106 THEORY OF COLLINEATIONS. 
Expanding this equation we get 
oe —(1+k+k’) Dp? (kk'+k+k’) D’p —kk’ D? =0,. (18) 
A B 1 | 
where ID) = OT 
All B’ il | 
This equation factors at once into (p — D) (pe —kD) (p —k’D) 
= 0; the three roots are therefore D, kD, k’D. 
THEOREM 23. The roots of the characteristic equation of the 
normal form of Z are D, 4D, kD, where D is twice the area of the 
invariant triangle of 7. 
134. Properties of Type II. A collineation of type II may 
be regarded as the limiting case of a collineation of type I 
when two vertices of the invariant triangle approach coinci- 
dence. Let T’ be a collineation of type II leaving invariant 
the figure ABI, Fig. 19. Along the invariant line AB and 
Fic. 19. 
in the pencil through A there are one-dimensional transfor- 
mations whose cross-ratios are respectively k and k,. Along 
the invariant line / and through the invariant point B there 
