GEOMETRIC CONSTRUCTION. 83 
coo’ pairs of conics. Among these ~‘ pairs of conics there are 
oo” pairs which give the same collineation 7. To show this 
let us choose any conic C touching a, b, and ¢; the collineation 
T transforms C into C’ touching a, b,c, and some other com- 
mon tangent m. Corresponding tangents to C and C’ are 
concurrent with m. These conics C and C’ and the line m 
may be used to construct the collineation T in the same way 
that K, K’, and 1 were used. It is evident that to each of 
the conics touching a, b, and ¢ there is a corresponding 
conic; hence there are ~? different constructions of the same 
collineation. 
THEOREM 13. A collineation 7 can be constructed by means of 
a pair of conics in o? different ways. 
100. A Second Construction. From the self-dualistic char- 
acter of a plane collineation it is evident that the construction 
in the plane dualistic to that developed in the last paragraph 
also holds. This new construction may be deduced from the 
last by the principle of duality, or it may be developed inde- 
pendently from first principles. We shall take the latter 
course for the sake of the methods employed, and also for the 
sake of the wider view of the whole subject thus obtained. 
101. Two Intersecting Conics, K and K,. Suppose that a 
collineation T transforms a point S into S,and S, into S,. The 
pencil of lines through S is transformed into the pencil of 
lines through S,. The original pencil through S and the de- 
rived pencil through S, are projectively related, and hence the 
locus of the intersection. of corresponding rays of the two 
pencils is a conic K passing through both Sand S,. In like 
manner the pencil through S, is transformed into the pencil 
through S, and the locus of the intersection of the corre- 
sponding rays in these two pencils is a second conic, K,, pass- 
ing through both S, and S,, Fig. 13. Since the pencils 
