bluuebunny/crossref_metadata_2025 huggingface dataset:
Anaphora and Relativization in Child Second Language Acquisition
Authors: Flanigan Beverly Olson
Published: November 2008

The focus of this study is the development of control over anaphoric reference and relativization by children learning ESL in a pull-out classroom employing little overt grammar instruction. Twenty-three children aged 6.5–14 representing ESL proficiency levels 3–5 on the Bilingual Syntax Measure (BSM)were given paper-and-pencil tests to determine comprehension of anaphoric (reflexive and pronominal)reference in English; in addition, they were tested on both comprehension and production of restrictive relative clause types (SS, SO, OO, and OS)in English. Scores were higher on reflexives than on pronominals, with length of residence significant in ambiguous references. Relative clause interpretation varied significantly with proficiency level, but production was not predictable from general proficiency except at BSM level 5, and then only on SO and SS relativization. Transfer from the L1 was minimal. It is concluded that exposure and overall L2 proficiency, rather than age or L1 background, are the most significant factors in the development of these generally untaught and untested “late-learned” rules.
Acquiring Nothing?
Authors: Polio Charlene
Published: November 2008

Some earlier SLA research suggests that zero pronouns should be easily acquired by second language learners, yet this has not been carefully examined because studies of nominal reference are usually limited to the acquisition of languages that do not have zero pronouns. This study examines how speakers of languages with zero pronouns (Japanese) and without them (English) use zero pronouns when acquiring a language that has them (Mandarin Chinese). The findings show that second language learners do not use zero pronouns as often as native speakers and that their use increases with proficiency. When examined more closely, it can be seen that the speakers have no difficulty using zero pronouns when there is a syntactic or semantic restriction, but they do have difficulty at the discourse level. It is claimed that this underuse of zero pronouns corresponds with other research that shows that second language learners tend to avoid pronouns in favor of full noun phrases. Some explanations for these phenomena are postulated.
On the Number of Lines which meet Four Regions in Hyper-Space
Authors: Vaidyanathaswamy R.
Published: November 2008

The problem of finding the number of r-dimensional regions which are situated in a space Sn of n dimensions, and which satisfy a suitable number of conditions of certain assigned types (called “ground-conditions”) has been investigated by Schubert. A special class of such problems arises when the r-dimensional region is merely required to intersect k regions Pλ of rλ dimensions (λ = 1, 2 … k) situated in general position in Sn where for the finiteness of the sought number we must have
An early formulation by Stokes of the theories of the rotatory polarizations of light
Authors: Larmor Joseph
Published: November 2008

The two letters now communicated are from G. G. Stokes to W. Thomson, of dates Dec. 12–13, 1848, three years after Faraday's great magneto-optic discovery. They formulated already the permissible types for general equations of propagation, virtually on the basis of the very modern criterion of covariance,—relative to all changes of the spatial frame of reference in the case of active fluids, but having regard to the fixed direction of the extraneous magnetic field in the Faraday case. Their form was elucidated in each case by correlation with a remarkable and significant type of rotational stress in a propagating medium.
The remainder in a gap Tauberian theorem for Abel summability
Authors: Luther Wolfram
Published: November 2008

Using Pitt's Tauberian classes and a theorem of Pitt (3), Krishnan(2) has recently obtained a gap Tauberian theorem for (Aα) summability. We recall briefly the notation introduced in (2). Let

and assume that the series a(t) and A(t) converge for all t > 0.
Innovation projections of a jump process and local martingales
Authors: Elliott Robert J.
Published: November 2008

Square integrable and local martingales on a family of σ-fields generated by a basic jump process are shown to have representations as stochastic integrals with respect to a family of martingales associated with the jump process by using the idea of an innovation projection and the associated Lévy system, which is a local characterization of the jumps.
On universal unfoldings of certain real functions on a Banach space
Authors: Magnus R. J.
Published: November 2008

The aim of this article is to prove a result which has been thought true for some time. Roughly speaking, if you take a universal unfolding of a germ in finitely many variables, and add to it a non-degenerate quadratic form on an infinite-dimensional space, you still have a universal unfolding.
The strength of metals
Authors: Bragg Lawrence
Published: November 2008

1. When crystals of a metal are highly perfect, their elastic limit is low, plastic flow taking place when the applied shear stress is very small. The elastic limit reaches higher values as the perfect structure is progressively broken up by cold-work. A steady state is finally reached where further distortion of the metal does not increase the elastic limit. The stress at which a metal passes beyond the elastic limit and begins to yield is ill-defined, as in general the crystal begins to ‘creep’ appreciably near this point and the time element enters into the definition of the stress-strain curve. Nevertheless, there is a fairly well-defined stress beyond which a metal, which has been brought into a steady state by being saturated with cold-work, ceases to behave elastically and undergoes permanent deformation. An attempt is made in this paper to derive an expression for the ultimate elastic limit or yield stress of a cold-worked metal, in terms of its structure and elastic constants.
The scroll of tangents of an elliptic quartic curve
Authors: Edge W. L.
Published: November 2008

1. The equation of the scroll R of tangents of the curve Γ of intersection of two quadrics in general position in [3] was found by Cayley in 1850 ((l), p. 50). Cayley also considered special positions of the quadrics but we shall not be concerned with these. was reproduced in Salmon's treatise ((8), p. 191) and Salmon, on this same page, provides an equation in covariant form and so applicable to pairs of quadrics whether in canonical form, or in general position, or not.
The asymptotic nature of the analytic spread
Authors: Brodmann M.
Published: November 2008

In (3), corollary, p. 373) Burch gives the following inequality for the analytic spread l(I) of an ideal I of a noetherian local ring (R, m):

In this paper we shall improve this by showing that the number min depth (R/In) may be replaced by the asymptotic value of depth (R/In) for large n (which exists) (see Section (2)). By its definition (see (6), def. 3)) the analytic spread is of asymptotic nature, i.e. depends on the modules In/mIn = Un only for large n. We shall prove a stronger result, Section (4), which also shows the asymptotic nature of l(I). This result might be interesting for itself, particularly as it is not of local nature. Once Section (4) is proved and once we know that depth (R/In) is asymptotically constant (which turns out to be an easy consequence of (1), (1)), our improved inequality is easily established: Indeed, replacing R by R/xR where x is regular with respect to almost all modules (R/In), we perform a change which affects only finitely many of the modules Un (see Section (8)).
Relative Galois module structure of rings of integers and elliptic functions
Authors: Taylor M. J.
Published: November 2008

Let K be a quadratic imaginary number field with discriminant less than −4. For N either a number field or a finite extension of the p-adic field p, we let N denote the ring of integers of N. Moreover, if N is a number field then we write for the integral closure of [½] in N. For an integral ideal & of K we denote the ray classfield of K with conductor & by K(&). Once and for all we fix a choice of embedding of K into the complex numbers .
Some link-polynomial relations
Authors: Lickorish W. B. R.
Published: November 2008

The polynomial invariants of classical links that relate to the Jones polynomial have various inter-connections. Here a new one is given: the square of the Jones polynomial is obtained from the F-polynomial by means of a certain substitution of variables. This explains and corrects some similarities between known evaluations of these polynomials. The proof uses the ‘Dubrovnik’ version of the F-polynomial; this is first proved to be equivalent to the F-polynomial. Finally a correlation between the F and P polynomials is included as a provocative curiosity.
On multi-dimensional annealing problems
Authors: Chan Terence
Published: November 2008

In [1] Chan and Williams considered a one-dimensional diffusion of the form

where F is a strictly increasing continuous function with F(0) = 0 and ε is a decreasing deterministic function such that ε(0) is finite and ε(t) ↓ 0 as t↑ ∞, and gave necessary and sufficient conditions for Yt →0 a.s. as t→∞.
The Electrical Resistance of Dilute Solid Solutions
Authors: Mott N. F.
Published: November 2008

1. As is well known, the electrical resistance of a metal is very greatly in-creased by the addition of a second metal with which it forms a solid solution. The increase Δρ in the resistivity due to the addition of a small percentage of the second metal is in general independent of the temperature (Matthiessen's rule), though there are oertain exceptions (e.g. Cr in Au). The quaritum-mechanical explanation of both these facts was first given by Nordheim, and may be expressed as follows: the electrical conductivity of any metal may be written in the form

where τ is the “time of relaxation”, equal to half the time between collisions, and N is the effective number of free electrons per unit volume: hence, for the resistivity, we have
On the Steinhaus billiard table problem
Authors: Croft H. T. , Swinnerton-Dyer H. P. F.
Published: November 2008

Let be a plane closed convex curve, and regard as the boundary of a billiard table. If a perfectly elastic billiard ball is set moving on the table, its path will be a polygon inscribed in ; and this will be closed if and only if the motion is periodic. We seek periodic paths which are triangles.
Uniform approximate solutions of certain nth-order differential equations. I
Authors: Heading J.
Published: November 2008

Exact analytical solutions of certain second-order linear differential equations are often employed as approximate solutions of other second-order differential equations when the solutions of this latter equation cannot be expressed in terms of the standard transcendental functions. The classical exposition of this method has been given by Jeffreys (6); approximate solutions of the equation (using Jeffreys's notation)

are given in terms of solutions either of the equation

or of the equation

where h is a large parameter. A complete history of this technique is given in the author's recent text An introduction to phase-integral methods (Heading (5)).
On the Polygons Inscribed in one Conic and Circumscribed to Another
Authors: Burnside W.
Published: November 2008

This problem is in general treated in connection with the division of the periods of the elliptic functions. It is the object of the present note to shew, from a purely algebraical point of view, that the condition of closing of the polygon depends solely on a difference equation of the form
On a problem of Littlewood concerning Riccati's equation
Authors: Swinnerton-Dyer H. P. F.
Published: November 2008

The study of the Riccati equation

plays an essential part in the ‘large parameter’ theory of the inhomogeneous van der Pol equation; see for example Littlewood(1), (2). The crucial result is Lemma B of (1), restated and proved as Lemma 5 of (2); for the present paper the relevant parts of it are as follows:

Lemma 1. Let z = z(x) be the solution of (1·1) which satisfies the initial condition z = 0 at x = 0, and assume α > 0. Then there is a unique β0 = β0(α) with the property that

(i) if β > β0 then z → − ∞ as x → + ∞;

(ii) if β < β0 then z → + ∞ at a vertical asymptote x = x0(α,β);

(iii) if β = β0 then z ≥ 0 in 0 ≤ x < + ∞ and z = x + β0 + o(1) as x → + ∞.

Moreover, β0(α) is a continuous monotone increasing function of α.
An integral involving Fox's H-function and Whittaker functions
Authors: Bajpai S. D.
Published: November 2008

1. In this paper we have evaluated an integral involving Fox's H-functions and Whittaker functions. One particular case of the integral has been employed to establish an expansion formula for the H-function involving Laguerre polynomials.
The average number of maxima of a random algebraic curve
Authors: Das Minaketan
Published: November 2008

Let g0, gl, g2,…be a sequence of mutually independent, normally distributed random variables with mathematical expectation zero and variance unity. In this work, we obtain the average number of maxima (minima) of the random algebraic curves with the equations

This average is (½(3½ + 1)) log N + O((log N)⅔ (log log N)½), when N is large.
Statistical processes of aggregation and polymerization
Authors: Whittle P.
Published: November 2008

We study processes in which units (particles) associate into clusters, which are then also capable of dissociation. Such processes are discussed generally in section 2, where a stochastic kinetic equation (10) is proposed which bridges the gap between the conventional kinetic equations (8) and the statistical equilibrium concept of the Gibbs distribution.

In section 4 we consider the equilibrium behaviour of a process for which the association rate of two units which are already bound tojandkother units respectively has the form (21). This is very much more general than the equi-reactive bond model usually discussed. The principal results are given in Theorem 1; from a single pair of equations (28) and (29) based on theHjof (21) one can determine critical points, expected number of bonds, the distribution and moments of cluster size, and most other quantities of interest. This is without reference to any other consideration, such as kinetic or stoichiometric relations.

Some particular cases are worked through in section 5. The classic Flory-Stockmayer results for units withfequi-reactive sites are obtained systematically and economically, with all parameters in terms of physically given quantities. Another type of example seems to indicate the existence of a second critical point.

Corresponding results for the case of several types of unit are stated and illustrated in section 6.
Statistical electrodynamics
Authors: Marshall T. W.
Published: November 2008

The random pure radiation field postulated in an earlier paper is set up in a relativistically invariant manner. The requirement that this field be isotropic in three dimensions makes much of the formalism identical with the theory of isotropic turbulence, as has been noted by a previous author. It is found, however, that the more stringent requirement of invariance under the Lorentz group, together with the fact that the field components satisfy Maxwell's equations, mean that, to within a multiplicative constant, the entire random process is uniquely specified.

In accordance with results obtained previously, the constant is effectively identified as Planck's constant. There then results a striking resemblance between the formalism of the present theory, and that of the quantized pure radiation field, with random variables taking the place of operators, and correlations between these random variables taking the place of commutator brackets.

The infinite field energy, which arises also in quantum field theory, where, however, it is usually considered to be purely formal, remains a central difficulty of the random theory. It can be shown to be a natural consequence of assuming Maxwell's equations, and it therefore becomes necessary to consider non-linear modifications of these.

The connexion between the present results and the inconsistencies of classical and quantum electrodynamics is discussed. Also some new information is obtained concerning the behaviour of thermodynamic quantities under a Lorentz transformation.
Unsteady heat transfer in compressible boundary layers
Authors: Riley N.
Published: November 2008

The flow following a sudden change in the wall temperature in a laminar boundary layer, which was originally incompressible and steady, is studied. The temperature to which the wall is raised is supposed large enough to bring about density changes in the boundary layer. The details of the flow immediately after the change in wall temperature are analysed for a general boundary layer. Two specific examples are considered and for these the manner in which the new steady state is achieved is also investigated.
On the higher singularities of plane algebraic curves
Authors: Scott C. A.
Published: November 2008

1. During the last sixty years the principal questions presented by the higher singularities of plane algebraic curves have been completely solved, and definite results obtained. The two most successful lines of research have been by expansions and quadratic transformation. By each method it has been shown that a higher singularity may be looked upon as containing concealed or “latent” multiple points or lines in addition to those immediately recognized; and from each, with the help of small quantities, has been constructed a topological explanation of these latent multiple elements, which are accounted for as situated in the immediate vicinity of the point and line base of the singularity. Further, by each method it has been proved that as regards the numerical relations known as Plvicker's equations a singularity produces the same effect as a definite number of nodes, cusps, bitangents, and stationary tangents.
On elliptic quartic curves with assigned points and chords
Authors: Welchman W. G.
Published: November 2008

1. The problem which I propose to solve is that of finding the number of quartic curves of intersection of two quadrics which pass through p points and have q lines as chords, where p + q = 8. There are ∞16 elliptic quartics in space; to contain a line as a chord is two conditions, and to pass through a point is two conditions, so we should expect a finite number of solutions. Throughout this paper I shall refer to an elliptic quartic curve in space of three dimensions simply as a “quartic.” I shall denote the number of solutions for a particular value of p by np.
On the Operational Solution of Linear Finite Difference Equations
Authors: Milne-Thomson L. M.
Published: November 2008

The application of the operational method of Oliver Heaviside to the solution of linear differential equations has been fully described in a recent Cambridge Tract by Dr H. Jeffreys.
On simplexes doubly incident with a quadric
Authors: Thaswamy R. Vaidyana
Published: November 2008

(1) The simplex in a space Sn of order n (or n − 1 dimensions) is a set of n points a1 a2 …an which do not all lie in a subspace of Sn.

The elements of the simplex a1 a2 …an are the n points a1 a2 …an whieh are elements of order one, the lines ar as which are elements or faces of order two, etc., etc., and finally the n primesa1 a2 … ar−1 ar+1 … an which are elements or faces of order n – 1.
The absorption of energy by a wireless aerial
Authors: Ratcliffe J. A.
Published: November 2008

If a receiving aerial is placed in the field of a wave produced by a distant transmitter, currents are induced in the receiving aerial and these dissipate energy in the ohmic resistances of the aerial. This energy may be thought of as being absorbed directly from the incident wave, but this viewpoint gives no insight into the mechanism by which the absorption takes place. In this paper the following alternative way of regarding the phenomenon is considered. The currents induced in the aerial cause it to re-radiate a secondary wave. The electric and magnetic fields of this secondary wave combine with those of the unabsorbed incident wave to give resultant fields which have, in general, a direction different from those of the incident wave. These resultant fields give rise to energy flows which are different from those in the incident wave, both in magnitude and direction. In this way a flow of energy into the aerial is produced, by the superposition of the incident and the re-radiated fields. It is important to notice that if this viewpoint is adopted we must add the secondary re-radiated wave to the unabsorbed main wave, the addition, in itself, producing the requisite absorption.
Note on the Gibbs Phenomenon, II
Authors: Verblunsky S.
Published: November 2008

1. This note is a sequel to a previous one. Our first object is to rectify some mistakes in I. For these corrections I am indebted to the kindness of Prof. J. R. Wilton. In the enunciation of Theorem A (I, p. 159) the equation

should be altered to

The mistake occurs at the end of I (p.169) where it is erroneously assumed that

the correct being
A portable apparatus for measuring the magnetic field strength in an electromagnetic wave
Authors: Vedy L. G. , Wilkins A. F.
Published: November 2008

A portable apparatus is described which is capable of measuring directly, by means of a loop aerial, the magnetic field in an electromagnetic wave. Accurate measurements are possible of magnetic fields corresponding to field strengths of 0·2 millivolts per metre. Special means of providing small known calibrating E. M. F. S are described. The apparatus can be used to measure signals over the range 6 microvolts to 300 millivolts. Used in conjunction with a small portable vertical aerial, field strengths down to 2 microvolts per metre can be measured.
Some properties of line congruences
Authors: Roth L.
Published: November 2008

In a previous paper the writer has obtained various formulae for surfaces in higher space by means of Cayley's functional method; and recently the same method has been applied to surfaces on a quadric form in [5] with the object of investigating the properties of line congruences in [3]. The purpose of the present note is two-fold: to indicate a few applications of the formulae mentioned above, and to give direct and (it is believed) novel proofs of some results obtained elsewhere by the functional process. The demonstrations given here differ, save in two cases, from those based on Schumacher's four-dimensional representation which, as it stands, applies only to congruences without double rays and therefore lacks generality. It was James who first noticed that a congruence without singularities has in general a finite number of double rays and is accordingly defined by five, instead of four, independent characters. These characters have been considered in previous work; but for present purposes it is convenient to describe them afresh.
On the desorption of gases from the walls of a closed system in which the pressure is independently changing
Authors: Pringle G. E. , Van Praagh G.
Published: November 2008

The following calculations were made in connection with recent work on the kinetics of the attack on platinum by iodine vapour at low pressures. A platinum filament was heated in a glass bulb to which iodine was admitted at a pressure of 0·027 mm. of mercury from a reservoir of iodine kept at 0°C. The tap to the reservoir was opened for a period of two minutes, which was evidently sufficient to allow the iodine to saturate the walls of the bulb, since on closing the tap the pressure remained constant. On heating the filament, reaction ensued, causing a steady fall in pressure. This does not however truly represent the “clean up” due to the reaction, as the gas initially adsorbed on the walls of the bulb is continuously desorbed as the pressure falls. To obtain the true pressure decrease due to the reaction, a quantity representing the gas desorbed must be subtracted from the pressure reading at any moment.
Note on the instability of a surface of discontinuity
Authors: Rosenhead L.
Published: November 2008

In a paper recently published I found it necessary to extend those investigations of Lord Rayleigh which deal with the instability of the surface common to two streams of fluid. The Rayleigh investigation is equivalent to a first order approximation to the flow and concludes that the initial form of the disturbance of the common surface is maintained but that its amplitude increases exponentially with the time, the form of the surface at time t being, with particular values inserted,

where a is the initial amplitude, 2π/k the wave-length, U the velocity of the upper stream, – U the velocity of the lower stream, and ρ the density of both fluids. A second order approximation to the flow for the above particular case shows that the original disturbance does not grow symmetrically, the ordinate of the common surface being given by

(A slightly amended form of this solution is given byequation III(8).) The solution suggests the initial stages in the formation of vortices. The problem was further discussed by a numerical step-by-step method and showed the formation of vortices quite clearly.
The interval between the departure of the disintegration particle and the emission of the gamma radiation
Authors: Wright P.
Published: November 2008

Previous work on the existence and period of radium C′ is discussed with reference to an experiment of Jacobsen which provides evidence that a γ ray transformation of period comparable with that of radium C′ precedes the expulsion of α particles. It is shown that, from Jacobsen's results, part of the γ radiation from a source of recoil atoms should originate in the space surrounding the source.

A rough calculation is made which shows that the γ rays above the source should be detectable by ordinary methods, and a description is given of an ionisation method capable of detecting the effect. The γ rays predicted by Jacobsen's experiment were tested for by using specially prepared sources of radium C. Phenomena associated with α recoil were also investigated for sources of radium (B + C) and thorium (B + C).

No evidence of a γ ray emission from the space above any of the sources was obtained. The negative result indicates that the interval between the departure of the disintegration particle and the emission of the γ ray quantum is considerably less than 10−5second.
On double integrals in the calculus of variations
Authors: Gillespie R. P.
Published: November 2008

The problem of the double integral in the calculus of variations, when expressed in the parametric notation, was first fully discussed by Kobb. In this paper Kobb finds conditions for the minimising of the double integral

taken over the surface
Energy distribution curves of the disintegration electrons
Authors: Sargent B. W.
Published: November 2008

With the aid of experimental results on the absorption of homogeneous β-rays in aluminium, a method is developed for calculating the absorption curve of β-rays forming a continuous spectrum. When applied to certain known spectral distributions reasonable agreement is obtained with the experimental absorption curves of these heterogeneous β-rays. Distribution curves with momentum of the β-rays from actinium C″, uranium X2, thorium C and thorium C″ are found, which lead to similar agreement.

The distribution curves with energy and with momentum for eight β-ray bodies and a table of average energies emitted in thése spectra are shown.
On the canonical curve of genus five
Authors: Richmond H. W.
Published: November 2008

This paper contains an attempt, begun several years ago and only partially successful, to do for the canonical curve of genus five something similar to what W. P. Milne had done for that of genus four. A fundamental feature of Milne's work was the use of a rational normal curve of order three drawn through the six points of contact of a quadric with the sextic curve; here it is not in general possible to pass a rational curve of order four through the eight points of contact. (Exceptionally it may be possible to do so and then the development follows much the same lines as in Milne's paper: only the general case is discussed in what follows.)
Note on the trisecants and quadrisecants of a space curve
Authors: Welchman W. G.
Published: November 2008

1. Formulae obtained by enumerative methods are always liable to break down owing to the appearance of an infinity of degenerate solutions, and when this happens it is usually necessary to make a fresh start. Enumerative arguments have been used to prove the two following results for a curve of order n and genus p in space of three dimensions:

(i) The order of the ruled surface formed by trisecant lines of the curve is

(ii) The number of quadrisecant lines is
Photo-electric absorption in hydrogen-like atoms
Authors: Dirac P. A. M. , Harding J. W.
Published: November 2008

If light of a frequency which corresponds to an energy greater than the ionisation potential falls on an atom, an electron may be ejected and energy absorbed. To calculate the absorption coefficient, or the rate of absorption of energy per unit intensity of incident radiation for a given frequency, one must first choose a model for the atom. If we confine ourselves to the inner K electrons there will be two electrons in this shell for the heavier atoms, and a fairly good model of the atom is obtained by considering each electron to be moving independently in a central field of force due to the charged nucleus: i.e. we neglect electronic interaction and assume that the wave functions for the system are hydrogenic. Some writers make a partial correction for this neglect of interaction by modifying the central charge through the introduction of a screening factor which is so chosen that the minimum calculated energy required to remove one of the K electrons will agree with the experimental value provided by the K absorption edge. In general, however, the approximation is fairly good, and this is particularly so in the interior of a star where the atoms are highly ionised. It is not so good when the atom is bound as in a metal, and, of course, most of the laboratory work has been carried out on atoms in this bound state.
What determines the Resistance and the Tilt of an Aeroplane?
Authors: Larmor Joseph
Published: November 2008

It is recognized that the main hope for advance in a rational theory of aeroplane propulsion lies in extensive use of the general principle of conservation of resultant momentum, even more than on considerations of energy; for energy can be dissipated by friction, while the relation of momentum to force is absolute.
Rational normal octavic surfaces with a double line, in space of five dimensions
Authors: Babbage D. W.
Published: November 2008

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.
The upper limit of energy in the β-ray spectrum of actinium B
Authors: Sargent B. W.
Published: November 2008

Using a thin-walled electroscope, absorption curves in paper and aluminium were obtained for the β-rays of actinium (B + C) and actinium C″. From these curves the range of the β-rays of actinium B was found to be 0·08 ± ·02 gm./cm.2 The energy required for this range is 300,000 ± 50,000 volts, which must be the end-point of the continuous spectrum.

On account of the overlapping of the β-ray spectra of actinium B and actinium C″ the range method can never yield an accurate value of the end-point of the former. Magnetic analysis appears to be the best means of finding this accurately, but the weakness and rapid decay of sources are formidable difficulties.
A general canonical expression
Authors: Bronowski J.
Published: November 2008

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:

Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that

(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;

(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..
A condition for inverting the order of integration in a repeated integral
Authors: Pollard S.
Published: November 2008

When the inner integrals in a pair of repeated integrals are absolutely convergent, inversion of the order of integration is a not too difficult problem and there are a number of general theorems which, under comparatively unrestricted appropriate conditions, can be applied to give a solution. When, on the other hand, the inner integrals are only conditionally convergent, the problem is of a very different nature and the solution, if it exists at all, is usually only to be obtained by special devices.
On the Lens Interferometer
Authors: Steward G. C.
Published: November 2008

The interferometer designed by Michelson has been adapted in recent years to the testing of various types of optical systems, one form being used for the testing of prisms and another form—the “lens interferometer”—for the examination of the symmetrical optical system. The details of the instrument may be found in a paper by Mr F. Twyman, in which are given some photographs of the results of its application to certain lenses of known aberration. Essentially the interferometer ensures that a plane and undistorted wave of light falls upon the lens or system to be examined, after refraction through which it falls upon a convex mirror whose centre is at the focus of the converging wave; by which, therefore, the wave is returned through the optical system and emerges as a plane wave if the system under examination be ‘perfect,’ or free from aberration. Otherwise the final wave is not plane, but is distorted by the aberrations impressed upon it by the system under test—through which, it will be noticed, it passes twice; and this emergent wave is superposed upon a portion of the incident (plane) wave, obtained by previous passage through a partly silvered mirror; with which, therefore, interference effects are observed. And these effects depend only upon the aberrations produced by the system under examination, for they measure, in fact, the departure of the finally emergent wave from the ideal plane form. Various interference contours are obtained in this way, and their form and size enable the observer to deduce the aberrations of the system and to measure their magnitude.
The second solution of the Mathieu equation
Authors: Ince E. L.
Published: November 2008

The fact that the Mathieu equation, and, as was proved in the preceding paper, the more general Hill equation, can admit of but one solution of period π or 2π, suggests the problem of determining how far the second solution deviates from periodicity when the first solution has either the period π or the period 2π
A figure in space of seven dimensions, and its sections
Authors: Edge W. L.
Published: November 2008

Several papers have recently been published which include proofs of the following:

Theorem 1. Those trisecant planes of a rational normal quartic curve Cu which meet a second rational normal quartic curve Cv having six points in common with Cu also meet a third rational normal quartic curve Cw through these same six points. The three suffixes may be permuted in any way, the three curves forming a symmetrical set.
An attempt to detect a neutral particle of small mass
Authors: Chadwick J. , Lea D. E.
Published: November 2008

It has long been realised that there are difficulties in understanding the continuous β-ray spectra emitted by certain radioactive bodies, if the conservation of energy, as applied to nuclear processes, is to be retained. In the case of radium E, for example, the energy of the disintegration β particles varies from quite low values to about a million volts, the mean being nearly 400,000 volts.
The vector representation of a sample
Authors: Bartlett M. S.
Published: November 2008

The geometrical or vectorial representation of a sample as a vector with n mutually perpendicular components corresponding to the n observations in the sample was introduced into statistics by Fisher (1), and led to the solution of many theoretical problems of statistical distributions. Subsequently Fisher (2) gave an alternative algebraic method applicable to finding distributions in connection with regression and the analysis of variance. The actual use of symbolic vector notation in deducing some of the properties of the sample vector—besides tending to stress the complementary character of these two different methods of proof, and the common principle underlying the analysis of any sample into its components—indicates also some points in connection with the assumption of the normal law in which it seems almost essential to retain the geometrical side of this vector representation. Moreover, it will be seen that the vector theory used here in reviewing briefly the analysis of a sample of one dependent variate can readily be extended to cover the case of correlated variates.
The flow due to a rotating disc
Authors: Cochran W. G.
Published: November 2008

1. The steady motion of an incompressible viscous fluid, due to an infinite rotating plane lamina, has been considered by Kármán. If r, θ, z are cylindrical polar coordinates, the plane lamina is taken to be z = 0; it is rotating with constant angular velocity ω about the axis r = 0. We consider the motion of the fluid on the side of the plane for which z is positive; the fluid is infinite in extent and z = 0 is the only boundary. If u, v, w are the components of the velocity of the fluid in the directions of r, θ and z increasing, respectively, and p is the pressure, then Kármán shows that the equations of motion and continuity are satisfied by taking
The heat of adsorption of hydrogen on tungsten
Authors: Roberts J. K. , Whipp B.
Published: November 2008

A method is described for measuring the heat of adsorption of hydrogen on a clean tungsten filament. The value obtained is 2·8 × 104 calories per mol.
Ramanujan's Continued Fraction
Authors: Watson G. N.
Published: November 2008

The best of the theorems on continued fractions, to be found in Ramanujan's manuscript note-book may be stated as follows:

where there are eight gamma-functions in each product and the ambiguous signs are so chosen that the argument of each gamma-function contains one of the specified numbers of minus signs. Then

provided that one of the numbers β, γ δ, ε is equal to ± n, where n is a positive integer; and the products and sums on the right range over the numbers α β, γ δ, ε.
On Subdivisions of Complexes
Authors: Whitehead J. H. C. , Newman M. H. A.
Published: November 2008

1. This paper is supplementary to an article by M. H. A. Newman on the superposition of manifolds. Guided by his main idea, I extend Newman's theorem that two equivalent manifolds have a common subdivision from equivalent manifolds to equivalent complexes of an arbitrary nature. The theorem is also sharpened by the possibility of leaving a certain subcomplex unaltered, and by substituting a partition for a general subdivision.
On a Uniqueness Theorem
Authors: Pryce M. H. L.
Published: November 2008

1. The purpose of this note is to prove the uniqueness of the solutions of two closely related differential equations, under suitable boundary conditions. The first is the equation satisfied by the potential in the electrostatics of the new field theory proposed by Born, namely,

where øx stands for etc.
On the generalizations of the theorems of Parseval and Riesz-Fischer
Authors: Pollard S.
Published: November 2008

Any two real numbers p and q will be called conjugate if
Summation of the integral conjugate to the Fourier integral of finite type
Authors: Grimshaw Margaret Eleanor
Published: November 2008

1. The complete Denjoy-Fourier integral is of the form

where

the generating function ƒ(t) being integrable in the general Denjoy sense in every finite interval, and the integrals (1·2) being interpreted as principal values or in some other suitable way.
Eddy currents in thin circular cylinders of uniform conductivity, due to periodically changing magnetic fields, in two dimensions
Authors: Carter F. W.
Published: November 2008

The paper deals with the eddy currents in thin circular cylinders of uniform conducting material, due to periodic currents in conductors lying parallel to the axis of the cylinder, or to the rotation of the cylinder in a two-dimensional field of force. The first of these problems was discussed by Mr M. B. Field in a paper entitled “Eddy current losses in three-phase cable sheaths,” read before the British Association at their Cambridge meeting in 1904. The solution proposed, however, although probably sufficient for the object, is mathematically defective, in that the field due to the current carried by the cable is assumed as the total field, the effect of the eddy-current field on the eddy currents themselves being left out of account.
Trajectories of small horizontal velocity in a resisting medium
Authors: Littlewood J. E.
Published: November 2008

The results of the present paper were developed some years ago for quite trivial purposes. It seems possible, however, to judge by a recent publication, that they may sooner or later have scientific utility. They have also a certain slight intrinsic interest.
On the Applicability and Deformation of Surfaces
Authors: Sen B. M.
Published: November 2008

Beltrami's discovery that corresponding to any ruled surface there is another applicable on it having corresponding generators parallel but with the parameter of distribution of opposite signs, has brought into prominence the distinction between the applicability of two surfaces and the continuous deformation of one into the other. It is pointed out that since the parameter of distribution differs in sign, the tangent plane rotates in opposite directions as the point of contact proceeds along the two corresponding generators on the respective surfaces. One surface cannot, therefore, be deformed into the other. It is proposed in the present paper to investigate the distinction between applicability and deformability of two surfaces in general, and to connect this isolated fact with the general theory of deformation.
Note on the Doppler principle and Bohr's frequency condition
Authors: Dirac P. A. M.
Published: November 2008

When an atomic system emits or absorbs radiation, the frequency of the radiation is connected with the change of energy of the system by Bohr's frequency condition δE = hv. This equation is usually applied to the transition of an atom from one stationary state to another, the frequency of the radiation being measured in a frame of reference in which the atom as a whole is at rest, both before and after the transition. It has been pointed out by Bohr that difficulties arise when no natural frame of reference presents itself in which to measure the frequency, which is the case for non-isolated systems, or even for isolated systems when the process of radiation is connected with a change of momentum.
On Differentiation with respect to a Function
Authors: Francis E. C.
Published: November 2008

Thirty years ago, in a paper on continued fractions, Stieltjes published a definition of the integral which bears his name. His replacement of the variable of integration x by a more general “base function” φ(x)—a change which throws so much light upon other theories of integration—received at first little attention, but has later sprung into greater prominence; so much so that Professor Hildebrandt, in summarizing these various theories in a paper to the American Mathematical Society, makes the statement that “it [the Stieltjes Integral] seems destined to play the central rôle in the integrational and summational processes of the future.” Yet even now the integral and the allied theory of differentiation with respect to a function have been subjected to little detailed analysis, and the possibilities of extension have been only touched upon. It is the object of this present paper to establish certain results which are of some value in themselves and which prepare the way for an attack upon the integral.
The Practical Calculation of Unit Algebraical Numbers
Authors: Pocklington H. C.
Published: November 2008

The object of this paper is to give a method of finding the units of a set of algebraical numbers that shall be convenient practically. Little attention is given to theoretical considerations. The case of numbers composed with the cube root of an integer is treated in § 2, and a few theoretical remarks are added in § 3. In § 4 we show how to test whether the unit found is a fundamental one or not. In § 5 the method is extended to other cases.
On Four Mutually Orthogonal Circles
Authors: Vaidyanathaswamy R.
Published: November 2008

If A0, A1, A2, A3 are four arbitrary circles, it is well known that there are eight circles which cut them all at equal angles, and that these eight circles fall into two tetrads B0, B1, B2, B3, C0, C1, C2, C3 which are in desmic position to each other and to the tetrad of circles orthogonal to threes of A0, A1, A2, A3.
On the Convergence of a certain Multiple Series
Authors: Mordell L. J.
Published: November 2008

Any one who has dealt with the convergence of multiple series soon becomes aware that simple and interesting examples are very scarce. It may therefore be worth while placing on record the following result which is not altogether trivial.
A Note on the Stability of a Valve Amplifier
Authors: Moullin E. B.
Published: November 2008

This paper considers the stability of a valve amplifier which has an oscillatory circuit in the grid and in the anode circuit. An exact condition for stability is obtained for circuits which have no resistance, and it is shown that when both circuits have the same natural frequency instability is then impossible if the anode inductance exceeds μ times the grid circuit inductance. This condition is believed to be new; in practice it is unnecessarily severe, but it is believed that stability should be sought by increasing the anode inductance and the grid circuit capacity. The stability of circuits which have resistance is too cumbersome to express generally, but it is discussed on broad principles; the stability and amplification of multistage amplifiers are also considered briefly. The following papers should be compared with the present one.
The Polarisabilities of Atomic Cores
Authors: Swirles Bertha
Published: November 2008

If it is assumed that the series electron of an atom polarises the core, then it has been shown by Born and Heisenberg that the polarisability α of the core in a given state may be calculated from the corresponding term value by means of the approximate formulae, where q is the quantum defect, δν is the difference between the term and the corresponding hydrogen term, R is the Rydberg constant in cm.−1, α1 is the radius of first hydrogen orbit, n is the principal quantum number, k is the azimuthal quantum number, For terms with small quantum defect either of the formulae (1) and may be used, but for terms with large quantum defect (1) gives a higher degree of accuracy.