This paper discusses the extension of the discriminant function to the case where certain variates (called the covariance variates) are known to have the same means in all populations. Although such variates have no discriminating power by themselves, they may still be utilized in the discriminant function. The first step is to adjust the discriminators by means of their `within-sample' regressions on the covariance variates. The discriminant function is then calculated in the usual way from these adjusted variates. The standard tests of significance for the discriminant function (e.g. Hotelling's $T^2$ test) can be extended to this case without difficulty. A measure is suggested of the gain in information due to covariance and the computations are illustrated by a numerical example. The discussion is confined to the case where only a single function of the population means is being investigated.

Unified and simplified derivations are given for the limiting forms of the difference (1) between the empirical distribution of a large sample and the corresponding theoretical distribution and (2) between the distributions of two large samples.

The application of integral equations to renewal theory in population analysis and problems of industrial replacement is beset with certain difficulties which have been particularly discussed by W. Feller (these Annals 1941 vol. 12 pp. 243-267). Some of these difficulties are avoided if the data of the problem are introduced into the analysis directly in the discontinuous form (tabulated by class intervals) in which they are usually supplied in a concrete case. A numerical example based on population statistics is presented, illustrating how, using discontinuous data, a recurrent series takes the place of the integral equation, and a finite exponential series appears in place of the Heaviside expansion of the previous solution. There is close analogy with the procedure previously presented, but with factorial moments appearing in place of ordinary moments. The fundamental data being given for values of the replacement function at discrete intervals only, some question arises as to the applicability of the solution as an "interpolation" formula for non-integral values of the time $t$, and as to the effects of subdividing the class interval of the original data. In the actual computation of the factorial moments a shift of origin by one-half class interval becomes necessary. An algorithm for effecting this shift is presented.

The present paper deals with the numerical solution of equations by the combined use of Newton's method and inverse interpolation. In Part I the case of one equation in one unknown is discussed. The methods described here were developed by Aitken [1] and Neville [2], but do not seem as widely known as they should be, perhaps because the original papers are not readily available. (A short summary of Aitken's work will be found in a recent paper by Womersley [3].) Mention should also be made of an interesting paper by Spoerl [4], which treats the same problem from a somewhat different viewpoint. In Part II these methods are extended to sets of simultaneous equations.

Necessary and sufficient conditions are given for the existence of a uniformly consistent estimate of an unknown parameter $\theta$ when the successive observations are not necessarily independent and the number of unknown parameters involved in the joint distribution of the observations increases indefinitely with the number of observations. In analogy with R. A. Fisher's information function, the amount of information contained in the first $n$ observations regarding $\theta$ is defined. A sufficient condition for the non-existence of a uniformly consistent estimate of $\theta$ is given in section 3 in terms of the information function. Section 4 gives a simplified expression for the amount of information when the successive observations are independent.

The use of the repeated Cauchy principal value affords greater facility in the application of inversion formulae involving characteristic functions. Formula (2) below is especially useful in obtaining the inversion formula (1) for the distribution of the ratio of linear combinations of random variables which may be correlated. Formulae (1), (10), (12) generalize the special cases considered by Cramer [2], Curtiss [4], Geary [6], and are free of some restrictions they impose. The results are further generalized in section 6, where inversion formulae are given for the joint distribution of several ratios. In section 7, the joint distribution of several ratios of quadratic forms in random variables $X_1, X_2,\cdots,X_n$ having a multivariate normal distribution is considered.

The weighing problem is discussed from the point of view of factorial experimentation. The paper contains a brief description of the fractional replication of the $2^n$ factorial system. It is shown that optimum designs for the weighing problem may easily be obtained with this approach. This approach is valuable in indicating the structure of weighing problem designs, and the limited conditions under which such designs can give results of value.

A multiple (sequential) sampling scheme designed to test certain hypothesis is discussed with the following assumption: $X$ is a random variable with density function $P(x)$ which is piecewise continuous and differentiable at its points of continuity. Formulae are derived for the probability of acceptance and rejection of the hypothesis and for the expected number of samples necessary for reaching a decision. These formulae are found to depend on the solution of a Fredholm Integral equation. Explicit solutions to the problem are obtained for the case when $P(x)$ is rectangular by reducing the fundamental integral equation to a set of differential-difference equations. Several examples are given.

An explicit formula for the characteristic function of the deviation $\frac{1}{n} \sum_n {k=1}\|X_k - \bar X\|^\alpha,\quad\alpha > 0,$ is derived for samples from a normal population. For $\alpha = 1$ one can calculate the probability density function but the result does not seem to be in complete agreement with a recent formula of Goodwin [1].

Exact means in samples of size $\geq 3$, and exact second moments and product-moments in samples of size $\leq 4$, are given in Table 1 in terms of $\pi$ for order statistics selected from the normal distribution $N(0, 1)$. The derivation employs multiple integration and some general properties of the moments.

It is well known that, if the fourth moment about the mean of a frequency distribution equals the square of the variance, then the frequencies are piled up at exactly two points, namely, the two points that are one standard deviation away from the mean. In this paper is developed a general inequality which describes the piling up of frequency around these two points for the case where the fourth moment exceeds the square of the variance. In a sense, it is shown how "U-shaped" a distribution must be according to its second and fourth moments.