204 THEORY OF COLLINEATIONS. 
[A B, G | 
|As’ Bo! keCx! | = CC, 
| As’ Boll Kol Col! 
Expanding these three determinants along the top row and 
collecting coefficients we get 
Al B! A; B, 
fore alee (49) 
where m,, etc., are all second order determinants in A,’, B,’, 
etc. Solving the above equations we have A,=0 and B, = 0 
unless |" ™|=@0. But the value of |” ,..| is readily found 
to be (1 = k.) (1—k,’) ae a *, which does not vanish so long 
as T, is of type I. Hence we have A,=0 and B,=0. There- 
fore T, also leaves invariant the point (0, 0, C) and its normal 
form reduces also to the same reduced form as T and T,. 
Equations IX and X reduce respectively to 
| Ao! Boy | __ ~|Al Bil pv |Ay By 
Gi Ao! Bal =—— (Gi All Br Cc A,! By!’ 
, IX’ 
and 
CEN aS oe pe ioe Clare eon te ee 
Dividing X’ by the cube of IX’ we get 
hike Keke tee. (43) 
which is the same k-relation that we found in Art. 231. 
241. Reduced Normal Form of G,(l). If T and T, have a 
common invariant line, then T, will have the same invariant 
Ime. Let the line /”, joining A’ and A”, be the common in- 
variant line of T and T, and let this line be taken as the line 
z = 0 of the triangle of reference. Thus we have C’ = C” =0 
and C,' = C,’’ = 0 in the normal forms of T and T, respectively. 
Substituting these values in equations I, IX, we find that the 
right-hand sides of equations VII and VIII vanish and that IX 
reduces to the same form as in Art. 240. Solving as before 
equations VII, VIII, and IX, for C,’, C,’, we find C,’/=C,=0. 
Equations IX and X reduce to IX’ and X’ respectively of 
Art. 240, and hence we have the same k-relations as before, 
WAZ [Gee —— CC eer 
Aj! By 3 / 
A,’ B.| xX 
