In this paper an asymptotic formula for the cumulative distribution function (c.d.f.) of Hotelling's generalized $T_0^2$ statistic is derived for general values of the number of dimensions in the non-null case. This result includes as a special case the previous result of the author for the null distribution of $T_0^2$ [5], and shows certain properties of $T_0^2$ which help determine the power of the test based on the statistic for moderately large samples. Both of the author's results provide an approximate complete analysis of the $T_0^2$ test, although the exact null and non-null distributions of $T_0^2$ are not available at present.