**Proposition 8.1.7**: *In the herd space of a flock F the functions associated with three collinear points of W = 0 are linearly dependent over GF(q). In particular, if the herd space contains two functions based at points P and Q which are scalar multiples of each other, then all the functions based at points on the line PQ are scalar multiples of the same function and F is a star flock.*

*Pf*: Let P = (x_{P},y_{P},z_{P},0) and Q = (x_{Q},y_{Q},z_{Q},0) be two distinct points of W = 0, and let be the plane determined by V, P and Q. Let R be any point of the line PQ, other than P. The coordinates of R are given by (x_{Q} + ax_{P}, y_{Q} + ay_{P}, z_{Q} + az_{P}, 0) for some a in GF(q). For each fixed t in GF(q), the plane of a flock F meets in a line m_{t} which does not pass through V. m_{t} meets the lines VP and VQ at the points (x_{P},y_{P},z_{P},-f_{P}(t)) and (x_{Q},y_{Q},z_{Q},-f_{Q}(t)), respectively. The points of m_{t}, other than the intersection of m_{t} and VP, have coordinates of the form (x_{Q} + bx_{P}, y_{Q} + by_{P}, z_{Q} + bz_{P}, -f_{P}(t) - bf_{Q}(t)), as b ranges through GF(q). The point of intersection of m_{t} and the line VR is thus (x_{Q} + ax_{P}, y_{Q} + ay_{P}, z_{Q} + az_{P}, -f_{P}(t) - af_{Q}(t)) = (x_{Q} + ax_{P}, y_{Q} + ay_{P}, z_{Q} + az_{P}, -f_{R}(t)). So, for each t we have f_{R}(t) = f_{P}(t) + af_{Q}(t), and the functions f_{R}, f_{P}, f_{Q} are linearly dependent for each choice of R. If f_{Q} = kf_{P}, then f_{R} = (1 + ak)f_{P} for some a in GF(q). If k = 0, then F is a star flock by Proposition 8.1.3 , otherwise let R' be the point on PQ with a = -1/k, then f_{R'} is the zero function and F is a star flock.