NORMAL FORMS. 107 
are one-dimensional parabolic transformations whose charac- 
teristic constants are respectively t and t’. We must first 
find what relations exist between these constants i and k, and 
t and t’ respectively. 
135. k and k, are equal. In the invariant triangle ABC 
of type I the product of the cross-ratios k,, k,, k, is unity 
when they are reckoned in the same order around the 
triangle. In the triangle let C be moved to coincidence 
with A, then k,, along AC, reduces to unity (Art. 19) and 
we have k,k,=1 or k, = = . Now k,=A(CBPP,), hence 
S = A(BCPP,)=k,. Hence the characteristic cross-ratio 
of the transformation along AB is the same as that of the 
pencil through A, the order of the elements being as follows : 
k=(BAxxz,) = A(ULPP,), where x and x, are a pair of cor- 
responding points in the line AB or I’. 
THEOREM 24. Ina collineation of type Il the one-dimensional 
transformations along the invariant line 4 & and through the inva- 
riant point A are both characterized by the same cross-ratio 4. 
1 
aes B 
Fic. 19a. 
136. Relation between t and t’. If we consider the figure 
ABI as the limiting form of the triangle ABC, we ean find 
expressions for the parabolic constants ¢ and t’. Let the 
angle 1 AB be denoted by #, Fig. 19a. Along the side AC 
or Al we have t= lim. tae as C approaches A. In the pen- 
