1. In a recent paper in these Proceedings by Mr H. Lob, and iu an earlier paper by Mr F. P. White, it has been shown how the well-known chains of theorems in plane geometry discovered by Morley and Clifford may be proved by projection from higher space. A curve of order n in space of n dimensions and certain derived loci are projected from one, or two, or three, …, or n − 2, out of n + 1 chosen points of the curve upon a plane which contains two further points of the curve. The n + 1 lines of the plane which form the starting-point of each chain of results as originally proved (and which are obtained in various ways in the course of the projections) are actually the lines in which the plane of projection is met by the primes containing n out of the n + 1 points. Now, with the standard equations of such a curve in [n], viz.

1. As long ago as 1878 Neumann gave a formula expressing the product of two Legendre polynomials as a sum of such polynomials. In the same year Adams gave an inductive proof, and obtained the result in the form

1. A general threefold in [5] has an apparent double surface whose projection on to a [4] is a surface which contains a triple curve and which is characterised by the fact that it has no improper nodes; such surfaces have been considered in a previous paper. In the present note we consider a class of surfaces in [5] which possess triple curves and project into surfaces in [4] having improper nodes: these are the surfaces which represent the chords of a general curve of ordinary space upon a quadric primal Ω of [5]. For the representation of a line congruence of [3] by the points of a surface on Ω, reference may be made to a previous note, the results of which are used in the present work.

1. The classification of surfaces whose sections are hyperelliptic has been given by Castelnuovo and that of surfaces with sectional genus π = 3 by Castelnuovo† and Scorza. For higher values of π the work is naturally more complicated and the possible types far more numerous, but the problem is simplified if we restrict ourselves to the investigation of surfaces without singularities, generally lying in space Sr (r > 3). In the present paper we obtain all such surfaces having sectional genus four.

1·1. The points, tangents, osculating planes, …, osculating primes of a curve may be said to form a system which is characterised by the number of these elements which are incident with a prime, …, line, point, respectively. For the normal rational quartic curve the system is (4, 6, 6, 4); projection of this system from a general point gives the system (4, 6, 6) in [3]; section by a general prime gives the system (6, 6, 4). These two systems in [3], which are the systems with which we are concerned in this paper, are duals of one another, and will be called systems of the first and second kinds respectively.

In a previous paper in these Proceedings the problem of the double integral

was discussed when the function F had the form

where

It is proposed in the present paper to extend the method to the general problem, where F may have any form provided only that it satisfies the necessary condition of being homogeneous of the first degree in A, B, C.

If F(x0, x1, x2, x3) = 0 is the equation of a surface in space of three dimensions which has an ordinary isolated s-ple point O, then by means of the substitutions

where Φ0 = 0, Φ1 = 0, …, Φr = 0 are the equations of r + 1 linearly independent surfaces passing simply through O, F is transformed into a surface F′ in [r], on which to the point O of F there corresponds a simple curve γ. The points of γ arise from the points of F in the first neighbourhood of O, and in this simple case the genus of γ is ½ (s − 1) (s − 2). In the study of properties which are common to all members of an infinite family of birationally equivalent surfaces no distinction is made between O and γ, O being regarded as a curve which has become infinitesimal on the particular surface of the family in question.

Gauss gave a well-known proof that under certain conditions the postulate that the arithmetic mean of a number of measures is the most probable estimate of the true value, given the observations, implies the normal law of error. I found recently that in an important practical case the mean is the most probable value, although the normal law does not hold. I suggested an explanation of the apparent discrepancy, but it does not seem to be the true one in the case under consideration.

1. The work of this paper was undertaken with a view to finding out what ruled surfaces can be determined by incidences, i.e. generated by the lines which meet a certain set of spaces which I shall call a base. Such ruled surfaces I shall call incidence scrolls. In [3] the lines which meet three lines generate a quadric surface. In [4] it is easy to show that a base consisting of a line and three planes gives the general rational quartic scroll, while the lines which meet five planes in [4] give the general elliptic quintic scroll. One might be tempted to think that at least all the rational normal scrolls could be obtained as incidence scrolls by taking for base a suitable number of spaces containing directrix curves, but unfortunately there is a residual surface except in the case of the rational scrolls of general type and of those with a directrix line.

1. In this paper I discuss the expression of m + 1 general forms F, F′, …, F(m), each of order n and homogeneous in r + 1 variables z0, z1, …, zl, …, zr, as the sums each of the nth powers of (the same) h + H linear forms in these variables. I take h > 0 of these linear forms to be undetermined, namely the forms

whose coefficients are undetermined; and I take the remaining H ≥ 0 linear forms to be assigned, namely the forms

whose coefficients are assigned.

Let be any transitive permutation group on the n symbols 1, …, n. Let be the subgroup of whose elements leave i fixed. Let ′ be the normalizer of , i.e., the subgroup of the symmetric group on 1, …, n transforming into itself. Let G′, G′1, G′2, etc., denote elements of ′. Finally, let ″ be the centralizer of , i.e., the subgroup in transforming every element of into itself.

1. In a previous paper (1) the authors used the theory of the moment generating function to deduce the known random sampling distributions of the estimated variance and co-variance in a system of variables following the normal law of frequency. In this paper we shall go much further. By means of the same general method the simultaneous distribution of the ½p (p + 1) second order moment statistics in a normal system of p mutually correlated variables will be deduced. It will be shown to be completely independent of that of the p sample means, and incidentally the method of proof to be developed will be found, in the special cases p = 1 and 2, to be an improvement on that given in the earlier paper, being independent of previous results reached by other authors.

1. In this note we give a direct evaluation of the integral

whose value has been inferred from the theory of statistics. Here A = Ap = (αμν) and C = Cp = (Cμν) are real symmetrical matrices, of which A is positive definite; there are ½ p (p + 1) independent variables of integration tμν (1 ≤ μ ≤ ν ≤ p), and tμν is written also as tνμ for symmetry of notation; in the summation ∑ the variables μ, ν run independently from 1 to p; k is a real number. A word of explanation is necessary with regard to the determination of the power |A − iT|−k. Since A is positive definite and T real and symmetric, the roots of the equation

Solutions of the bi-harmonic equation valid in the region bounded externally by parallel lines and internally by a circle midway between the lines have been given by one of the Authors in a recent paper [2]. These solutions were adapted to the requirements of certain problems in the theory of elasticity, but modified solutions satisfying the boundary conditions characteristic of viscous fluid motion are easily derived. These modified solutions will here be given and will be used to find the stream function corresponding to the slow rotation of a cylinder placed symmetrically between parallel walls.

The effect of changes in magnetic moment and in the earth's field on the period of invar pendulums used for relative gravity determinations is discussed, and methods of eliminating them by the use of Mumetal screens are described.

It is well known that a beam of light falling on a reflecting mirror forms standing waves. This effect has been very beautifully made use of in Lippmann's colour photography process. The standing light waves, in this case, produce a periodic effect in the emulsion of the photographic plate which, when developed, scatters light and produces a similar colour effect. Instead of using a beam of light, it would seem possible to scatter electrons from the emulsion and obtain a reflection of electrons similar to that of a space grating. But it seemed to us that it would be of much greater interest to consider an experiment in which electrons are reflected from the standing waves of light. The direct scattering of free electronic waves by light has strictly never been observed, and it was thought possible that by this method, owing to the interference of the electrons and to the fact that the scattered electrons are focussed to one spot, the magnification of the phenomenon would be sufficient to make it observable. From the theory developed below, it will be seen that the experiment is just on the verge of possibility, and would be very difficult to carry out. The main interest of the experiment would come from the possibility of observing stimulated scattered radiation which up to the present has never been verified experimentally.

Experiments are made on wave-lengths between 200 and 500 ms. for distances of transmission less than 200 kms. It is found that the variations of downcoming wave intensity are uncorrelated on two receivers separated by about one wave-length, and it is shown that this implies a considerable amount of lateral deviation of the waves. A special receiver is used to confirm the occurrence of lateral deviation, and an estimate of the angle of deviation is made. The possible causes of intensity variation are considered in the light of these experimental results, and it is suggested that a major cause of “fading” is the interference, at the ground, of waves “scattered” from a series of diffracting centres distributed over an area of radius at least 20 kms. in the present experiments. The possible results of such a mechanism are discussed.