REAL GROUPS. 251 
Since the parameters k and k’ assume all real values, there 
are * collineations in the group hG,(AA’A”). By setting 
k’ = k” we see that hG,(AA’A’’) contains ©’ one-parameter 
subgroups h G,(AA’A’’),; since k and k’ are both real, 7 must 
also be real. The constant 7 may assume the six critical val- 
ues (1, 0, © ) and (— 1,2, 4) which give the three perspective 
subgroups of hG,(AA’A’’), and the three subgroups whose 
path-curves are conics. 
312. The Elliptic Group eG,(AA’A”). Let A bea real 
point and A’ and A” a pair of conjugate imaginary points; 
then /’ joining A’ and A” isa real line and / andl’ joining 
AA’ and AA” respectively are a pair of conjugate imaginary 
lines. The cross-ratios k and k’ may be shown to be conju- 
gate imaginary numbers. The implicit normal form of T 
was given in Chapter II, Art. 129, by the equations 
1 Of, dl x 7] 1 ert Ch at! x y ij 
AM EB rt AR Bier st Ag Bd ABS ei) 
PEGG Gey dl A i| A Aly Je 4 
ae pp, Tl) “Va mG OO ee oe al x Oh wall (86) 
|A’ Bo 1 |A’ Be | ABN Al Bh ni | 
AY! Bh YX |Au BY A” BY 4 A” BY 
Since A’ and A’’, (B’ and B”) differ only in the sign of 7, 
being conjugate imaginaries, k and k’ can differ only in the 
sign of 7 and are also conjugate imaginaries. Hence k’ is not 
independent of k; but k may be written in the form of pe?’ 
and depends therefore upon two independent quantities, 
and 4; and hence there are ~’ elliptic collineations leaving 
the triangle (AA’A”) invariant. These form the elliptic 
group eG,(AA’A’). There is one collineation in eG,(A A’A”) 
corresponding to each value of the complex number k, or, 
speaking geometrically, to each point in the complex plane. 
To show how eG,(AA’A’’) breaks up into one-parameter 
subgroups we proceed as follows: Let k=pe?# = e(e+*)"; 
since k, = kk,, we have 
@ (e2 +1) t, — eco+tah+i(4 +h). 
=17 
