The statistics of the rectangular lattice with first and second neighbour interactions is discussed in detail. Several terms of the low-temperature series for the partition function are obtained for the case when the interactions are ferromagnetic, and for the 'no-field' lattice reciprocal transformations are used to transform to series valid for antiferromagnetic interactions. The transition points for different signs and magnitudes of the interaction energies are estimated. An exceptional value of the ratio of the first and second interaction energies is found for which the lattice has no transition point and long-range order offers no energy advantage. The series are used to predict the behaviour of the first derivatives of the partition function, and the results are applied to models of a ferromagnet, binary alloy, antiferromagnet and adsorbed monolayer.