The subjoined is an easy geometrical proof of the following theorem (which derives its importance from being part of the method of Chasles of constructing geometrically the ninth point when eight points of an “associated system” are given).

The purposes of the following note are these:—(1) To show the relation between Whittaker's Cardinal Function and Fourier's Repeated Integral; (2) to give a new derivation of Fourier's Integral Formula; and (3) to extend the notion of the Fourier Integral to the case in which the variables involved are complex.

Diophantus (Heath, 1885, page 122) is said to have the porism—“The diffeience of two cubes can be expressed as the sum of two cubes.”

§1. Let

denote a linear substitution of non-vanishing determinant; and let the roots k∈ of its characteristic equation

be for the present assumed distinct. Then with each root k∈ is associated an invariant point or pole P∈, and a linear invariant, or invariant (n−2)-plane ξ∈. If the n points P∈ do not lie on an (n−2)-plane, the determinant of their coordinates,

In Mathematische Annalen, Vol. 32 (1888) Peano discusses the solution of a system of homogeneous linear differential equations

where rij denotes a real function of the variable t, and shows how, by a series of repeated substitutions, this system of equations may be replaced by the equivalent equation

where X denotes the complex [x1, x2, … xn] and R the matrix

of which equation the solution X can be represented as a sum of integrals.

The linear differential equation of the second order

is not in general integrable by any method at present available. At the same time, several equations of this type have been integrated, either in terms of finite functions or by means of expansions in series. Some properties of the integrals of the general equation have also been obtained. It is the object of this paper to develop some general properties of these integrals, which throw some light on the nature of the solutions, even if not obtainable in explicit terms.

The theorems which furnish in C.F. form the roots of a quadratic equation, and the similar process which leads to particular integrals of an ordinary differential equation of the second order, may be applied to certain types of difference equation. The types which suggest themselves for examination are

the bilinear equation, and

a special form of the linear equation; the coefficients are functions of r, and s is constant.

§1. The general theorem underlying the subject of this paper is as follows:—

If through c1, c2, c3, … points on AB, a side of ΔABC, rays be drawn from two vertices O1, O2, the former meeting AC in b1, b2 b3, …, and the latter meeting BC in a1, a2, a3, …, then the lines a1 b2, a2b2, … envelope a conic touching the sides AC and BC. This follows since the ranges a1 a2 a3 …, b1 b2 b3 …, are homographic.

In the Bulletin of the American Mathematical Society Whittaker defines the confluent hypergeometric function Wk,m(x) by the equation

where the path of integration begins at t = + ∞, and after encircling the point t = Q in the counter-clockwise direction, returns to t = + ∞ again.

Most of the solutions of Laplace's equation in common use in mathematical physics have been expressed in the integral form given by Whittaker, viz

An analytic function is generally given either directly as a power-series, or at anyrate in some form which can readily be converted into a power-series. The power-series is not an altogether satisfactory method of representing the function, on account of the failure of its convergence outside its circle of convergence; in this respect it is decidedly inferior to the method of representing the function by a continued fraction, as this latter expression in many cases converges over the whole plane, except on certain curves.

The base transformations which transform the chord system of one twisted cubic into the chord system of another.

The present paper is based on a method attributed to Euler of expressing as a continued fraction the logarithmic derivate of a solu tion of a linear differential equation of the second order. The method is particularly applicable to equations of hypergeometric type, and, in that connection, was previously employed by the present author as a means of adding to the number of known transformations of continued fractions.

The following interesting system of points presented itself while I was investigating certain properties of the cubic curve. What I have called a “Determinantal System of Nine Points” is really a generalisation of the nine points defined by two systems of three parallel lines crossing one another

I have not seen the following properties of the Polar Conic and the Polar Conic of the Hessian given in treatises on the Cubic Curve. The results can be extended to space of n dimensions.

The present paper is an extension to the general cubic of certain relations connecting two special cubics previously considered. In what follows the term “birational” refers exclusively to the quadric transformation through three fixed points, the correspondence therefore being a whole-plane one for each transformation, and not confined to points on the transformed cubics. Also, the notation is that used in the paper referred to above.

The determination of the harmonic functions of elliptic and hyperbolic cylinders depends on the solution of Mathieu's differential equation. This equation, it has been remarked by Professor Whittaker, is the one which naturally comes up for study after the hypergeometric equation has been disposed of. Its solution presents difficulties which do not arise in connection with the hypergeometric equation or its degenerate cases, and it cannot, I think, be said that any discussion of the equation has yet been given with which the student of analysis can rest content. The treatment given below, though certainly incomplete at some points, seetns to follow the lines along which a thoroughly successful theory may be hoped for.