Theorem 11.2.1: A flock F of a non-empty cone is of semi-field type if, and only if, it has a normalized representation as F(f(t),t,h(t)) with f and h additive functions over GF(q).
Pf: If F is a flock of a non-empty cone, we can recoordinatize W = 0 and reparameterize F so that F = F(f(t),t,h(t)) (see section 8 ). If F is a semi-field type flock, each secondary baseline must coincide with a primary baseline. The primary baselines of F are the lines f(t)X + tY + h(t)Z = 0 in W = 0, for t0. The secondary baselines are those with equations (f(t)-f(s))X + (t-s)Y + (h(t)-h(s))Z = 0, with ts and neither t nor s equal to 0. For secondary baselines to coincide with primary baselines, for each pair t,s with ts, there must exist an r so that f(t) - f(s) = f(r), t - s = r and h(t) - h(s) = h(r). Thus, we have f(t) - f(s) = f(t-s) and h(t) - h(s) = h(t-s) for all ts. Together with the condition that f(0) = h(0) = 0, this implies that both f and h are additive functions.
Conversely, if f and h are additive functions, then the secondary baselines of F = F(f(t),t,h(t)) are (f(t)-f(s))X + (t-s)Y + (h(t)-h(s))Z = f(t-s)X + (t-s)Y + h(t-s)Z = 0, and so, are also primary baselines.