The roots of the equation in λ

were named by Sylvester the latent roots of the determinant | a11a22 … ann |. So early as 1852, Sylvester showed that if any determinant D is given, we can at once write down a determinant whose latent roots are the squares of the latent roots of D: this determinant is in fact the square of D, the process of squaring being performed by multiplying rows into columns : so that, e.g. if the latent roots of are λ1 and λ2, then the roots of are and . Spottiswoode had also shown in 1851 that the latent roots of the reciprocal of a determinant are the reciprocals of the latent roots of the determinant itself. Both these theorems were soon found to be particular cases of a general theorem which was enunciated by Sylvester thus: The latent roots of any function of a matrix are respectively the same functions of the latent roots of the matrix itself.

In the present paper I propose to investigate the fundamental geometrical properties of the Apolar Locus of two tetrads of points in a plane.

The Apolar Locus of two tetrads of points K, L, M, N and P, Q, R, S is defined in the locus of the point X, moving so that the pencils X[K, L, M, N] and X[P, Q, R, S] are apolar.

It is a well-known fact and easy to prove that if Γ and Γ′ be any two class-cubics and P a variable point, such that the pencil of lines from P to Γ apolarly separate the pencil of lines from P to Γ′, then the locus of P is a cubic curve G called the “apolar locus” of Γ and Γ′. Also, Γ and Γ′ are said to be co-apolar class-cubics” of G, or simply “co-apolars” of G. The problem of finding the general system of co-apolars of a given cubic curve has not yet been completely solved, but particular and more important cases have been investigated by me in several papers contributed to the London Mathematical Society. In the present communication I propose to deal with the most general solution of the above problem.

Whittaker has shown that a general solution of Laplace's equation, ∇2V = 0, may be expressed in the form

Since the harmonic property of a function is in no way dependent upon any particular set of axes it follows that the same solution must be capable of being expressed in the form

where X, Y, Z are any second set of rectangular coordinates.

The continued fractions treated in this paper are of the general form

where s0, s1, s2, … are real integers (positive, negative, or zero). An arbitrary real number can, of course, be developed in such a fraction in an infinite variety of ways. The continued fractions discussed here have a number of striking properties and present numerous contrasts with the ordinary continued fraction usually employed.

The present paper is a further contribution towards the object defined in my former paper, namely, to derive the principal known results regarding Continued Fractions, and some new theorems, by transforming the functions considered from infinite series to Continued Fractions, by use of the theory of determinants.

In the study of rational approximations to irrational numbers the following problem presents itself: Let ω be a real irrational number, and let us consider the rational fractions satisfying the inequality

how small can the positive quantity k be chosen with the certainty that there will always be an infinite number of fractions satisfying the inequality whatever the value (irrational) of ω?

At the close of the preceding meeting of the Society a discussion arose concerning the effect of a uniform rise in prices upon the amount of small money necessary for the transaction of business. It was clear that the total amount of money must be increased, but in the case of small money, which is used only when fractional parts of the larger unit are involved, the effect was less obvious. It is the fact, however, that more small money required. The present paper attempts an explanation of that fact.

§1. The following three illustrations of cyclant substitutions are supplementary to those given in § 5 of a previous paper.

In the cyclant substitution in three variables with equimodular multipliers

in which the coefficients are real and satisfy the relations

the following expressions, as may at once be verified, are absolute invariants:

Interpolation is one of the most frequent processes in calculation, and yet it is the process in which most computers find the ordinary methods least satisfactory and most troublesome. Indeed, whenever linear interpolation is not practicable, it is usually worth while to find out a method depending on the nature the functions involved in the calculation, and use it in preference to the ordinary difference or Lagrangian formulae. In interpolation by differences there is the want of adequate tables of the coefficients, and worse than that, the necessity for watching the signs and the decimal points, a necessity which in these days calculating machines is relatively a great trouble. There is usually, moreover, a lack of system about interpolation by differences that makes it peculiarly susceptible to slips of working. In this connection I might mention a useful and not too well-known arrangement of the work for Newton's formula which Legendre gives in his Traité des fonctions elliptiqueg.

The theorem of Heilermann can be stated thus:—

If the series

is converted into a continued fraction of the form

then the elements of the continued fraction are

where

and is obtained from this determinant by deleting the (r + 1)th column and the last row.