bluuebunny/crossref_metadata_2025 huggingface dataset:
The Last of The Shireburnes: The Art of Death and Life in Recusant Lancashire, 1660–1754
Authors: Callow John
Published: July 2016

When J. M. Turner came to make his sketches of Stonyhurst Hall and the neighbouring church at Great Mitton, for the first time in 1799, he was immediately struck by the melancholia and faded splendour of that part of ‘darkest’ rural Lancashire. Perched high upon the brow of Longridge, the mansion commanded sweeping views of the valley beneath, of Pendle Hill and of the distant market town of Clitheroe; while the thirteenth century church of All Hallows—almost lost in the folds of the countryside—sat squatly on the borders of Lancashire and Yorkshire, at the confluence of the Rivers Calder, Ribble and Hodder, and served as a stubborn reminder of an earlier and less secular age. Relatively untouched by the forces of industrialisation, these buildings proved a delight to the Gothic imagination of the young artist.
Lisbon College—The Final Years
Authors: Williams Michael E.
Published: July 2016

A previous article gave an account of the difficulties besetting Lisbon College at the beginning of the twentieth century. These were due both to the unsettled state of Portugal where the new Republic was hostile to the Church and to the failure of Mgr James Warwick to win the support of his staff in revitalising the college after the long and rather monochrome presidency of William Hilton. Warwick’s failure led to an unfortunate difference of opinion among the staff of the college and in the absence of an Apostolic Nuncio in Lisbon, the traditional Protector of the College and the court of appeal, Cardinal Bourne sent Mgr Bernard Ward of Old Hall, Ware, to make an official visitation and report on the state of affairs in Lisbon.
Hiddenness, holiness, and impurity
Authors: KYLE BRENT G.
Published: July 2016

John Schellenberg has advanced the hiddenness argument against God's existence, based on the idea that an all-loving God would seek personal relationships. This article develops a reply to Schellenberg's argument by examining the notion of moral impurity, as understood by Paul the Apostle. Paul conceptualized moral impurity as a causal state that transfers from person to person, like a contagious disease. He also believed that moral impurity precludes divine–human relationship. The goal of this article is to develop these ideas into a problem for one of Schellenberg's key premises.
I. Sites Explored
Authors: Wilson D. R.
Published: July 2016

Cae Metta (SH 536650), 1.5 km south-west of Llandeiniolen: preliminary examination of the polygonal enclosed hut-group suggested occupation from the mid third into the fourth century.

Caernarvon fort (Segontium, SH 485624) :3 excavation outside the north-east gate (porta decumana) showed that the original Flavian causeway and inner ditch remained in use unaltered until shortly after 330. The causeway was then dug through and the ditch re-cut from a W-profile to the normal V-section, while a new outer ditch was constructed some 10 m out from the gate. Access to the gate was now over wooden bridges, for which structural evidence was found at the inner ditch. Fresh details were also obtained of the Flavian timber gate.
II. Inscriptions
Authors: Wright R. P. , Hassall M. W. C.
Published: July 2016

Wanborough, Wilts. (SU 194853). For this defixio (curse) (FIG. 22) see the report by Mr. J. Rea printed on pp. 363-7 as an Appendix.

Cirencester {Corinium, SP 028012), Gloucestershire (PL. XXIX, A). Oolitic limestone tombstone 0.65 by 1.21 by 0.26 m found in 1971 with two others by contractors carrying out demolition in School Lane on the site of the Stepped Stairs development (see above, p. 339). The three came from the rear of the earth rampart forming part of the defences of the town and may have been intended as building material (for the construction of bastions), which was then discarded. The tombstone, which tapers slightly towards the top, is surmounted by a triangular pediment within which is a lunette in relief. Pediment and die are surrounded by a plain border which has suffered some damage, particularly to the right of the lower part of the die.
IV.—On an Elementary Solution of a Partial Differential Equation of Parabolic Type. Part I
Authors: Copson E. T.
Published: July 2016

Professor E. T. Whittaker has recently discovered a Third Quantum-Mechanical Principal Function R(q, Q, t - T) and has worked out the theory of this function in detail when the Hamiltonian is

By using the Sturm-Liouville theory of linear differential equations and the properties of Green's function, it is shown that the function is an elementary solution of the adjoint of the Schrodinger wave equation associated with the Hamiltonian H.

It is pointed out that the modified Planck constant ħ arises solely from the commutation relation and may, from the analytical view-point, be any constant, real or complex. In particular, if ħ = i, the use of an algebra with the commutation relation leads to an elementary solution of the real equation of parabolic type
XXV.—Axiomatic Treatment of Kinematical Relativity: A Reply to Dr G. C. McVittie
Authors: McVittie Dr G. C. , Whitrow G. J.
Published: July 2016

In his recent article McVittie has criticised in most disparaging terms the analytical theory of time-keeping developed by Milne and myself in the last ten years. Milne has replied at length (see preceding paper), and it is my purpose in this note merely to touch on one or two points which he has not covered.

However, before doing so, I should like to take this opportunity of remarking that the “Kinematical Relativity,” about which McVittie has written, both in his recent article and in his monograph, “Cosmological Theory,” is not the Kinematical Relativity of Milne and myself, but is something much slighter, based, perhaps, on an incomplete understanding of the nature of the kinematical theory.
XXVI.—The Monomial Expansion of Determinantal Symmetric Functions
Authors: Aitken A. C.
Published: July 2016

The quotient of alternants

the denominator being the difference-product of the arguments a, β, γ, …, is an important symmetric function, introduced into algebra by Jacobi in 1841.
VI.—Atomic Wave Functions for Ground States of Elements Li to Ne
Authors: Duncanson W. E. , Coulson C. A.
Published: July 2016

Wave functions have been obtained for the ground states of the atoms Li to Ne. These are of the simple analytical type proposed by Morse, Young, and Haurwitz. Some errors in the latter work are corrected.
XVI.—Studies in Practical Mathematics. IV. On Linear Approximation by Least Squares
Authors: Aitken A. C.
Published: July 2016

R. Frisch, in a paper (Frisch, 1928) on correlation and scatter in statistical variables, made an extensive use of matrices, and in particular of the moment matrix, as he called it, of a set of variables. The matrices were square arrays, with an equal number of rows and columns. This paper of Frisch pointed the way to an even more extensive use of the algebra of matrices in problems of statistics.

What Frisch called the moment matrix may perhaps be more suitably called, nowadays, the variance matrix of a set or vector of variates, since the moments in question are all variances or covariances. In the present paper, which is illustrative of matrix methods, we explore the familiar ground of linear approximation by Least Squares, making full use of the properties of the variance matrix. We also study the linear transformations that convert crude data into smoothed or graduated values, or into residuals, or into coefficients in a linear representation by chosen functions.
XXII.—Two Numerical Applications of Chebyshev Polynomials
Authors: Miller J. C. P.
Published: July 2016

In this paper two computational processes are outlined in which the table of Chebyshev Polynomials Cn(x) = 2 cos (n cos−1 ½x) given in the preceding paper may be used with effect; these processes are (a) interpolation and (b) Fourier synthesis. A brief outline is also given of the idea behind the process of “Economization of Power Series” developed in Lanczos, 1938; this is related to (a). Finally the application of (b) to the calculation of Mathieu functions is considered.
XXIX.—On a Problem in Correlated Errors
Authors: Aitken A. C.
Published: July 2016

The problem with which this paper is concerned arose in the discussion of a series of chronometric observations, but it is of more general application, and is capable of wide extension. Pairs of readings (xi yi) were taken at times ti, i = 1, 2, …, n. These readings were known to be affected by respective errors (ξi ηi) from sources different but possessing some common part. It was important to have an estimate of the consequent correlation and to assess its precision. The assumptions made in the particular experiment were that x and y were both linear in t, representable by x = a0 + a1t, y = b0 + b1t, and that the distributions of error in x and y were normal. The parameters a0 and a1, b0 and b1 were therefore obtained from two separate sets of normal equations, and the unknown correlation was then estimated from the sum of products of corresponding residuals ui, vi, one from each set. In the corresponding situation in n samples (xi,yi) from a bivariate normal distribution the mean value of is (n − 1) ρσ1σ2, where σ12, σ22 are the variances of x and y and ρσ1σ2 is their product moment. One might therefore anticipate, by analogy, that in the present case the mean value of Σuivi would be (n − 2)ρσ1σ2. So indeed it proves to be, and the sampling variance of Σuivi conforms likewise with standard results; but it is desirable, by an extension of the problem, both to see why this is so and to take notice of cases where the analogy fails to hold.
XLV.—The Internal and External Fields of a Particle in a Gravitational Field
Authors: Clark G. L.
Published: July 2016

The gravitational field of a system of particles was investigated by de Sitter as far back as 1916. A minor alteration to the analysis was made by Eddington and Clark in 1938. The amended value of the potential g44 is the same as that derived by Einstein, Infeld and Hoffmann without making use of the energy-tensor; this agreement suggests that the revised de Sitter argument is correct. In this paper we show that this is not the case, for the de Sitter analysis completely overlooked any possible interaction terms in the stress components of the energy-tensor. We find the value of these terms, pmn, and show that the agreement mentioned above is due to the fact that the volume integral of pu vanishes.
VI.—Unbiased Statistics with Minimum Variance
Authors: Bhattacharyya A.
Published: July 2016

Synopsis

The problem considered is that of the estimation of a statistical parameter from a sample of values of the variate or variates concerned. Reference is made to the method of unbiased statistics with minimum variance, developed by Aitken and Silverstone. The principal result obtained by these authors is generalized, and an inequality involving the variances of unbiased statistics is obtained. Several examples illustrating the theory are appended.
XIII.—Studies in Practical Mathematics. VI. On the Factorization of Polynomials by Iterative Methods
Authors: Aitken A. C.
Published: July 2016

Synopsis

The method of iteration of penultimate remainders, introduced by S. N. Lin for approximating by stages to the exact factors of a polynomial, is subjected to theoretical analysis. The matrix governing the iterative process is obtained, and its latent roots and latent vectors are found. Incidental theorems yielding further factorizations are proved, and processes are developed for accelerating convergence. Numerical examples illustrate varying situations likely to arise in practice.
IV.—Expansion of a Finite One-dimensional Gas Cloud into a Vacuum
Authors: Mackie A. G.
Published: July 2016

Synopsis

An investigation is made of the motion of a one-dimensional finite gas cloud which is initially at rest and is allowed to expand into a vacuum in both directions. The density of the gas at rest is assumed to rise steadily and continuously from zero at the boundaries to a maximum in the interior of the cloud.

If the subsequent motion is continuous, it is completely specified by analytical solutions in seven different regions of the x-t plane joined together along characteristics. The motion of one of the boundaries is discussed, and conditions found for it to have (i) an initial stationary period or (ii) a final constant velocity of advance into the vacuum. The gas streams in both directions from a dividing point at zero velocity. This point ultimately tends to the mid-point of the initial distribution.

The possible breakdown of the continuity of the motion is discussed, and a condition on the initial density distribution found for shock-free flow to be maintained.
V.—Interpolatory Methods for Theorems of Vitali and Montel Type
Authors: Bowen N. A. , Macintyre A. J.
Published: July 2016

Synopsis

The classical theorems of Vitali and Blaschke are shown to be simple consequences of an inequality of an interpolatory character due to J. M. Whittaker.

Theorems generalising one of Montel relating to functions bounded in a half-plane and tending to zero at a sequence of points are established by similar methods.
VIII.—Riemann Extensions which have Kähler Metrics.
Authors: Patterson E. M.
Published: July 2016

Synopsis

Certain types of 2n-dimensional Riemannian spaces admitting parallel fields of null n-planes are studied. These are known as Riemann extensions of conformal, projective or other classes of spaces of affine connection. The circumstances under which a 2n-dimensional Riemannian space admits two non-intersecting parallel fields of null n-planes are also discussed. Such spaces satisfy a condition similar to Kähler's condition in the theory of complex manifolds, and hence are called Kähler spaces. Necessary and sufficient conditions are found for a Kähler space to be a Riemann extension with respect to one of the parallel fields of null n-planes, and canonical forms are found for the metrics in the cases of Riemann extensions of conformal and projective spaces.
I.—Focusing of Electron Beams in Crossed Static Electric and Magnetic Fields
Authors: Farago P. S.
Published: July 2016

Synopsis

In a crossed static homogeneous electric and magnetic field charged particles describing trochoidal orbits in a plane perpendicular to the direction of the magnetic field are focused, but a beam emitted by a point source in a finite solid angle spreads out indefinitely in the direction parallel to the magnetic field. The essential characteristics of trochoidal orbits can be preserved if the superimposed magnetic and electric fields are two-dimensional and orthogonal, such as derived from a vector potential, say, Ax = A(y, z), Av = Az = O, and a scalar potential Ф(y, z)= const. A(y, z). The focusing properties of such a field combination depend on the distribution of the magnetic field only. Following some general considerations, specific examples of double focusing field distributions are given, and the electron motion in one of them is treated in detail.
IX.—Possible Applications of Coincidence Counter Techniques in Optical Intensity Correlation Studies
Authors: Sillitto R. M.
Published: July 2016

Synopsis

Methods which are widely used in the mathematical analysis of random noise are used here to obtain expressions for the coincidence-counting rates which could be obtained in studies of the intensity correlations between plane-polarized, parallel beams of nearly monochromatic light. It is shown that, with presently available circuit techniques, delayed coincidence measurements could provide information about the breadths of lines radiated by atomic beam light sources, and in prompt coincidence experiments it should be possible to observe interference between beams of incoherent light from sources whose line widths are comparable with those of the Hg198 electrodeless discharge.
X.—Remarks on Coincidence Experiments with Visible Light
Authors: Bay Z. , Farago P. S.
Published: July 2016

Synopsis

After the detection of correlations in two coherent light beams by Hanbury Brown and Twiss, objections were raised by Brannen and Ferguson on the basis of the experiments of Adam, Janossy and Varga and their own experiments in which no correlations were detected. It is pointed out here that the different groups were looking for two entirely different effects, one being quadratic, the other one linear in the number of photons involved; the quadratic effect (discovered by Hanbury Brown and Twiss) is in agreement with quantum theory while the linear effect is not. It was shown by Purcell and by Hanbury Brown and Twiss that the choice of parameters in the experiments which gave negative results was inadequate to show the quadratic effect. It is shown in this paper that their experiments were also inadequate to decide between the existence or nonexistence of the linear effect.
XII.—Paraboloidal Co-ordinates and Laplace's Equation
Authors: Arscott F. M.
Published: July 2016

Synopsis

In this paper we examine the general paraboloidal co-ordinate system, in which the normal surfaces are elliptic or hyperbolic paraboloids, including as special cases the “parabolic plate” and the “plate with a parabolic hole”. We then show that normal solutions of Laplace's equation in these co-ordinates are given as products of three Mathieu functions, and apply this to the solution of boundary-value problems for Laplace's equation in these co-ordinates. In a subsequent paper the corresponding treatment of the wave equation will be given.
XXIII.— Dual Series Relations. V. A Generalized Schlömilch Series and the Uniqueness of the Solution of Dual Equations involving Trigonometric Series
Authors: Srivastav R. P.
Published: July 2016

Synopsis

The methods employed in papers I–IV of this series are modified to provide the solution of certain dual equations involving trigonometric series. It is necessary to introduce a modified form of the conventional operators of fractional integration and to discuss their relation with generalized Schlömilch series expansions of an arbitrary function. These general methods are illustrated by detailed reference to a particular special case.
VIII.—An Objective Criterion of the Limit of Useful Sophistication of the Semi-empirical Mass Equation
Authors: Feather N.
Published: July 2016

Synopsis

A “roughness parameter”, first used by the author in 1953 (Feather 1953 b) has been re-calculated for 348 points on the mass surface. Systematic features are identified in relation to the variation of this parameter with charge number (Z) and isotopie number (D). In the region of small Z these regularities provide evidence for the persistence of some degree of alpha-unit structure at least as far as Ca. In the region of greater Z (20 < Z < 50) they provide evidence for neutron-proton interactions among the last-added nucleons. Overall, they indicate that the “residuals” characterizing the various semi-empirical mass equations currently in use very probably arise in large part from sub-shell effects which it would be impracticable to attempt to include in the equations.
IX.—Orthogonal Boolean Matrices
Authors: Rutherford D. E.
Published: July 2016

Synopsis

Many results concerning real orthogonal matrices have their counterparts in the theory of orthogonal Boolean matrices. In particular, the analogues for Boolean matrices of certain theorems due to Kronecker are established. The structure of the group of orthogonal Boolean matrices of order n is determined in the case where the underlying Boolean algebra is finite.
X.—Nitrogen Dioxide Vapour and the Creation of Condensation Nuclei
Authors: Evans G. R. , Watson A. A.
Published: July 2016

Synopsis

An investigation has been made of the condensation nuclei created by an electric field in a N2 – H2O mixture. These nuclei are distinguished by the fact that they induce condensation in a vapour which is only 4 per cent supersaturated.

An explanation of these phenomena is found in the presence of nitrogen dioxide vapour, one of the products of reactions induced by the field, exerting a small pressure PNO2<1O−6 mm. Hg.

The observations are consistent with the assumption that the nuclei are created in the reaction, 2NO2 + H2O ⇌ HNO2 + HNO3. It is believed that the reason for the requirement of a more than critically supersaturated vapour is that this must be the condition for the nuclei forming reaction to proceed.

Once the nuclei have been created, any additional quantity of NO2 collected from the vapour forms acid molecules which promote condensation from a vapour which is not necessarily supersaturated.

Drops formed by these nuclei contain a significant quantity of HNO2 + HNO3, so that, unlike drops of pure water, they are stable against reevaporation in a vapour, the relative humidity of which is <100 percent.
XIII.—The Generating Function of Solid Partitions
Authors: Wright E. M.
Published: July 2016

Synopsis

The generating function for the number of linear partitions was found by Euler, the method being almost trivial. That for plane partitions is due to Macmahon, but, even in a simplified form found by Chaundy, the proof is far from trivial. The number of solid partitions of n, i.e. the number of solutions of

is denoted by r(n). It has often been conjectured that the generating function of r(n) is , but this is now known to be false. We write η(a, b, c) for the generating function of the number of solutions of (1) subject to the additional condition that

Macmahon 1916 found n(a, 1, 1) for general a. Here we find η(a, b, c) for general a, b. c.
XXI—The Whittaker-Hill Equation and the Wave Equation in Paraboloidal Co-ordinates
Authors: Arscott F. M.
Published: July 2016

Synopsis

The problem considered is that of obtaining solutions of the Helmholz equation ∇2V + k2V = 0, suitable for use in connection with paraboloidal co-ordinates. In these co-ordinates the Helmholz equation is separable, and each of the separated equations is reducible to Hill's equation with three terms (the Whittaker-Hill equation). The properties of solutions of this equation are developed sufficiently to make possible the formal solution of simple boundary-value problems for paraboloidal surfaces, principally for the case k2 < 0.
XXII.—Tensor Fields and Connections on Cross-Sections in the tangent bundle of a differentiable manifold
Authors: Yano Kentaro
Published: July 2016

Synopsis

Tensor fields and linear connections in an n-dimensional differentiable manifold M can be extended, in a natural way, to the tangent bundle T(M) of M to give tensor fields of the same type and linear connections in T(M) respectively. We call such extensions complete lifts to T(M) of tensor fields and linear connections in M.

On the other hand, when a vector field V is given in M, V determines a cross-section which is an n-dimensional submanifold in the 2n-dimensional tangent bundle T(M).

We study first the behaviour of complete lifts of tensor fields on such a cross-section. The complete lift of an almost complex structure being again an almost complex structure, we study especially properties of the cross-section as a submanifold in an almost complex manifold.

We also study properties of cross-sections with respect to the linear connection which is the complete lift of a linear connection in M and with respect to the linear connection induced by the latter on the cross-section. To quote a typical result: A necessary and sufficient condition for a cross-section to be totally geodesic is that the vector field V in M defining the cross-section in T(M) be an affine Killing vector field in M.
XXIV.—The Spirit of Science
Authors: Bragg Lawrence
Published: July 2016

I Thank you very much indeed for the compliment you are paying me in inviting me to give the James Scott Lecture. It is always an honour to be asked to give a lecture which is, as the nursery-man would say, a “named variety”, and in addition I welcome this invitation from a Society of which I am proud to be an Honorary Fellow.
A Sceptical Chemist in a Biological Wonderland
Authors: Davidson J. N.
Published: July 2016

It is the custom of our Society that the President of the day should at some period in his term of office address the Fellows. The manner in which he interprets this charge will, of course, vary according to individual preference, but for obvious reasons, he usually elects to give his personal interpretation of that area of science in which he is himself most deeply concerned.

Our Society was founded in the days before science was split into highly specialized sections and long before the advent of the specialist societies restricted to one scientific discipline, such as the Chemical Society, the Biochemical Society or the Society for Experimental Biology. This, in the opinion of many of our Fellows, is its main strength, since, as a general Society concerned with all aspects of science, it can exert an integrative function; in fact, it serves many of the purposes of the Academies which are so well known in other parts of Europe, as is made clear in the Letters Patent granting our Armorial Bearings.
XI.—On Unconditional Convergence in Topological Vector Spaces
Authors: Robertson A. P.
Published: July 2016

Synopsis

For a series of elements of a topological vector space, necessary and sufficient conditions are found, in terms of the set of finite partial sums, for unconditional convergence and for the corresponding Cauchy condition. The extent to which these results remain valid for topological groups is investigated. A new and direct proof, for locally convex spaces, is given of the theorem of Orlicz.
XVI.—The Fission of Aromatic Ketones
Authors: Campbell Neil , Thomson Alexander
Published: July 2016

The acyl groups in aromatic ketones with hydroxyl or methoxyl groups in theorthoorparapositions are readily removed by hydrobromic acid in acetic acid. The results are correlated with the fine structure of the parent hydrocarbons. 5,8-Dimethoxy-I-tetralone and 1,3-dihydroxyxanthone fail to undergo ring-fission with the reagent. The mechanism is discussed and it seems probable that fission occurs by direct proton attack at nucleophilic carbon atoms bearing the acyl groups.
XVII.—The Tacnodal Form of Humbert's Sextic
Authors: Edge W. L.
Published: July 2016

Synopsis

Humbert's 5-nodal plane sextic first appeared in his 1894 paper. Its canonical curve C was identified in 1951, when it was shown that the sextic is the outcome of projecting C from one of its own chords on to a plane.

In this present paper it is remarked that there are 60 chords of C such that the projection has two tacnodes, each a confluence of two of Humbert's 5 nodes, and an equation is found for this tacnodal curve.

A certain specialization permits C to be invariant for a group of 32, not merely 16, projectivities. Further specializations, described in the proper place, permit groups of orders 64, 96, 160. The resulting tacnodal sextics have groups of birational self-transformations isomorphic to these.
XIX.—Asymptotic enumeration of connected graphs
Authors: Wright E. M.
Published: July 2016

Synopsis

The number of different connected graphs (with some propertyP) onnlabelled nodes withqedges isfnq. AgainFnqis the number of graphs onnlabelled nodes withqedges, each of whose connected components has propertyP. We consider 8 types of graph for which. We use a known relation between the generating functions offnqandFnqto find an asymptotic expansion offnqin terms of binomial coefficients, valid if (q– ½nlogn)/n→∞ asn→∞. This condition is also necessary for the existence of an asymptotic expansion of this kind.
XII.—Quasi-residuated Mappings and Baer Assemblies
Authors: Blyth T. S. , Hardy W. C.
Published: July 2016

Synopsis

We consider, for a given ordered set E with minimum element O, the semigroup Q of O-preserving isotone mappings on E and examine necessary and sufficient conditions under which an element fε Q is such that the left [resp. right] annihilator of f in Q is a principal left [resp. right] ideal of Q generated by a particular type of idempotent. The results obtained lead us to introduce the concept of a Baer assembly which we use to extend to the case of a semilattice the Baer semigroup co-ordinatization theory of lattices. We also derive a co-ordinatization of particular types of semilattice.
6.—Electron Diffraction and Electron Mobility in Non-polar Crystals
Authors: Cochran W.
Published: July 2016

Synopsis

It is shown that the mobility of electrons in silicon or germanium can be estimated in a relatively simple manner. The scattering scross-section of a ’beam’ of electrons in the conduction band is evaluated in the same way as for a beam of X-rays or of slow neutrons which is scattered by phonons. It therefore involves the atomic scattering factor for electrons rather than the deformation potential introduced by Bardeen and Shockley. Predicted mobilities are in satisfactory agreement with observation.
12.—The National Physical Laboratory—Aspects of work on the primary standards of measurement
Authors: Dunworth J. V. , Dean P.
Published: July 2016

One of the traditional activities of the National Physical Laboratory is its work on the maintenance and improvement of the primary standards of measurement. Although one may possibly visualise such work, because of its long history and its association with calibration services, as of a largely routine character, this is certainly far from the case at the present time. The present is a period of considerable activity and change in fundamental metrology, with the classical material standards of measurement being superseded by atomic or quantum standards. The past decade has seen a change to atomic standards for the units of length and time, and there seems little doubt that the future will see an extension of atomic-based standards to other areas, notably that of the electrical quantities. Some of the changes which may come about as a consequence of adopting the most accurate and convenient quantum methods have interesting implications. For example, a possible outcome of the new techniques being developed for the accurate measurement of very high (infrared) frequencies is that the standards of length and time may become unified, with the velocity of light taking the role of an agreed defined constant rather than an experimentally determinable quantity.
11.—The Instability due to Acoustic Radiation striking a Vortex Sheet on a Supersonic Stream
Authors: Jones D. S. , Morgan J. D.
Published: July 2016

Synopsis

The propagation of sound across a vortex sheet separating two fluids, with the same density and sound speed, which are in supersonic relative motion is examined for both harmonic and impulsive sources. It is found that instability waves are present when M < 2√2, M being the Mach number, and it is suggested that these may account for the phenomena observed in certain experiments. For M > 2 there are also neutral stability waves and these lie in regions in the supersonic stream which are otherwise silent when the excitation is an impulsive source.
13.—On Some Results of Everitt and Giertz
Authors: Atkinson F. V.
Published: July 2016

Synopsis

The differential expression Mf = −f″+qf, on a half-line [a, ∞), is said to be ‘separated’ in L2(a, ∞) if the collection of all functions f ∈ L2(a, ∞) such that Mf is defined and also in L2(a, ∞), has the property that both the terms f″ and qf are separately in L2(a, ∞). When q is positive and differentiable on [a, ∞) this paper obtains sufficient conditions on the coefficient q for M to be separated; these take the form of bounds for q′q−3/2 on [a, ∞).
3.—A Note on Quadratic Systems
Authors: Hille Einar
Published: July 2016

Synopsis

‘Quadratic system’ is here used in the sense introduced by W. A. Coppel for a pair of simultaneous Riccati equations with constant coefficients. The problem is to study the occurrence of movable singularities for solutions of a system

under suitable restrictions on the lines

The aim of the restrictions is to obtain systems where the trajectories have four finite and three infinite topological singularities which form specified configurations. For the movable (function theoretical) singularities the infinitary (topological) singularities are decisive. They are the endpoints of the coordinate axes and of the line F: L2–L1 = 0. A detailed qualitative discussion is given for the trajectories. The system forces various properties of monotony on the integral curves.

The qualitative discussion is completed by quantitive data in the form of series expansions for the solutions. Here the value of the parameter

is critical.

If q is not an integer, then there is one and only one solution whose trajectory is associated with the upper end of the line F which has a simple pole at t = 0 (or any other preassigned value). All other trajectories in this family belong to solutions with expansions in terms of non-logarithmic psi-series of the form

for a singularity at t = 0. Here – 1 ≦ m, 0 ≦ n and c is an arbitrary constant.

On the other hand, if q is a positive integer, then we have logarithmic psi-series of the form

where Pn(s) and Qn(s) are polynomials in s of degree ≦ [n/q].

Solutions associated with the ends of the axes have non-logarithmic psi-series. One coordinate becomes infinite like 1/t at zero, the other goes to zero like tλ where λ = –b1/a1 for the x-axis and –a2/b2 for the y-axis. The psi-series have exponents of the form j + Kλ, where j and k are integers, –1 ≦ j, 0 ≦ k.

The results are used to construct new convergence proofs for the solutions of Emden's equation at a movable singularity.
8.—On Second-order Differential Inequalities
Authors: Atkinson F. V.
Published: July 2016

Synopsis

This paper is devoted to a study of differential equations and inequalities of the form

and

The results are mainly concerned with the existence of positive solutions, their uniqueness in the case of (*), and bounds for these solutions.
18.—The Isotropic Turbulent Dynamics of a Maxwell Fluid
Authors: McComb W. D.
Published: July 2016

Synopsis

Using the constitutive relationship for a Maxwell body, as an example of a viscoelastic fluid, the equations of motion are derived in the Fourier wavenumber-time domain, and specialised to the case of isotropic turbulence. It is shown that, for grid-generated turbulence, the model predicts increased spectral intensity levels, reduced decay rates and steepening of the spectrum in wavenumber, relative to the Newtonian case. These forms of behaviour have been observed in dilute solutions of drag reducing polymers. The key factor in this is found to be the presence of an ‘elastic’ non-linear term in the equations of motion: this term reverses the normal direction of turbulent energy transfer in wavenumber.
26.—Singularities of Solutions of Certain Plane Autonomous Systems
Authors: Smith Russell A.
Published: July 2016

The paper discusses the nature of the singularities at any finite pointt=t0of real solutionsx(t),y(t) of the system of differential equationsx′ =P(x,y),y′ =Q(x,y) in whichP,Qare polynomials in bothxandy. Its main interest is in cases when the leading terms ofP,Qare of the same degreen. Conditions are given under which the only possible singularities are algebraic poles and pseudo-poles. Conditions are also given under which the only possible singularities are of wildly oscillatory type.
29.—Finiteness of the Order of Meromorphic Solutions of some Non-linear Ordinary Differential Equations
Authors: Hille Einar
Published: July 2016

Synopsis

It is well known that the existence of transcendental meromorphic solutions of non-linear ordinary differential equations puts severe restrictions on the equation, the most striking example being the theorem of Malmquist [3]. The value distribution theory of R. Nevanlinna was applied to such questions by K. Yosida [8] who used it to prove Malmquist's theorem as well as important generalisations. An alternate approach was given by H. Wittich [4,5,6] and in his argument the finiteness of the order played an essential role. Wittich estimated the corresponding enumerative and proximity functions via the calculus of residues. In this note a geometric argument is proposed instead (closest packing of small discs in a bigger circle or on its rim). This method seems to generalise more readily to Yosida's extensions.
I.—On the Geometry of Metrisable Lie Algebras
Authors: Ruse H. S.
Published: July 2016

Synopsis

Metrisable Lie algebras have been defined by Tsou and Walker (1957). Their definition is adopted below in § I.

The object of the present paper is to display something of the geometrical background of such algebras, particularly for those taken over the field of real numbers.

§ I is introductory. In § 2 appears a statement, made as brief as possible because it is wholly classical, of the relationship between the vector-space of the Lie algebra L and the associated affine and projective spaces A and P. Some properties of metrisable Lie algebras are then examined in terms of the geometry of P, which provides an (n –I)-dimensional map of the n-dimensional algebra L. It is assumed throughout the paper that the Lie algebras under discussion are non-abelian, since the projective map of an abelian algebra presents nothing of interest.

As the present work is intended as no more than a preliminary, it is confined, so far as its applications are concerned, to a discussion of metrisable algebras of dimensions 3 and 4 and to one example of an algebra of dimension 6.

I have been privileged in preparing the paper to have access to the typescript of the paper by Tsou and Walker referred to above, and also to the doctoral thesis (1955) of the former. I am greatly indebted to them both, and also to Dr Paul Cohn and to a referee for suggestions regarding certain details of presentation.
XX.—On the Numerical Evaluation of a Class of Multivariate Normal Integrals
Authors: Ruben Harold
Published: July 2016

SYNOPSIS

The probability that each ofnequally correlated normal random variables shall not fall short of a given valuehis obtained as the product of the joint density function of the variables at the cut-off point (h, h,…, h) and an infinite power series inh. The coefficients in the latter series may be interpreted geometrically as the moments of a regular (n– l)-dimensional spherical simplex with common dihedral angles arc cos –prelative to a certain plane of symmetry. These moments may in their turn be expressed as linear functions of the measures of regular hyperspherical simplices of various dimensionalities, tables of which are available elsewhere.
XXVIII.—The Synthesis of 7-Substituted Fluoranthenes
Authors: Campbell Neil , Crombie Douglas A.
Published: July 2016

The synthesis of 7-bromo- and 7-chloro-fluoranthene from the readily accessible 1, 2, 3, 1ob-tetrahydro-3-oxofluoranthene is described. Oxidation of the chloro-compound yields 1-chlorofluorene-9-one-8-carboxylic acid and thus opens up a method for the preparation of 1, 8-disubstituted fluorenes and fluorenones.

Thanks are expressed to the Department of Scientific and Industrial Research for the award of a Maintenance Grant to one of us (D. A. C).
Saint Margaret Clitherow and Alexander Pope: An Unexpected Link
Authors: Longley Katharine M.
Published: July 2016

FATHER JOHN MUSH, biographer of Saint Margaret Clitherow, the quater-centenary of whose martyrdom has been commemorated this year, recorded within three months of her death that after her condemnation her kinsfolk and friends ‘laboured much’ to persuade her to claim a respite on account of a pregnancy of which she was uncertain. He had no need to identify these anxious and distressed persons, for his earliest readers in York and its neighbourhood would be aware of their identity. When, however, his True Report began to spread secretly in manuscript to other parts of the country where its subject was unknown, the circumstances of the saint's life and the almost unique nature of her constancy were less well understood and appreciated. Fr. Mush tells us that Margaret, whose mother was ‘a rich widow’, had ‘many worldly friends’; her maternal kinsfolk, it has now been discovered, were among the élite of York at a time when the Queen's Council in the North Parts, which had its headquarters in that city, was one of the most active instruments in the country for the suppression of the ‘old religion’.
Four Gallican Bishops in Exile: Concepts of Episcopacy in the Face of Revolution
Authors: Bellenger Dominic
Published: July 2016

WHEN THE BISHOPS of the French Church were confronted with the final form of the Civil Constitution of the Clergy in 1790 they presented an almost united front of opposition to the new order. Their ideological and social solidarity remained substantially intact in the protracted exile which, with the victory of the revolutionary party, was the almost inevitable consequence of their shared refusal. This was nowhere more apparent than in London where the largest single concentration of exiled bishops gathered. But, given the general concord, there was a subtle variety of response, a continuation of the debate leading up to the Civil Constitution in which many of the bishops had taken part, to the events beginning with the Civil Constitution and ending with the Napoleonic Concordat, which throws some light on the attitudes of these men to their office of bishop. This paper focuses attention on four of the London bishops: La Marche, Dillon, Thémines and Boisgelin; and suggests, from their example, four possible views of, or approaches to, the episcopate.
James I and his Catholic Subjects, 1606–1612: Some Financial Implications
Authors: La Rocca, S.J. John J.
Published: July 2016

MUCH recent research has examined Catholicism in early Stuart England and some has discussed the contribution of anti-Catholicism to the outbreak of the Civil War—but how well-founded were the fears underlying the rhetoric which surfaced in parliament? This paper1 addresses one aspect of that question by looking at certain financial features of Catholic non-conformity (as demonstrated by absence from Anglican services and/or refusal of the oath of allegiance) in the first half of James I’s reign, chiefly between 1606 and 1612. The significance of this period is that it begins just after the attempt to blow up king and parliament and ends with the death of Robert Cecil, Earl of Salisbury who, during his term of office as Lord Treasurer from 1608 onwards, was grappling with an intractable royal indebtedness which he managed to curtail but not to cure and whose own effectiveness waned following the failure of the Great Contract in 1610.2
The 1906 Education Bill: Catholic Peers and Irish Nationalists
Authors: Cashman John
Published: July 2016

ON 20TH DECEMBER, 1906, the Liberal Government of Sir Henry Campbell-Bannerman withdrew from the Order Book the Education Bill which had received its First Reading on 9th April, 1906. The Bill had proposed a drastic revision in the generous settlement which the voluntary schools had welcomed with the passing of the 1902 Education Act. The new proposals had sought to make all public elementary schools into ‘council schools’ controlled by the Local Authority. The denominations would receive an agreed rental for the use of their school buildings, and the upkeep of these buildings would become the responsibility of the Local Authority. But the power to appoint the teachers would also pass to the Local Authority. The only concession to the denominations was the proposal that extended facilities for denominational instruction on each school day would be made available in certain elementary schools if three-quarters of the parents whose children attended the schools asked for them. Head-teachers would not be allowed to give the denominational instruction, but assistant teachers would be allowed to give the instruction if they volunteered to do so. Failing this, the denominations would be responsible for providing the teacher to give the instruction. The controversial Clause Four excluded from this concession elementary schools in areas in which the only school was a denominational school, and in areas with a population of less than 5,000.
A literature overview of knowledge sharing between Petri nets and ontologies
Authors: Cheng Haitao , Ma Zongmin
Published: July 2016

Knowledge representation is a subarea of artificial intelligence concerned with using formal symbols to represent a set of facts within a knowledge domain. Two popular knowledge representation languages, namely Petri net and ontology, are promising knowledge sharing and reusing methods in knowledge engineering. The combination of Petri net and ontology can facilitate achieving complementary advantages. Currently, many efforts have been done on knowledge sharing between Petri nets and ontologies. To investigate these issues and more importantly serve as identifying the direction of knowledge sharing between Petri nets and ontologies, in this paper we give a comprehensive literature overview of knowledge sharing between Petri net models and ontology models to satisfy the obvious need. In detail, we discuss the knowledge sharing from two aspects: the different knowledge representation approaches of ontology to represent and reason Petri net and issues of constructing Petri net from ontology. In addition, other important issues on applications and directions for future research are discussed in detail.
Six unlikely intersection problems in search of effectivity
Authors: HABEGGER P. , JONES G. , MASSER D.
Published: July 2016

We investigate four properties related to an elliptic curveEtin Legendre form with parametert: the curveEthas complex multiplication,E−thas complex multiplication, a point onEtwith abscissa 2 is of finite order, andtis a root of unity. Combining all pairs of properties leads to six problems on unlikely intersections. Using a variety of techniques we solve these problems with varying degrees of effectivity (and for three of them we even present the list of all possiblet).
Bifurcation in the Duffing equation with independent parameters. I
Authors: Hale Jack K. , Rodriques Hildebrando M.
Published: July 2016

Synopsis

In the standard treatment of the harmonic solutions of Duffing's equation with small non-linearities and small forcing, all small parameters are assumed to be common multiples of some small parameter. As a consequence, the parameters do not vary in a full neighbourhood of zero and the bifurcation surfaces are not obtained. It is the purpose of this paper to give a complete description of the number of harmonic solutions for the parameters varying in a full neighbourhood of the origin in the parameter space.
dg near rings on the infinite dihedral group
Authors: Malone J. J.
Published: July 2016

Synopsis

It is shown that exactly six non-isomorphic distributively generated near rings can be defined on the infinite dihedral group. One has the null multiplication, two have trivial multiplications in which the group of left annihilators has index two, and three are distributive and have their product sets equal to a subgroup of order 2.
Harmonic tetrads and harmonic elliptic quartic curves
Authors: Dye R. H.
Published: July 2016

Synopsis

The group of a harmonic elliptic quartic has exceptional action on the edges of the self-polar tetrahedron, singling out one pair of opposite edges. Geometrical explanations are given. One concerns the possession by each of these two edges of a certain harmonic tetrad constructible in three ways. The same constructions yield, for a general quartic, three distinct tetrads on each edge of its tetrahedron, with a related fourth. The cases of exceptional overlap of these tetrads are examined: they occur for quartics of moduli 0, ∞ and — 32/49.
Stability of deficiency indices
Authors: Behncke Horst , Focke Heinz
Published: July 2016

Synopsis

Many known results about the stability of selfadjointness are extended to results about the stability of the deficiency index of closed symmetric operators on Hilbert space under perturbation.
26.—On Shapiro's Cyclic Inequality
Authors: Bushell P. J. , Craven A. H.
Published: July 2016

Synopsis

The inequality, for suitable sets of non-negative numbers x1, x2, …, xn,

is undecided in the cases n = 11, 12, 13, 15, 17, 19, 21 and 23. In this note we present the results of numerical analysis supporting the conjecture that the inequality is valid in these cases. A new counter–example for n = 25 and a new elementary proof when n = 7 are given.