Two new methods to find the basic vector correspondence matrix of the coincidence site lattice for two phases have been developed by means of theory of matrix and elementary theory of numbers. Based on these methods the three equations which are applied to find a coincidence site lattice have been incoporated into a single one so that the problems concerning the composite lattice of two phases with a coincidence site lattice relationship are greatly simplified. It has been verified that the whole of selfcoincidence problems in the cubic system, the most of self-coincidence problems in non-cubic systems and a part of inter-coincidence problems can be simplified by the incorporating of discriminant equations.
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