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The method of plotting observations on probability paper becomes clear when the curve on probability paper is regarded as an integral of the histogram. The observations are divided into intervals, which may often conveniently be the natural interval of measurement, and the total frequency of observations up to and including the nth interval is plotted against the upper end of the nth interval. It is shown that in drawing a straight line to fit the observations, less weight should be attributed to deviations from the line towards its ends than in the middle regions.

The Barkas analysis is applied to a series of nine papers of known visual gloss. With some reservations, the analysis is found to hold approximately and the distribution of mirror facets so derived is found to accord well with the surface structure of the papers as found by microscopic examination. It is concluded that, whilst the analysis is of negligible value for practical gloss determinations, it appears to present a useful method of investigating the surface structure and "finish" of paper and similar delicate materials. The gloss of the papers examined appears to be assessed solely by the intensity of the light reflected at the specular angle and not by the contrast noted by examination at different angles nor by the ratio of specularly-reflected to diffusely-reflected light. Some suggestions for future research are given.

Examination of recent papers on dimensions and physical quantities suggests a need for more careful definition of these quantities. This matter was dealt with in an earlier paper, but is now expanded and treated in more detail. It is maintained that if definitions were more precise there would not be the same confusion about the dimensions of certain quantities. There may still be differences of opinion on the most appropriate definition of a particular quantity, but if the definition is adequate then there is no doubt about the dimensions of the quantity as deduced from the definition. In this connection there is discussed the use of quantities defined as ratios, such as specific gravity (or relative density), and their more fundamental equivalents, such as density. Further, a logical set of definitions leads to the conclusion that the introduction of certain universal constants is unnecessary - for example, the Newtonian constant N in the equation for force, F=Nmf. Finally, there is discussed briefly the definition of the intensive quantity, temperature, in terms of the extensive quantity, energy.

The performance of the Bausch and Lomb and Sheen glossmeters is examined. Comparison of gradings of nine printing papers according to visual and instrumental means shows that, despite high correlation between them, there are nevertheless important discrepancies not attributable to sampling, instrumental or personal errors. The effect of angle of incidence and viewing on grading is investigated; best results are noted at 45°, and the discrepancies become more pronounced at greater angles. Gradings of the same papers according to the Jones, Askania, Goerz, Ostwald and Klughardt methods are examined; all are found unsatisfactory. It is concluded that, for the limited range of papers tested, the most satisfactory grading is given by measurements of the intensity of light received when angles of incidence and viewing are 45°.

For a range of papers of luminance factor varying from 20% upwards and of gloss varying from almost complete matt to gloss high enough to give fair mirror images the equation G = S(a + 10/L) is proposed, where G is a measure of visual gloss, S is the relative intensity of reflected light under 45° - 45° conditions of illuminating and viewing, L is the luminance factor expressed as a percentage and a is a constant which, under the conditions used by the authors, is approximately 0.20. The visual gradings of eight papers out of ten are not markedly affected by the conditions of observation, but for the remaining two there is a highly significant shift in grading according to whether these conditions permit formation of mirror images or not. Gloss is considered to be a gestalt, corresponding to no single physical property of a surface, being judged by appraisal of a physical situation taken as a whole. The validity of the method used in establishing the psychological gloss scale is checked by experiments in weight grading.

A method is described for making a projection drawing from any point of view in which the drawing is set out directly from the dimensions of the object without the use of intermediate plans and elevations. Seven key figures are given with the necessary numerical factors which give directly the proportions for forty-two points of view. By the use of square ruled paper and these figures, aided by the slide rule, figures may be drawn in which the dimensions are found by means of the rulings on the paper only, so that scales of length are not needed. An example of the use of the method is an illustration of two pentagonal dodecahedrons, one that of Euclid and the other that of a crystal of iron pyrites. The last part of the paper deals with the solution of the spherical triangles on which the method depends.

A conducting paper analogue has been employed to provide a rapid and inexpensive method for the determination of values of stress in fields of uniaxial tension and of pure torsion, to which the equations of Laplace and Poisson apply. The reliability of the equipment has been verified by comparison of selected experimental results with those of known theories. Applications of the technique to the determination of the stress distributions in thin sheets containing geometric openings are given, and also to the determination of the stress distribution in a circular shaft containing a keyway, in pure torsion. The relevance of of Inglis' theory for equivalent elliptic holes is examined. Results are presented in graphical form.

Three documents referring to the development of the telescope are discussed. They are arranged with respect to the gradual development of this optical device and its production in London. Document I is a letter written by Beal on 24th November, 1656. It refers only to the Dutch or Galilean telescope and gives interesting details regarding the hand-glass used by Gustavus Adolphus and the taste shown by the English opticians of 1656 in decorating the bodies of Galilean telescopes. Document II is a photographic reproduction of J. Wiesel's price list of 1647. It is by far the most important from the historical standpoint. From it we learn some details of the industrial development of Wiesel's telescope factory in Augsburg at the surprisingly early date of 1647. Document III is a letter written by the well-known London optician Richard Reeves on 11th March, 1660. It informs us of the state of telescope manufacture in London at that time. The prices given for the three kinds of telescopes - Dutch or Galilean, Keplerian, and Schyrlean - are quite reasonable and much lower than those quoted in Document II.

This paper contains several additional observations which serve to revise and further unify the proof of the approximation theorem.

Edited by D. ter Haar Oxford: Pergamon Press. 1964. Pp. xvi + 503. Price 140s. For the seventy-year-old Peter Kapitza the publication of this volume of his collected works must be a source of much satisfaction. I had the pleasure of seeing a good deal of him during his early days at the Cavendish Laboratory when I realized, as all who associated with him realized, that we had among us an experimenter and a thinker of no mean order.

It is shown that an arbitrary locally flat imbedding of one piecewise linear manifold into another of codimension greater than two can be changed to a piecewise linear imbedding by means of an arbitrarily small isotopy of the ambient manifold. Bibligraphy: 5 entries.

Though the notation is identical with that of the standard Hartree-Fock formalism, the `Hartree-Fock parameter' of Sharma and Coulson, 12, cannot be taken as an off-diagonal Lagrangian multiplier for the general case of non-orthogonal orbitals. An alternative formulation of the singlet and triplet equations is derived in terms of standard Lagrangian multipliers in which 12 may be taken as an off-diagonal Lagrangian multiplier for either orthogonal or non-orthogonal orbitals.

In a recent paper Furth has proposed an experiment in which he predicts that a positive result of first order in v/c will be observed, caused by the difference in local time of two observers in uniform relative motion. It is shown that, if the special theory is correct, the phenomenon predicted by Furth would not be observed.

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Starting from the experimental fact that any colour can be uniquely expressed by a trichromatic equation, provided negative coefficients may enter, it is shown that all problems of colour mixture are amenable to an exact system of geometrical calculation. The particular methods described in the paper are such as to obviate the introduction of stereographic projection and other geometrical complications which enter into previously published methods. This simplification is effected by conducting the actual colour mixture part of any calculation in the quantity units of one trichromatic system, leaving the relative magnitudes of the various systems of units, where more than one system is involved, to be accounted for by the introduction of suitable coefficients in the purely arithmetical part of the work. The problems dealt with are: Mixtures of two colours. Mixtures of three colours. Mixtures of four or more colours. Transformation of data from one trichromatic system to another, either directly or by means of general equations of transformation. Transformation of data between the trichromatic and monochromatic systems of specification. Calculation of colour from energy distribution of stimulus. Graphical solutions are described for all these problems, non-graphical solutions being also given for all cases in which they are applicable. The graphical solutions are described in relation to rectangular co-ordinates so that they may be carried out on ordinary graph paper without the necessity of specially prepared charts. The methods are illustrated by a number of worked examples specially designed to show how various types of data which may enter into a practical problem may be reduced to a form suitable for solution, and how the result of the colour calculation can be reconverted to the form in which the answer is required.

The transmission error of a pair of spur gears is detected by an optical method and recorded continuously as a function of time. Measurement of the alternating component of the error is repeatable within ?5 ? 10-5 in., and of the mean value within ?1 ? 10-4 in., at base circle radius, at any speed in the range 0-6000 rev min-1 and at any load within the capacity of the gears. Records are taken on photographic paper wrapped round a drum which rotates with one of the gears being tested. In this way the wavelength of record per cycle of tooth-engagement is made independent of speed and comparison of static and dynamic behaviour is facilitated. The method of measurement is applicable only to a pair of gears with the same number of teeth.

A mechanical device for plotting characteristic curves of photographic emulsions is described. Details of construction are given, and readings from the instrument are compared with those obtained from an orthodox plot on graph paper. Considerable saving of time, and of material, may be effected by its use in instances where values of co-ordinates are to be entered from curves of similar character drawn through a series of experimental points, for which one set of co-ordinates has fixed values.

Traditional breadmaking methods have remained essentially unchanged since the introduction of leavening by yeast some 6000 years ago, despite the introduction of process mechanization. Twelve years ago, a typical British bakery would have gently mixed together batches of the basic ingredients – flour, yeast, salt and water – and set the dough aside in bowls to ferment for several hours before dividing it into pieces, moulding these to the desired shape, allowing them to rise and baking them into loaves.

That the factor space of a semisimple Lie group by an arithmetic subgroup has finite volume with respect to Haar measure is well known. In this paper we study results related to the converse of this theorem. In particular, under some rather weak assumptions on a semisimple Lie group G we prove that every discrete subgroup of G with a non-compact factor space of finite volume that satisfies some natural irreducibility conditions, is an arithmetic subgroup of G. In this paper we also study various results from the theory of algebraic groups and their arithmetic and discrete subgroups. In the proof of one theorem we use a construction from representation theory that is of independent interest. At the end we state some unsolved problems in the theory of discrete subgroups.

In this paper the methods of M. Karoubi (MR 41 #6205) are generalized to the case of equivariant K-theory. The sets of Fredholm operators in certain (Hilbert) spaces of representations of finite groups G are described which are classifying spaces for equivariant K-functors. The results were announced in the paper MR 46 #2702. Bibliography: 16 items.

In this paper we give necessary conditions for the local solubility of linear differential and pseudodifferential equations with smooth coefficients. Our results generalize those of H. Lewi, Hormander, Nirenberg, Treves, Egorov, and others, described in the survey paper [16] (see also [17], [18]).

Two problems are considered in the present paper as examples of the integral method formulated in part I (the preceding paper). The first concerns arcs in uniform flow, and it is shown that the only form of steady solution, if the conductance shape factor is taken to be a function of power level, is the fully developed arc. A known solution of the full differential equations provides a test of the adequacy of the result. Free recovery is briefly discussed. The second problem concerns steady arcs in nozzles. With the assumptions of constant shape factors and negligible radiation, a simple model is derived which provides insight into behaviour as a nozzle-blocking condition is approached.

In this paper the mathematical methods of quantum field theory are applied to some problems that arise in the statistical description of mechanical systems with very many (in the idealized case, infinitely many) degrees of freedom. This application is based on a graphical representation of the individual terms of the formal perturbation series in powers of the coupling constant in the form of Feynman diagrams. A variety of properties of such diagrams makes it possible to sum partially the perturbation series with a view to obtaining closed integral equations that contain the required quantities as unknowns. The approach is treated in more detail in connection with the statistical hydrodynamics of a developed turbulent flow, which is similar to the theory of a quantum Bose field with strong interaction. The functional formulation of statistical hydrodynamics makes it possible to obtain integral equations of turbulence theory, which can also be derived by means of diagram methods. At the end of the paper, some closed equations of statistical hydrodynamics are considered.

The theory of Koster and Slater for the electronic states around an impurity in a metal is formulated in terms of the Wannier functions of the host lattice. These functions are not uniquely defined, however, because they depend on how the arbitrary phase factors exp{iα(n, k)} associated with the Bloch functions are chosen. In this paper it is shown that the frequently used `localized approximation' of the Koster-Slater theory has the effect of fixing the relative phases of the Bloch functions, and consequently in this approximation the Wannier functions are uniquely defined. In the second part of the paper an approximation is described, valid in the region close to the impurity nucleus, which leads to expressions in terms of matrix elements of Bloch functions rather than Wannier functions. For two examples, the Knight shift at impurity atoms and the x-ray emission intensity, results are obtained which seem to be more acceptable physically than those previously derived.

This paper is devoted to the substantiation of a criterion for the quasisymmetric conjugacy of an arbitrary group of homeomorphisms of the real line to a group of affine transformations (the Ahlfors problem). In a criterion suggested by Hinkkanen the constants in the definition of a quasisymmetric homeomorphism were assumed to be uniformly bounded for all elements of the group. Subsequently, for orientation-preserving groups this author put forward a more relaxed criterion, in which one assumes only the uniform boundedness of constants for each cyclic subgroup. In the present paper this relaxed criterion is proved for an arbitrary group of line homeomorphisms, which do not necessarily preserve the orientation. Bibliography: 4 titles. Introduction A homeomorphism g : R → R is said to be quasisymmetric [1] if it satisfies the condition M−1 g g(x+ t)− g(x) g(x) − g(x− t) Mg . (1) If g is a quasisymmetric homeomorphism, then the homeomorphism g−1 is also quasisymmetric. For arbitrary quasisymmetric homomorphisms g1 and g2 their composite is also a quasisymmetric homeomorphism and Mg1g2 Mg1Mg2 [1]. Since the constant Mg in condition (1) for a homeomorphism g is not unique, this means that there exists a constant Mg1g2 for the homeomorphism g1g2 such that the inequality holds. We say that a group G consisting of quasisymmetric homeomorphisms is quasisymmetric. The following basic result for a quasisymmetric group of line homeomorphisms was obtained in [2]. Theorem 1. Let G be a group of line homeomorphisms. Then a quasisymmetric homeomorphism η such that η ◦G ◦ η−1 is a group of affine transformations exists if and only if G is a quasisymmetric group such that Mg = M for all g ∈ G, where M is a fixed constant. This research was carried out with the support of the Russian Foundation for Basic Research (grant no. 53-01-00174) and the Programme of Support of Leading Scientific Schools of RF (grant no. NSh-457.2003.1). AMS 2000 Mathematics Subject Classification. Primary 54H15; Secondary 20F38, 28D99.

In this paper a classification of infinitely differentiable functions is developed, based on derivatives introduced earlier by the author. In terms of the classes Cγ, κ (0, u] and Cγ, κ (0, ∞) considered in this paper, not only is an interrelation between the different classes given, but also the problem of representation of the functions of these classes, in the case of quasi-analyticity (in the sense of Denjoy and Carleman) as well as in the case of non-quasi-analyticity of the class, is completely solved. Some applications of these results are given.

The paper deals with the problems of localization and convergence of Fourier series with respect to a so-called fundamental system of the Laplace operator. (As understood by the author in this paper, a fundamental system of functions includes the eigenfunctions of all boundary-value problems and is characterized by the absence of any kind of boundary conditions).Chapter 1 of the paper contains a survey of all the important results on the problems of localization and convergence of Fourier series, both for concrete systems of eigenfunctions of the Laplace operator (and, in particular, for the multiple trigonometric system) and for arbitrary fundamental systems of functions of this operator.In Chapters 2-5 detailed proofs are given for the recent results of the author concerning general fundamental systems of functions of the Laplace operator, including: 1) a comprehensive solution of the localization problem for an arbitrary -dimensional domain in the Sobolev classes (with non-integral ), 2) a comprehensive solution of the localization and convergence problem for an arbitrary odd-dimensional domain in the Holder classes , 3) almost definitive conditions for localization and convergence for an arbitrary even-dimensional domain, 4) a proof of the result that in the class of all -dimensional domains the smoothness conditions prescribed for the function to be expanded are best possible even for an arbitrary rearrangement of the terms of the Fourier series.

The chief aim of this paper is to show that the tendency, which has been developing among optical designers in recent years, to follow in the footsteps of (mostly) German optical workers and test the performance of optical designs by exact ray tracings by trigonometrical methods on paper instead of by inspection of the actual performance of the embodied idea, involves a needless waste of both time and working costs. While such exact ray tracings may give valuable data concerning the optical performance of a lens system, if carried out thoroughly enough, yet the results so obtained do not and cannot give any indication of the direction in which changes are required, and only an appeal to the general algebraic formulae, such as those, for instance, founded upon Coddington's work, can afford the designer the enlightenment necessary to enable him to bring the system to the best possible perfection. Therefore, I have fully come to the conclusion that ray tracing methods are, generally speaking, unnecessary and superfluous for any designer who cares to master the general implications of the much more philosophical and illuminating system of optics founded on Coddington and other workers of the British school. This conclusion I seek to illustrate and justify by a few illustrative examples from my own experience.

This paper describes the application to three areas of physics of computer programs that carry out formal algebraic manipulation. The application areas discussed are celestial mechanics, general relativity and quantum electrodynamics. The paper describes typical problems from each of these disciplines which can be solved using algebraic manipulative systems and presents sample programs for the solution of these problems using several algebra systems. For each discipline a review of published work acknowledging the use of algebra programs is presented and the most advanced applications are discussed in detail. In particular the Lie transform, Petrov classification and Kahane's simplification procedure are reviewed from the standpoint of algebra programs. A number of simple examples are used to introduce the reader to the capabilities of an algebra program and a brief review of the technical problems of algebraic manipulation is given. Further applications of such systems to mathematics, chemistry and engineering are briefly mentioned in the text and relevant work is referenced in the bibliography but the main emphasis is placed on applications in theoretical physics. However, the simple examples indicate, and the applications in the physical sciences confirm, that algebra systems are capable of exploitation over a much wider area than is covered in the present review. This review was completed in December 1971.

The equations of boundary layer theory were derived by Prandtl in 1904. These equations are the basis of boundary layer theory, which has been intensively developed for more than half a century and is one of the important branches of contemporary hydromechanics. Boundary layer theory is the subject of numerous monographs (see [3], [4] and others), written in the USSR and abroad, and also of a large collection of articles setting out the results of theoretical and experimental investigations. In connection with engineering problems (in particular, the important problem of determining the resistance of a medium to a body moving in it), the most important methods are numerical methods for solving Prandtl's system. Problems of the approximate solution of systems of boundary layer equations were investigated by Karman, Polhausen, Kochin, Dorodnitsyn, Loitsyanskii, and many others. The present paper deals with mathematical problems in boundary layer theory. One of these is the problem of conditions for which a solution of Prandtl's system exists. This mathematical problem is important, since in the boundary layer there may occur the so-called boundary layer separation, which leads to the appearance of singularities in the solutions of Prandtl's system and to their noncontinuability. In the paper we also discuss the problem of uniqueness and stability of solutions of Prandtl's system, the problem of unsteady flow in the boundary layer resulting in steady conditions as , we construct approximate solutions of Prandtl's system and we prove their convergence. The treatment of all these problems in this paper follows that of [7], [8]. Solutions of all principal two-dimensional problems in boundary layer theory are constructed here for steady and unsteady flow of an incompressible fluid. The corresponding uniqueness and stability theorems are proved for these solutions.

Methods are given for the rapid calculation of the mutual inductance of two coaxial circular currents. Maxwell's first formula is converted into theta-functions, and then expanded in a Jacobian q series. The logarithmic values of this series for various values of q have been tabulated in a previous Paper by the author. When the circles are near one another a series for M is given in terms of q1, where q1 is the complement of q. In this Paper the author treats Maxwell's second formula in a similar way. A table of the values of these series found, computed to six decimal figures by T. Tishima, is given. The chief advantages of this table are that nearly all practical cases are included within a short range of the argument, and the calculation is simple, as the numbers in the difference columns are small. By the help of these tables and series the mutual inductance between two coaxial currents can be easily computed to a high degree of accuracy.

We have earlier proved (Dokl. Akad. Nauk SSSR 164 (1965), 40-42; Mat. Sb. 70 (112) (1966), 132-144) a theorem on the representation of an arbitrary function analytic in a closed convex region by a Dirichlet series in the open region . In this paper we prove that any function analytic in an open convex finite region and continuous in can be represented by a Dirichlet series with coefficients which can be computed by means of specific already-known formulas.We also prove that if the convex region is bounded by a regular analytic curve, then any function analytic in can be expanded in a Dirichlet series in . These two theorems are based on the following theorem from the theory of entire functions:Let be a finite open region, the support function of , , and a function satisfying the conditions Then there exists an entire function of exponential type with growth indicator and completely regular growth, which satisfies the following conditions:1) All the zeros of are simple, and 0$ SRC=http://ej.iop.org/images/0025-5734/9/1/A05/tex_sm_2048_img10.gif/>.2) We have the estimate r_0.$ SRC=http://ej.iop.org/images/0025-5734/9/1/A05/tex_sm_2048_img11.gif/>3) The sequence is part of a sequence , , which depends on the region but not on the function .In this paper we prove an analogous theorem for entire functions of arbitrary finite order .

This paper is a sequel to a previous one (containing an error which is corrected) in which the general hydrostatics of an electrically conducting liquid under an electrostatic field was treated, taking into account the electric double layer, the theory being considered as rigorous within the framework of macroscopic physics. In that paper a set of equations was obtained governing the shape of a liquid boundary. Two problems were solved in the first approximation, but no general method of solution was given and a procedure of guess-work was used at an important stage. In this paper a general method is given with forms pertaining to axisymmetric two- and three-dimensional situations, respectively, and it is applied to find the second approximation to the solution of one of the previous problems, that of the two-dimensional sessile drop. The treatment of this problem involves the solution of an interesting electrostatic problem and the tabulation of a function.

In this paper the group of homeomorphisms of an arbitrary topological manifold is considered, with either the compact-open, uniform (relative to a fixed metric), or majorant topology. In the latter topology, a basis of neighborhoods of the identity is given by the strictly positive functions on the manifold, a homeomorphism being in the neighborhood determined by such a function if it moves each point less than the value of this function at the point. The main result of the paper is the proof of the local contractibility of the group of homeomorphisms in the majorant topology. Examples are easily constructed to show that this assertion is false for the other two topologies for open manifolds. In the case of a compact manifold the three topologies coincide. In conclusion a number of corollaries are given; for example, if a homeomorphism of a manifold can be approximated by stable homeomorphisms then it is itself stable.

Polyanalytic functions of a complex variable (a class which is one of the most important natural generalizations of that of analytic functions) have been the subject of a considerable number of papers in recent years. The present paper is a fairly complete survey of results obtained so far in this branch of complex analysis.

Two networks are said to be dual to each other if the following correspondence exists between them: - Each element of one network has a counterpart in the other one. The currents through the elements in one network and the voltages across their counterparts in the other network are proportional to each other. A network dual to a given one is usually constructed by connecting elements of the appropriate kind in a circuit the diagram of which forms the topological counterpart to the diagram of the original circuit. If this original circuit is a non-planar one (i.e. if it cannot be drawn without crossings between some of its branches), then there exists no topological dual, and for this reason it has been said that there exists no dual to a non-planar electrical network. However, if we are interested in an electrical network not because its circuit diagram has certain geometrical properties, but because it constitutes an assembly of a number of passive and active elements which perform in a certain manner (on account of their impedance properties and of certain relations of mutual constraint imposed by Kirchhoff's mesh and junction relations), then there is really no restriction of this kind. We need only convert the original network into another with the same elements and the same performance, but with a planar circuit diagram and with additional constraints implemented by ideal transformers. This equivalent network can be then converted into its dual counterpart, which is another planar network subject to the dual counterparts of the transformer constraints of the first case. In this way we arrive at an assembly which performs in the required dual manner. Introducing thus the concept of physical duality as opposed to geometrical duality, the paper explains in an elementary manner a number of ways in which this conversion into equivalent planar networks can be achieved. These methods are: - A method originally published by Julia. The dual counterpart of this method. The method of fictitious junction points (an electrical analogue to F. Schur's method of fictitious junction points in pin-jointed frameworks). The method of the transferred terminal. The general method of separation. Some of these methods require a smaller number of ideal transformers than Julia's original method. The paper ends with a short discussion of the errors introduced when the dual network is set up on a network analyser and when the ideal transformers have to be replaced by real ones; the numerous methods available for the realization of an equivalent network enable a choice to be made so as to minimize these errors, or possibly eliminate them altogether. The existence of a dual to a non-planar network is of some interest in connection with electro-mechanical analogies as it removes a restriction to which the direct analogy would otherwise be subjected.

In circuits possessing constant inductance, resistance and capacity the differential equations for the currents and the voltages are linear with constant co-efficients, and may therefore be solved by aid of the known simple properties of symbolic operators. The symbolic operator method proves to be very compendious in problems concerning the determination of the primary and secondary currents and voltages of transformers whenever the applied E.M.F. can be expressed as an exponential function of the time, or when it consists of a sudden application of a constant or an exponential function, and also when it is impulsive. In the Paper the method is first applied to a pair of indirectly coupled circuits - i.e., circuits that are insulated from each other. The cases of pure magnetic coupling, pure electric coupling and of mixed magnetic and electric coupling are treated in turn and shown to be operationally all of the same form. Then general operational equations are obtained for direct coupling and for mixed direct and indirect coupling - this latter, of course, includes all types of autotransformers, both high and low frequency. Here again the operational equations are of the same form as in the case of magnetic coupling. The equations readily show that if the primary applied voltage be a pure sine or cosine alternating E.M.F. the well-known circle diagram for determining the instantaneous magnitudes and phases of the primary and secondary currents in an ordinary transformer is applicable to the most general type of transformer, and is capable of certain extensions. It is then shown that a similar circle diagram can be used for obtaining the primary and secondary currents in any general transformer under the application of the quasi-steady conditions of a slowly-damped alternating voltage. A further new theorem arises here, which affirms that a similar graphical method of computation can be used for deducing the voltages at the terminals of the primary and secondary condensers in a high-frequency transformer under steady or quasi-steady alternating current. In the second section of the Paper it is shown how to calculate accurately the free periods and the damping factors of a coupled system with two degrees of freedom, and approximate formulae are also deduced. The third section passes to the investigation of the currents and voltages produced in the circuits by sudden application or removal of a constant, a harmonic, or a damped harmonic E.M.F., and of those produced by the application of electromotive impulses. For this purpose formulae expressing the effect of the various types of symbolic operator occuring in the Paper, on impulsive E.M.F.s and on sudden applications, are deduced and tabulated. Using these tables of integrals it is very easy to write down quickly the solutions appropriate to any given transformer under the action of any of the types of E.M.F. specified. These tables include all the cases likely to arise in wireless telegraphy and in ordinary transformers. As an example of the application of the methods the problem of a primary spark-circuit and its antenna is worked out under resonance conditions, and for various couplings and time graphs are made of the primary and secondary currents and voltages. A further example is the solution for the incidence of a natural electric wave (a stray) on a receiving antenna and its resonant secondary.

In this paper we formulate operators which constitute an essential generalization of the Riemann-Liouville fractional integro-differentiation operator. With the aid of these operators fundamentally new analogues of the classical formulas of Cauchy, Schwarz, and Poisson for the representation of analytic and harmonic functions in the interior of a circle are established. These formulas enable us to give a complete structural representation for the broad classes of harmonic and analytic functions associated with the generalized operators. Finally, in this paper we also establish sufficient conditions for the solvability of the Hausdorff and Stieltjes moment problems for some general families of sequences of positive numbers.