bluuebunny/crossref_metadata_2025 huggingface dataset:
A House Divided: The Fall of the Herberts of Powis, 1688–1775
Authors: Murphy Martin
Published: July 2016

In addition to his major work on the English Jesuit province as a whole during the 18th century, Geoffrey Holt has made many valuable contributions to more local studies. One such is his article on ‘Jesuits in Montgomeryshire, 1670–1873’, published in 1993. In it he noted the Jesuit practice of depositing funds with a wealthy family of the district, as with a bank. Thus the funds of the Residence of St. Winefrid (North Wales) were deposited with the Marquess of Powis. Sebastian Redford, the Jesuit chaplain at Powis Castle in the 1740’s, had reason to complain of the Marquess’ dilatoriness in paying the interest due, and feared that the Society might come to regret having entrusted large sums to a family ‘whose circumstances are more and more on the decline’. His fears were justified.
The Jesuits and the Rebirth of the Catholic Church in Bristol
Authors: Hankins Kenneth
Published: July 2016

Jesuit archives refer to Bristol as ‘a very ancient mission’ of the Society of Jesus and as ‘one of the Society’s first class missions’. This article traces briefly the early development of the Society in that part of the old Western District which included Bristol and which for their own administrative purposes the Jesuits called the College (District) of St. Francis Xavier, and then seeks to show how in the first half of the eighteenth century they established a permanent mission in Bristol itself—a city strongly Protestant, by the standards of the time wealthy and cosmopolitan in character, and for a while second in importance only to London.
The Relationship between Father John Morris, S.J. and the Institute of the Blessed Virgin Mary
Authors: Kirkus M. Gregory
Published: July 2016

The common ground trodden by Father John Morris of the Society of Jesus and members of the Institute of the Blessed Virgin Mary was at first a shared interest in the acts of the English martyrs. This widened to a study of the history of the Institute and Father Morris’s involvement in its current problems—the removal of the Church’s three-century old ban, the vexed question of Mary Ward’s title of foundress, the desirability of union of all the members, and the drawing up of the constitutions acceptable to all. These intellectual explorations and their practical application led him to the Bar Convent in York, to Haverstock Hill and Ascot in the south, and to Nymphenburg, Altötting and Augsburg in Germany.
‘Our Patriarch’: Bishop Bonaventure Giffard, 1642–1734. An Introductory Sketch
Authors: Williams J. Anthony
Published: July 2016

The accolade above, applied to Bishop Giffard in the 1720s, was bestowed by Edward Dicconson, then chaplain at Chillington Hall in Brewood, Staffs., the Giffards’ principal seat and hints at the combination of affection and veneration with which he was regarded by clergy and laity alike. The bishop, described by Archbishop Matthew as ‘a very old gentleman of delightful manners and deep piety’, was then about eighty years of age, having been born at Wolverhampton, probably in the opening year of the English Civil War, into a junior branch of that ancient family, referred to by Michael Greenslade as ‘perhaps the most considerable in the history of Staffordshire recusancy’. Frequently known as Joseph but increasingly as Bonaventure, the future bishop was the third of four sons of Andrew Giffard (youngest brother of Peter Giffard of Chillington) and his wife Katherine, daughter of Sir Walter Leveson, kt. of Wolverhampton, a place where the two families, also linked by earlier marriage, possessed substantial influence but which by the 1640s, as Dr Rowlands has shown, was hardly living down to its reputation as a hotbed of popery and a haven for ‘Rome’s snaky brood’, though in the following decade it was still ‘by many styled “Little Rome’”, echoing the title earlier applied to the Viscountess Montagu’s recusant establishment in Sussex.
Clergy Funds and Episcopal Control—Was John Stanford Maligned?
Authors: Turner Richard H.
Published: July 2016

John Stanford, missioner with the Hunlokes at Wingerworth in Derbyshire from 1692 to 1737, is nowadays remembered for one event. ‘In a huff and a pett’, the received story goes, he—being unable in 1725 properly to account as administrator for the Common Fund of the secular clergy of the Midland District—handed it over to his vicar apostolic, John Talbot Stonor.

To those familiar with the traditions of clergy fund independence, this peevish act must appear self-evident treachery. However—just how soundly based is the story?
Liverpool’s Catholic Mercantile and Maritime Business Community in the Second Half of the Eighteenth Century, Part Two
Authors: Pope David J.
Published: July 2016

Another Catholic who for a number of years was in business as a timber merchant, was Robert Watmough, who was born in 1762 or 1763, the son of Thomas Watmough, husbandman of Litherland, a village five miles north of Liverpool. Though much of Watmough’s career as a timber merchant falls into the early nineteenth century, he is mentioned here as he provides a link with another business connected with the operation of the port, namely shipbuilding. He was a partner for a number of years of John Dwerryhouse, who was possibly the leading local Catholic shipbuilder of the eighteenth and early nineteenth centuries.
John Henry Newman and the Roman Oratory
Authors: Foley Brian C.
Published: July 2016

[It is with great regret the journal records the death, in his ninetieth year, of the longest-serving Vice-President of the Catholic Record Society. Bishop Brian Foley, D.Litt., became Vice-President in 1978 after fourteen years as President. Throughout his priestly life he was a keen supporter of the historical enterprize of the CRS. His most recent book was a study of the Jubilee Years from 1300 to 1975, published in anticipation of the Great Jubilee of the millennium. In tribute to Bishop Foley, Recusant History reissues an article by him that first appeared in the Venerabile in 1989 (vol.29, no.3). Acknowledgement is made to the Rector of the English College in Rome and to the editor of the Venerabile for their kind permission to reproduce the article in its original form.]
Great Aunt and Great Niece: Two 17th/18th Century Members of the Bedingfield Family
Authors: Kirkus M. Gregory
Published: July 2016

The portraits in the Great Parlour of the Bar Convent, York, include two of identical size and frame. The sitters both bear the name of Bedingfield and share the same illustrious ancestry. Both were members of the Institute of Mary and are seen wearing the same religious habit. They successively held the office of Superior of the convent and together they experienced imprisonment in the city’s infamous Ousebridge Gaol. There the similarities end, for two more different characters would be far to seek. It is the purpose of this paper to examine them.
The Irish Diaspora
Authors: Gilley Sheridan
Published: July 2016

The fifth volume in Patrick O’Sullivan’s ground-breaking series The Irish World Wide (1996) is devoted to Irish religion. In his choice of contributors and contributions, the editor has achieved a careful balance between Catholic and Protestant, the latter being a category often too ill-researched to appear in such collections. O’Sullivan’s introduction opens with a retelling of the tale of a confused sixteenth-century Irish Catholic lad who conformed to Protestantism in England, became a sailor and fell victim to the Mexican Inquisition. The introduction concludes with another American tale, also told in Janice Tranter's essay on the Sisters of St Jospeh in Australia, of a Sister from the order who was executed by Shining Path guerillas in Peru. Yet another moving narrative is Anne-Maree (sic) Whitaker’s story of the Irish convict priests and rebels who founded the Catholic Church in Australia.
Buddhism, meditation, and ‘the inner world’
Authors: PURTON CAMPBELL
Published: July 2016

Buddhist meditation seems to involve giving attention to one's thoughts and feelings, to one's ‘inner world’. In this article I explore what is involved in such talk of one's inner world, with special reference to Wittgenstein's claim that ‘the inner’ is a delusion. The article explores the nature of thoughts and feelings, and suggests that we cannot fully understand what is involved in meditation without some consideration of its ethical and religious context. I conclude with some reflections on how the pictures we naturally employ in thinking about meditation can lead us into a misleading metaphysical view of ‘the inner’.
V.—On an Elementary Solution of a Partial Differential Equation of Parabolic Type. Part II: The Nature of the Singularity.
Authors: Copson E. T.
Published: July 2016

It is shown that the elementary solution Γ(x, ξ; t – τ) of the equation

behaves, as t → τ + O, in very much the same manner as the elementary solution of the equation of heat.
IX.—On a Certain Variation of the Distributive Law for a Commutative Algebraic Field
Authors: Robinsohn Abraham
Published: July 2016

A quasi-field is denned by the postulates of a commutative algebraic field, except that the distributive law a(b + c) = ab + ac is replaced by a(b1+ … +bn)=ab1+ … +abn for a fixed integer n.

The properties of quasi-fields are investigated. The study of their ideals is reduced to the study of the ideals of a certain type of ring. A particular quasi-field is constructed formally by means of polynomial domains modulo a natural number, with addition specially defined.

Quasi-fields are connected with multiple fields—another generalisation of the conception of a commutative field, in which a fixed number of elements (> 2) co-operate symmetrically in the formation of any sum or product.
XXII.—Some Applications of Marcel Riesz's Integrals of Fractional Order
Authors: Copson E. T.
Published: July 2016

An account is given of Professor Marcel Riesz's generalisation of the Riemann-Liouville integral of fractional order. It is shown that the new ideas introduced by Riesz may prove valuable in the theory of partial differential equations and in the theory of the wave-equation in momentum space.
XXIV.—On the Equation of Motion of a Free Particle in the Expanding Universe of Kinematical Relativity
Authors: Milne E. A.
Published: July 2016

In a recent paper in these Proceedings, Dr G. C. McVittie has published some criticisms of kinematical relativity. These criticisms are to a large extent based on his formula (4.10), namely,

It must be stated at the outset that McVittie's interpretation of his derivation of (1) as a derivation of “Milne's formula for the acceleration of a ‘free particle moving in the presence of a substratum,’ for the special case of one spatial co-ordinate only” is wrong. McVittie does not derive the result, as he claims, from what he calls the “axioms of kinematical relativity” alone; he deduces it from these axioms together with an additional assumption, which is equivalent to begging the answer to the whole problem it was my object to solve. Instead of considering a free particle, as I did—that is, a particle whose motion we do not a priori know—he prescribes a priori the motion of his particle as being constrained to obey the rule, in his notation,
II.—The Fundamental Concepts of Natural Philosophy
Authors: Milne E. A.
Published: July 2016

From the time of Galileo, experiment has been the core of Natural Science. Before him, of course, observation alone had in the development of astronomy played a fundamental part. Besides the great workers of the ancient civilisations, who knew the path of the sun amongst the fixed stars and could predict eclipses, and besides the fruits of Greek astronomy associated with the names of Hipparchus and Ptolemy, the more modern observational work of Tycho Brahe, analysed by Kepler, had vindicated the self-consistency of the Copernican theory of the solar system and had led to its remarkable refinement in the form of Kepler's three quantitative laws—the law of the ellipse, the law of areas, and the law connecting periodic times and major axes. This was a triumphant example of the execution of the programme then being put forward by Francis Bacon for discovering all natural laws—the method of induction from a number of instances. But it was reserved for Galileo to make a start with the process of ascertaining as far as might be, by controlled experiment, the particular nature of motion. The metaphysical questions associated with motion had not escaped the attention of the Greeks; but Zeno was apparently content with stating paradoxes, and did not resolve them. Galileo, first, experimented with moving bodies; and established that in falling they received equal increments of velocity in equal times—a kinematic theorem, like Kepler's laws. Huyghens was perhaps the first person to establish dynamical-theorems; that is to say, to infer a kinematic result from a stated physical principle—as, for example, his proof of the approximate isochromism of the pendulum based on the principle of vis viva, or, as we should now say, the conservation of energy. Huyghens, together with some of the early Restoration men of science in this country, dealt also with the collisions of bodies. The peerless Newton went further. Assuming outright three primitive “laws of motion,” he showed how the results of Galileo, Huyghens, and their contemporaries could be actually deduced; and by the addition of a fourth law, the law of universal gravitation, already conjectured by some thinkers, he arrived at the laws of Kepler as inferences. Not only so, but the four highly general and abstract laws introduced by Newton have been found sufficient to deduce an enormous complex of dynamical theorems, to express their relationships in the subsequent beautiful systems of Lagrange and of Hamilton, and to derive all but every detail in the motions both in the solar system and in distant binary stars. The basic principles laid down by Newton remained unaltered till our own day, when Einstein modified simultaneously the laws of motion, the law of gravitation, and the background of space and time which had been explicitly adopted by Newton as the scene in which his laws were to play their parts.
V.—On Whittaker's Solution of Laplace's Equation
Authors: Copson E. T.
Published: July 2016

In 1902, Professor E. T. Whittaker gave a general solution of Laplace's equation in the form

where f is an arbitrary function of the two variables. It appears that this is not the most general solution, since there are harmonic functions, such as r−1Q0(cos θ), which cannot be expressed in this form near the origin. The difficulty is naturally connected with the location of the singular points of the harmonic function. It seems therefore to be worth while considering afresh the conditions under which Whittaker's solution is valid.
VIII.—A Measurement of the Velocity of Light in Water
Authors: Houstoun R. A.
Published: July 2016

In the Proc. Roy. Soc. Edin. (Houstoun, 1941) I described a new method of measuring the velocity of light. Owing to the difficulties of the times progress has been slow, but I have now succeeded in measuring the velocity of light in water and communicate the result here.

Previous Measurements.—Every elementary textbook on Light explains how Newton's, corpuscular theory required the velocity of light in water to be μ times the velocity in air, whereas Huygens's wave theory required it to be I/μ times the velocity in air, and how Foucault's determination decided between the two theories. As a matter of fact it did, but Foucault's determination was not an accurate one and there have been only four previous determinations of the velocity of light in water, none of which can lay claim to accuracy. These are the determinations of Foucault (1850), Fizeau and Bréguet (1850), A. A. Michelson (1891), and Gutton (1911).
XII.—Quantum Mechanics of Fields. II. Statistics of Pure Fields
Authors: Born Max , Peng H. W.
Published: July 2016

The quantum mechanics of fields recently developed by us leads to a modification of statistical mechanics of elementary particles which seems to overcome some of the difficulties (divergence of integrals) occurring in the usual quantum theory of fields. The main difference between the new theory and the usual one is as follows.

In the usual theory the wave-vector k is introduced classically and, so to speak, kinematically by the Fourier analysis of the field. The Fourier coefficients of the field components are then treated according to quantum mechanics as non-commuting quantities; those belonging to the wave-vector k describe the corresponding “model” mechanical system, namely the kth radiation oscillator. But the statement that the Fourier coefficients belonging to a certain k all vanish, which statement classically is significant, is now meaningless because there is a lowest state with zero-point energy for each radiation oscillator. The field is thus made to be equivalent to the assembly of radiation oscillators of all possible wave-vectors which, being necessarily infinite in number, contribute an infinite zero-point energy for the pure field and lead to other divergent integrals for the interaction between different fields.
XXXI.—Thermal Diffusion in some Aqueous Solutions
Authors: Docherty Archibald C. , Ritchie Mowbray
Published: July 2016

Thermal diffusion in aqueous solutions of raffinose, sucrose, glucose, xylose, glycerol and acetone was studied in respect of variation of concentration and of temperature. Initial rates of separation were determined as produced by an apparatus consisting of two vertical concentric glass tubes of length 130 cm. and mean annular separation o·68 mm.

With a solution of glucose of 1·7 gm. mol. per litre, the initial rate of separation for the extremes of temperature which could be experimentally applied was found to be independent of the column length for lengths greater than 100 cm.

With the exception of acetone, all the solutes concentrated at the column foot. In singlesolute solutions a maximum rate was observed at an intermediate concentration, which in the sugar series was at a lower molar concentration the higher the molar weight of the solute. With the two-solute solutions sucrose-glucose and sucrose-glycerol little or no relative separation was obtained, contrary to expectation based on the single-solute data.

Initial rates increased with rise in mean temperature and with increase in the applied temperature gradient.
XXXII.—An Elementary Treatment of Thermal Diffusion in Gaseous and Liquid Systems
Authors: Ritchie Mowbray
Published: July 2016

An elementary theory of thermal diffusion applicable to gaseous and liquid systems has been developed. This is based on the difference of diffusional characteristics of a molecule considered as diffusing through two different temperature regions, when the pressure is constant throughout.

For gaseous systems, the resultant expression is shown to be in general accordance with experimental variation of temperature, mass, and diameter factors, and is further developed to include isotopic separation, change of sign of separation with concentration, and general force law considerations.

A similar approach to thermal diffusion in solution, combined with the convection effect of a “cascade” system, gives an expression which is in general agreement with the results of experimental variation of mean temperature and temperature gradient for aqueous solutions of sucrose, glucose and glycerol. The simple expression does not account rigidly for the sign of separation or the effect of altered concentrations. These discrepancies are discussed in relation to the general formula; it is concluded that in addition to the diffusion diameters, the relative thermal expansions of solute and solvent are of importance in this connection.
XXXVIII.—On the Estimation of Many Statistical Parameters
Authors: Aitken A. C.
Published: July 2016

In an earlier paper (Aitken and Silverstone, 1941) the problem of estimating from sample a parameter θ of unknown value was treated by adopting two postulates for the estimating function: (i) that it should be unbiased in the linear sense; (ii) that its sampling variance should be minimal.
XLIII.—On the Gravitational Mass of a System of Particles
Authors: Clark G. L.
Published: July 2016

In classical mechanics the mass of a system of gravitating particles can be denned to be the mass of an equivalent particle which gives the same field at great distances, or alternatively the mass can be defined by means of Gauss' Theorem. Reference to the former procedure was made by Eddington and Clark (1938) in a discussion on the problem of n bodies. The relativistic extension of Gauss' Theorem has been investigated by Whittaker (1935) for a particular form of the line-element and for more general fields by Ruse (1935). The latter, treating the problem from a purely geometrical point of view, expressed the integral of the normal component of the gravitational force as the sum of two volume integrals. The physical significance of one of these integrals was quite obvious but the meaning of the other was far from clear. In this paper the terms in Ruse's result are examined as far as the order m2 in the case of a fundamental observer at rest and the 1938 discussion modified to bring the two investigations into line. It is concluded that the surface integral of the normal component of the gravitational force taken over an infinite sphere is –4π × the energy of the system.
XLVII.—Non-Associative Arithmetics
Authors: Etherington I. M. H.
Published: July 2016

Introduction and Summary

The systems of “partitive numbers” introduced in this paper differ from ordinary number systems in being subject to non-associative addition. They are intended primarily to serve as the indices of powers in algebraic systems having non-associative multiplication, or as the coefficients of multiples in systems with non-associative addition, but are defined more generally than is probably necessary for these purposes. They are essentially the same as root-trees (Setzbäume) with non-branching knots other than terminal knots ignored, with operations of addition and multiplication defined.

Partitive numbers are of two kinds, partitioned cardinals and partitioned serials, defined respectively as the partition-types of repeatedly partitioned classes and series. For each kind, multiplication is binary (i.e. any ordered pair has a unique product) and associative. Addition is in general a free operation (i.e. the summands are not limited to two, and indeed, assuming the multiplicative axiom, may form an infinite class or series); but it is non-associative, which means that for example a + b + c (involving one operation of addition) is distinguished from (a + b) + c and a + (b + c) (involving two operations). A one-sided distributive law is obeyed:

Partitioned cardinals are commutative in addition.
V.—Les transformations asymptotiquement presque périodiques discontinues et le lemme ergodique. (Première Note.)
Authors: Fréchet Maurice
Published: July 2016

Synopsis

With the aim of establishing, under wide conditions, the ergodic theorem of G. D. Birkhoff, the author extends the class of asymptotically almost-periodic functions, considering now not only continuous functions, as he had already done in 1943, but discontinuous functions. Definitions and properties of the extended class of functions are set out, some comparisons being made with almost-periodic functions in the sense of Bohr, Stepanoff, Weyl and Besicovitch. Applications to the ergodic theorem are adumbrated.
VII.—Parallel Planes in a Riemannian Vn
Authors: Ruse H. S.
Published: July 2016

Synopsis

In an earlier paper a description was given, in terms of classical projective geometry, of some of the properties of parallel fields of vector spaces (parallel planes) in a Riemannian Vn, and a detailed analysis was made of the case n = 4. The present paper contains the corresponding formulae for any n, though omits their projective interpretation. A parallel þ-plane is said to be of nullity q when the þ vectors of any normal basis contain q null and þ − q non-null vectors. The conditions of parallelism, namely that the co-variant derivatives of the basis-vectors should depend linearly upon these vectors, are examined for any þ and any q(<þ), and attention is thereafter mainly confined to the cases (i) n even, q = ½n − 1, p = ½n − 1 or ½n; (ii) n odd, q = ½(n − 3)) ,p = ½(n − 1), which possess exceptional features. In the former of these cases light is thrown upon the curious circumstance, noted in the previous paper, that the existence in a V4 of a null parallel i-plane necessitates the existence of parallel planes other than its conjugate. For a general n similar situations arise in the cases indicated.
VIII.—A Further Note on a Problem in Factor Analysis
Authors: Lawley D. N.
Published: July 2016

In a recent paper by the author several unfortunate omissions and misstatements occurred which it seems desirable to correct. I am indebted to Professor T. W. Anderson for pointing these out to me.
X.—The Reciprocity Theory of Electrodynamics
Authors: Green H. S. , Cheng K. C.
Published: July 2016

Synopsis

This paper represents the application of the Principle of Reciprocity, formulated in a previous communication, to the outstanding problems of classical and quantum electrodynamics.

The first step consists in the formulation of a reciprocally invariant Lagrangian function for a system of electrons in interaction with the electromagnetic field. A study is made of the unaccelerated motion of an electron, and this is subsequently extended to embrace the problem of an electron in arbitrary motion. It is found that the usual difficulties of classical electrodynamics do not appear. The methods of the earlier paper are applied to the derivation of the Hamiltonian energy of electron and field, and this enables a quantized formulation of the theory to be given, which also does not lead to the usual divergence difficulties.
XI.—Application of Relaxation Methods to Compressible Flow past a Double Wedge
Authors: Mitchell A. R. , Rutherford D. E.
Published: July 2016

Synopsis

The relaxation technique of R. V. Southwell is shown to be applicable in certain cases to transonic problems. For a uniform stream with a low subsonic velocity impinging on a symmetrical 2-dimensional double wedge, an asymmetrical supersonic region can be isolated in the neighbourhood of the corner of the wedge, and the streamlines and the values of the Mach number within this supersonic region can be determined with the aid of relaxation methods. Difficulties must be expected to occur in the neighbourhood of the sonic line, but in the present problem these have been surmounted.
XXIII.—Studies in Practical Mathematics. VII. On the Theory of Methods of Factorizing Polynomials by Iterated Division
Authors: Aitken A. C.
Published: July 2016

Synopsis

The division of one polynomial by another is studied with the object of ascertaining the errors produced in the coefficients of successive remainders by small errors in the coefficients of the divisor. It is shown that the matrix which effects this transformation of errors is a polynomial in the rational canonical matrix for which the divisor polynomial is characteristic. The theory gives rise to a numerous class of iterative processes for finding an exact factor, such as the extant method based on the penultimate remainder, Bairstow's iterative method of finding a quadratic factor, and many others. Some new suggestions are made for accelerating convergence.
XXVIII.—A Molecular Sum Rule
Authors: ter Haar D.
Published: July 2016

Synopsis

A derivation is given of a sum rule which may be useful in the discussion of line intensities of molecular spectra.
VI.—The Maximum Term of an Entire Series
Authors: Shah S. M. , Singh S. K.
Published: July 2016

Synopsis

The relation between the maximum term and the maximum modulus of an entire function is exhibited by means of general theorems and specific examples. Functions of zero order and of infinite order are mainly considered.
VII—Tidal Charts Based on Coastal Data: Irish Sea
Authors: Doodson A. T. , Rossiter J. R. , Corkan R. H.
Published: July 2016

Synopsis

The object of this paper is not to produce a chart of co-tidal and co-range lines which is more accurate than an existing one, but to investigate methods of computing such charts on the supposition that there are no observations of tidal streams such as were used to produce the existing chart. Only coastal observations of tidal elevations are supposed to be known, for such conditions would exist in many parts of the world. The methods used are similar to the so-called “relaxation methods”, using finite differences in all variables and attempting to satisfy all the conditions of motion within the sea, proceeding by successive approximations. There are many difficulties, peculiar to the tidal problem, in the application of these methods, due to the very irregular coast-lines and depths, gaps in the coasts, shallow water near the coasts, frictional forces, and the very serious complication due to the fact that the tides are oscillating and thus require two phases to be investigated simultaneously owing to their reactions one upon the other. One very important point in testing the methods is that no use whatever should be made of existing charts in obtaining first approximations of heights to commence the processes, not even where there are wide entrances to the sea. The resulting chart is shown to be very closely the same as the existing chart, thus proving the validity of the method.
IX.—On the Geometry of Ɛ-Matrices
Authors: Ruse H. S.
Published: July 2016

Synopsis

The two regular representations of quaternions give rise to a classical set of sixteen 4 × 4 matrices that have fairly recently reappeared in a paper by S. R. Milner. He uses them as the basis of a calculus of “Ɛ-numbers”, which he develops for the purpose of making physical applications. The covariantive nature of his calculus is, however, not always fully apparent, and raises some points of interest of which an examination is made in the present paper in terms of 3-dimensional projective geometry. The theory that emerges is the classical one of the collineations of projective 3-space that transform a quadric into itself, but the formulation is different from that of existing theories based on the same set of matrices and having the same or a similar geometrical background. For example, the present theory is quite different from that of 4-component spinors. The constants of multiplication γijk of quaternion algebra make their appearance in a generalized form and in a geometrical setting. In the final section an indication is given of possible generalizations to Riemannian geometry, and of the connection of the present work with the theory of Kähler manifolds of two complex dimensions.
XIII.—A Statistical Examination of the Centroid Method
Authors: Lawley D. N.
Published: July 2016

Synopsis

A modified form of the centroid method used in factor analysis is described. Various large sample results are obtained, including a test of significance of the residuals. The method is compared with the corresponding form of maximum likelihood estimation and its efficiency is investigated. A numerical illustration is given of some of the foregoing theory.
XIV.—Studies in Practical Mathematics. VIII. On the Iterative Methods of Lin and Friedman for Factorizing Polynomials
Authors: Aitken A. C.
Published: July 2016

Synopsis

The methods of S. N. Lin (1943) and B. Friedman (1949) for approximating to the factors of a polynomial by iterated division are studied from the point of view of convergence. The general theory, hitherto lacking, is supplied. The matrices which transform the errors in coefficients from one iterate to the next are explicitly found, and the criterion of convergence derived. Numerical examples are given. The tentative conclusion is that the methods are less simple in theory and less adaptable than the method of penultimate remainder, which admits of accelerative devices.
XV.—Bounds for the Indices of Nilpotent and Solvable Lie Algebras
Authors: Patterson E. M.
Published: July 2016

Synopsis

By examining certain connections between the derivatives and the powers of a Lie algebra, bounds are obtained for the indices of nilpotent Lie algebras over an arbitrary field. The results are used to obtain bounds for the indices of solvable Lie algebras over a field of characteristic zero.
XVIII.—Recent Oceanographical Investigations in the Faroe-Shetland Channel
Authors: Tait John B.
Published: July 2016

Synopsis

The Faroe-Shetland Channel is the threshold from the north-eastern Atlantic Ocean to north-west European seas. Through it passes the main bulk of the oceanic water-mass which is the predominant influx, among several other water-masses, to these seas.

The following research into the dynamics and general hydrography of the region is based on numerous observations of temperature and salinity, from surface to bottom, taken mainly on two vertical cross-sections of the Channel between the years 1927 and 1952 inclusive, excepting the war years 1940 to 1945.

The research reveals very large scale seasonal and long-term variations in the northeastward volume-transport of oceanic water, suggests the existence on occasions of what appear to be horizontal tortional currents within the oceanic water-mass, and demonstrates (a) the intrusion of Gulf of Gibraltar (extra-Mediterranean) water into this mass over a period of years, (b) the formation of heavy oceanic water and (c) of a sub-oceanic watermass. The last-mentioned may sometimes almost entirely displace the bottom Norwegian Sea water-mass which normally underlies the oceanic mass.

One or other, or both, of two types of Arctic water may also sometimes displace bottom Norwegian Sea water as the bottom water-mass of the region, the process, like that of the above-mentioned Gulf of Gibraltar water influx, waxing and waning over a term of years and thus exemplifying the phenomenon of marine climatic change.
XXIII. —Marine Research—The Work of the National Institute of Oceanography
Authors: Deacon G. E. R.
Published: July 2016

Synopsis

The National Institute of Oceanography owes its formation to a growing need for more detailed and systematic knowledge of the physics as well as the biology of the oceans, and the feeling that the United Kingdom, in co-operation with the Commonwealth countries, should play a part in marine science more in keeping with its tradition of interest in the oceans and their navigation. The Institute is controlled by the National Oceanographic Council, which was granted a Royal Charter in 1949 and operates with the aid of Government grants.

The emphasis of the programme is on long-term research under the main headings:

Interchange of energy between the atmosphere and oceans

Response of the sea surface to wind and pressure changes

General circulation of the oceans

Free and forced oscillations in the oceans

Distribution of marine organisms

Organic production in the oceans

Bionomics of whales.

There are many other activities and as much as possible is done to work out the practical implications of the research, and to help other marine scientists. The headquarters of the Institute are at Wormley, near Godalming, Surrey, and its research ship, the R.R.S. “Discovery II”, is based at Plymouth.
II.—Spontaneous Fission Rates and the Role of the “Excess” Neutrons in Heavy Nuclei
Authors: Feather N.
Published: July 2016

Synopsis

Empirical regularities are sought amongst the experimental data relative to the spontaneous fission rates and the nucleon-pair binding energies of the heavy even-even nuclei. Certain regularities are found which have not hitherto been noted; in particular it appears that the isotopic number D is a significant parameter in relation to these various quantities. The speculation is made that possibly the saddle-point deformation of the protons in the nucleus is greater than that of the neutrons. Predictions are made concerning the spontaneous fission rates of the even-even isotopes of thorium.
V.—A Search for n-γ Coincidences from a Po-Li Neutron Source
Authors: Galloway R. B.
Published: July 2016

Synopsis

Nemilov and Pisarevskii (1957) have reported a 390 keV γ-ray in coincidence with neutrons from a Po-Li source, and interpret this as evidence for a level in B10 at 390 keV. A search for confirmatory evidence of this γ–ray has been unsuccessful, and it is estimated that such n–γ coincidences would have been detected in the experiment here described if they involved 8 per cent or more of the neutrons from the Po-Li source.
VI.—Endomorphisms of Abstract Algebras
Authors: Evans Trevor
Published: July 2016

Synopsis

A study is made of the generalization of the entropic law (xy)(zw)—(xz)(yw) to an identity connecting two operations. It is shown that such an identity is equivalent “in the large” to the condition that the set of endomorphisms with respect to one operation is closed with respect to the other. Furthermore, for such entropic operations, each may be regarded as a generalized endomorphism of the other and various generalizations of elementary properties of endomorphisms are obtained. Examples of entropic pairs of operations are quite common in mathematics and a number of these are discussed. An important aspect of the paper is the matrix notationintroduced to facilitate what would otherwise be extremely cumbersome computations with entropic operations.
VIII.—Theoretical Expressions for Relaxation Effects in the Thermal Conductivity of Gases as Measured in the Modified Co-axial Cylinder Apparatus of Callear and Robb
Authors: Wright P. G.
Published: July 2016

Synopsis

The apparent thermal conductivity of a polyatomic or isomerizing gas (as measured in a given apparatus) may decrease at low pressures for two distinct reasons. There may be accommodation effects at boundary surfaces, and there may be relaxation effects arising because molecules with excess energy do not yield it up fast enough to maintain local “chemical” equilibrium. If the apparatus is such that the temperature measured at points in the gas and not in the walls, relaxation effects may be observed directly, and accommodation effects are (in theabsence of relaxation effects) absent.

A detailed analysis is made of the apparent thermal conductivity measured in such an apparatus with cylindrical symmetry. Expressions are obtained in closed form. Numerical calculations show that, for a gas of relatively long relaxation time in an apparatus of reasonable dimensions, the apparent thermal conductivity would decrease appreciably at readily-attained pressures.
XVIII.—The Emission of Long-Range Alpha Particles in Fission
Authors: Feather N.
Published: July 2016

It is generally agreed that the long-range alpha particles of fission are set free before the fragment nuclei have acquired more than a small fraction of their final energy of separation, but whether the alpha particle is liberated before the instant of scission, at that instant, or from one of the fragment nuclei very shortly thereafter, has remained an open question. Each of these views has been seriously advocated. These various hypotheses are examined in relation to recently published information regarding the distribution of mass in low-eneigy ternary fission, and other considerations, and it is suggested that the hypothesis having the strongest claim to attention is that which assumes that the alpha particles originate in the heavy fragments exclusively, being liberated, very shortly after the instant of scission, with probability not much less than unity, from fragment nuclei of low yield and small neutron excess. Conclusions which would follow, if this hypothesis were accepted, are indicated, and possible experimental tests of these conclusions are suggested.
XXI.—A New Approach to the Statistical Thermodynamics of Liquids
Authors: Fürth R.
Published: July 2016

An attempt is made to develop the statistical mechanics of the liquid state based not on the usual concept of a “radial distribution function” but on that of a “next neighbour distribution function” which is closely linked up with Bernal's ideas on the characteristic features of liquid structure. Making certain simplifying assumptions it is indeed possible to construct a partition function for an atomic liquid in this way and from this to derive the thermodynamic properties of the system according to the principles of classical statistical mechanics. It is shown that the free energy, the equation of state, the specific heat and entropy as obtained from the theory are consistent with the expected behaviour of such liquids. It is further shown that the computed next neighbour distribution function for close packing is in good agreement with the one derived empirically from a model by Bernal and Mason.
XIV.—On Eigenvectors of Boolean Matrices
Authors: Blyth T. S.
Published: July 2016

Synopsis

Throughout this paper we shall be concerned with n × n matrices X=[xij] whose elements xij belong to a given Boolean algebra B(≤, ∩, ∪, ′). In tne first part of the paper we show that the set S(λ, x) of matrices with a given eigenvector x and eigenvalue λ is a subgerbier (i.e.), γ-semi-reticulated sub-semigroup of Mn(B), the Boolean algebra of all n × n matrices over B. We also determine the greatest element M(λ, x) of S(λ, x) and consider some of its properties. In the second part of the paper we consider the structure of matrices which possess a given primitive eigenvector x and show in particular that, for any given λ ∈ B, there is a matrix, namely M(λ, x), having x as the maximum primitive eigenvector associated with the eigenvalue λ. We also determine necessary and sufficient conditions under which the matrix M(λ, x) is the same for all λ in any given sublattice of B.
XXIII.—Autocorrelation in Black Body Radiation and in Density Fluctuation Waves in Liquids
Authors: Fürth R.
Published: July 2016

Synoposis

General formulae are derived from first principles for the temporal and spatial autocorrelation functions of stochastic parameters which are defined in terms of superposed, uncorrelated waves with known spectral density distribution. These formulae are first used for obtaining expressions for the autocorrelation functions of the components of the electromagnetic field strength and the electromagnetic energy density in black body radiation fields. The general theory is further applied to compression waves in liquids, and expressions are derived for the temporal and spatial autocorrelation of thermal density fluctuations in liquids, in particular near their critical point. Finally the spectrum of the fluctuations in the total radiation emitted by a thermal source, owing to the fluctuations in the energy supply to the source, is obtained from the appropriate Langevin equation, and the temporal autocorrelation function of the radiation intensity due to this cause is derived from the spectrum.
VIII.—Some 9-Fluorenyl and 9,9′-Bifluorenyl Derivatives.
Authors: Birnie J. M. , Campbell Neil
Published: July 2016

9-Carbamoylfluorene with lithium aluminium hydride, 9-cyanofluorene with this reagent and aluminium trichloride, or (on one occasion) treatment of the oximes of 9-formylfluorene with thionyl chloride yield 9,9′-dicyano-9,9′-bifluorenyl. 9-Bromofluorene and ethanolic potassium cyanide yield 9-cyano-9,9′-bifluorenyl, and 9-formylfluorene when kept in ether for a month gives 9,9′-diformyl-9,9′-bifluorenyl. The so-called α-oxime of 9-formylfluorene described in the literature contains about 33 per cent of the higher melting β-oxime. Reduction of the oximes with zinc and acetic acid yields di(9-fluorenylidenemethyl)amine, previously obtained by other methods. A new method for the preparation of 9-aminomethylenefluorene is described and its structure has been confirmed. Many 9-substituted and 9,9′-disubstituted fluorenes exhibit characteristic absorption at 1920–1880 and 1960–1940 cm.−1.
XII.—The Hexapole Magnet as a Velocity Selector and Polarizer for Neutral Spin-½ Particles
Authors: Brash H. M. , Campbell D. M. , Farago P. S. , Rae A. G. A. , Siegmann H. Chr. , Wykes J. S.
Published: July 2016

Synopsis

The general paraxial particle-optic properties of the hexapole field are summarized and lens formulae derived. Formulae are given for the transmission as a function of particle velocity and system geometry for geometries appropriate to the case of neutral atom beams and of slow neutron beams respectively. These formulae are used to evaluate the hexapole magnet as a velocity selector and polarizer for atomic beams and as a spin polarizer for neutron beams. Some experimental observations on potassium beams are quoted in support of the theory.
XX.—Finite Hilbert Transform Technique for Triple Integral Equations with Trigonometric Kernels
Authors: Srivastava K. N. , Lowengrub M.
Published: July 2016

In this paper, we shall be concerned with an investigation of the solution of triple integral equations involving sine and cosine kernels. These type of equations arise in the study of certain two-dimensional mixed boundary value problems in infinite planes and infinitely long strips.
III.—Theory of the Whittaker Hill Equation
Authors: Urwin Kathleen M. , Arscott F. M.
Published: July 2016

When the Helmholtz equation ∇2V + k2V = o is separated in the general paraboloidal coordinate system, the three ordinary differential equations obtained each take, after a suitable change of variable, the form of the Whittaker Hill equation. For the case k2 < o, a considerable amount is known about the periodic solutions of this equation. The theory for k2 < o does not carry over immediately to the case k2 < o, and so far only perturbation solutions have been obtained. This paper gives, in sections 1–5, explicit solutions for the case k2 < o, in the form of trigonometric series determined by three term recurrence relations. In sections 6, 7 some important relations and orthogonality properties are discussed, which are of particular significance in respect of application to boundary value problems. Section 8 discusses some degenerate cases, and in the Appendix an important property is established of continuity of the solutions with respect to the parameters of the equation.
IV. A Modified Hilbert Transform
Authors: Jones D. S.
Published: July 2016

The solution of certain integral equations, which are related to the Hilbert transform, is discussed from the viewpoint of generalized functions. The behaviour of the corresponding transforms in the complex plane is examined for generalized functions which effectively vanish at infinity. A space of generalized functions which need not be small at infinity is introduced and the integral equations solved in this space. Various possible extensions of the theory are also mentioned.
V.— On the Spectrum of Second Order Partial Differential Operators.
Authors: Brown K. J. , Michael I. M.
Published: July 2016

Synopsis

In a recent paper, Jyoti Chaudhuri and W. N. Everitt linked the spectral properties of certain second order ordinary differential operators with the analytic properties of the solutions of the corresponding differential equations. This paper considers similar properties of the spectrum of the corresponding partial differential operators.
XI.—Correlations in alpha-Particle-accompanied fission
Authors: Feather N.
Published: July 2016

Synopsis

The attempt is made to exhibit the necessary correlations between fragment excitation (or prompt neutron number), fragment kinetic energy, and a-particle energy, in α-particle-accompanied ternary fission. Treating a single mode of mass and charge division on the assumption that the ternary process develops directly out of an intermediate binary phase, and using the mutual electrostatic potential energy of the nascent binary fragments of this phase and the additional kinetic energy developed at the moment of α-particle emission as independent variables, various formal results are obtained and discussed in the light of the experimental evidence.

In terms of a classical description, it appears likely that (for a given mode of division) the nuclear configuration at a-particle release in ternary fission is subject to much smaller variations than is the nuclear configuration at scission in binary fission in the corresponding mode. Possible inadequacies of this classical description are very briefly discussed.