198 THEORY OF COLLINEATIONS. 
We may eliminate /, m, and n from these equations, as in 
Art. 217, and thus get: 
la,|—7 bi, |e1| 
A(1)=|le|  aol-z lal | =o, (42) 
jas, bs es|—1 
where |a,| etc., are the elements of the matrix of the normal 
form of T. But this relation is no other than the character- 
istic equation of T, with p replaced by 7. Hence, by Art. 
133, it becaks up into three factors as follows: 
(A’—1) (kA’—1) (k’A’—1) =0, (42’) 
where A’ is the determinant of the invariant triangle of T. 
We wish to know the meaning of each of these factors. 
The form of equations (41) shows that the point (/, m, 1) is 
transformed into itself. But this can only be true for one of 
the invariant points of 7. Suppose that it is the invariant 
point (A, B,C); substituting A, B, C for |, m,n in equations 
(41) we get A’ A=A, A’/B=B, A’C=C. Hence we have 
A’=1. If the point (1, m, 1) is (A’, B’, C’), this leads to the 
condition A’ =—7 sci (loam) asi Ave BC”). then kai 
The conditions for the group G,(A) require that one fac- 
tor of (42’) must vanish. If A’ = 7, the invariant triangles 
of the collineations in G,(A) all have the point (A, B,C) in 
common which is therefore the invariant point of the group. 
Le GAS 1 CAT BC is the invariants point) teks Asie 
( A’, B’, C’) is the invariant point. 
231. The k-Relation. Suppose that the invariant point of 
G,(A) is the point (A,B,C). Since the relations R, hold 
for (-7- and Gh inuG CA"), wwe Nayeg Ne ee leenel 
A,’ =1. Equation X of Art. 1794 may be written in the form 
Rese Nl 2 aie Ne hen N eee xX 
Substituting the values of the A’’s in X we get 
Jeske Tek lester (43) 
