**Proposition 11.2.7**: *The Penttila-Williams flock in PG(3,3 ^{5}) is a flock of a quadratic cone.*

*Pf*: Let q = 3^{5}, and F = F(t^{3},t,t^{27}). To show that this flock is a flock of a quadratic cone, we must find a conic in the complement of the baselines of the flock. The baselines of F are the lines in W = 0 satisfying the equations t^{3}X + tY + t^{27}Z = 0 for t0 in GF(q). Consider the conic in W = 0 given by X^{2} + YZ = 0. The conic is in the complement of the baselines if, and only if, X^{2} + (-t^{2}X -t^{26}Z)Z = 0 has no non-trivial (i.e., X = Z = 0) solutions in GF(q). This quadratic equation has no non-trivial solutions provided (-t^{2})^{2} - 4(-t^{26}) is a non-square for all t0. Now,