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The aim of this paper is to compare the two more standard geometrical formulations of gauge systems: the so-calledpresymplectic formulation and the formulation bygroup actions. We summarize the main features involved in them and prove that, at least locally, every presymplectic formulation can be interpreted in terms of group actions. The converse is also proved.

In this paper our aim is to give refinements of Jensen?s type inequalities for the convex function defined on the co-ordinates of the bidimensional interval in the plane.

In this paper we study a class of sub-spaces of loop spaces which have appeared in the calculus of variations. Generalizing a result of Smale, we show that the space of loops tangent to a distribution satisfying Hδrmander's condition is weakly homotopic to the space of all loops. If the distribution is fat, we resolve the end point map from the space of horizontal paths. This resolution has two applications: (1) the proof that the cut-locus on an analytic fat Carnot-Caratheodory manifold is sub-analytic; (2) a study of the singularity of the horizontal loop space. At the end we study the geometry of left-invarian t Carnot-Caratheodory metrics on fact nilpotent groups.

In this paper we consider one dimensional random Jacobi operators with decaying independent randomness and show that under some condition on the decay vis-a-vis the distribution of randomness, that the distribution function of the average spectral measures of the associated operators are smooth.

In this work, we introduce ABC-Caputo operator with ML kernel and its main characteristics are discussed. Viral diseases models for AIDS and Zika are considered, and finally, as third model, the macroeconomic model involving ABC fractional derivatives is investigated, respectively. It is presented that the AB Caputo derivatives satisfy the Lipschitz condition along with superposition property. The numerical methods for solving the fractional models are presented by means of ABC fractional derivative in a detailed manner. Finally the simulation results obtained in this paper according to the suitable values of parameters are also manifested.

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For an n-dimensional spherical unit speed curve r and a given point P, we can define naturally the pedal curve of r relative to the pedal point P. When the dual curve germs are singular, singularity types of pedal curves depend on singularity types of the n-th curvature function germs and the locations of pedal points. In this paper, we investigate sigularity types of pedal curves in such cases.

The main result of this paper is a quantified version of Ingham's Tauberian theorem for bounded vector-valued sequences rather than functions. It gives an estimate on the rate of decay of such a sequence in terms of the behaviour of a certain boundary function, with the quality of the estimate depending on the degree of smoothness this boundary function is assumed to possess. The result is then used to give a new proof of the quantified Katznelson-Tzafriri theorem recently obtained in [21].

THERE appeared in your pages some three years ago (vol. xv. p. 515) an article of mine “On the Simplest Continuous Manifoldness of Two Dimensions and of Finite Extent.” In a succeeding number a correspondent (Mr. Monro, of Barnet) propounded a query which may be shortly stated as follows:— “How does it happen that the perpendicular on a geodesic from a point moving along another geodesic changes sign without passing through either the value zero (0) or the value infinity (∝)?” The problem here suggested is a peculiarly knotty one. In the case of the Euclidian plane the perpendicular of course changes sign by passing through the value ∝, while in the case of a spherical surface it is equally obvious that the perpendicular passes through zero, since the two geodesies intersect twice. But what are we to say of the strange hybrid surface which formed the subject-matter of my paper? Your correspondent appeared to insinuate that the problem was insoluble, and that the definition of the surface must therefore involve a logical contradiction. For a while I was greatly puzzled by this unforeseen difficulty, but after a little thought came to the conclusion that the perpendicular changes sign by passing through the value l/2√−1, where l is positive and represents the absolute length of a complete geodesic. In other words, I conceived that the sign of the perpendicular changed from + to − by a continuous variation of the real numbers a and b in the complex number a + b√- 1. I conceived a to diminish continuously till, passing through o, it became − -a, while b at the same time increased with simple harmonic motion from o to a maximum, and then decreased from a maximum to o.

IN a paper published in the Messenger of Mathematics (May, 1878), Prof. Sylvester has given a rule for abbreviating the calculation of (w:i,j) (w -1:i,j) ; where, to fix the ideas, let (x:i,j) be regarded as the number of modes of coinposino x with j of the numbers 0, 1, 2, . . . The abbreviation consists in rejectino from the partitionis of w-all partitions whose highest number is not repeated and rejecting from the partitions of w -1. all partitions which do not contain i; the number of partitions thus rejected being shown to be the same in the two cases. This becomes even more obvious if we convert the above (i, j) partitions into (j, i) partitions: that is, replace each of the above partitions by a corresponding one consisting, of i of the numbers 0, 1., . ..j. This, as is well known, can be done by decomposing each number into a column of l's and then recomposing by rows. Now, if we do this, it is plain that those partitions whose highest number was not repeated become partitions containing,r 1; and that those partitions which did not contain i become partitions having less than the full number of parts, or, in other words, partitions containing 0. So that Prof. Sylvester's abbreviation is equivalent to rejecting from the partitions of w those partitions which contain 1, and from the partitions of w 1 those which contain 0. And it is plain that the number of partitions of w which contain 1 is equal to the number of partitions of w 1 which contain 0; for the two sets of partitions are interchanged by the interchange of 0 and 1. Obviously, instead of rejecting the partitions of w which contain 1 andi those of w -1 which contain 0, we may reject the partitions of w which contain m (where qn is any one of the numbers 1, 2, . i (or j)) and those of w -1 which contain m1; the reason being the same as above. 187

IN order to avoid a clumsy circumlocution, I have ventured to use the word quantification to denote in general that class of operations expressed in the several special cases by the terms, rectification, quadrature and cubature. Some of the quaternion formulae for quantification were given in a paper entitled "Investigations in Quaternions," communicated to the American Academy of Arts and Sciences, 9th January, 1878. They are here reproduced in a more general form together withi the formnulae for barycentres.

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This paper presents a lower and upper solution approach for singular second order boundary value problems on the half line and establishes the existence of positive, unbounded and monotone solutions.

If we apply to a set of quaternions the definitioni of a group as given in the Theory of Substitutions, then a quaternion group of the mth order means a set of m quaternions (scalar unity always included) whose products and powers are also quaternions of the same set or group. The object of the present paper is to determine all the possible finite quaternion groups. These groups, or rather their analogues in the ordinary Theory of Functions, have usually been interpreted geometrically as the linear transformations of the plane of complex variables in itself-automorph1dc traisformnations. The forimulae for these linear transform-nations were first given by Professor G0ordan. in his paper "Ueber endliche Gruppen linearer Traiisforinationen einer Veriinderlichen." * The quaternion formulae, although they have their exact correspondents in Gordan's algebraic formulae, have, when interpreted geometrically, a more general character, in that they represent certain automorphic linear transtbrmations of a three-dimensional infinite homoloidal space, or what is the same thing, of a three-fold extended sphere. The determination of these formulae might be made to depend upon Gordan's solution, but the following is a shorter and simpler solution than that of Gordan, and includes it as a special case.

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In the year 1849 the great paper of Stokes "On the Dynamical Theory of Diffraction " was read before the Cambridge Philosophical Society, and this has remained until the present day the standard upon this important subject. The method of Stokes was based upon the old elastic solid theory of light, and gave the following conclusions: First. That when the incident light was plane polarized, the diffracted light from a small orifice was also plane polarized in such a manner that the displacement was in the same plane as that of the medium at the orifice. So that if a sphere was drawn with the orifice as a center and meridians drawn on the sphere with the axis in the direction of the vibration at the orifice, then these meridians represented the direction of displacement in the diffracted light. Second. The intensity of the polarized light was represented as follows: Let 3 represent the angle between the incidenlt ray prolonged and the diffracted ray, and let q be the angle between the diffracted ray and the direction of the displacement at the orifice. Then the intensity of the diffracted light around a very small orifice will be proportional to

The notation adopted in this paper for the triangle ABC is: G the centroid. I the centre of the inscribed circle. I 1 , I 2 , I 3 the centres of the escribed circles within angles A, B, C. O the centre of the circumscribed circle. D, E, F; D 1 , E 1 , F l ; D 2 , E 2 , F 2 ; D 3 , E 3 , F 3 the points of contact of the inscribed and escribed circles with BC, CA, AB. The Ds lie all on BC, the Es on CA, and the Fs on AB. H, K, L the mid points of BC, CA, AB. X, Y, Z the feet of the perpendiculars from A, B, C, on BC, CA, AB. A′, B′, C′ the vertices, opposite to A, B, C, of the triangle formed by drawing through A, B, C parallels to BC, CA, AB.

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In my former paper on Fourier's double-integral I remarked that Poisson's form of the integral gave the same incorrect; result as Fourier's form in an example by which I tested it, and seemed subject to the same limitations.

The communication on this subject, as originally made to the society, consisted of a series of theorems, giving (1) expressions for the radii of a great many sets of circles, (2) identities connecting several sets of these radii, and (3) miscellaneous identities closely related thereto. As, however, the paper culminated in a general theorem which may be looked upon as fundamental, and the proof of which makes evident the mode of arriving at the said expressions for radii, and as the relations connecting sets of radii are easily found when attention has been directed to their existence, I have thought it best to print little more than the fundamental theorem and a few auxiliary notes.

1. In the first part of this paper we have considered merely the contours of curves, that is, contour points, and the method of obtaining the various physical diagrams. In this part we shall consider chiefly the contours of surfaces; that is, contour lines. If any curve be cut by planes parallel to that of ( x, y ) and if the various points of intersection be projected on any one of these planes, say z = 0, the contour points so obtained will evidently lie on a definite line, and the line will be more accurately indicated in proportion as the number of intersecting planes is greater and their mutual distance is less. It will be given without any break in continuity by projecting every point of the curve upon the plane z = 0. But such a line may be regarded as the intersection, by the plane z = 0, (see fig. 48) of a cylindrical surface whose generating lines are parallel to the z -axis and are drawn from the given curve to meet that plane. We have here then the intersection of a given surface by a surface over which z is constant. But this satisfies our definition of a contour line. This case of a cylindrical surface supplies the simplest system of contour lines by giving z different values. The contours are all superposed in the diagram, but are not in general conterminous. The only case in which they would be conterminous is that in which the same values of the x and y co-ordinates of a point on the curve correspond to different values of the z -co-ordinate.

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hQ(x) of bounded one-dimensional sectional curvature. In this paper, such metrics are referred to as supporting functions of a convex set Q. If Q is a finite convex polyhedron in the Lobachevsky space, then the following formula holds: hQ(x) = mini{h i(x)}, where h i(x) are supporting functions of the (n − 1)-dimensional faces of the polyhedron Q. The process of computing h i(x) is recurrent and is reduced to the case where i are kdimensional simplexes in Hn κ (k < n). Such functions, called conformal splines, are computed by the procedure “MatLab.”

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The following paper contains little that can be regarded as new mathematical information. It aims only at showing, or rather at emphasising, the correspondence which exists between two geometrical theories which are related to each other in the same way as the arithmetical theories of multiplication and division. Such value, therefore, as it possesses is primarily pedagogical.

The following note may be considered as an addendum to the paper by me on pp. 42–47 of this volume of the Proceedings . In that paper it is shown how to inscribe in a triangle ABC, a triangle DEF, such that the perpendiculars to the sides of ABC, drawn through the points D, E, F, shall be concurrent in a point P. This is done by constructing on each of the sides of ABO a triangle similar to DEF; then O the point of concurrence of the three lines joining the vertices of ABC to the vertices of these triangles is the point “inverse” to P. The question, then, naturally arises, What must be the shape of the triangle DEF in order that the point P may be one of the Brocard points, and, as a consequence, O the other one? and the answer is easily seen to be that DEF must be similar to ABO. Hence the following construction:—

[At the first meeting of this Session a paper was read on the value of cos 2π/17, which evidently may be made to depend on the solution of x17 – 1 = 0.* The present paper is the outcome of a suggestion then made, that a sketch of Gauss's treatment of the general equation might prove interesting. To give completeness to the subject the necessary theorems on congruences have been prefixed. The convenient notation introduced by Gauss is here adopted; thus, when the difference between a and b is divisible by p, instead of writing a = Mp + b, we may write a ≡ b (mod p), the value of M seldom being of importance. It is evident that if a ≡ b, then na ≡ nb, and an ≡ bn, n being any positive integer, and the same modulus p being understood throughout. Also a/n ≡ b/n provided n be prime to p. Other properties (similar to those of equations) are easily seen, but only the above are needed here.

[The substance of this paper will appear in the second volume of Professor Chrystal's Algebra.]

In this paper, quantitative bounds in high-frequency central limit theorems are derived for Poisson based $U$-statistics of arbitrary degree built by means of wavelet coefficients over compact Riemannian manifolds. The wavelets considered here are the so-called needlets, characterized by strong concentration properties and by an exact reconstruction formula. Furthermore, we consider Poisson point processes over the manifold such that the density function associated to its control measure lives in a Besov space. The main findings of this paper include new rates of convergence that depend strongly on the degree of regularity of the control measure of the underlying Poisson point process, providing a refined understanding of the connection between regularity and speed of convergence in this framework., Comment: 19 pages

In this paper we study the action of certain integral operators on spaces of holomorphic functions on some domains in Cn: These integral operators are defined by using powers of the Szego kernel as integral kernel. We show that they act like differential operators, or like pseudo-differential operators of not necessarily integral order. These operators may be used to give equivalent norms for the Besov spaces Bp of holomorphic functions. As a consequence we prove that, when 1 p < 1; the small Hankel operators hf on Hardy and weighted Bergman spaces are in the Schatten class Sp if and only if the symbol f belongs to Bp: The type of domains we deal with are the smoothly bounded strictly pseudoconvex domains in Cn and a class of complex ellipsoids in Cn: Our results for strictly pseudo-convex domains depend on Fefferman's expansion of the Szego kernel. In this case, its powers act like a power of the derivation in the normal direction. The ellipsoids we consider are the simplest examples of domains of finite type. In this case, the symmetries of the domains can be exploited to use methods of harmonic analysis and describe the pseudo-differential operators involved.

§1. Since the former paper on this subject was read, Prof. Cantor has published the second volume of his history of Mathematics. This has necessitated various additions to the paper, which can perhaps be best given as an appendix.On page 413 Prof. Cantor says that the construction of Dürer's pentagon is found in a book called Geometria deutsch, which was lately discovered in the town library at Nürnberg, and gives 1487 as the upper limit to its date. The construction is said to be “mitunverrücktem Zirckel,” the same expression that Schwenter applies to Dürer's solution.

§ 1. It is proposed to discuss in this paper partial differential equations involving two independent variables x and y, and a dependent variable z. The method of reduction which is explained can be applied to certain equations involving more than two independent variables, but such application is subject to too many restrictions to be of much general utility.

In this paper the laws of addition and subtraction of vectors were considered, and examples of their extreme usefulness in geometrical applications were given.

1. My only justification for presenting this paper to the Society lies in the fact that, so far as I am aware, the uniform convergence of the Fourier Series is nowhere alluded to, and far less discussed, in any English textbook; while the precautions that are necessary in differentiating the series are hardly ever mentioned even in treatises which give a very thorough treatment of its convergence. I have confined myself almost exclusively to what may be called ordinary functions, as a complete discussion of what has been done in recent years for functions that lie outside the category of “ordinary” would make the paper much too long. For information as to the original authorities, I would refer to the paper which I communicated to the Society last session On the History of the Fourier Series. It is sufficient to say here that the proof I now give is simply an adaptation of that of Heine (Kugelfunctionen, Bd. I. 57–64, Bd. II. 346–353) and of that of Neumann (Uber die nach Kreis … Functionen fortsch. Entwickelungen, 26–52).

The notation employed in the following pages is that recommended in a paper of mine on “The Triangle and its Six Scribed Circles”* printed in the first volume of the Proceedings of the Edinburgh Mathematical Society. It may be convenient to repeat all that is necessary for the present purpose.

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In this paper the following expansion will be obtained: in which n and r are positive integers. The Series in the square brackets are Hypergeometric Series with a finite number of terms.

The graphical representation of a group given by Cayley* leads to a diagram consisting of several lines of different colors, a so-called color-group, which affords a very clear insight into the structure of the group. Cayley himself applied his method only to groups of comparatively low orders, and it seems that the inethod has never been used for more complicated cases.t The purpose of the present paper is to show how readily Cayley's method can be applied to the construction and investigation of numerous groups of higher orders. In particular, the color diagrams for the rotation groups of the regular bodies can be arranged in such a way that they lend themselves miuch easier, at least in some respects, to a study of the groups concerned, than even the models of the regular bodies. The most prominent feature of these diagrams, to which their high degree of perspicuity is due, consists in the fact that their color lines do not intersect each other, so that the diagrams, when described on the sphere, constitute convex polyhedrons. I determine, in the first part of the paper, all the color-groups thus defined and show that, apart from two other cases, they are identical with the rotation groups of the regular bodies. In the second part I study in detail the connection between the rotation groups and the corresponding diagrams. The third part of the paper contains some extensions of the

The following paper was written last summer, and was submitted to Dr Mackay with a view to eliciting his opinion particularly on the curious passage referred to in § 3, and on the remarks contained in § 8. I was not aware of the intention of Mr T. L. Heath to follow up his excellent edition of Apollonius by an edition of Archimedes on similar lines, and when I saw the announcement of his Archimedes in the month of October, I at once concluded that the notes I had made would have been anticipated by him. Since reading his masterly work, however, I am disposed to think there is still sufficient interest in the notes I have written to justify me in laying them before the Society; I therefore submit them in their original form, although I should have omitted certain details had I been acquainted with Mr Heath's work before writing the paper.

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In the first two sections of this paper we provide a brief account of the Cayley–Dickson algebras and prove that the radius on these algebras is given by the Euclidean norm. With this observation we resort to three related topics: a variant of the Gelfand formula, stability of subnorms, and the functional power equation.

It is to be expected that the systematic introduction of irrational covariants will enrich geometry with curves and surfaces, not previously observed or discussed, allied to any given fundamental system by projective relations. Well known irrationalities also must be expected often to reappear. In 1888 Dr. HILBERT t remarked the existence of two nets of conics covariantly related to a general curve of third order in a plane. Two additional systems, nets of curves of the second class, can easily be defined by anl equation closely analogous to Hilbert's. These systems of conics and the four cubics whose polars they are prove to be not entirely unknowvn hitherto, for their dually equivalent loci and envelopes form the basis of the one-to-one correspondences upon the point-cubic and the line-cutbic respectively. That only two of every three such correspondences are found here is due to the domain of rationality that is assumed. By employing the irrationality that occurs in HESSE'S canonical fornm of the cubic I ani able to identify Hilbert's two systems of irrational covariant conics and to exhibit their relation to the other two systems just mentioned. As a consequence it is found possible to give explicitly covariant equations of definition for the two cubics which have the same Hessian and for those which have the same Cayleyan as a given fundamental cubic. These results are here derived by the aid of a canonical form of the cubic containing Hesse's irrationality. The desirable invariantive proofs of these results are given by Professor GORDAN in a paper presented by him to the Americani Mathematical Society for publication in its Transactions.4

Introduction. In 1772 LAGRANGE published a celebrated meinoiron the Problem of Three Bodies, which contained all the solutions in which the ratios of the mutual distances of the bodies are constants. He found two distinct configurations. In the one, the three bodies always lie in a straight line; in the other, they are always at the vertices of an equilateral triangle. Their distribution upon the line depends upon their masses, being explicitly defined by the real positive root of a certain quintic equation. The equilateral triangular configuration is possible for all distributions of the masses. In both cases the three bodies move in the same plane, in conic sections with respect to each other or with respect to their common center of gravity, and in such a manner that the law of areas is true for each body considered separately. No other periodic solutions of the motion of three or more bodies were discovered for more than a century, although many splendid papers appeared on the Problem of Three Bodies. A new impetus was given to the subject by the celebrated memoir of Dr. HILL on the Lunar Theory,t in which he discussed a new species of per iodic solutions. He made the restrictions that one body should be infinitesimal, and that the finite bodies should describe circles around their center of gravity. For the purposes of the applications to the Lunar Theory he neglected the ratio of the distances of the iiioon and the sun, thus introducing errors in the numerical results which in certain cases became important. Professor DARWIN has more recently taken up the subjectt by a method not essentially different from that employed by HILL, and discussed a great many orbits in detail, not neglecting the solar parallax. He has considered only cases in which the infinitesimal body moves in the plane of the finite bodies with direct motion. To these must be added the masterful researches ? of POINCARP, who has

In order to construct an algebraic surface containiing as a part the unilateral paper-strip of MOEBIUS,t let a straight line L move in space along a circle C, perpenldicular to the tangents of C and in such a way that, when the point of intersection Q of L with C has described the full circle, the initial position of L makes with its final position an angle of 1800. The condition that L meets C at right angles is equivalent to the condition that L meets a straight line A passing through the center M of the circle and perpendicular to its plane; let P be the movable point of intersection of L and A. If now we add the further condition that the range P on A be projective to the range Q on C (e. g., by taking the angle QPE always half the angle of the arc described by Q on C) then L describes, according to a general theorem,4 a ruled surface of the third order. Conversely: take anly ruled surface R of the third order, particular cases excepted, pass a plane section through one of the generators L which will meet R besides L in a conic section X7, anid describe a curve T on R the points of which have along the generators a sufficiently small constant distance from K; then T' will cut out of R a unilateral Moebius surface.

Introduction. In any field of geometric investigation the curves fall roughly into two classes, constituted respectively of the curves ordinarily investigated and of the other curves; these unusual curves are in positive designlation the crinkly curves. In this paper we are to investigate by interplaying graphic and analytic methods (in I) the continuious surface-filling way-curves: _ (t), y = (t): of PEANO and HILBERT and (in II) the continuous tangentless yt-curve: y (t): connected with PEANO'S ctirve. We define the various curves A as point-forpoint limit-curves for n = oo of certain curves k (t = 1, 2, 3, ** ); these curves K are broken-line curves derivable each from the preceding by processes simple and such that the (nodal) extremities of the various n-links of K1 persist as corresponding points and also nlodes of the KW1; thus, the nodes of IT are points of K; the set of all these nodes (for all n's) is on K everywhere dense. The curves K are continuous and approach their point-for-poilnt limit-curve IT uniformly; K is accordingly continuous, a conclusion however which is geometrically evident. From the continuity of K and the presence of the set of nodes the properties of IT follow in such a way as to appeal vividly to the geometric imagination. Indeed the yt-curve from the simplicity of its geometric definition and from the intuitive clearness of its properties appears to be fit to replace the classical WEIERSTRASS curve as the standard example of continuous curves having no tangents, since, further, we develop closer knowledge of its progressiveand regressive-tangential properties (II ?? 8, 11). The basal notions of this paper were cominunicated to Chicago colleagues in February and March, 1899. Part II has certain relations of content with the interesting paper by STEINITZ, Stetigk1eit und Di/frentialquotienten, Mathematische Annalen, vol. 52, pp. 58-69, May 1899. These relations are indicated in the foot-note of II?7. STEINITZ determines a class of continuous functions having for no argument a derivative; he does not broach the question of progressive and regressive derivatives. [Jan. 17, 1900. Part 1I has relations of method, but neither of origin nor of contenlt, with the memoir of

A number of years ago several investigators f published independently this theorem: Any screw motion-the most general mechanical motion of space -can be decomposed into the product of two semi-rotations. By a semi-rotation is meant that transformation which consists of rotating space through 180O about a fixed axis 1. For generalizing however it is more convenient to look at this transformation as reflection in a line. From this point of view a semirotation may be defined as that transformation which replaces each point P of space by a point P' such that the line PP' cuts orthogonally a fixed line I and is blsected by it. If we turn to the projective group of three dimensionis we find in it, as a transformation corresponding to the semi-rotation of the mechanical group, the skew reflection which may be defined as follows: A skew reflection is that transIobrmation qf space which replaces each point P by a point P' such that the line PP' cuts each qf two non-coplan ar i nes 1, 1' and is div ded harmonically by them. The lines 1, 1V are called directrices. The purpose of this paper is to ask and answer the question: Is it posstble to decompose the general collineation of space into the product qf a number of skew reflections; and if so, what is the least number o/ skew reflections involved in such a decompos tion? The question will be answered by giving

is unaltered by any infinitesimal transformation of a certain group of the tenth order, and that all stuch transformations which leave the equation unaltered are contained in the above group. It is at once evident that any equation which by a point transformation can be reduced to the above formi will also be unaltered by the transformatiolns of a group of the tenth order of like composition with the above. In the present paper a more general proposition is considered, viz., the form to which linear partial differential equations, of the second order in three independent variables, can be reduced which have the property of beinlg unaltered for some infinitesimal transformations. Such equations form a class by themselves, the potential equation above and equations redulcible to it by point transformation being only particular types of this class; it is here shown that the infinitesimal transformations which leave unaltered the equations of this class form in all cases a finite group of the eleventh order at highest; and certain types are tabulated to which all equations of the class may be reduced.

The point of departure for this paper is furnished by the article of Professor F. MORLEY t in the April number of the T r a n s a c t i o n s. For the convenience of the reader, the notation of that memoir has been as much as possible followed; but it will be perceived that even where, in the opening sections of the present essay, the consequent resemblance to portions of the former rises to the point of identity of formulae, the geometric meaning which underlies these is quite distinct, while in the later portions of the article it has been necessary to find forms of statement unlike those suited to the preceding work. In the last section of the article quoted (p. 184) its author points out that the problems to which that memoir is mainly devoted arise from an initial combination of n lines by pairs, while a grouping by threes, fours, or higher numbers is possible. The present paper is concerned with the case in which the lines are originally grouped in threes, and has for its basic element, analogous to the intersection of two lines as treated in the former article, the center of a circle tangent to the three. I shall briefly indicate a new series of theorems which thus arises from an altered interpretation of formulhe practically identical with those of Professor MORLEY, including analogues to his own theorems upon center-circles, as well as to the chain of propositions associated with the names of STEINER, MIQUEL, KANTOR, and CLIFFORD; I shall then develop a relation by which the new theorems are connected with those of the foregoing case; and finally I shall devote some space to the simpler aspects of the added multiplicity of forms resulting from that character by which the case here treated is chiefly distinguished from the preceding, to wit, the assignment to each line of a definite direction, the reversal of which in any instance, while leaving the original configuration of n lines apparently unaffected, entirely changes the diagram of circles built upon it.

It has been observed by SOPHUS LIE that the stationary motion of a fluid can serve as a perfect picture of a one-parameter group in three variables. So far as I know, neither he nor any of his followers utilized this fact for the purposes of hydrodynamics. It is the purpose of the present paper to do this. One of the advantages gained for hydrodynamics by this standpoint lies in the general conception. But another advantage is, as is always the case when a class of problems is investigated from a new standpoint, that from the group-theoretical point of view, certain special cases are of exceptionial interest, simplicity, and im portance, cases which otherwise would appear difficult and unpromising.

Surfaces enveloped by spheres cutting orthogonally a fixed sphere were first studied by MOUTARDt in the paper: Sur la transformation par rayons vecteurs reciproques, Nouvelles Annal es de Mathematiques, ser. 2, vol. 3, 1864. In the discussion there given, the transformation of space named in the title is all important. The volume of DARBOUX, Sur une class remarquable de courbes et de surfaces, 1873, treats the surfaces and curves of MOUTARD more at length, with especial reference to the case of order 4, the well-known cyclides and bicircular quartics. These surfaces belong to the general class to be studied in this paper, viz.: Surfaces enveloped by spheres belongilng to a linear spherikcal complex. The configuration of o3 spheres, the Kugelcomplex, owes its origin to SOPHUs LIE.* In the surfaces of MOUTARD the complex involved consists of all spheres intersecting a fixed sphere orthogonally. Other special cases are: 10. Complex of all spheres of constant radius. The surface is now either a parallel or tubular surface. 20. Complex of all spheres cutting a fixed plane under constant angle. This case has been treated in a paper by the author, Ont a transformation of Laguerre, Annals of Mathematics, ser. 2, vol. 1, July, 1900. As in the investigations of MOUTARD and DARBOUX the familiar transformation, inversion in a sphere, serves as the main instrumenit, so in the following pages the properties of a more general contact-transformation under which spheres remain spheres, suffice for the derivation of the principal theorems.

The main object of this paper is the determination of all the possible groups whose group of isomorphisms is either the symmetric group of order 6 or the synimetric group of order 24. We shall also determine the infinite system of groups whose group of cogredient isomorphisms is the former of these two symmetric groups. It will be proved that this system includes one and only one group (which is not the direct product of an abelian and a non-abelian group) for every power of 2. It is well known that every simple isomorphism of a group G with itself may be obtained by transforming G by means of operators that transform it into itself.t In what follows we shall generally employ this method of making G simply isomorphic with itself. In a few cases it will be convenient to employ two special methods, which 'we proceed to explain. The first of these two methods may be employed when G contains a subgroup H' which is composed entirely of operators which are selfconjugate under G and which is also simply isomorphic to a quotient group of G with respect to a selfconjugate subgroup which includes H'. In this case we may evidently multiply all of the operators of each one of the various divisions of G with respect to this quotient group by the corresponding operator of H' and thus obtain a simple isomorphisin of G with itself.-To illustrate this method we may employ the direct product G12 of' the symmetric group of order 6 anid an operator s1 of order two. If 'we multiply each of the six operators of G12 which are not contained in its cyclical subgroup of order 6 by s1 we obtain a simple isomorphism of G12 with itself. It is evident that this isomorphism corresponds to the selfconjugate operator of order two in the group of isomorphisms of G12 t It is important to observe that any operator t1 of the group of isomorphisms of G which is obtailned in this manner is selfconjugate under this group of isomorphisms whenever H' is composed of characteristic operators

Of the many theorems contained in STURM'S famous memoir in the first volume of Liouville's Journal (1836), p. 106, two, which I have called the Theorems of Comparison, may be regarded as most fundamental. I have recently shownt how the methods which STURM used for establishing these theorems can be thrown into rigorous form. In the present paper I propose to prove these theorems by a simplert and more direct method. This method was suggested to me by a passage, to which Professor H. BURKHARDT kindly called my attention, in one of D'ALEMBERT' S papers on the vibration of strings.? D'ALEMBERT'S fundamental idea, and indeed all that I here preserve of his method, consists in replacing the linear

without the assumption of the continuity of the derivative f'(Z) on the closed region R bounded by the curve of integration C, and thereby he has laid deeper foundations for the CAUCHY-RIEMANN theory of fuinctions of the complex variable. An abstract of these memoirs is to be found in the Bulletin of this Society for June, 1899, pp. 427-429. GOURSAT set out by a direct process t to evaluate the integral in question. In the present paper, by an indirect process, I prove that the integral has the value 0. The essential elements of the proof are those of GOURSAT'S first paper; by the modification indicated, and by the imposition on the curve C of a certain condition fulfilled by all the usual curves, one avoids the necessity of introducing the lenmma to which GOURSAT'S second paper is devoted. The necessary preliminary definitions and theorems are given in some detail in ? 1, in which connection I refer especially to JORDAN'S Cours d'Analyse, 2d ed., vol. 1, 1893, and to HURWITZ'S address at the Zurich Congress of 1897 entitled.: Uber die Entwickelung der (illyemeinen Theorie der analytischen Functionen in neuer er Zeit (Verhandlungen des iilathematiker Kongresses inZiirich . .; Teubner, 1898). Then in ?? 2 and 3 I state and prove the two

The transformations discussed in the present paper are, like the isogonal and isotomic transformations, particular cases of the general birational quadratic transformation, in which points correspond to points, and lines to conics passing through three fixed points. They seem to possess some interest in connection with the Geometry of the Triangle.

It is well known that the properties of the orthocentre and of the nine-point circle of a triangle may be most symmetrically stated when the triangle and its orthocentre are looked upon as the vertices of a four-point, the opposite sides of which intersect at right angles. This point of view leads naturally to a generalisation of the ninepoint circle, by consideration of any four-point in place of the orthic four-point—a generalisation which was first given in detail by Beltrami in the year 1863; though the theorems involved had been previously stated by T. T. Wilkinson. A number of papers have since been written on the nine-point conic; but they have for the most part merely given Beltrami's results over again, and have generally been written in ignorance of his work. In this paper I propose giving the properties of the nine-point conic from a different point of view, associating them with the triangle instead of the four-point. There are certain advantages belonging to each point of view. If, for instance, we consider a triangle ABC with its orthocentre H as an orthic four-point, any proof that shows that the nine-point circle touches the inscribed (or an escribed) circle of the triangle ABC, will, in general, also show that it touches the inscribed (and escribed) circles of the triangles HCB, CHA and BAH. On the other hand, as the nine-point conic circumscribes the diagonal triangle of the four-point, if the four-point is given, the nine-point conic is definitely determined; whereas, if the triangle be considered, as the fourth vertex of the four-point may be taken arbitrarily, a number of nine-point conics are obtained, touching the same inscribed conic.

On reading Dr Peddie's paper, the following modification of the proof, which avoids summation, occurred to me:— If in Figure 7 we take a point S on the circle BQD such that PQ + PS = 2 a , where a is the radius, and a corresponding point S′ such that PQ′ + PS′ = 2 a , then it is clear by Dr Peddie's construction that the potential at P due to the zone of the spherical surface lying between planes through Q and Q′ perpendicular to BD is given by 2πσ(PQ′−PQ) · a /CP, and is therefore equal to that due to the corresponding zone between S and S′, since

The fact that for two or more real positive quantities there exist three well-known algebraic means, the Arithmetic, the Geometric, and the Harmonic, which stand in a fixed order of magnitude independent of the quantities operated on, suggests the question whether there may not be other algebraic means that stand in a definite order of magnitude with reference to those just named and to one another. The following paper supplies an affirmative answer to the question. The results given in the first section are, so far as I know, novel; some of those in the second section are well known, but I hope some freshness may be apparent in their treatment here.

The subject of this note is that dealt with in Mr Tweedie's paper in the Proceedings , vol. XVII., 33–37, and my only reason for bringing it before the Society is to call attention to a slightly different method of presenting the same order of ideas. The method is that adopted by Peano, Lezioni di Analisi Infinitesimale , vol. I., §23, but as the book is not readily accessible to teachers, there may be some interest in having the method reproduced in our Proceedings . I add one or two remarks.

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SCHEFFERSt has divided all number systems into the quaternion and nonquaternion systems aiid has shown that the n fundamental units of a nonquaternion system may be so chosen that the multiplication table takes a particularly simple form, which is in turn characteristic of the non-quaternion systenms. In this paper I shall show that the choice of the units may be so regulated that the multiplication table becomes still simpler. SCHEFFER'S form, which we shall call the regular Jbrm, has the following characteristic properties: 10. The units are divided into two essentially different classes, the e's and the n's, with the notation

where p alnd q are throughout an interval a cx ' c real alnd colntinuous f unietions of the real variable x . t One of the most important of STURM'S results (Li o u ville's Jourlnal, vol. 1 (1836), p. 106) is that, if y1 and Y2 are linearly independelnt, between two successive roots of one lies one and only olle root of the other. The followilng generalizationi is (implicitly at least) contained in STURM'S paper, alnd from it what I have called STURM'S theorems of comparisonl for a single equation : follow at once. It is my object in the present note to prove this theorem by a simple anid elementary method which makes use only of a silngle property of y1 anid y2, namely that a necessary and sufficient conditioll for their linear dependelnce is that y1ly -2y should vanish at some point of the interval cib. ? The theorem in question may be stated as follows, and wheni it is so stated the method of proof is at once suggested: SUppose that y1 vanishes neither at a nor, at b , and that y2 it does not vaanish at a, satisfies the relation

Of the literature of the geometry of hyperspace that has accumulated in recent years the following papers are cited as having points of contact with the ideas here set forth: CLIFFORD: Prelinminary Sketch of Biquaternions in P r o c e e d i n g s o f t h e London Mathematical Society, vol. 4 (1873), pp. 381-395. CLIFFORD'S theory of parallels in elliptic space is identical with the theory of isoclinal systems of planes in four-dimensional space; namely, planes that pass through a fixed point and make equal dihedral angles with any transversal plane through the same point. (See ?? 30-3 2 of this paper.) CHARLES S. PEIRCE: Reprint of the Linear Associative Algebra of BENJAMIN PEIRCE in the American Journal of Mathematics, vol. 4 (1881). In the foot-note of page 132 attention is called to the fact that in four-dimensional space two planes may be so related to one another that every straight line in the one is perpendicular to every straight line in the other. (See ? 28 (3) of this paper.) I. STRINGHAM: (1) On a Geometrical interpretation of the Linear Bilateral Quaternion Equation; (2) On the Rotation of a Rigid Systenm in Space of Four Dimensions; (3) On the Measure of 4Inclination of two Planes in Space qf Four Dimensions. Papers presented to Section A of the American Association for the Advancement of Science, the first two at the Philadelphia neeting of 1884, the third at the Cleveland meeting of 1888. Abstracts printed in Proceedings of the Association, 1884, pp. 54-56, and privately, 1888. These papers form the nucleus of the present investigation. A. BUTCHHEIM: A Mfemoir on Biqataternions, in the A m e r i c a n J o u r n al

Up to the present time few theorems of a general character for the conivergence of continued fractions with complex elements have been obtained, and these few are of very recent date. In the first section of this paper such theorems upon the subject as are known to the writer are brought together for the purpose of indicating the present state of our knowledge, and the scope of the paper is also explained. Some new criteria for convergence are then deduced in the succeeding sections. The results obtained are summed up in theorems 1-10, which may be read independently of the rest of the paper. The demonstration of these theorems is based upon certain equations, Nos. 3-8, 11, and 12, which seem to be new and of a fundamental character.

The primitive continuous groups of point-transformations in two variables can, by a proper choice of the variables, be transformed into projective groups of the plane, a result LIE obtains after determining the canonical forms of the primitive groups.t This fact canl, however, be established from the general properties of such groups, and its use leads to a new determination of these primitive groups, to show which is the object of this paper. A primitive group will be defined as a group which does not leave invariant a differential equiation of the first ordert Such a group is at least three-parametric, as a two-parametric group possesses a differential invariant of the first order, J say, and therefore an invariant differential equation of the first order, f(J) = constant.?

In this paper I consicler the improper simple definite integrals of HARNACK (1883, 1884). In the introduction I wish to characterize somuewhat clearly the theories of the improper simple and multiple integrals recently given by JORDAN (1894) and STOLZ (1898, 1899), and in this introductory paragraph I summarize the contents of the whole introduction. These theories for the simple integrals have intimate relations with the HARNACK theory. The definition adopted for the multiple integrals is inore exacting than that for the simple ilntegrals. The miiultiple integrals converge or exist (as limits) only absolutely. For the simple integrals we have then two theories, on the one hanld, of the integrals with the milder definition, and, on the other hand, of the integrals with the stronger definition and so with a larger body of properties. The first class of integrals includes the second class of integrals. The HARNACK theory relates to the first and general class of integrals; this theory has not received systellmatic development; however, for the theory of the absolutely conivergent HARNACK integrals this is nlot true, and these initegrals conistitute the second and special class of integrals. I discuss both classes of simple integrals simultaneously and by uniform process; this is made possible by suitable determinations of the definitions; the absolute convergence of the integrals of the seconid class appears only at the conclusion, and hence it is desirable to introduce terms of discriminiation conlnoting the two definitions, the milder ancl the stronger; the terms chosen, "narrow," " broad," connote the geometric form of the definitions, and likewise the fact that the class of narrow integrals lhas a less extensive body of properties than the (included) class of broad integrals. There has been a tendency to do away with the non-absolutely convergent HARNACK integrals; I hope to show that this tendency rests uponi misconceptions.-The tlheory of DE LA VALL-fE POUSSIN (initiated in 1892) is in form distinct from the HARNACK theory and

(2) X= f(t), y =a(t), + f (Q) , z= ry ()+ 8(q)4, where f, a , 3, y, 8 are arbitrary functions of {, subject only to the condition that a8 fly must not vanish identically. The present paper, besides deducing some new theorems, will be mainly concerned with geometrical interpretations. We shall again confine ourselves to the special-case of equations (1) for two reasons. In the first place this will enable us to make use of the concrete results of our former paper, and in the second place we can thus avoid the consideration of configurations in hyperspace. It will not be difficult to generalize our considerations so as to include the general case, if only a space of the proper number of dimensions be employed.

1. Since the announcemnent t by KRONECKER, in 1882, of his now well-known theorem regarding linear relations between the minors of an axisymmetric deterininant various papers t have appeared treating of the subject. Dr. MUIR in his paper of 1888 showed that a similar relation exists between the muinors of a centrosymmetric determinant and in his paper of 1900 he gives the following two theorems: THEOREM A: If , and v be any integers, , being the less, taken fromn the series n, n + 1, n + 2, .., 2n and a, 3, ry, *.., w be what the series becomes when , is removed, and a, /3, ry, * , 4 what it becomes when both are removed; then in connection with any even-ordered determinant 112 2n we have

10. In this paper I wish to define a system of types of improper simple definite integrals, a system embracing in particular the four current types; of the theory of the general type I give at present merely the elements, the methods employed, however, being characteristic. The four current types are compared in ? 1 2-13o. By way of generalization of their diversities the new types arise (1lo-17o). As the desirable basis for the new types I propose (160) an extension of the notion of the proper simple definite integral; this involves likewise an extension of the notions of the four current types. In ? 2 I state in convenient notations a body of elementary properties of the general type of integrals. These properties with two definitional processes of induction developed in ?? 3, 4 serve as the basis for the definition in ? 5 of the system of types of improper integrals related to the (extended) type of proper integrals defined in 160.

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PEANO in Mathesis, vol. 9 (1889), p. 75 and p. 110 seems to have been the first to point out that the identical vanishing of the Wronskian of n functions of a single variable is not in all cases a sufficient condition for the linear dependence of these functions.t At the same time he indicated a case in which it is a sufficient condition, j and suggested the importance of finding other cases of the same sort. Without at first knowing of PEANO'S work, I was recently led to this same question, and found a case not included in PEANO'S in which the identical vanishing of the Wronskian is a sufficient condition.? It is my purpose in the present paper to consider these cases and others of a similar nature. By far the most important case in which the identical vanishing of the Wronskian is a sufficient condition for linear dependence is that in which the ftunietions in question are at every point of a certain region analytic functions, whether of a real or complex variable is, of course, immaterial. This case requires no further treatment here. We shall therefore be concerned exclusively with the case in which the independent variable x is real. This variable we will suppose to be confined to an interval Iwhich may be finite or infinite, and if limited in one or both directions muay or may not conltain the end points. In some of the proofs we shall use a subinterval a x_ c b of I; 11 this subinterval we call I' T Whether the functions are real or complex is immaterial. We use the symbol --to denote an identity, i. e., an equality which holds at every point of the interval we are considering.

It should be noticed that, as a consequence of the assumptions made above, these functions are continuous and have finite first derivatives at every point of the interval (I). The roots of functions of this form were considered by STURMt by a method which, however, yields results distinctly less general than those we shall obtain. The relation of our methods and results to those of STURM will be indicated at the proper places. The third section of the present paper may appear at first sight to be of slight importance, the generalizations of the results of the first two sections which are contained in it being in themselves not very far-reaching. The method used in establishing these generalizations is, however, I believe, a valuable one apart from the special application here made of it. I have indicated one application to other questions in the footnote on p. 435.

The present paper is written to bring together and, in some measure, to add to the results which have been obtained concerning curves L of triple curvature. Equations of motion for systems in a four-dinmensional space have been deduced and used to introduce the notion of an instantaneous plane of rotation. The derivation of the equations of motion is not new but it seemed best for the sake of clearness to retain it. By constructing the principal tetrahedroid at a point of a curve of triple curvature and studying its motion by means of the kinematical equations obtained, geometrical interpretations of the six rotations and also a set of formulae corresponding to the Serret-Frenet formulke for curves of double curvature have been arrived at. These formula have been applied to the study of curves L and, in particular, of the osculating hypersphere and the locus of its centers.

DR E. WARREN'S recent paper onl inheritance in Aphis incidentally draws attention* to the need for determining further correlations between grandparent and offspiing. At the suggestion of Professor Karl Pearson I have worked out two further cases for the inheritance of coat-colour in thoroughbred horses. Using his index of coat-colours for the chief sires I extracted frorn Weatherby's Studbooks the coat-colours of 1000 colts and their paternal grarndsires, and of 1000 fillies and their paternal grandsires. The correlation Tables I. and II. were then formed in the manner described in Pearson and Bramley-Moore's memoirt on inheritance of coat-colour in thoroughbred race-horses. These tables were then reduced to the fourfold division

Introduction.-This paper is in continuation of a previous paper t under nearly the same title. The notation given there is used in this paper with the exception that u, v are here used instead of ,u, v. We are concerned with a doubly infinite system of given curves: (1) f3(Au V) + Af2(u, v) + Bfi(u, v)=O, of which the differential equation is t (2) a du3 + a4 dv3 + a2 dU2 dv + a3du dv2 + a5 (du d2v dvd2u) = 0.

The congruence n_ x2(mod k) is possible, and n is therefore a quadratic residue of k, when n is a quadratic residue of each prime factor of k, so that in order to determilne the possibility of the congruence in all cases we must be able to determine its possibility when k is any prime number. The case k 2 is simple, but when k is an odd prime the problem presents some difficulties, and it has perhaps received more attention than any other in the theory of numbers. LEGENDRE introduced the symbol (n/k) = i 1 E= ny(k-) (mod k), the sign being + or as n is or is not a quadratic residue of the prime number k, and since his time the problem has consisted in determining the sign of (n/k) for any given values of n and k, n being prime to the odd prime k. The method of evaluation, or algorithm, of LEGENDRE, improved by JACOBI, is still the standard solution. It requires the use of the law of quadratic reciprocity formulated by LEGENDRE, though perceived earlier by EULER: theoremafundamentale, as it was called by GAUSS, who first supplied for it a satisfactory demonstration. The derivation of this law has attracted uniusual attention from many mathematicians, eight demonstrations having been prodcuced by GAUSS alone. The chief improvement since the time of JACOBI consists in an observation made independently by SCHERING and KRONECKER4 namely, that "4 GAUSS'S characteristic," ,, is available for the proof of the law of reciprocity when k is not prime. The definition (n/k) (1)', employed by TANNERY in his proof of the usual algorithm, is one of two employed in the present paper, and is herein extended and applied to wider purposes, with only the slightest reference to the law of reciprocity. I find great advantage in substituting for the symbol ,t the broader symbol , (n, k), so as to be able to discuss the function , for different values of n and k, and thereby to develop relations of the fulnctions ,t(n, k) im

If every operator of an abelian group is put into correspondence with its ath power, an isomorphism of the group with itself or with one of its subgroups is obtainled for any integral value of a.t If a is prime to the order of every operator in the group, the resultinlg isomorphismn is simple; otherwise it is multiple. To avoid an unnecessarily cumbrous phrase, let us denote by a-isomorphism any isomlorphism obtained by putting each operator of a group into correspondence with its ath power; and let us say a-holomorphism whenever the resulting isomorphism is simple. It has been shown that the a-holomorphisms of an abelian group G constitute the totality of invariant operators in the group of isomorphisms of G, and that their ilumber is equal to the number of integers less than and prime to the highest order occurring among the operators of G. Every group admits an a-holomorphism, when a 1 (mod n), where m denotes the lowest common multiple of the orders occurring among the operators of the group. The questions naturally arise: (1) Under what conditions do nonabelian groups admit a-holomorphisms other than the identical? (2) What are the properties of the corresponding operators in the group of isornorphisms? The present paper concerns itself with these questions. The writer is indebted to Professor G. A. MILLEPR for suggestions and criticisms during the preparation of this paper.

On the occasion of the GAUSS-WEBER celebration in 1899 HILBERT published an important memoir, Grundlagen der Geometrie, in which he devoted chapter V to the consideration of DESARGUES's theorem. He summarized the results obtained in this chapter as follows :t The necessary and sufficient condition that a plane geometry fulfilling the plane axioms 1 1-2, II, III may be a part of (or set in) a spatial geometry of more than two dimensions fulfllling the axioms I, II, III, is that in the plane geometry Desargues's theorem shall be fulfllled. The proofs of the necessity and of the sufficiency of the condition are given by HILBERT in ? 22 and ?? 24-29 respectively. In ? 23 he proves that DDESARGUES'S theorem is not a consequence of the axioms I 1-2, II, III,t and states (theorem 33) that it cannot be proved even though the axioms IV 1-5 and V be added. His method is to exhibit a non-desarguesian geometry fulfilling the axioms in question. His example is of a somewhat complicated nature, involving in its description the intersections of an ellipse and a system of circles (euclidean) which are defined so that no circle intersects the ellipse in more than two real points. The demonstration that the geometry fulfills the axioms in question, the details of which are not given by HILBERT, depends upon the real solutions of simultaneous quiadratic equations. Moreover, HILBERT's example does not fulfill all of the axioms enumerated, the exception being IV 411 -which, in connection with the definition which precedes it, requires that the angles (h, Ik) and (k, h) shall be congruent, while the angles in HILBERT'S geometry whose vertices are on the ellipse depend upon the order in which their arms are taken for their non-desarguesian congruence relations. HILBERT'S final theorem, stated at the beginning of this paper, does not involve IV 4 and was completely

By properly modifying the set of postulates considered in the preceding paper, we can construct two different sets of postulates such that every assemblage which satisfies either of these new sets will be equivalent to the svstem of positive integers, when a o b = a + b.* In the first set (? 1), postulates 1-5 are left unchanged, while 6 is replaced by a new postulate 6'. In the second set (? 2), postulates 1-3 are retained, while postulates 4, 5 and 6 are replaced by a single postulate, 4". Both of these sets are complete sets of postulates in the sense defined on p. 264, although one contains six postulates and the other only four. A problem is therefore at once suggested, to which no satisfactory answer has as yet been given, viz., " when several complete sets of postulates define the same system, which shall be regarded as the best 9 " By a further modification of the postulates, in which 1-3 are still retained, while 4, 5 and 6 are replaced by 4"' and 5"', we obtain (? 3) a complete set of postulates for the theory of positive rational numbers.

In two papers,t each having the title " On the Continuous Group that is defined by any given Group of Finite Order," BURNSIDE establishes certain results of decided interest and importance. among them being the theoremst of FROBENIUS on the irreducible factors of group-determinants. The object of this paper is the development of the theory of analogous groups in any arbitrary field or domiiain of rationality. In particular, when the field is the general Galois field of order pn, we obtain a doubly-infinite system of finite groups corresponding to each given finite group. An exceptional case not treated here is that of a field having a modulus which is a factor of the order of the given finite group. ? BURNSIDE bases his woik upon several theorems proved by means of the LIE theory of continuous groups. The corresponding theorems for an arbitrary field are here derived by simple rational processes (?? 2, 3). The auxiliary theorems on invariant-factors (the i; Elementartheiler" of WEIERSTRASS) are established in ? 4 by mneans of the canonical form of a linear transformation in any field. The later developments (?? 5-7) run parallel to the corresponding parts of BURNSIDE'S treatment, but include essential modifications. The results find application in the problem of the representation of a given finite group as a group of linear transformations in a given field upon the smallest number of variables. That the introduction of the concept of a field gives rise to a generalization of KLEIN'S normnal problem may be illustrated by the fact that a given group may be represented as a modular group upon a smaller number of variables than is possible for a representation as an algebraic linear group.