It is familiar that if on an algebraic surface there is an exceptional curve, that is an irreducible rational curve of virtual grade − 1 when no points of it are assigned as base points, and if there is on the surface a canonical system containing some actual curves, so that pg ≥ 1, then the exceptional curve is a fixed constituent of every curve of the canonical system, generally a simple constituent, and in that case has no intersections with the residual constituent. More generally, if there is on the surface a reducible exceptional curve, i.e. a set of curves which can be transformed into the neighbourhoods of a family of simple points (some of which are in the neighbourhoods of others) on a surface birationally equivalent to the given one, then the canonical system has as a fixed constituent of all its curves at least that combination of the curves which corresponds to the sum of the total neighbourhoods of the points, and generally just this combination, in which case this fixed part has no intersection with the residual variable part.

A ternary form of degree n can be expressed as a symmetrical determinant, of n rows and columns, whose elements are linear forms; furthermore, not only is such a mode of expression known to be possible, but A. C. Dixon, in 1902, gave* a process by which the determinant can be obtained when the ternary form is given. This process, however, although it admits of such a straightforward theoretical description, cannot be carried through in practice, for a general ternary form, without the introduction of complicated algebraical irrationalities, even if we restrict ourselves to forms of the fourth degree; consequently no application of Dixon's process to an actual example seems to have been published. If then a choice can be made of a quartic form for which the reduction to a symmetrical determinant can be carried out without undue complication, it seems fitting to give some account of it. The following pages are therefore devoted to the study, from this aspect, of the form x4+y4+z4, for which the reduction can be accomplished without introducing any irrationality other than the fourth root of − 1.

A Cremona transformation Tn, n′ between two three-dimensional spaces is said to be monoidal if the surfaces of order n in one space which form the homaloidal system corresponding to the planes of the second space have a fixed (n − 1)-ple point O. If the surfaces of order n′ forming the homaloidal system in the second space have a fixed (n′ − 1)-ple point O′, the transformation is said to be bimonoidal. A particularly simple bimonoidal transformation is that which transforms lines through O into lines through O′, and planes through O into planes through O′. Such a transformation we shall call an M-transformation. Its equations can, by suitable choice of coordinates, be expressed in the form

where φn−1(x, y, z, w) = 0, φn(x, y, z, w) = 0 are monoids with vertex (0, 0, 0, 1).

In discussing Hecke's function

where {t} = t − [t] − ½ Hardy and Littlewood* made use of the identity

Here

θ is an irrational number between 0 and 1, and pν/qν is the νth convergent to θ (beginning with p0 = 0, q0 = 1). The proof of (1) given by Hardy and Littlewood is of an analytic character.

The Hankel transform of a function x−½νf(x) can be determined by the help of the following rule, obtained for ν = 0, 1 by B. van der Pol and K. Niessen(1), for positive integral values of ν by K. Niessen(2), and for arbitrary values of ν, [R(ν) > − 1], by F. Tricomi(3). If*

F(p) ≒ f(x),

then

1. Particular series of conal and toroidal functions have been studied by various authors*, but so far no method has been given of expanding an arbitrary function in such a series. Ordinary methods fail, since conal functions (and also toroidal functions) are not orthogonal, although they form a complete set. In this paper I give an expansion theorem.

This paper may be regarded as a sequel to a previous papers(1) in these Proceedings. The vector and matrix notation of that paper used for a statistical sample is systematized somewhat further, so that while a sample S refers as before to the matrix of nm values (a sample of m observations in one variate only being a row vector), we write

for the linear regression formula between the dependent and independent variates into which a sample is supposed partitioned (in place of equation (12) of (1)). More generally, a third submatrix S0 is partitioned off, and its effect eliminated (corresponding to equation (13) of (1)), but without loss of generality we assume that S2 in equation (1) above can always stand for S2.0 if necessary.

The notion of fiducial probability was introduced by R. A. Fisher and is now widely used in statistical work involving a single unknown parameter. Fisher has also considered the joint fiducial distribution of several parameters for which there exists a sufficient set of statistics, and has derived the fiducial distribution of the two parameters of the one-variate normal law*. Since Fisher's account is brief, we give a more extended description of the fiducial distribution of several independent parameters possessing a sufficient set of statistics.

The velocity distribution in the wake behind a body of revolution calculated by Miss Swain on the momentum transfer theory with l constant over a cross-section of the wake is not in accordance with experiment.

On the generalized vorticity transfer theory the equation for the first approximation to the velocity defect u far downstream is

There are two methods of simplifying this equation on the analogy of the successful simplification of the corresponding equation for the two-dimensional case. One method gives the same equation as if the turbulence were symmetrical; the other gives the equation of the modified vorticity transfer theory.

For the former there is no real solution if l is constant over a cross-section; for a real solution l must be infinite on the axis of symmetry. The equation was integrated with l ∝ r−1 over a cross-section; this variation of l was suggested by putting the change in the area enclosed by a circular vortex-line proportional to the area of the section of the wake. The agreement with experiment is fairly good over the middle portion of the wake, but not near the edge, where the theoretical value of ∂u/∂r is not zero. If l2 ∝ rp, ∂u/∂r vanishes at the edge of the wake only if p = − 3. The solution is carried through for this case; it shows only fair agreement with experiment over the middle of the wake, and none at the edge.

The calculation on the modified vorticity transfer theory is carried out with l constant over a section; in this case there is no definite edge to the wake, nor does u fall off sufficiently rapidly to make the momentum integral for the drag converge. (u ∝ r−2 for large r.) Even over the middle of the wake there is a discrepancy with experimental results, rather like that shown by the momentum transfer theory.

The temperature distribution behind a heated body was calculated according to each of the foregoing theories.

The temperature distribution was also calculated from the experimental velocity distribution, without any assumption being made concerning the kinematic coefficient of turbulent diffusion, which was eliminated between the equations for the velocity and temperature. The vorticity transfer theory with symmetrical turbulence gives an impossible result. The modified vorticity transfer theory gives θ/θmax < u/umax when the experimental velocity distribution is used.

By using Bragg and Williams' approximations and Bethe's interaction energies between nearest neighbours, it has been shown that, for alloys of the AB3 type, (1) the maximum value of the critical temperature Tc must occur when c = ½ (2) for all temperatures within the range 0 < T < (Tc)max there is always a concentration range within which the alloy exists in a two-phase state, the one phase being partially ordered and the other disordered. This two-phase region is small at all temperatures and vanishes both at T = 0 and at T = (Tc)max. In particular for T/(Tc)max = 0·82, which is the value of Tc for c = ¼, the limits of the two-phase region are c = 0·245 and c = 0·256. The theoretical phase diagram has therefore qualitatively the shape, given in Fig. 5. For a fixed concentration there is thus a certain temperature range in which a disordered phase and a partially ordered phase exist in equilibrium with one another. The temperature at which the disordered phase completely disappears is that usually given by experimental determinations of the critical temperature. It is somewhat higher than Tc, but its maximum value also occurs when c = ½, so that the discrepancy between theory and experiment for this type of superlattice still remains.

6. Bethe's theory of order and disorder in alloys is generalized so as to include transitions in which the lattice structure changes from cubic to tetragonal. It is shown that, with the assumptions made, there is a discontinuity in the order at the transition temperature and also a latent heat. An expression is derived for the degree of tetragonality as a function of the order.

The present discussion of the conduction levels of copper and silver was undertaken in view of the experimental investigations of the absorption and emission bands of metals, in the soft X-ray region, which have been carried out in this laboratory and elsewhere*. It is a well-known result of the electron theory of metals that, for cubic crystals, of lattice constant a, if k is the wave vector of an electron moving in the crystal lattice, there are discontinuities in the energy E of the electron for values of k such that

(n1, n2, n3) being the Miller indices of the planes for which Bragg reflexion of the electron waves takes place. If we draw the energy E as a function of k for a particular direction in the lattice, then we obtain, using the usual notation, curves such as those shown in fig. 1, d being the half-distance between the corresponding planes for which Bragg reflexion takes place.

The conductivity of thin films of the alkali metals has recently been measured in the H. W. Wills Physical Laboratory, Bristol*. It was found that as the thickness of the film is decreased to that of a few atomic layers the conductivity drops below that of the bulk metal. In the papers quoted the hypothesis was put forward that this effect is due to the shortening of the mean free paths of the conduction electrons of the metal by collisions with the boundaries of the film. The experimental results were compared with a formula derived on the basis of this hypothesis. This formula was, however, obtained subject to a number of simplifying assumptions, and it is the first purpose of this paper to obtain a more accurate formula. I also compare this formula with experiment, and make certain deductions about the surfaces of thin films.

It is shown that the soft X-ray emission bands from Be and A1, present in small quantities in solid solution in Cu and A1, have a form without a sharp highenergy edge, the feature most characteristic of these bands in pure metals. The diffuse form of the bands indicates (1) that all the valence electrons of the impurity atom are given up to the lattice, (2) that the impurity atom refuses to take within it those electrons of the lattice which have energies close to the maximum energy of the conduction electrons.

The absorption limits for the primary β-particles of mesothorium 2 and uranium X2 have been determined using modified tube counters of various types. Results in good agreement with previous single determinations with gold-leaf electroscopes, and at variance with deductions from expansion chamber observations, show the need for caution in fixing limiting energies in β disintegrations from measurements on a relatively small number of tracks.

No ionization was found in hydrogen leaving palladium, either when the gas diffused through the metal or was evolved by warming palladium previously charged with hydrogen by electrolysis in dilute sulphuric acid.