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.

1. Let f(x) be a real function of period 2π, integrable L over (0, 2π), and let

By sn(x) and σn (x) we denote respectively the partial sums and the first arithmetic means of the series (1·1). Similarly, by and we denote the partial sums and the first arithmetic means of the series

conjugate to (1·1). By we mean the function conjugate to f(x), that is

where the integral is taken in the principal-value sense.

1. Die unendliche Folge der Partialsummen {sn} der Fourierschen Reihe einer integrierbaren und nach 2π periodischen Funktion f(ξ) der reellen Variabeln ξ ist an einer Stelle ξ = x, wo die Funktion stetig ist, (C, 1) limitierbarzu f(x) = s als Grenzwert, d.h. es ist

This note is a sequel to two in earlier volumes of the Proceedings, the first by myself and the second by Wilton†.

Suppose that

for |z| <1; that x > 0; that

forr ≥ 0; and that

where τ(x) is to mean 0 if x is not an integer. Thus

where the dash shows that the last term is to be halved when x is an integer;

forr > 1; and

Suppose that k(x) is of bounded variation in (− ∞, ∞) and that s(x) is bounded and continuous in the same range. Let

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.

It has been shown by Kulakoff that if G is a group, not cyclic, of order pl, p being an odd prime, the number of subgroups of G of order pk, for 0 < k < l, is congruent to 1 + p (mod p2); and by Hall that if G is any group of finite order whose Sylow subgroups of G of order pk, p being odd, are not cyclic, then, for 0 < k < l, the number of subgroups of G of order pk is congruent to 1 + p (mod p2). No results were given for the case p = 2. In the present paper it is shown that analogous results hold for the case p = 2, but that the role of the cyclic groups is played by groups of four exceptional types: the cyclic groups themselves, and three non-Abelian types. These groups are defined as follows:

(1) The dihedral group, of order 2k, generated by A and B, where

(2) The quaternion group, of order 2k, generated by A and B, where

(3) The "mixed" group, of order 2k, generated by A and B, where

1·1. Es bezeichne für nichtnegative ganze n und k

das n-te Cesàro-Mittel der Ordnung k der Reihe μ0 + μ1 + … Insbesondere ist also die n-te Partialsumme dieser Reihe.

The object of this paper is to complete the proof of the

Theorem. Let P(√d) be the quadratic field of discriminant d > 0. Then Euclid's algorithm does not hold in P(√d) if d is sufficiently large.

It is shown that slight departures from independence and equal weight of the sample observations introduce a bias into Fisher's z, but that the distributions of the variances “within and between groups”, and of Fisher's z would appear to maintain their form to the first order, although the extent to which the approximations hold in practice has not been considered.

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).

On any algebraic variety of d dimensions there exist certain systems of equivalence which are relative invariants under birational transformation. These systems include the canonical systems (Todd(3,4)) which can be defined for each dimension from zero to d − 1, and the systems defined by the intersections of these canonical systems among themselves. It is natural to enquire what is the precise behaviour of these invariant systems under birational transformations. Relatively few results of this kind are known. For threefolds, Segre has recently (2) investigated the transformations undergone by the invariant systems in any algebraic (not necessarily birational) transformation whose fundamental points and curves are of general character. More recently, A. Bassi (1) has obtained by topological methods the relations between the Zeuthen-Segre invariants of two Vd in an (α, α′) correspondence. I have recently (6) discussed the transformation of the invariant systems on a Vd for birational transformations on the assumption that the fundamental points are isolated and of general character.

The present author recently published what he believed to be a proof and extension of Campbell's theorem on fluctuations. More recently a paper has been published containing a criticism of this proof. It therefore seems desirable that some parts of the proof be amplified. Further explanation of the foundations of the theory seems required, in order that it shall not appear that the theory ignores the randomness of the events, and a more detailed account of the derivation of the mean value of functions of pairs of events (i.e. sums of the type ) is necessary.

Suppose that we have a set of observed frequencies classified in groups, the number in a typical group being nr, and that we wish to estimate the expectations in the groups according to some law with adjustable parameters. If these expectations are mr we take

Σ denoting summation over the groups, and according to the law in question the probability that one observation should come in the rth group is mr/N. The joint probability, for given mr, that there should be n1 observations in the first group, n2 in the second and so on, is therefore

It is well known that solutions of partial linear differential equations of the second order and of elliptic type are uniquely determined by their boundary data, and that they assume their maximum and minimum values on the boundary. The usual proofs make use of the principle of superposition and are therefore not applicable to non-linear problems. But recently Pryce has proved the uniqueness theorem for the non-linear equations of minimal surfaces and of Born's electrostatics. These equations are the Euler equations of the variational problem

k = + 1 corresponds to the case of minimal surfaces in n + 1 dimensions; k = − b−2, n = 3 corresponds to Born's electrostatics. Pryce's procedure depends essentially on the notion of conjugate variables in the calculus of variations for multiple integrals and can therefore be extended to a wide class of differential equations arising from variational problems (for several functions of several variables) as we show in § 3.

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 technical importance of constructing valve amplifiers disturbed by as little background noise as possible has led to a very searching investigation of the causes of this defect. Some causes—bad contacts, changes in the surface of the filament, secondary ions from the grid, and so forth—though technically speaking not trivial, are of minor physical interest; but it has long been recognized that there are two which cannot be dismissed so easily.

In this paper the condition for the existence of a secondary flow in a straight non-circular pipe is determined according to the modified vorticity transfer theory, with Goldstein's assumed form for the tensor It is shown that a secondary motion arises if the mixture length is not constant on the curves along which | grad u | is constant, u being the velocity parallel to the pipe axis.

In problems of turbulent flow treated by means of the modified vorticity transfer theory, the quantity where is the mean value of the square of the velocity fluctuation and p the mean pressure, plays a part analogous to the pressure in laminar flow. In two-dimensional flow through a channel the theory shows the existence of a gradient of across the channel from the central plane to each wall. Qualitative arguments such as are used to explain the existence of a secondary flow for laminar motion in a curved pipe are applied here to show that a secondary flow is to be expected near the short sides in the turbulent flow through a straight rectangular pipe of large length/breadth ratio.

The equations to determine the secondary flow through an almost circular elliptic pipe are discussed, the mixture length being assumed constant on ellipses similar to and concentric with the pipe section. For a first approximation the problem is reduced to the numerical solution of three simultaneous ordinary linear differential equations.

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. Taylor has shown (1) that two characteristic lengthsλ and λη may be defined for turbulent fluid motion. The length λ, which is connected with the dissipation of energy, is, for isotropic turbulence, given by

where is the mean rate of dissipation of energy per unit volume and represents the mean square value of any component of velocity. The length λη can be defined in terms of thus

where For isotropic turbulence Taylor assumed that

where B is a constant. Since the turbulence is isotropic,

and so, from (1), (2), (3) and (4) we have