Pair correlations on two one-dimensional Ising models with next-nearest-neighbor interactions are calculated exactly. For an appropriate range of values of the interaction energies, each of these models exhibits a disorder point, at which the nature of the short-range order changes abruptly, and at which there is a cusp in the graph of range of order vs. temperature. The zero-field susceptibilities are calculated exactly, and are found to have quite smooth temperature dependence.