A repeated measures design occurs in analysis of variance when a particular individual receives several treatments. Let $X_i = (x_{il}, \cdots, x_{ip})'$ be the vector of observations on the $i$th individual. It is assumed that the $X_i$ are independently normally distributed with mean $\mu_i$ and common covariance $\sum > 0$. The researcher wants to test hypotheses about the $\mu_i$. Let $\varepsilon_i = (\varepsilon_{i1}, \cdots, \varepsilon_{ip})' = X_i - \mu_i$. For this paper, in order to get powerful tests, the simplifying assumption that the $\varepsilon_{i1}, \cdots, \varepsilon_{ip}$ are exchangeable is made. We assume that the design is given and use a coordinate-free approach to find optimal (i.e., UMP invariant, UMP unbiased, most stringent, etc.) procedures for testing a large class of hypotheses about the $\mu_i$.