A pair of test statistics is proposed for the null hypothesis $\mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$ when the data consists of a sample from each of the $p$-variate normal distributions $N(\mathbf{u}_1, \mathbf{\Sigma}_1)$ and $N(\mathbf{u}_2, \mathbf{\Sigma}_2)$. These tests are motivated in Section 1 and defined explicitly in Section 2. Section 3 proves a theorem which includes the null hypothesis distribution theory of the tests. Section 4 gives some details of the computation of the test statistics. An appendix describes the shadow property of concentration ellipsoids which facilitates the geometrical discussion earlier in the paper.