142 THEORY OF COLLINEATIONS. 
‘ 100 A Al Al 
But V factors into V=llo « o|| X ||B B Br l_=T'S. 
00k Cue 
Since U-'=S, we have U=S™ Hence 
Serre Ss 
We see that the collineation S transforms the triangle of 
reference into the invariant triangle of 7; thus the point 
(0, 0,1) goes over into (A”, B”, C’’), the point (0, 1,0) into 
(A‘B’, C’),-and(7,0;0) into (A,B,C). Hence Ss trans- 
forms the invariant triangle of T into the triangle of refer- 
ence. The collineation of 7 is consequently decomposed into 
three operations acting in the following order: S~ trans- 
forms the invariant triangle of T into the triangle of reference, 
T’ leaves the triangle of reference invariant, but transforms 
every other point in the plane, S transforms the triangle of 
reference back into the invariant triangle of T. 
T’ is a collineation in its canonical form, Art. 148, and T is 
the result of operating on JT’ by S. T is thus the so-called 
transform of T’ by S; T’ and T are equivalent collineations. 
In like manner the normal forms of each of the other types 
of collineations may be broken up into the product S‘T’S, 
where S is a collineation depending on the invariant elements 
of T, and T’ is the canonical form of its type. 
THEOREM 4. Every collineation T in its normal form may be 
factored into S~‘7’S , where 7’ is the canonical form of its type. 
177. Resultant of T and its Inverse. Another interesting 
application of the determinant form of the resultant is its use 
in finding the resultant of a collineation T and its inverse 77’. 
Let T be written in the form : 
Pr=anx+biy+ez, 
.: Pyi = aou-+ boy + c2z, (11) 
Pa =azx+bsy + 82, 
