The subset of the herd space of *F* consisting of all of those functions which are permutations is called the ** Herd of F**. Note that a herd may be the empty set. Also note that while the actual functions in a herd space depend upon the parameterization of

Let C = C(V, S) be a cone and let PG(3,q) be coordinatized in the standard way (). If F is a flock of C, then for each point P in S, since no two planes of F can meet on VP, f_{P} is a permutation of GF(q). The set of these permutations, one for each point of S, is called the ** Herd of F relative to C**. The herd of a flock F relative to cone C is thus a subset of the herd of F.

It is immediate, from our normalization, that:

**Proposition 8.1**: If F is a flock of a non-flat cone, then the coordinate functions of F are permutations of GF(q).

[In the cases of point or flat cones, we can only say that g, or f and g, respectively, are permutations.]

By reparametrizing the planes of the flock, we can arrange to have any coordinate function be any permutation we like (for non-empty non-flat cones). We shall normally then take g(t) = t (for all non-empty cones). Other normalizations are at times useful, in particular, for quadratic cones in even characteristic, the normalization g(t) = t^{1/2} brings to light some additional structural properties of the herd [forward reference: oval herds].

Given a flock *F*, let S be the set of points of W = 0 associated to the herd of *F*. Clearly, the cone C(V,S) has *F* as a flock, and is the critical cone of that flock. Rephrasing this observation (without reference to herd spaces) gives a
fundamental result, which has been alluded to in earlier work and has been proved in many special cases:

**Theorem 8.1: (The General Herd Theorem)** Let f, g and h be permutations of GF(q) with f(0) = g(0) = h(0) = 0. Define a point set S to consist of those points with coordinates (x, y, z, 0) such that xf(t) + yg(t) + zh(t) is a permutation of GF(q). Then {xf(t) + yg(t) + zh(t) | (x, y, z, 0) is in S} is the herd of the flock F, whose planes have the equations f(t)X + g(t)Y + h(t)Z + W = 0, of the cone with carrier S and vertex V = (0, 0, 0, 1).

Conversely, any flock of any non-empty non-flat cone admits such a representation.

The herd concept has evolved over time and there are several variants in the literature. [more reference: concerning herds]

Herds have proved to be very useful in the study of flocks, simplifying many proofs and opening a new direction of investigation. They can also be used to classify flocks for small q .