"The author starts with the assertion that the distributions obtained from samples from normal populations under various types of null hypotheses can be reduced to the generalized beta distribution, with parameters Q and R, of 0i(i = 1, .* * , s), where 0 < 01 < 02 < * * * < 0. < 1, and where the 0's are nonzero roots of certain determinantal equations associated with Wishart matrices of certain degrees of freedom. When s = 1 the generalized beta distribution reduces to the ordinary beta distribution. Student's t distribution, the F distribution, and Hotelling's generalized T2 distribution can be transformed to the ordinary beta distribution and the Wishart distribution to the generalized beta distribution. The author shows how various univariate and multivariate tests based on these and other statistics can be made by the use of a single table, that of percentage points of the sum V(8) of s roots, which is included as Table 1 of the book. He divides these tests into three groups: (1) univariate (s = 1); (2) multivariate (s = 1); and (3) multivariate (s > 1). The first group includes tests of hypotheses that a simple correlation coefficient (squared) is equal to zero, that the variances of two populations are equal, that the variances of more than two populations are equal, that the means of two populations are equal, and that two variances whose sample estimates are chi-square variables are equal (with applications to analysis of variance in cases of single classification, two-way classification, and factorial experiments). The second group includes tests of hypotheses that a vector mean is equal to 0 (one-sample problem) and that the difference of two vector means is equal to 0 (two-sample problem). The third group includes tests of hypotheses that canonical correlations between one set of p variables and another set of q variables are equal to zero, that the covariance matrices of two populations are equal, and that the mean vectors of k p-variate normal populations are equal. Table I gives percentage points of the sum V(') of s roots. For s = 1, upper and lower 0.5%, 1.0%, 2.5% and 5.0% points are given for parameter values Q and R, where Q + 2 = 1(1)12, 15, 20, 30, 40, 60, 120 and R + 2 = 1(1)10, 12, 15, 20, 24, 30, 40, 60, 120. For s = 2(1)50, upper and lower 1% and 5% points are given for parameter values Q = -1(1)10, 20, 30, 40, 50, 60, 80, 100, 120 and R = 10(10)200. The sum V(1) of one root has the ordinary beta distribution. The lower percentage points in the case s = 1 have been reproduced from a table by Catherine M. Thompson ("Percentage points of the B-distribution," Biometrika 23 (1942), 168). Thomp- son's 2a and 2b are equal to Q + 2 and R + 2, respectively. The upper percentage points for s = 1 and all of the percentage points for s > 1 of the sum V(') of s roots were obtained by fitting a Pearson-type curve to the distribution of the first elementary symmetric function of the roots 0i . Table II gives the Beta-function parameters (a, b) of the mean sum (sic) V(' /s of s roots, where a and b are functions of the first two moments v[ and v4 about the origin of V() /s, and where v, and v are themselves functions of Q, R, and s. This gives exact values of a and b for s = 1 and approximate values for s > 1. The latter enable one to obtain approximate upper and lower 0.5% and 2.5% points for s > 1 by entering Table I for 2a = Q + 2, 2b = R+ 2, and s = 1, which may require a two-way interpolation. Professor Mijares is to be congratulated for his unified treatment of the theory of testing univariate and multivariate statistical hypotheses and for the computation of the tables needed in the application of the unified theory."

• H. Leon Harter