In a series of papers W. F. Sheppard (1912, 1914) has considered the approximate representation of equidistant, equally weighted, and uncorrelated observations under the following assumptions:–(i) The data beingu1, u2, …, un, the representation is to be given by linear combinations(ii) The linear combinations are to be such as would reproduce any set of values that were already values of a polynomial of degree not higher than thekth.(iii) The sum of squared coefficientswhich measures the mean square error ofyi, is to be a minimum for each value ofi.

The aim of the present paper is to extend Daniel Bernoulli's method of approximating to the numerically greatest root of an algebraic equation. On the basis of the extension here given it now becomes possible to make Bernoulli's method a means of evaluating not merely the greatest root, but all the roots of an equation, whether real, complex, or repeated, by an arithmetical process well adapted to mechanical computation, and without any preliminary determination of the nature or position of the roots. In particular, the evaluation of complex roots is extremely simple, whatever the number of pairs of such roots. There is also a way of deriving from a sequence of approximations to a root successive sequences of ever-increasing rapidity of convergence.

The problem of statistical “selection” is concerned with the alteration induced in a frequency distribution in several variables by an alteration of the parameters in a subsection of the distribution. It may be illustrated by a simple trivariate case, as follows:From a population characterised by variables x, y, z, correlated and normally distributed, with means 0, 0, 0, variances and product variances r12σ1σ2, r13σ1σ3, r23σ2σ3, a sub-population is extracted by selection in x alone, in such a way that after selection x is still normally distributed, but with mean h and variance s2 . It is required to determine the new values, in the selected population, of the means and variances of y and z, and of the product variances.

In many branches of applied mathematics problems arise which require for their solution a knowledge of the latent roots of a matrix A, sometimes only the root of greatest modulus but often the second and other roots as well, and the corresponding latent vectors. A few examples, among many that might be cited, are problems in the dynamical theory of oscillations, problems of conditioned maxima and minima, problems of correlation between statistical variables, the determination of the principal axes of quadrics, and the solution of differential or other operational equations. It is important, therefore, to have a choice of methods for obtaining latent roots and latent vectors without undue labour, and the object of the present paper is to augment the existing store of such methods.

Linear interpolation between two values of a function ua and ub can be performed, as is well known, in either of two ways. If the divided difference (ub−ua)/(b−a), which is usually denoted by u (a, b) or u (b, a), is provided, or its equivalent in tables at unit interval (the ordinary difference), we should generally prefer to use the formulawhich is the linear case of Newton's fundamental formula for interpolation by divided differences. If differences are not given, but a machine is available, then the use of proportional parts in the form of the weighted averagethe linear case of Lagrange's formula, is actually more convenient, since it involves no clearing of the product dials until the final result is read.

The quotient of alternantsthe denominator being the difference-product of the arguments a, β, γ, …, is an important symmetric function, introduced into algebra by Jacobi in 1841.

The object of this paper is to communicate a general formula for elementary symmetric functions of any assigned degree of a given number of the roots, in ascending order of magnitude, of an algebraic or transcendental equation with complex coefficients. In virtue of the simple relations that exist between these symmetric functions, the formula gives a literal expression in terms of infinite series for all the roots of an equation. In practice it is generally desirable to transform the equation first into one with roots widely separated in value (e.g. by the root-squaring process) in order to make the convergence of the series sufficiently rapid for satisfactory numerical computation.

Notations and jwoperties.The matrices which form the subject of the present communication have been considered* from the point of view of determinants by Cauchy, Schlafli, Sylvester, Zehfuss, and various later writers, and from the point of view of matrices, their latent roots being in question, by Rados, Franklin, Metzler, Stephanos, Burnside, and others. Our present object is to go beyond this and to investigate the elementary divisors of the characteristic determinants of the matrices, which specify completely their invariance under transformations of the type HAH~X, in brief, to obtain the classical canonical form of the matrices.The matrices which we shall consider are related either to a single generating matrix A = [a (j ], in some prescribed field and of given canonical form, or else to a set of such matrices A it j -1, 2, ..., m, not necessarily all of the same order. Interest in anj^ matrix will centre throughout upon its classical canonical form C = HAH-1 , \H\ =£ 0, and it will be convenient to represent this by the various notations, e.g. C == diag[C 3 (A 1 ), C 2 (A 2 ), C 1 (A 3 )]A list of references is given at the end of this paper. 1933.] COMPOUND AND INDUCED MATRICES. 355 x (0 = { x u> X 2i> • • • > x n) (* = 1, 2, ..., m), let the m-th compound be constructed, namely X m = {| # 1 1 # 2 2 ••• x m m » •••> x r -m + l > r -m + 1 ••• x r r \ } -