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.

1. If a class consists of n members of which r have the property φ, and we wish to estimate r by the process of sampling, the distribution of probability among values of r, given the composition of the sample, necessarily depends to some extent on our previous knowledge. This is expressed by saying that the prior probability of a particular value of r is f(r). According to Laplace's theory f(r) is constant. It is obvious that there are cases where the previous knowledge is such as to ensure that f(r) is not constant; but the question remains whether such cases should be considered the exception or the rule.

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.

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.

1. In a previous note the author has examined the systems of tangent planes to a degenerate surface in S3 consisting of n planes, regarded as the limit of a general surface of the same order. It is well known that a pair of space curves which are the limiting form of a non-degenerate curve must have a certain number of intersections; hence, if a surface in higher space degenerates into a number of surfaces, these must intersect in curves of various orders. In the present paper we consider the nature of the envelope to a surface consisting of two general surfaces of S4 having in common a single curve of general character, the degenerate surface being regarded as the limit of some surface of general type. The same conclusions hold for a similar degenerate surface in Sr (r>4).

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.

In a recent paper we have discussed certain general principles underlying the determination of the most efficient tests of statistical hypotheses, but the method of approach did not involve any detailed consideration of the question of a priori probability. We propose now to consider more fully the bearing of the earlier results on this question and in particular to discuss what statements of value to the statistician in reaching his final judgment can be made from an analysis of observed data, which would not be modified by any change in the probabilities a priori. In dealing with the problem of statistical estimation, R. A. Fisher has shown how, under certain conditions, what may be described as rules of behaviour can be employed which will lead to results independent of these probabilities; in this connection he has discussed the important conception of what he terms fiducial limits. But the testing of statistical hypotheses cannot be treated as a problem in estimation, and it is necessary to discuss afresh in what sense tests can be employed which are independent of a priori probability laws.

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).

This paper gives experimental results for the scattering of 4 and 2 volt electrons in nitrogen, which are compared with Stier's theoretical curves. An approximate theory of the scattering of electrons in nitrogen is given which is applicable when the wave length of the electron is small compared to the internuclear distance. The results are compared with experiment.

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.

The following paper arises from a remark in a recent paper by Professor Baker. In that paper he gives a simple rule, under which a rational surface has a multiple line, expressed in terms of the system of plane curves which represent the prime sections of the surface. The rule is that, if one system of representing curves is given by an equation of the form

the surface being given, in space (x0, x1,…, xr), by the equations

then the surface contains the line

corresponding to the curve φ = 0; and if the curve φ = 0 has genus q, this line is of multiplicity q + 1.

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

Some experiments are described in which the efficiency of ionisation of potassium at a hot molybdenum surface was investigated. The degree of ionisation observed is consistent with the most recent measurements of the electron work function of molybdenum.

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. It is known that, in [3], a ruled surface of order n and genus p has in general a double curve of order ½ (n − 1) (n − 2) − p and genus ½ (n − 5) (n + 2p − 2) + 1, 2(n + 2p − 2) torsal generators, 2(n − 2)(n − 3) − 2(n − 6)p generators which touch the double curve, and triple points.

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.

The effect of a monomolecular film on the reflexion of light from a plane surface is considered. The film is treated as a two-dimensional assembly of scattering centres, and the possibility of molecular orientation is allowed for. Experimental results, when available, should indicate whether the scattering properties of groups of molecules in the “two-dimensional” state differ from those of molecules in bulk.

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 three-phase star-connected alternating-current motor of simplest type, connected to busbars maintained at sinusoidal potentials e1, e2 and e3 with respect to the star-point and connected to a shaft load which exerts a reactive torque f on the rotor, is dynamically specified by the following functions:

where the are the armature currents in the three phases, i is the current in the amortisseur circuit, Q the current in the field circuit, θ the rotor position angle, T the kinetic energy, V the potential energy, S the Rayleigh dissipation and U the activity of the external forces on the machine. The self-inductances of the armature circuits and the mutual inductances between them are assumed to be constant.