86 THEORY OF COLLINEATIONS. 
collineation 7. For example, the line AB cuts K in A and B 
and the corresponding line cuts K, in the corresponding 
points; but these are also A and B; hence the line AB is 
transformed into itself. 
104. «’* Different Constructions of the Same Collineation. 
There are ~* conics passing through the three invariant points 
A,B,C. From these ©” conics one can form ~‘ pairs of con- 
ics. Out of these ~’ pairs of conics, ©” pairs give the same 
collineation. Let us choose any conic L passing through A, B, 
and C. The collineation 7 transforms L into L, intersecting 
Lin A,B,C and V,. The course of reasoning used in art. 101 
shows that corresponding points on L and L, are collinear 
with V,. The two conics L and L, and the point V, may be 
used to construct the collineation T in the same way that 
K, K, and S, were used. It is evident that to each of the ~’ 
conics through A, B, and C there is a corresponding conic, and 
hence there are ~’ different constructions of the same col- 
lineation T. 
THEOREM 16. The same collineation 7 can be constructed by 
means of a pair of intersecting conics in o* different ways. 
105. Comparison of the Two Constructions. The two 
methods developed in this chapter for constructing a collinea- 
tion are dualistic to one another. In order to render this 
dualism more apparent we give here the principle properties 
of both methods in the following form : 
THEOREM 17. A collineation 7 is completely determined and 
constructed by means of two conics K and Kk, having a common 
angent / P tangents 
ae Hs corresponding SOL K and K;, are 
{ coneurrent with J ) r { tangents to } 
2 A 5 - » er "ee © 4 a=! * . 
| collinear with S, _ { The other three common | points on | 
Kand K;, are the three 1 Sue ae of the invariant triangle of 
the transformation 7. 
