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

is known by the theory of linear differential equations to have a general solution of the type

φ and ψ being periodic functions of z, with period 2π.

The science of the solution of Differential Equations has been in great measure systematized by the aid of ideas borrowed from the Theory of Functions, the equations being classified according to the singularities possessed by their solutions. In the case of linear Differential Equations of the second order

the solutions can have no singularities except at the singularities of the functions q(x) and r(x) (and possibly also at x = ∞ ): these equations may therefore be classified simply according to the number and nature of these singularities.

The proof usually given of Von Staudt's Theorem is entirely analytical in character. The following proof is geometrical:—

If A, B, C, D be the vertices of a tetrahedron whose opposite faces are α., β, γ, δ, and if l be any line, to prove that l [ABCD] = l [αβγδ] where lA denotes the plane through the line l and the point A and lα. denotes the intersection of the line l with the plane α.

Periodic solutions of Mathieu's equation*

where a is a suitable function of q have recently been discussed in several papers in these Proceedings. An elegant method of determining these solutions, which are written

was given by Whittaker, † who obtained the integral equation

which is satisfied by periodic solutions of Mathieu's equation.

Most lecturers on Analytical Statics only consider the following catenaries: the common catenary, the parabolic catenary of the suspension bridge, and, less frequently, the catenary of uniform strength. Of these the second only represents the form of the suspension bridge chain when the weight of the chain is neglected in comparison with that of the roadway, while the first represents the opposite extreme when the weight of the chain alone is taken into account.

In a paper which appeared in the Proceedings of the Edinburgh Mathematical Society, Vol. XXXII., Session 1913–14,* I showed how the application of Laplace's transformation to certain linear differential equations enables us to solve some homogeneous integral equations of the first and second kinds, and I obtained, by an extension of this method, the solution of integral equations whose nucleus is of the form f(zt) or ef(z)f(t).

According to Feuerbach's theorem, the inscribed and the three escribed circles of a plane triangle are all touched by a circle

Hart's Theorem extends the proposition to a spherical triangle, or, which comes to the same thing, to a plane triangle formed by three circular arcs.

In the present paper the disposition of the roots of the confluent hypergeometric functions — denoted by Wk, m(z) — as affected by changing the parameters k and m is investigated. The results are then shewn in a graphical form, and various typical illustrations of the functions are given. By giving special values to k and m it is then exemplified how the roots of other functions expressible in terms of Wk, m(z) may be studied. The zeros of the parabolic cylinder functions are then discussed. Some of the properties of an allied class of functions, denoted by ψn(z), are then given, and finally, it is shewn how the properties of Abel's function φm(z) may be obtained from results already given.

Ellipsoidal harmonics are defined to be those solutions of Laplace's equation

(where x, y, z are rectangular coordinates) which are useful in problems relating to ellipsoids. If the equation

represents a family of confocal quadrics, it is known that the ellipsoidal harmonics belonging to the family are products of the form

where l1, l2… are constants: one term is to be picked out of the square brackets as a multiplier of the other factors. Now if we consider the case in which two of the principal axes of the ellipsoids are equal, the latter become spheroids. If then we put b = 0 in (1) the family of confocal spheroids has the equation

and belonging to this family there will be spheroidal harmonics of the form given by (2) with b zero.

Let O1, and H1 be two points, in the plane of any triangle of reference ABC, so related that if 01P, O1Q, O1R be the perpendiculars drawn to the sides of ABC, then AP, BQ, CR meet in H1. We shall find that O1, and H1 describe respectively two cubics which are related to each other in a remarkable manner. We shall show, for instance, that points in each curve may be derived from each other by two sets of three alternative rational quadric transformations, and that the join of correspondents passes through a fixed point as in plane projection. We shall then discuss the homographic relation between corresponding pencils formed by rays through pairs of related points—not direct correspondents—and investigate the relation between these latter points.

§ 1. The famous theorem of the pedal line of a triangle in ordinary geometry can be stated as follows:—“Given a triangle ABC and a point P such that the feet of the perpendiculars X, Y, Z, dropped from P on the sides of the triangle, are collinear, then the locus of P is the circumcircle.” In noneuclidean geometry this locus is not a circle or even a curve of the second degree, but a cubic; and in both cases the envelope of the line XYZ is a curve of the third class. The explanation of the inconsistency in ordinary geometry is that the complete locus consists of the circumcircle together with the straight line at infinity.

In a recent number of the American Mathematical Monthly (December 1914), Professor E. V. Huntington calls attention to the inconclusive, and in some cases erroneous, discussion of the uniplanar motion of a rigid body as it is presented in the more elementary books on mechanics. In the simpler problems considered the solution is usually found by use of the rule that the rate of change of moment of momentum about some convenient point is equal to the torque or turning moment about the same point. In many cases there is no difficulty choosing this convenient point; but apparently there is confusion as to the points which can be legitimately chosen.

This paper extends some familiar theorems concerning the relations between the roots of a polynomial and those of its first derivative to the more general case of the rational function with a pole at a single point.

The contents of this paper were suggested by a discussion of the equation:

in a paper by Glaisher which appears in the Philosophical Transactions, 1881, Part III.

The solutions in series of (1) are:

and in the paper referred to it is shewn that the coefficients of hp+1 in the expansions of and of satisfy equation (1) when p is a positive integer.

In a paper recently read before this Society, Mr E. Blades obtained a general formula for spheroidal harmonics in the form of the general solution of Laplace's equation given by Professor Whittaker,

If spheroidal coordinates r, θ, φ are defined by

the result obtained is

Various solutions of Mathieu's equation, or the equation of the elliptic cylinder functions

have recently been discussed in an elegant series of papers in these Proceedings. These papers have dealt with the periodic and quasi-periodic solutions, but the present paper merely considers determinants which give the infinite series of relations between a and q, so that the solutions are purely periodic, i.e. the solutions denoted by Professor Whittaker

In this paper are introduced what we shall term “successive oscillation functions.” These functions are derived from functions of a real variable. The word “function” as here used has its widest meaning. We say y is a function of x in an interval of the the x-axis, if given any value of x, in the interval one or more values of y are thereby determined. The values of the function may be determined by any arbitrary law whatsoever. We shall deal with discontinuous functions; the theorems will be true for continuous functions, but will be trivial, except in the case of functions which are discontinuous and whose points of discontinuity are infinite in number. We shall assume in what follows that the values of the function lie between finite limits.