**Proposition 8.1.6**: *The herd space of a flock F contains three functions based at non-collinear points that are scalar multiples of each other, if and only if, F is a linear flock.*

*Pf*: The necessity of the condition follows immediately from the characterization of the herd space of a linear flock given in Proposition 8.1.4 .

To prove the sufficiency, we may assume that f_{Q} = uf_{P}, and f_{R} = vf_{P} for three non-collinear points P, Q and R. If f_{P} is the constant function, then F is linear by Proposition 8.1.4 , so we assume that it is not constant. Similarly, we may assume that u and v are not both zero. Now, since three scalar multiples of the same function are linearly dependent, we have by Proposition 8.1.5 , that F is a star flock. To prove that F is linear, we need to show that there are at least two sets of constants, a,b,c and a',b',c' with the second set not a scalar multiple of the first set, which express the linear dependence of the functions, i.e. af_{P} + bf_{Q} + cf_{R} = 0 and a'f_{P} + b'f_{Q} + c'f_{R} = 0. This follows from the proof of Proposition 8.1.5 , since each such set gives rise to a common point on all the planes of the flock, and only sets which are scalar multiples of each other give rise to the same point. Since we have,

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