(a) W=TI{1/(l+cj)}, (b) T-=n nc3, (c) V=n n{cj/(l+cj)}. The test (a) is the likelihood ratio criterion (Wilks, 1932). Bartlett (1939) proposed (b) and (c) as reasonable alternatives to (a) for testing the general linear hypothesis. The test (b) was studied by Hotelling (1947, 1951) and (c) has been studied intensively by Pillai (1964, 1966). Asymptotic formulae have been proposed for the distribution functions of W and T2; see, for example, Sugiura & Fujikoshi (1969), Ito (1956, 1960), an unpublished report of M. Siotani, and Lee (1971). In this paper asymptotic formulae are derived for the distribution function of V in the general case and the percentage points for V in the null case. Using the asymptotic formulae, the powers of the three tests (a), (b) and (c) are compared numerically.