In an attempt to construct an analytic theory of anisotropic random flight, the need has arisen to construct phase-functions for which the expansion in spherical harmonics has only a limited number of terms, but which have a high value for the asymmetry parameter g. The authors describe the procedure to find the phase-function which has a maximum value of g for given N, where N is the number of spherical components of the phase-function. It appears that in order to attain g = 0.9, one needs a phase-function composed of at least nine spherical components, or equivalently a polynomial of degree nine.