116 THEORY OF COLLINEATIONS. 
A were equal. Hence it follows in the degenerate form S 
that the characteristic cross-ratio along any invariant line is 
equal to that in any invariant pencil. 
THEOREM 28. A collineation of type I degenerates into one of 
type IV whenever the cross-ratio along any side of the invariant tri- 
angle is unity and the two vertices on that side do not coincide. 
A collineation of type Il degenerates into one of type [V when the 
parabolic constant along A/ is zero. 
144. Normal Form of Type IV. The equations of the 
normal form of type IV are gotten from those of type I by 
making k’=k, k=1 or k’=1. In the first case the line 
joining (A’, B’) and (A”, B’’) is the line of invariant points. 
If k’=1, the line of invariant points is that joining (A, B) 
and (A”, B’); if k=1, the line of invariant points is the 
line joining (A, B) and (A’, B’). 
The same equations may also be gotten by making ¢=0 in 
the normal form of type II. The line A/ of Fig. 19 then be- 
comes the line of invariant points and B is the isolated inva- 
riant point. The last equations are: 
mo OW tf | @ i @) 
AS fel Al ak Yay IP 183 
A! Bi ft kA! PAV Bela 3,4) 
c ce 0 c¢ le ec Oc | 
a, = —, y= ————.._ (18) 
x 0: in hp th) 
7 dep ak al YAN dep UN 
A’ B1sk Al OB! 1 ke 
Bo Gg Oo Oo 8 Y 
145. Properties of Type V. In the case of an elation the 
invariant figure consists of all points on a line / and all lines 
through a point Oon/. An elation S’ transforms a point P 
of the plane into P, some other point on the line OP. On 
each of the invariant lines through O there is a one-dimen- 
sional parabolic transformation having its single invariant 
point at O. Every pencil of lines having its vertex at A, any 
point on J, is an invariant pencil of the collineation. On each 
