The functions that represent directional properties of a line array of acoustic elements can be related to a general class of polynomials that includes those of Jacobi, Gegenbauer, Chebyshev, Legendre, Laguerre, and Hermite. The significance of these polynomials for array design is that the response amplitudes are given in terms of parameters that varied continuously, produce continuous and predictable changes in the directivity pattern. For a symmetrical distribution of amplitudes, the functions are essentially Gegenbauer polynomials that, expanded in a trigonometric series, give amplitudes explicitly in terms of a single parameter. With proper normalization, this parameter can be varied from zero to infinity, producing the pattern of an unshaded line for value unity; the Chebyshev distribution, with equal side lobes, for zero value; and the binomial distribution, with vanishing side lobes, in the limit of large values. A value of one-half gives Legendre polynomials. Choosing a representative pattern with a particular distribution of side lobes, the introduction of an additional parameter, which amounts to a change of scale, permits a uniform reduction of all side lobes by a fixed ratio relative to the main beam.