NORMAL FORMS. 1138 
a p 
and #2’, respectively ; also «’*+ 3’*=1, and |, ,,|=1. The 
a | 
normal form of type III may be written 
i ay tk 
ABiA 
a § 0 At+a 
a! 0 A (at?+ht)+2ata-+ a! 
¢, = ; 
«sy (0) 
AN Bes oly eT 
a B 0 t 
a Bf 0 at?+ht (17) 
a R]Y a gl 0 
AieBs TB 
a 6 0 Bt+ ) 
a ff 0 B(at?+ht) +2at3+ (7 
Ug = 
(Same denominator.) 
140. Properties of Type IV. A collineation S of type IV 
leaves invariant a point O and every point on l, a line not 
passing through O, and the pencil of rays through O, Fig. 21. 
Fig. 21. 
—8 
