The partition function $Z$ of a two-dimensional Ising lattice is evaluated by expressing it in the form $Z=\Sigma i^{}\lambda _i^{}{}_{}{}^{\mathrm{nm}}$ where ${\lambda }_i$ are the eigenvalues of a $2^n$-dimensional matrix M, $n$ is the number of rows and $m$ the number of columns of the lattice. M is a generalization of the V-shaped matrix studied by Kramers and Wannier in connection with the same problem. It is shown that M can be expressed in terms of $2^n$-dimensional representations of $2n$-dimensional orthogonal matrices. The eigenvalues of M are determined for $n$ large by making use of the known relations between the eigenvalues of a $2n$-dimensional orthogonal matrix and the eigenvalues of its $2^n$-dimensional representative.

The lattice considered differs only slightly from that treated by Onsager and Kaufman. In the lattice considered here, the boundary effects are eliminated by introducing an interaction between the last lattice point of each row and the first point of the next row. The matrix V considered by Onsager and Kaufman is, except for the boundary effects, equivalent to the matrix ${\mathbf{M}}^n$. $Z$ is shown to differ from the partition function calculated by Onsager and Kaufman only in order $n^{-2\mathrm{}}$.

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