180 THEORY OF COLLINEATIONS. 
it may be shown in the same way that the group thus defined 
is a three-parameter group of perspective collineations whose 
geometric invariant is a point A and all lines through it. 
We might now go on and determine all groups of perspec- 
tive collineations defined by sets of linear relations on the 
elements of M. But this problem will be taken up later, and 
the results reached by other methods are only what the pres- 
ent method would yield. 
215. The Limiting Case D'=0. In discussing the group 
G,(K) defined by &, we assumed Art. 210, that the determi- 
nant 
Lp q 
Dae r 
G) iP WB 
0. 
If D’ = 0, the quadratic function breaks into factors; thus 
la? +my?+nz2z*+ 2pay + 2quz+ 2ryz= (Ae+lyt+~2) (A/x+Hly+”/2); 
whence we have /=22', m=uu', n=vy', p=Apn'=2'u, 
G=Ai! =)'v, r=pr'=u'v, The six relations, of (24) sre- 
duce to 
(Aqi+/a2+%a3) (Wai + d2+¥/a3) =27', 
(Ab; +b2 +4b3) (A! by + 2!b2 +4 b3) =H, 
(Ac, +H e2 +¥%e3) (Aer +H en +4 e3) =, 
(Aq, +Fa2+¥%a3) (7/6) +2! b2+4/b3) =2 /or7r, (34) 
(Aaqi+Ha2+/%a3) (Aa+M!e,+-c3) =A” or 2/¥, 
(2.6; +b. +%b3) (le, +2! e2+%!e3) =”! or LY, 
These relations define a mixed group mG,(Ill’) which will be 
discussed in § 3 of Chapter IV. Equations (84) may be 
factored in two ways, as follows: 
20, + Has+ ¥a3 =A, Wa, +H/a.+/a3=7', 
26; +b. +%b3 =F, and by + /b2+b3=L!, (34’) 
A¢, +e. +%C3=%", Mey +P 2 + Ye, =”, 
or 
hay + Hay + ¥a,=2, Hay + Ha» + als =2, 
Ab; + bs + ¥b3=P!, and Wb) + bo + /b3 = by ( oAu ) 
Ae; + Hes + ¥e3=/, Merch Mera wes —Y- 
Equations (34’) define a group G,(/l’); this is already included 
in the list of Art. 209. Equations (34’’) define a system 
of collineations which interchange the two lines 2% + wy-+ 1z = 0 
