Gates and Penrose have given criteria under which classical gases with weak long-range interactions fail to be described by the van der Waals equation with Maxwell's rule. Unfortunately, -examples of equations of state for such systems have not yet been produced. This paper examines the Gates-Penrose class of interactions — i.e., Ur(r)=q(r)+ 7q)(Tr), in the limit 7 — + 0, where the Fourier transform ~(p) has a minimum at a nonzero value of p — for the spherical model on a one-dimensional lattice. Free energy and magnetization isotherms are computed; it is seen that there is a phase transition, but that the zero-field spontaneous magnetization is always zero (a parahelicoidal phase). However, the pair-correlation function may exhibit either long-range order or the appearance of oscillation.