1. In this note we investigate, from a new point of view, some properties of cubic primals in [4]. These are enunciated by Fano as follows.

In a recent publication the problem of determining the abundance of two interacting species of animals was investigated, and the fundamental equations were established.

The well-known methods of classical mechanics, based on the use of a Lagrangian or Hamiltonian function, are adequate for the treatment of nearly all dynamical systems met with in practice. There are, however, a few exceptional cases to which the ordinary methods are not immediately applicable. For example, the ordinary Hamiltonian method cannot be used when the momenta pr, defined in terms of the Lagrangian function L by the usual formulae pr = ∂L/∂qr, are not independent functions of the velocities. A practical case of this kind is provided by the electromagnetic field, considered as a dynamical system with an infinite number of degrees of freedom, since the momentum conjugate to the scalar potential at any point vanishes identically. Again, for the very simple example of the relativistic motion of a particle of zero rest-mass in field-free space, the Lagrangian function vanishes and the usual Lagrangian method is not applicable.

The theory of the exact difference equation in the general linear case has been fully developed, but the corresponding theory for the non-linear equation of the first order does not appear to have been considered. In this paper necessary and sufficient conditions for the difference equation of the first order to be exact and the form of the primitive are obtained. It appears that two conditions are required for a difference equation to be exact, one of which is identically satisfied in the limiting case of the exact differential equation. These conditions are applied to determining the primitive in some cases where the conditions for exactness are not satisfied.

Nöther in his classical paper Zur Theorie des eindeutigen Ent-sprechens algebraischer Gebilde has given a set of sixteen examples of the computation of his invariants pn, p(2), p(1) for algebraic surfaces in ordinary space. In the following I discuss, as a matter of some interest, the birational transformation of his surfaces into surfaces which are non-singular.

The bisecant curves of a ruled surface, that is to say the curves on the surface which meet each generator in two points, are fundamental in the consideration of the normal space of the ruled surface. It is well known that if is a bisecant curve of order ν and genus π on a ruled surface of order N and genus P, then

provided that the curve has no double points which count twice as intersections of a generator of the ruled surface.

In the Wöhler rotating bar fatigue test, in order to examine the behaviour of a specimen under alternating stress, a load is suspended from the end of a cantilever bar, and the bar is then rotated, the load remaining stationary. It is evident that the rotation will set up additional stresses in the bar, so that the limits between which the stress at any point will alternate are not the same as the stress at the corresponding point of a statically deflected bar.

In a recent paper in these Proceedings on the rational normal octavic surfaces with a double line in [5] I found four such surfaces, , and representable on a plane respectively by the systems of curves C5 (22, 19), C6(26, 14), and C7 (3, 28), with the base points in each case lying on an elliptic cubic. Inadvertently I overlooked a solution of certain indeterminate equations which leads to a fifth type represented by the plane system C9(38, 1).

In the following pages I have developed the theory of matrices by resolving them into parallel components arranged diagonally, rather than into the usual rows and columns. This treatment is natural in view of the fundamental fact that the resolution is undestroyed when matrices are formed into products (Theorem 2). It is closely related to the theory of continuants and of continued fractions. Certain features stand out in such a presentation—the distinction between the length and range of a diagonal (§ 4), that between regular and irregular diagonals (§ 6), and the use of equable partition (§ 7). The exact conditions for the existence of an rth root of a given singular matrix are examined in § 9 and summarized under the title, the condition of equability.

I am indebted to Mr L. Kassel of the Department of Commerce, U.S.A., for pointing out an error of computation in a recent paper in these Proceedings.

The asymptotic expansion for the partition function should read

where

An instrument is described which provides an accurately monochromatised beam of X-rays by reflection at a crystal and permits the wave-length to be changed without moving either the X-ray tube or the measuring apparatus. This is effected by an automatic adjustment which moves a second crystal into the correct position and orientation to reflect the monochromatised beam along a fixed emergent direction. Twenty wave-lengths well distributed in the range 0·8Å.—4Å. are available, the time required to select any one being about one minute. The whole instrument is contained in a small metal box which is evacuated.

I am indebted to Professor A. M. Tyndall, F.R.S., for extending to me the facilities of his laboratory, and for his interest in this work.

A convenient form of cylindrical condenser consists of a stack of thin circular discs of suitable insulating material, interleaved with discs of metal foil of somewhat smaller diameter. Numbering the foils from one end, all the odd foils and all the even foils are connected together. The total capacity is then n times that between two adjacent foils, if the total number of foils is n + 1.

The algebraical foundation of this representation is very simple: it depends on the theorem

If a coordinate system in [m] is given, then in a general [k] of the [m] a unique set of k + 1 points can be found whose coordinates form the scheme

where h = m − k − 1, and n = (m − k)(k + 1) = kh + m.

In the spectrum formed by a grating the red is more deviated than the blue, while in that formed by a prism, which does not exhibit anomalous dispersion in the visible spectrum, the reverse is the case. If, then, white light is passed first through a grating and then through a prism, the nature of the spectrum produced will depend on the relative dispersions of the grating and prism. If throughout the spectrum the dispersion of the prism is less than that of the grating, the sequence of colours will be the same as that in the spectrum due to the grating alone, red being more deviated than blue. If, on the other hand, the dispersion of the prism is throughout the spectrum greater than that of the grating, the sequence of colours will be reversed. On account of the rapid increase in the dispersion of glass near the blue end of the spectrum, it may, however, be possible that the dispersion of the prism exceeds that of the grating in the blue while in the red the reverse is the case. Under these conditions both red and blue will be more deviated than some intermediate colour for which the dispersions of the grating and prism are equal. The spectrum will thus have a sharp edge of this colour at the end of minimum deviation, while in the direction of greater deviation it will at each point consist of two superimposed colours. It is as if an ordinary white light spectrum was folded back on itself about the sharp edge.

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.

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.

1·1. 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.