k-RELATIONS. 199 
If the invariant point of G,(A) is (A’, B’,C’), we get in the 
same way the relation 
kel kl ke! ; 
Keo? a ke2key2 ‘ (43 ) 
If (A”, B”, C’’) is the invariant point, we get 
kes Keke, 
Ke? 7 jRE kiy2 ° (43’’) 
In all three cases we see that the product of the two cross- 
ratios in T, along the two sides of the invariant which meet 
in the common invariant point, both cross-ratios being taken 
in directions away from the point, is equal to the product of 
the corresponding cross-ratios of T and T,. 
232. This shows that we have a very simple relation 
among the k’s of T, T, and T,. The ‘‘k-relations” 48, 43’, 
43”, are due to the fact that T, T,, T,, were written so as 
to have one root of their characteristic equations = 7. Since 
division of all coefficients a,, b,, etc., by a root of the char- 
acteristic equation allows us to throw any collineation T 
into this form, we may state: 
THEOREM 27. The 4-relations 43, 43/ and 43” hold for the en- 
tire group Gs, provided its transformations be written as here di- 
rected. 
233. Normal Form of R,in G,(l). Using the normal form 
of T the relations R, defining the group G, (1) become 
Bua CurA BeeiG.h,B B Ae ChaG 
ABC! AG ee Be Co Bly, BIC! ake! | — 7, , 
|BUY CV RAY |B’ Cv kB B’ Cc’ WC" 
Auer Al AG IB A Cee . 
A\A co kA’ |+ ula Cc ke i|+yijac r|=p, (44) 
| All (Olt! ki A” All Cc" kB" | A” (Ou kicv 
Ai BeasA AL 1B 18} Ae SBeaCr «| 
7a Alee Biwi Ale 1 |All. B! EB a Ale Reser Clab—=77. 
Al Bl kiA” All B" kB" All Br kc” 
Again we may eliminate 2, «,7 from these equations and 
obtain, as in the case of R,, the condition A(1)=0. We 
shall find as before the significance of each factor in (42’). 
