ABSTRACT
If t1=0, then z1=0, However, for any b if z1=0,
then z2=b'd(b'S2b) - 1/2, and z2 is maximized if Similarly if t2=0, the
solution assumed in the theorem is optimum. Now consider t1>0, t2<0, and t1-t2=1. Any hyperbola
for k>0 cuts the z1 axis at ±(t1k) 1/2. The rule assumed in the theorem has z1>0 and
z2<0. From (9.48) we get
The maximum of this expression with respect to c for given b is attained for c as given in (9.54). Then z1, z2 are of the form (9.54), and (9.61) reduces to (9.55). The maximum of (9.61) is then given by b=(t1S1+t2S2)- 1d. It is easy to argue that this point is admissible because otherwise there would be a better point which would lie on a hyperbola with greater k.
