A class of distributions is defined and studied which includes as particular cases (cf. Section 13) the ordinary $\beta$-distribution, the (univariate) triangular distribution, the uniform distribution over any nondegenerate simplex, and a continuous range of other distributions over such a simplex, called basic $\beta$-distributions (Section 6) and immediately analogous to the ordinary $\beta$-distribution. Our class also includes (Section 13 (vi)) various (univariate and other) distributions which arise in connection with the random division of an interval. The main results are given in Section 2 and further results for the univariate case are given in Section 8. This paper is exclusively concerned with the mathematical theory. One application may, however, be mentioned, which will be considered in more detail elsewhere. Suppose we wish to test the hypothesis $H_0$ that $n - 1$ numbers $y_1, \cdots, y_{n-1}$ (all lying between 0 and 1) were drawn independently from a rectangular distribution over (0, 1). Let $u_1, \cdots, u_n$ be the lengths of the $n$ intervals into which the $y_j$ divide the interval (0, 1). Then $H_0$ is equivalent to the hypothesis that the point with vector-coordinate $\mathbf{u}$ is distributed uniformly over a certain non-degenerate simplex $S$, and a useful set of alternative hypotheses is the set of basic $n$-dimensional $\beta$-distributions. Hence (using Section 4) this theory can be used to find the power-functions of certain tests of the hypothesis $H_0$.