**Theorem 8.1.1**: *If two functions in the herd space of a flock F are scalar multiples, then F is a star flock.*

*Pf*: Suppose that f_{P'} = kf_{P} for distinct points P and P'. We can assume that these are not constant functions (k0) and that any functions which appear in this proof are not constant by Proposition 8.1.3 . Let T be any point of W = 0 which is not on the line PP'. Choose points Q on PT, and R on P'T which are distinct from P, P' or T (this is always possible since q2). If any of f_{Q}, f_{R} or f_{T} is a scalar multiple of f_{P}, then F is a linear flock by Proposition 8.1.6 . By Proposition 8.1.7 , there exist scalars a and b so that f_{T} = af_{P} + bf_{Q}, and scalars c and d so that f_{T} = cf_{P'} + df_{R} = ckf_{P} + df_{R}. Since f_{T} is not the constant function, nor a scalar multiple of f_{P}, b and d are not zero. Equating the two expressions for f_{T} gives, (a-ck)f_{P} + bf_{Q} - df_{R} = 0, and so, f_{P}, f_{Q} and f_{R} are linearly dependent functions associated to non-collinear points. That F is a star flock now follows immediately from Proposition 8.1.5 .