In the following paper I give a complete list of the types of covariants belonging to the concomitant system of three quaternary quadrics, where covariant is used in its restricted sense and refers solely to a concomitant involving the variable x alone. A complete list of the types, 62 in number, is given in §1. In §§(6–10) the covariants are determined, and in §§(11–12) a list of the identities used in the reduction of the covariants is given, along with typical examples of the process.

and

denote two double binary (2–1) forms in (x, ξ). It is proposed to discuss the geometrical significance of their simultaneous covariant complete system, which is here quoted without proof.

Casey, in an extensive and complete mémoire, has discussed very thoroughly the bicircular quartic. Among the unfinished notes of Dr Clifford the properties of the curve are established by means of a tetracyclic system of coordinates. Such a system of coordinates has been extensively used by Coolidge. The object this paper is to present a valid discussion of the curve in terms a tricyclic system of “power” coordinates.

Let be a linear differential expression involving n independent variables xi the coefficients Aik Bi, and C being functions of the independent variables but not involving the dependent variable u. Associated with F(u) is the adjoint expression

The differential equation in question is of the second order and has three regular singular points. It is usually denoted by

where a, b, c are the singular points, a, a′ etc. the exponents at those singularities. The solution when no one of the numbers α – α′, β – β′, γ – γ′ is an integer or zero is well known; all types of solution are expressible in terms of the hypergeometric functions.

where (Am m) represents a square of m rows and m columns of a's; (Bm r) represents a rectangle of m rows and r columns of b's; (Cr m) represents a rectangle of r rows and m columns of c's;, and (Orr) represents a square of r rows and r columns of zeros.

The problem of the conduction of heat in one dimension is usually concerned with the propagation of a thermal disturbance along a bar or rod of uniform cross section. The solution of the problem is required for a given initial distribution of temperature, and given boundary values, usually at each end of the rod. In most cases this solution is found by assuming a series solution and then proving that the series satisfies the equation of the disturbance well as all the assigned conditions. Other methods, for example the contour integral method developed by Carslaw, also introduce this arbitrary element of choice in choosing the integrand and the contour of integration. The object of the present paper is to develop the application of Heaviside's Operational method to the solution of the problem, and to show that it leads in all cases to solutions equivalent to the known forms, although initially no assumptions are made regarding the nature of the solution.

B. Pascal and others have studied the properties of the table:

Very little information is now to be found regarding James Gray, the author of the famous Scottish Arithmetic, so universally in use in Scotland during the 19th century as to make his name synonymous for Arithmetic, just as Cocker's was England during the 18th century.

The possibilities under rearrangement of terms in a complex series were discussed by Levy and by Steinitz. A reference the Steinitz paper as first disposing of the questions raised made by Bieberbach. It is the purpose of the present paper give an independent treatment by methods somewhat resembling those of the Levy paper.

The chief aim of this note is to investigate directly the relation between the inverse factorial series

and the binomial coefficient series, or Newton's interpolation formula,

which may also represent Ω(x), the sum of the series (1).

Binet shewed that the function

can be expanded as an inverse factorial series. This note furnishes a new and much simpler proof of his result, based on a formula which is an analogue of the Binomial Theorem for factorials.

This formula is that, if we denote by [x]n the ratio

then

where denotes the coefficient of xr in the expansion of (1 + x)m.

Theorem 1 needs very little explanation. It is the converse of the well known theorem that the indefinite integral F(x) of a function f(x) possesses a derivate on the right at every point at which f(x + 0) exists. If f(x + 0) does not exist, nothing can be said as to the existence or otherwise of F+(x); but in a general way we might expect that the integral of a function which oscillates comparatively slowly, say sin (log x) at x = 0, would be more likely to possess a derivate than that of a function which oscillates more rapidly, say . It appears from Theorem 1 that this is not by any means the case. In fact the integral of sin (log x) has not a definite derivate at x = 0 while that of has such a derivate.

Sr denotes an infinite space of r dimensions. Such a space is divided into two regions by a Sr−1. An infinite line is divided into n + 1 regions by n points. An infinite plane or other surface topically equivalent to it is divided into two regions by an infinite line.

Let f(x) and F(x) be polynomials which are supposed to be decomposed into a series of n terms as

Further let θ be a distributive operation and φ(x) be a given function. Then the functional equation

has a solution of the form

where yν (x) (ν = 1, 2, … n) is the solution of the equation

The function whose properties are discussed in this note, is a special form of Whittaker's Confluent Hypergeometric Function, Wkm(z). It is the general solution of the Differential Equation

and can be obtained in the form of a series, terminating only when k is half a positive odd integer, viz.,