Hermite, in 1864 (Comptes Rendus, vol. 58) introduced into analysis the polynomials defined by the relation

where n is a positive integer. He showed that they satisfied the differential equation

that they were orthogonal functions, and that an arbitrary function f(x) could be expanded in the form

The proposition that if AA′‖BB′ then BB′‖AA′ appears at first sight so simple that it might be regarded as almost intuitive. This is because we already think of parallelism as a symmetrical relationship between two straight lines, in accordance with Euclid's definition of parallels as “straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.” If we take along with this definition Euclid's fifth postulate, or Playfair's equivalent, it defines a unique line through a given point parallel to a given line; but, without the postulate, it cannot be assumed to define more than a class of lines, and a stricter definition is required.

It is proposed in this paper to show now the well-known Laplace's transformation,

which is of great help in finding the solution of linear differential equations, gives also interesting results concerning the theory of integral equations. In §2 we shall study its application to certain differential equations, and find a large class of equations which remain unchanged by this transformation. Then, (§3), taking instead of eazt, a more general function of the product zt, we shall find a solution for some homogeneous integral equations ; in § 4 we shall describe a method of solving a very general type of integral equation of the first kind, namely,

a further extension to integral equations with the kernel ef(z)f(t) the object of §5. Then, studying an extension of Euler's transformation, we shall (§ 6) consider equations such as

which will prove to be singular; and finally, in §7, we shall give other examples of singular integral equations.

This paper contains

(a) an explicit solution of the problem of finding a function which is harmonic within a given sphere and takes at the surface the same value as a given rational integral homogeneous function of the rectangular coordinates of a point referred to the centre of the sphere as origin;

(b) a concise symbolical expression for the integral, over the surface of the sphere, of the product of any three rational integral spherical harmonics.

This method is associated with the well-known construction of a triangle equal in area to a given polygon.

In Fig. 1 let ABCDD1C1B1A be the polygonal circuit enclosing the area, and let DD1, be any side not adjacent to A.

In his article in No. 15 of Mathematical Notes, Dr Dougall has shown how by introducing series with n terms and a remainder, instead of infinite series, some well-known proofs of Newton's Theorems and others can be rendered more elementary in character. Once the idea has been suggested, it is of course easy to find further applications for it.

§ 1. This paper deals with certain formulae which, though probably not all new, have not appeared in the text-books. They were suggested to the writer while engaged in discussing the expression for the intensity of the transmitted beam in the Lummer Gehrcke Interference Spectroscope, viz.

(1). Let ABC be the given triangle; A′BC, B′CA, C′BA triangles described externally on its sides, and let the angles of these triangles be A′BC=μ1, A′CB=v1, B′AC = λ2, B′CA = v2 C′AB = λ3, C′BA = μ3, (Fig. 1).

The functions of Hermite, which are the same as the functions associated with the parabolic cylinder in harmonic analysis, may be defined* by the differential equation which they satisfy, namely,

where n denotes any constant.

The differential equation of Mathieu, or “equation of the elliptic cylinder functions,”

occurs in many physical and astronomical problems. From the general theory of linear differential equations, we learn that its solution is of the type

where A and B denote arbitrary constants, μ is a constant depending on the constants a and q of the differential equation, and φ(z) and ψ(z) are periodic functions of z. For certain values of a and q the constant μ vanishes, and the solution y is then a purely periodic function of z; but in general μ is different from zero.

Mathieu's differential equation

is the equation which arises out of those two-dimensional problems Mathematical Physics in which the boundary is an ellipse, such problems, for example, as the vibrations of an elliptic membrane, which was first discussed by Mathieu,* and the scattering electromagnetic waves by a wire of elliptic cross-section. A different use of the same equation is found in Celestial Mechanics the treatment of perturbations and oscillations about periodic orbits, and, in a more mundane connection, it has been shown to be the differential equation of the variety artiste who holds an assistant poised on a pole above his head while he himself is standing on a spherical ball rolling on the ground!

So far as I am aware, the methods, described below, of finding the attraction of a uniform shell of matter on a particle placed at external or an internal point, are new. They are particular applications of a method of finding the attraction of a thin shell uniform volume density bounded by similar and similarly situated ellipsoidal surfaces (what some have called a homothetic shell, and Thomson and Tait an elliptic homceoidœ) which I explained a paper, on the attraction of ellipsoidal shells and of solid ellipsoids, which was published in the Philosophical Magazine for April 1907. I give here also a short account of the more general problem with some additional notes and remarks. The solution depends on a geometrical theorem of some interest which occurred me in thinking over the problem of the ellipsoidal shell, and its solution by Poisson by a laborious and somewhat difficult process integration (Me'moires de I'Institut, t. xv., 1835).

The transformation of Continued Fractions into one another is a subject in which very little work has so far been done. Beyond the simple transformations given in works on elementary algebra, few transformation-theorems are known ; the best known being those connected with the “contraction” and “extension” of Continued Fractions, and the transformations of Euler, Bauer and Muir.

1. The following is an account of a theorem whose origin has been traced to Prof. Morley of Johns Hopkins University.

In the course of certain vector analysis, some 14 years ago, Prof. Morley found that if a variable cardioide touch the sides of a triangle the locus of its centre, that is, the centre of the circle on which the equal circle rolls, is a set of 9 lines which are three by three parallel, the directions being those of the sides of an equilateral triangle. The meets of these lines correspond to double tangents; they are also the meets of certain pairs of trisectors of the angles, internal and external, of the first triangle. This result was never published, and it was only the particular case of the internal trisectors that reached the present writers, the existence of the enveloping cardioides and the set of 9 lines being quite unknown to them.

In a previous paper (q.v.) by Mr W. L. Marr and the present writer, it was shown that, in accordance with Morley's Theorem, the angles A+2pπ, B+2pπ, C+2rπ of the triangles ABC be trisected, the three groups of six lines at the vertices give rise to 27 triangles DEF, the biangular coordinates of D with respect to BC being (B/3+2qπ/3, C/3+2rπ/3) or (βq, γr), and similarly for and F with respect to CA and AB.

In Morley's trisection theorem there are three triads of parallel lines which by their intersection with each other form equilateral triangles. The three lines EF, E11F11, E22F22 (v. Taylor and Marr) form one of these triads, and the equations are:—

where

and β γ β1, γ1, etc., are similarly defined.