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The authors consider the uniformly most powerful invariant test of the testing problems (Ⅰ)H0: μ′∑-1μ ≥ C H1: μ'∑-1μ < C and(Ⅱ) H00: ≥ C H11: <C under m-dimensional normal population Nm (μ, ∑ ) and normal linear model ( Y, Xβ, σ2 ) respectively.Furthermo