Formulae for multiple tangents to the general surface in ordinary space were obtained at different times by various writers, but the discussion remained incomplete until the advent of Schubert's enumerative method, which solves the whole problem by a purely mechanical process. Schubert later extended the method to general forms in [n], in the restricted case where the multiple tangents have only a single contact. In the present paper the results for three dimensions have been used to build up the formulae for forms in [4], by means of the correspondence theorem on which much of Schubert's work is based. It would no doubt be possible to evolve a complete set of incidence formulae for four dimensions and then to proceed as in Schubert's discussion of surfaces; but the present method is preferable for two reasons. In the first place, all the results have been obtained in a very simple manner; and secondly, a large number of minor results have been found in the process.

Let f(t) be a function periodic in 2π, and absolutely integrable in the interval (0, 2π). We write

Sommerfeld has shown that the electromagnetic field set up by a Hertzian oscillator placed on the infinite plane surface separating two media normal to that surface may be obtained from the Hertzian function of cylindrical coordinates r, z,

where the suffixes 1, 2 refer to the two media occupying the regions z>0, z<0,

and

where ɛ1, μ1, σ1 are the dielectric constant, permeability, and conductivity (in Heaviside units) of the first medium, k2, k2 like expressions for the second, c is the velocity of light, and n the frequency in radians per second; +(λ2−k2)½ is taken to have its real part positive.

The boundary layer equations for a steady two-dimensional motion are solved for any given initial velocity distribution (distribution along a normal to the boundary wall, downstream of which the motion is to be calculated). This initial velocity distribution is assumed expressible as a polynomial in the distance from the wall. Three cases are considered: first, when in the initial distribution the velocity vanishes at the wall but its gradient along the normal does not; second, when the velocity in the initial distribution does not vanish at the wall; and third, when both the velocity and its normal gradient vanish at the wall (as at a point where the forward flow separates from the boundary). The solution is found as a power series in some fractional power of the distance along the wall, whose coefficients are functions of the distance from the wall to be found from ordinary differential equations. Some progress is made in the numerical calculation of these coefficients, especially in the first case. The main object was to find means for a step-by-step calculation of the velocity field in a boundary layer, and it is thought that such a procedure may possibly be successful even if laborious.

The same mathematical method is used to calculate the flow behind a flat plate along a stream. The results are shown in Figures 1 and 2, drawn from Tables III and IV.

I think that some light may be thrown on a recent paper of Professor Whittaker's by giving concrete examples. The first example considered belongs to a transformation, which I gave some years ago, whereby a constant acceleration is incorporated in Einstein's form. Opportunity is taken to develop some features of the kinetic form of solution, in which all space coordinates are relative to a source, and the time of the latter is the only time appearing. This is carried as far as the treatment of energy.

Continued fractions were generalised to more than one dimension by Jacobi and others: later Perron gave an account of the existing state of the subject with a detailed discussion of periodic fractions. Quite recently the subject has been attacked afresh by Mr. Maunsell

It has been remarked in a former paper that perhaps a fuller understanding of the theory of linear groups may be arrived at by considering the real representations. It is sufficient that these be irreducible in the real field. In the present paper we continue this investigation and deal with the angles of rotation; in particular we find the form of the commutator

where S and T are substitutions of order p1 and p2 and have angles of rotation θ, θ′, θ″, … and φ, φ′, φ″, …. If certain conclusions regarding the orders of S, T, and C may be drawn we shall then be able to attack a very important problem—;that of Primitivity. This is in essence very similar to the method of Blichfeldt, since the commutator is the product of

which are both of order p2.

A continuous arc joining two points (x1, y1), (x2, y2) is the aggregate of points given by

where f and φ are continuous functions, and where

The usual methods of investigation of asymptotic expansions of the various types of Bessel Functions show that the remainder is less in absolute value than the first term neglected. A more refined result was obtained by Stieltjes for K0(x) and certain other functions of order zero; he found an asymptotic series for the remainder and showed that the error due to stopping at one of the smallest terms is of the order of half the first term neglected. Watson notes that it would be of some interest to obtain corresponding results for functions of any order, and this is the object of this note. The method is quite different from that of Stieltjes.

A series is said to be summable by Riemann's method of order k to S, or summable (R, k) to S, if

where k is a positive integer, and where we suppose that

is convergent for values of h in a neighbourhood of the origin.

We are concerned in this note with a problem which may be regarded as a generalisation of one which occurs in the theory of Fourier series.

This paper is devoted chiefly to the consideration of the surface oscillations of water contained in a vessel in the shape of a circular cylinder with its axis vertical, when the motion is slightly disturbed from a uniform rotation about the axis of the vessel. The work was undertaken with the hope of finding some indication of the effect of the depth of the water in the vessel on the period of the surface waves, and for the purpose a vessel of circular cross-section was naturally chosen. It is shown that a slight change of shape does not affect the periods of the oscillations. The solution of the corresponding problem when the surface oscillations take the form of “long waves” or “tidal waves” is well known, and the present paper deals only with “short waves,” for which the horizontal velocity is not the same at all depths.

The principal object of the present paper is to give a geometrical account of a certain Cremona transformation of four-dimensional space ([4]) into itself. The transformation in question is analogous to the cubo-cubic transformation of [3] which is obtained by drawing cubic surfaces through a twisted sextic curve of genus three, . A similar transformation, in space of any number of dimensions, has been described analytically in a note by Godeaux; and most of our work in the present paper extends to the general case. The case of four dimensions is studied here as being less abstract, and more easily susceptible of geometrical treatment; in the general case it is less simple to dispense completely with algebraical methods. We shall see that the various loci which arise in the course of the work are already well known.

1. It is known, from the theory of the Riemann theta-functions, that the canonical series of a general curve of genus p has 2P−1 (2P − 1) sets which consist of p − 1. points each counted twice. Taking as projective model of the curve the canonical curve of order 2p − 2 in space of p− 1 dimensions, whose canonical series is given by the intersection of primes, we have the number of contact primes of the curve. The 28 bitangents of a plane quartic curve, the canonical curve of genus 3, have been studied in detail since the days of Plücker. The number, 120, of tritangent planes of the sextic curve of intersection of a quadric and a cubic surface, the canonical curve of genus 4, has been obtained directly by correspondence arguments by Enriques. Enriques also remarks that the general formula 2P−1 (2p −1) is a special case of the formula of de Jonquières, which was proved, by correspondence methods, by Torelli§.

Consider the series

and let

The series

is not necessarily convergent. Suppose that its Poisson sum exists. This we denote by

The expression

when existent will be said to be the sum (, p) of the series (1).

The formulae considered in this paper deal with the multiple tangent lines and planes to surfaces in three dimensions; the functional method employed is that of a previous paper, in which a more restricted discussion was given for surfaces in higher space.

We now consider some properties of “almost all” series

where the numbers vary independently in the interval (0, 1). What we need is a definition of measure in the space of sequences or, what is the same thing, in the space

of infinitely many dimensions. Such a definition has been given by Steinhaus. He defines a correspondence between the cube (7.2) and the interval (0,1) in the following way:

The planes of a given S5 can be represented by the points of a locus V9 in space of nineteen dimensions. This locus is a double locus on a certain other manifold V14, and the tangent spaces of the V9 generate a W18. Segre has a memoir on the subject of these loci, in which he arrives at his results by a series of short steps, the argument being mainly geometrical.

The Heine-Borel Theorem for one-dimensional intervals may be enunciated as follows:

(A) If a set D of intervals, all in the closed interval (a, b), be such that every point of (a, b) is an interior point of at least one interval of the set D (the end points a, b being regarded as interior to an interval when either of them is an end point of such interval), then a finite set E of intervals all belonging to D exists such that every point of (a, b) is interior to at least one interval of E.

The Fourier series of the function defined by

is

and if

then

while

The expression (1) exceeds (2); this is the Gibbs phenomenon.