54 GEOMETRIC THEORY. 
at A and another at some point D such that DD’ is the tan- 
gent to K” parallel to AA’. By means of the conic K the 
range ABCD-- is transformed into A,B,C,D,-, by means of 
the conic K’ the range A,B,C,D,-- is transformed into A,B, 
C,D,--, by means of K’’ the range ABCD-- is transformed 
into A,B,C,D,. Consequently we have 
(ABCD) = (ATB:C,D;): 
But AR — An — Ales — sen — te eT oD) 
a (ABCD) — CABG). 
Expanding we get 
AC BD AC BoD AC. BoC B,D 
1 2 1 (40) 
BE” AD = BG AD? WHER’ Gera, — Ep - 
The characteristic cross-ratio of the transformation T due 
to Kisk=(ABCC,); that of the transformation T, due to 
K'isk,=(AC,BB,). Expanding and multiplying: 
eae , Ba _ ZB Gale AC.AB.B2G 
~ BC AG CE TAB BC. AC AB: 
AC. BG AB 
=(Se ee (Geel 
Substituting from equation (39) we get 
Le ABER) AB AR: 
ik, =e ein a DB DBA (UB (41) 
But (ADBB,) = k, the characteristic cross-ratio of the trans- 
formation due to the conic K”, thus kk,=k,. Hence the 
characteristic cross-ratio of the resultant in G,(A) is equal to 
the product of the characteristic cross-ratio of the components. 
68. The Perspective Subgroup, G,O. The two-parameter 
subgroup G,(O), which leaves the point O invariant, is made 
up of one-parameter subgroups each of which leaves O and 
some other point, as A, invariant. Fig. 6 shows the 
construction of a transformation which belongs to the one- 
parameter subgroup G,(OA). This transformation is 
determined by the degenerate conic OQ. All transforma- 
tions which leave the point O invariant are determined by the 
conics which touch J, l’, and O~ ; but since these three lines 
meet in a point, it follows that all these conics must be de- 
