The mean-field saddle-point theory for Ising spin models with long-range interactions is rewritten in terms of the eigenvalues and eigenvectors of the interaction matrix. This gives a natural division of long-range models into two classes: those where the rank of the interaction matrix is finite (in the limit N to infinity ); and the saddle-point integral for the partition function can be directly evaluated (subject to several other weaker conditions) and those where the interaction matrix has a divergent rank. In the latter case the saddle-point integral method cannot be directly applied to the partition function. In the former case of the simply solvable models there are natural order parameters associated with the eigenvectors of the non-zero eigenvalues which characterize the system. It will be shown that models in this class are also described by Curie-Weiss mean-field type equations.