“k 
t= tx+ (1-tA) . (19) 
This is the same form as (1). A and ¢ are found in terms of 
a, b, c, d, as before, by comparing coefficients and solving for 
A and t; thus, 
a—d 2c 
Ava on ane te— aaa" (20) 
Equation (18) is called the implicit normal form of type II. 
THEOREM 8. Every transformation of the form, j= 
belongs to one or the other of the implicit normal forms, 
rw—A x 1 1 
aA k x — Al Oe mo 7 mew + t. 
15. Geometrical Interpretation of the Normal Forms.— 
The normal form of type I may be written: 
b= ang = (A'Ame,) ; (21) 
i. e., k is the eross-ratio of the four points A’, A, x, #,, where 
A’ and A are the invariant points, and x and «, a pair of cor- 
responding points. Here « and «, are any pair of correspond- 
ing points, and k is a constant quantity. 
In the normal form of type II the expressions x—A and 
x,—A are the distances of a pair of corresponding points 
from the invariant point. The normal form of type II may be 
written : F 1 
Beka! ye ig aaa t (18’) 
