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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.

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.

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

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.

The equations of all analytic minimum curves may be written in terms of a complex parameter s and an arbitrary analytic function F(s). The curve will be algebraic when and only when F(s) is an algebraic function, and then the rank, order and class of the curve can be easily expressed as the orders of three functions, #(s), 4(s) and X(s), (see ? 1), of the general form f [s, F(s), F'(s), F"(s)]. It is the purpose of this paper to give the developments in series of

In the article t A ne?w method of determining the differential parameters and invariants of qutadraitic differential quantics I have shown that the application of a certain symbolic method leads very readily to the formation of expressions remaining invariant with respect to the transformation of quadratic differential quantics. The presentation in that article was only a preliminary one and the work practically confined to the case of two independent variables. In my paper 4 Invariants and covariants of quadratic differential quantics qf n variables a more complete treatment was intended and the investigation applied throughout to the case of n variables, leaving aside, however, simultaneous invariant forms of more than one quantic. The present paper contains in ?? 1-6 and ? 8 essentially the content of the last mentioned paper; the greater parts of ? 5 and ? 8, and all the remaining articles are new, in particular the extensive use of covariantive differentiation.

1. The group of all quinary orthogonal substitutions of determinant unity in the GF [p" ], p> 2, has a subgroup OQ of index 2 which is simple. The latter is simply isomorphic with the quotient-group Q of the quaternary abelian group anid the group composed of the identity and the substitution which merely changes the sign of each variable. The difficulty in the employment of Q is apparent, while for OQ there is unfortunately no known practical t criterion to distinguish its substitutions from the remaining quinary orthogonal substitutions. While the abelian form seems best adapted to the determination j of the subgroups of order a power of p, the orthogonal form is found to possess advantages in the study of the subgroups of order a power of 2. The case pn = 81 ? 3, namely, that in which 2 is a not-square in the GtF [pn ], is here treated on account of its simplicity (comnpare in particular ?? 2, 4, 5, 22) and in view of the applications to be made in subsequent papers in these T r an s a c t i o n s to the determination of all the subgroups when pn= 3 and pn = 5. There is established the remarkable result that, independent of the values of p and n (such that pn is of the form 814 3), the group Q contains the same number of distinct sets of conjugate subgroups of order each power of 2, one set of representatives serving for every OQ (compare the diagrammatic summary in ? 21, the group notations being given in earlier sections in display formule separately numbered). Moreover, except for the subgroups of orders 2, 4, and certain types of order 8, the order of the largest subgroup of OQ in which a group of order a power of 2 is self-conjugate is independent of p and n.

* Presented to the Society at the San Francisco meeting, April 30, 1904. Received for publication July 12, 1904. t BURNSIDE, On the Representation of a Group of Finite Order as an Irreducible Group of Linear Substitutions, etc., Proceedings of the London Mathematical Society, November, 1903, pp. 117-123. The quantities Xi have been called weights by the author in two papers published in these Transactions, vol. 4 (1903), p. 387, and vol. 5 (1904), p. 310. t Cf. equation (true for transitive groups) given by BURNSIDE, P r o c e e d i n g s o f t h e L o ndon Mathematical Society, NMarch, 1903, p. 122, bottom. 461

It has recelntly been proved that in every non-cyclic group of order pt", p being an odd prime, the number of cyclic subgroups of order pg, /3> 1, is always a multiple of p, and that this number is of the form 1 + p + kp2 when /3 = 1.t From this it follows alrnost directly that the number of cyclic subgroups of order pg in any group (G) is always of the form kp whenever the Sylow subgroups of order ptm in G are non-cyclic, and that the number of subgroups of order p in such a G is always of the form 1 + p + kp2. j ' hen p = 2, both of these theorems have exceptions. The present paper is devoted to an exhaustive study of the exceptions of the former theorem. Since the cyclic groups are so elementary we shall confine our attelntion to the noln-cyclic groups of order 2m. Moreover, every group of even order contains an odd

1. It would seem that the most effective method of determining all the subgroups of order a multiple of p of a l'inear group in the Galois field of order pn is that based upon a complete knowledge of the subgroups of order a power of p. This method has proved successful for the ternary groupsl and, as I will show on another occasion, also for the quaternary groups. The present investigation proceeds far enough to give a clear insight into the nature of the simple laws pervading the subject. It is hoped that the results are capable of extension by induction to all powers of p. To indicate the difficulty of this step, it may be remarked that its completion would give the means of deriving at once an explicit list of all groups of order a power of a prime, and simultaneously all the subgroups of each. A second aim of the paper was to furnish data for the problem of the determination of all m-ary groups for low values of m. Following Lie, I write (A, B) for the commutator A-B1AB. I employ the usual notation Bj for the transformation which alters only i, replacing it by 6; + afj. We have the simple relation ?

An abelian group may be defined by the property that, in an automorphism of the group, more than three fourths its operators may be placed in a one to one correspondence with their inverses.t It may be of interest to know the groups possessing the property that five eighths or more of the operators may be inade to correspond to their inverses. The principal object of this paper, however, is to establish the following elementary theorem (I) and to illustrate the use that may be made of it in certain problems. THEOREM I. A group that has two invariant subgroulps with nothing in common, but the identity can be set up as a nlzutiple isornorphisn?, between two groups of lower order. Let a group ( G ) of order k, k x have the two invariant subgroups K1 and K, of order k1 and k2 respectively. If K, and K2 have only the identity in comiimon, every operator of K, is commutative with every operator of K2. It may be assumed that G is not merely the direct product of K, ancl K2. Let 1, r2 r31 ** , be the operators of K-1 and 1 s2, s, ..*, those of K2. Now

BAIRE in his important thesis t has given a number of interesting theorems concerning pointwise discontinuous functions. For their demonstration he employs the concept of semi(i. e., upper or lower) continuity. t However necessary the concept may be for subsequent portions of his investigation, its introduction is not needed for the particular theorems referred to. When the unnecessary element is removed, the principles of their demonstration, although very different in foi!m, become equivalent to those used by OSGOOD ? in the derivatioin of his theorem relating to the convergence of series of continuous functions. (Cf. ? 5 below.) The object of the following paper is, in part, to establish the theorems without going so far as to introduce the notion of upper continuity, thus reducing the proof to somewhat lower terms. It is found that BAIRE has restricted f(x, y) more than is necessary in the following theorem: Jff( x, y) is continuous in x and y separately and has a partial derivative af/ax at every point of

n/a

In this paper I shall consider only nanlber systenls with UllitS el u e> , eX X whose constants of multiplications tj lie in BS a domain of rationality determined by an arbitrary aggregation of sealars whieh form a closed systenl with respect to addition subtraction multiplication and division5 such systems are saidto belowag to B. The vluits e e2 * en Will be so chosen that there shall exist between them no linear lelation with coeffleients in R. In gs3leral I shall consider ollly numbers

1. The systems of notation hitherto used by writers on the theory of correlation are somewhat unsatisfactory when many variables are involved. In the present paper a new notation is proposed which is simple, definite, and quite general, thus very greatly facilitating the treatment of the subject. The majority of the results given in the sequel were, in fact, first suggested by the notation itself. 2. Let x 1 x 2 ... x n denote deviations in the values of the n variables from their respective arithmetic means. Then the regression equation may be written :— x 1 = b 12.34... n x 2 + b 13.24... n x 3 + ... + b 1 n .23... n -1 x n

I have given elsewhere a necessary and sufficient condition that certain categories of abstract groups of finite order be simply isomorphie with irreducible homogeneous linear groups in the domain of all real and complex numbers.t In the present paper I establish a two-fold generalization of this result by showing that the same condition applies to all groups of finite order and to an arbitrary domain. The proof for the necessity of this condition rests upon a conclusion drawn from Theorem II of SCHUR'S lMeue.Begrundung der Cheorie der Gr?ppencharaktere.t. Suppose that we have a hologeneous linear group S of finite order whose coefficients belong to n (an arbitrary finite field or an arbitrary domain) and which is irreducible in . If P is a given substitution on the same / variables

The methods now adopted in the teaching of elementary geometry have made it most important that the teacher should have clear views upon the nature of the problems which are soluble by Euclid's methods: that is, with the aid of the ruler and compass only. With this general question I have dealt in another place. In this paper I give a short account of the argument by means of which Gauss proved that the only regular polygons of n sides, which can be constructed by Euclid's methods, are those in which n, when broken up into prime factors, takes the formm1, m2, m3,…mr being all different.

The ideal elements of projective geometly are usually introduced by means of the parallel and congruence axioms. The idea of defining the ideal elements without the assumption of the parallel axiom is due to KLEIN.+ It was developed by PASCH; :L by SCHUR; § by BONOLA; || and by VEBLEN.1| I11 all of these developments a three-dimensional geometry is assumed. The problem of defining the ideal elements in a plane geometry satisfying only order relations is closely connected with the probleln of finding the necessary alld sufficient condition that a plane may be a part of a three-space in which the axioms of order are satisfied. This condition is stated by HILBERT** to be the validity of the Desargues theorem in the p]ane. The Desargues theorenl may be proved in a three-dimensional space satisfying only order relations, but can not be proved in the corresponding plane geollletry, without additional assumptions, e. g., of the parallel and congruence axioms. In this paper plane axioms of order will be assumed in the form given them by VEBLEN, the undefined elements being taken as the point, and a relation among points, called order. The first eight axioms are identical, except for notation, with his. Another axiom, one of closure, is then introdueed, limiting the set of lvoints considered to a plane. Two more axiollls are then introduced, foruls of the Desargues theorem, and of its converse, in terms of the set of points satisfying only order relations.

If in the ordinary theory of rational numbers we consider a composite illteger m as modulus, alld if from among the classes of congruent integers with lespect to that modulus we select those which are prime to the lnodtllus, they form a well-knowll multiplicative group, which has been called by WEBER (Alyebra, vol. 2, 2d edition, p. 60), the most ialportant example of a finite abelian group. In the more general theory of numbers in an algebraic field we nsay in a corresponding manner take as modulus a composite ideal, which illeludes as a special case a composite principal ideal, that is, an integer in the field, and if we regard a11 those integers of the field which are congruent to one another with respect to the modulus as forming a class, and if we select those classes whose integers are prime to the modulus, they also will form a finite abeliall groupt under multiplication. The investigation of the nature of this group is the object of the present paper. I shall confine my attention, however, to a quadratic number-field, and shall determine the structure of the group of classes of congruent quadratic integers with respect to any composite ideal modulus whatever. Several distinct cases arise depending on the nature of the prime ideal factors of the modulus; for every case I shall find a complete system of independent generators of the group. Exactly as in the simpler theory of rational numbers it will appear that the solution of the problem depends essentially on the case in whicls the modulus is a prime-power ideal, that is, a power of a prime ideal. The most important {:ase, however, is probably that in which the modulus is a rational principal ideal or in otller words a rational integer; therefore a separate discussion will be given of this case. Allother interesting case is that in which the group is

Our purpose in this paper is first to discuss nonlinear operators and nonlinear projections in Banach spaces which are related to the resolvents of $m$-accretive operators and maximal monotone operators. Some of these operators in Banach spaces are new. Next, we discuss some properties for such nonlinear operators and nonlinear projections in Banach spaces. Further, using these properties, we prove strong convergence theorems by hybrid methods for nonlinear operators with equilibrium problems in Banach spaces.

In 1858, D. Bierens de Haan published as "Vierde Deel" of the Verhandelingen der Koninklijke Akademie van Wetenschappen (Amsterdam) his "Tables d'Int6grales Definies." In this volume there are 7300 formulas, and these are accompanied by complete bibliographical references. De Haan then undertook the revision of these formulas, and the consideration of the underlying theory of Definite Integrals. Accordingly, in 1862, his "Expos6 de la Theorie des Proprietes, des Formules de Transfornmation, et des Methodes d'Evaluation des Integrales D6finies" was issued as Volume VIII in the same Verhandelingen. In this is included all his critical work of the intervening four years. "Nouvelles Tables d'Integrales Definies par D. Bierens de Haan" were published in 1867 (Leide). These tables contain 8339 formulas, of which 4200 come from the original tables, 2620 from the " Expose " and the remainder from notes published at various times. In this paper it is proposed to examine from the modern rigorous standpoint certain evaluations in the "Nouvelles Tables," and the theorems of the "Expose" on which they depend. In nearly every case the formulas here considered will involve the principal value integral.

It is the purpose of this paper to demonstrate three theorems and two corollaries, whoseformal statement will be preceded by a brief explanation. The treatment is suggested bv recent papers of LEsEsGuE,t in which corresponding problems are discussed for the case of trigonometric series. A part of the work of LEBESGUE is reproduced at length in my thesis,: and where the developments of the present paper are closely parallel to those in the trigonometric case I shall occasionally refer the reader to one of those other presentations.

Suppose X X is a metric space, and Y Y is a complete metric space. In this paper a sufficient condition is given to insure that a sequence of continuous functions from X X into Y Y converge to an embedding from X X into Y Y .

In a previous paper (q.v.) by Mr W. L. Marr and the present writer, it was shown that, in accordance with Morley's Theorem, the angles A+2pπ, B+2pπ, C+2rπ of the triangles ABC be trisected, the three groups of six lines at the vertices give rise to 27 triangles DEF, the biangular coordinates of D with respect to BC being (B/3+2qπ/3, C/3+2rπ/3) or (βq, γr), and similarly for and F with respect to CA and AB.

Since Klein promulgated his famous Erlangen Programmet it has been known that the various types of geometry are such that each is characterized by a group of transformations. In view of the importance of the concept of transformation in nearly all mathematics and perhaps especially in geometry, geometers may properly seek to develop the various types of geometry in terms of point and transformation. For euclidean geometry this has been done by Pieri.:t This paper is devoted to a similar treatment of general projective geometry.§ One would naturally lay such postulates on the system of transformations so as to make the system form the group associated with the geometry. This was the scheme that Pieri used. His postulates make his transformations form the group of motions. In general projective geometry, however, this method is not necessary. If we are given the group of all projective traDsformations we can deduce the geometry from it but it will be shown in the sequel that we can also do that from a properly chosen semi-group belonging to that group. Our basis, to repeat, is a class of undefined elements called points and a class of undefined functions on point to pointll or transformations called collineations. For notation we will use small Roman letters to designate

The object of this paper is to study the groups formed by the residue classes of a certain type of Kronecker modular system and some closely related generalizations of well-known theorems in number theory. The type of modular system to be studied is of the form 9) = (mn, mn-1, , Mnl, m). Here m, defined by (ml, m2, ***, Mk), is an ideal in the algebraic domain Q of degree k. Each term mi, i = 1, 2, *.., n, belongs to the domain of integrity of Qi = ( Q, xl, x2, * , xi), and is defined by the fundamental system ((1t/, t'2j), ***, i,t')). The various 1(i), j = 1, 2, *.., jI, are rational integral functions of xi with coefficients that are in turn rational integral functions of xl, x2, ... , xj_i, with coefficients that are algebraic integers in Q . In every case the coefficient of the highest power of xi in each of the 't') shall be equal to 1. We shall see later that the developments of this paper also apply to modular systems where the last restriction here cited is omitted, being replaced by another admitting more systems, these new systems in every case being equivalent to a system in the standard form as here defined. Any expression that fulfills all of the conditions placed upon each {() with the possible exception of the last one, we shall call a polynomial, and no other expression shall be so designated. This definition includes all of the algebraic integers of Q. Throughout this paper we shall deal exclusively with polynomials as here defined. The first part of this paper will contain the introduction with the necessary definitions and a discussion concerning the factoring of the system 9). The second section will then be devoted to setting up necessary and sufficient conditions that a set of residue classes belonging to 9) form a group when taken modulo 91. In the third section we shall study the structure of such a group with respect to groups belonging to certain modular factors of 9), besides

1. Object and results. In this paper I define for every function f (x), bounded and summablet on an interval a x b, or a measurable subset thereof, two numbers, the measurable upper and lower bounds of f (x) on (a, b). These numbers are analogous to the extrema extremorum of a function which is continuous on (a, b), with which in fact they coincide when f (x) is continuous. I show that the measurable bounds can be determined by means of an enumerable set of constants (the " momental co1stants ") defined by the Lebesgue integrals

The analytic basis for any projective theory of congruences, in which the lines of the congruence are defined either by a pair of points or a pair of planes, consists of the invariant theory of a completely integrable system of four homogeneous linear partial differential equations with two dependent and two independent variables, two of the equations of the system being of the first order, and two of the second order. If the developables of the congruence are known, this system of differential equations can be written in a very simple form.t But the determination of the developables of a congruence requires the integration of two partial differential equations of the first order, and it seems highly desirable to possess a theory which will be immediately applicable to any congruence, whether its developables can actually be found explicitly or not. The considerations made by G. M. Greent show that the existing theory can actually be modified so as to cover all such cases. However there are various ways in which this can be done and the various methods which might be used are not all equally desirable. Green himself has indicated one such method for the theory of congruences.? But Green only indicated in a general way what was to be done without actually working out a complete theory. The great disadvantage which this particular theory would have, as compared with the one to which we are devoting this paper, is that only the final results would be of interest. The various types of invariants corresponding to the transformations of certain subgroups would

Various definitions of simple continuous arcs and closed curves have been given.t The definitions of arcs usually contain the requirement that the point-set in question should be bounded. In attempting to prove that every interval t of an open curve as defined in a recent papert is a simple continuous arc, while I found it easy to prove that t satisfies all the other requirements of Janiszewski's definition (modified as indicated below) it was only by a rather lengthy and complicated argument that I succeeded in proving that it satisfies the requirement of boundedness. In Lennes' definition the requirement of boundedness is superfluous.? However I found it difficult to prove that t satisfies a certain one of the other requirements of this definition, namely that the point-set in question should contain no proper connected subset that contains both A and B. In the present paper I will give a definition l of a simple continuous arc which stipulates neither that the set 111 should be bounded nor that it should contain no proper connected subset containing both A and B. I will show that, in a euclidean space of two dimensions, every point-set that satisfies this definition is an arc in the sense of Jordan. It is easy to prove? that every interval of an open curve satisfies this definition.

without changing the conjugate net under consideration. Bianchi also proved that the property of isothermal conjugacy is of a projective character.t That is, if an isothermally conjugate net is subjected to any projective transformation, the resulting net will again be isothermally conjugate. But Bianchi did not furnish any geometric interpretation of the analytic conditions (3) which serve to define such systems. Moreover, although the importance of this notion was becoming more and more apparent, because of a steadily increasing body of theorems which made use of it, no serious attempt seems to have been made to discover its true significance until 1915, when the author of the present paper discovered an algebraic relation, between certain completely interpreted projective in

1. Statement of the problem. Given two algebraic curves, C (xl, x2, x3) C C(x) = 0 of genus p in the plane (x), and C'(x') = 0 of genus p' in the plane (x'). Suppose that to a point (y) on C correspond n' points (y') on C', and that to a point (y') on C' correspond n points (y) on C. The two curves C, C' are then said to be in (in, n') correspondence. It is the purpose of this paper to give some methods of constructing curves having such correspondences, and of obtaining the equations which define them. For certain positions of the point (y), two of the n' images on C' may coincide. Such a point is called a branch-point, and the image point that is counted twice is called a coincidence. If the number of branch-points on C is denoted by 'j, and on C' by -q', then we have by Zeuthen's formula

has the smallest possible value, when m and n are given. The writer has recently proved the existence and uniqueness of Tmn (x) for any positive integral value of n and for any value of m i 1, whether integral or not, t and has given some indication of the behavior of Tmn (x), for fixed n, as m becomes infinite.+ The first purpose of the present paper is to discuss the convergence of Tmn (x) toward the valuef (x), when m is held fast and n is allowed to become infinite. It is found to be a sufficient condition for uniform convergence that lims=o co 3()/'l = 0, where co (8) is the modulus of continuity? of the function f ( x ),that is, the maximum of if ( x') -f (x" ) I for x' x"I ! c . The proof makes use of Bernstein's theorem on the derivative of a trigonometric sum. For m = 1, the condition as stated would require that f (x ) be a constant, but in this case it is sufficient that f (x) have a continuous derivative. One would be inclined to expect that convergence would be more likely for large than for small values of m, and the result obtained is favorable to this view, as far as it goes. The theorem is less general, however, even for large

The purpose of this note is to establish the truth of the same proposition in the case that m = 1, for which the proof is considerably less simple. The function f (x) in the earlier paper was, more generally, a linear combination, with constant coefficients, of an arbitrarily given set of linearly independent continuous functions. The existence of at least one minimizing function is proved with the same degree of generality here. The argument involves no new difficulty, and in fact is slightly simpler than for m > 1. For the proof of uniqueness, however, ? (x) is essentially a polynomial as indicated,t though a corresponding treatment could be given for other approximating functions, such as finite trigonometric sums, which are similarly determined by their roots. 2. Existence of an approximating function. Let po (x), pi (x), p, pn (x) be n + 1 functions of x, continuous throughout the interval a _ x _ b, and linearly independent in this interval. Let

* Presented to the Society, September 7, 1922. t Regarding the term "Boolean" cf. the paper by H. M. Sheffer, A set of five independent postulates for Boolean algebras, these T r a n s a c t i o n s, vol. 14 (1913). t Ernst Schroder, Vorlesungen uiber die Algebra der Logik, Leipzig, 1890-95 (3 volumes). ? Eugen Muller, Schroder's Abriss der Algebra der Logik, Leipzig, 1909, 1910. ** Following Schroder and others I say that a Boolean entity c lies "between" a and b, if a is wholly contained in c and c in b (or the same conditions with a and b interchanged). Josiah Royce, following Kempe, means by the statement c lies "between" a and b, that ab is wholly contained in c and c in a + b; cf. Josiah Royce, The relation of the principles of logic to the foundations of geometry, these T r a n s a c t i o n s, vol. 6 (1905). The former usage follows the analogy between the "inclusion-relation" and the "less-than relation." The latter, however, while at first a little surprising, is really the better. It includes the former whenever a is wholly contained in b (or vice versa). tt The symbol

A Study of the questions on this subject set in the recent geometry papers of the London Intermediate Examinations will show that there is considerable danger that what should be the chief aim in the subject may be subordinated to the subsidiary aim of affording practice in evaluating and plotting results.The chief aim should be, I think, a careful study of those important curves in Physics and in Higher Mathematics which best lend themselves to trigonometrical rather than to algebraic treatment. In the actual plotting of these curves much varied practice in the use of tables will, of course, be obtained; but this should be considered of distinctly minor importance.

In a former papert the author communicated a number of properties of every real function f (x) ; these were stated in terms of the successive saltus functions associated with a given function. The present paper makes no use of the saltus functions, and the new properties are direct qualifications of f (x). Since f (x) is entirely unrestricted, except, of course, that it is defined t and therefore finite for every real x-these qualifications are consequences of nothing else than that f (x) is a function. A new light is thus thrown upon the nature of a function. The new properties are of two types, descriptive and metric; the former are concerned with density, and the latter with measure (Lebesgue). For the sake of greater concreteness of exposition, we shall discuss, for the most part, planar sets and real functions of two real variables.

In a. previous papert I pointed out the existence of two operations with respect to each of which the elements of a boolean algebra form an abelian group. If we denote the logical sum of two elements a, b by a + b, theii logical product by a b, and the negative of an element a by a', then the two operations in question are given by ab' + a'b, ab + a'b'. In the present paper I determine all the operations with respect to which the elements of a boolean algebra form a group in general and an abelian group in particular. Postulates for groups.t A class K of elements a, b, c, . . . is a group with respect to an operation 0 if the following two conditions are satisfied: P1. (aOb)Oc = aO (bOc), whenever a, b, c, a0b, b0c, aO(bOc) are elements of K. P2. For any two elements a, b, in K there exists an element x such that aox = b. The group is abelian if the following condition also is satisfied: P3. aob = bOa, whenever a, b, b Oa are elements of K. Determination of group operations. We shall have all the operations of a boolean algebra with respect to which the elements form a group if we determine for groups in general all the boolean operations which have the properties P1, P2, and for abelian groups, all the operations which have the properties P1, P2, Ps. I proceed to effect this determination. If f(x, y) is any determinate function of two elements x, y of a boolean algebra, then f(X, y) = f(1, 1)xy+f(1, O)xy'+f(O, 1)x'y+f(O, O)x'y'

One of the most important of recent theorems in analysis is a theorem due to S. Bernstein, which may be stated as follows: If T, (x) is a trigonometric sum of order n, the maximum of whose absolute value does not exceed L, then the maximum of the absolute value of the derivative Tn (x) does not exceed nL. Bernsteint proved the corresponding theorem for polynomials first, and from it obtained the theorem for the trigonometric case. His conclusion was that IT.(x)l could not be so great as 2nL. Various proofs were given by later writers,+ leading to the simplified statement which appears above. The simplest proof was discovered independently by Marcel Riess? and de la Vallee Poussin.11 The purpose of this paper is to prove the corresponding theorem for Sturm-Liouville sums: The maximum of the absolute value of the derivative of a Sturm-Liouville sutm of order n(n ? 1) can not exceed np M, wvhere M is the maximum of the absolute value of the sium itself, and p is independent of n and of the coefficients in the sum. The proof to be given here is similar to one which de la Vallee Poussin?T

The purpose of the present paper is to exhibit the "complete existential theory" (in the sense of E. H. Moore+) of each of these sets. This requires the discussion, in the usual way, of 26 = 64 examples for each of the sets (1)-(8), and 27= 128 examples for each of the sets (9)-(11). The results show that sets (1)-(10) are comrpletely independent while set (11) is not. In the case of sets (1), (2), (3), (5), (6), (7), which happen to be the sets which do not contain either postulate 3 or postulate 7, the necessary examples are given in terms of a class K containing only four elements. In the case of sets (4), (8), (9), (10), and (11), some of the examples require the use of a class K containing five elements. These five-element

in which the number of postulates is reduced to five. Moreover, the new postulate 9 itself is easier to remember and more convenient to handle than any of the other postulates 1-8. The addition of this new postulate makes desirable an extension of the discussion of the earlier paper so as to include all thirteen of the basic postulates; and this extension has been made in the present paper. Finally, the postulates of the new set (12) are shown to be completely independent in the sense of E. H. Moore. (In regard to the other sets, a

Every power partial isometry (p.p.i.) in the Calkin algebra lifts to a p.p.i. in B(4). An element u in a C* algebra is a power partial isometry (p.p.i.) if ul is a partial isometry for every integer n > 1. This notion was introduced by Halmos and Wallen [5], and they characterized all p.p.i.'s in B(?). In a penetrating study [6], Herrero classifies p.p.i.'s up to their unitary orbits and up to unitary equivalence modulo the compacts. In this note, we show that every p.p.i. in the Calkin algebra lifts to a p.p.i. in B()). This is done by using a few straightforward computations involving the theory of C* extensions. Some of the results of [6] follow from this method as well. In this paper, Hilbert spaces are always separable. The ideal of compact operators will be denoted by K. The canonical quotient map of B()) onto the Calkin algebra B())/K is denoted by ir. Two elements 5 and t of the Calkin algebra are (strongly) compalent ( -t) if there is a unitary U in B()) so that t = ir(U)s7r(U*). They are weakly compalent ( -w t) if there is a unitary u in B())/K so that t = U5U*. For p.p.i.'s, these two notions may differ. Given a nuclear C* algebra A, Ext A denotes the group of extensions of K by A modulo compalence [1, 3, 8] and Extw A denotes the quotient of Ext A modulo weak compalence. We use little more than the fact that when 0 J -J :/,7-*0 is an exact sequence, and Ext J = 0 = Ext AI/J, then Ext A = 0. The first step is to classify p.p.i.'s up to algebraic equivalence by computing C*(T) for all p.p.i.'s. That is, given two p.p.i.'s S and T, when is there a C* isomorphism p of C*(S) onto C*(T) such that o(S) = T? For convenience, this will be written S T. Let Nk be the nilpotent Jordan cell of order k acting on a k-dimensional Hilbert space, and let S be the unilateral shift. If a belongs to No U {w}, let A(a) be the direct sum of a copies of A acting on 3((a), the direct sum of a copies of 91. By [5], every p.p.i. T represented on a Hilbert space can be decomposed as

Section I of the following paper, though using the postulational method, is motivated by the consideration of classes of vectors, on a finite range PI (1, 2, ...* n), whose elements belong to a general division algebra or, as we shall say, number system. Section II deals only with vectors oni a finite range. Section I is also of use as giviing a general basis preliminiary to tlle more intensive study of (a) classes of vectors on a general range, (b) number systems over a division numbersystem; that is, to the initiation of a theory analogous to that of an "algebra over a field," where the field is replaced by an associative division number system. Notation. Throughout the paper certain logical notationst will be used as follows: logical idenitity t logical diversity definitional identity definitional identity between statements implies is equivalent to .3. such that 31 there exists is unique, used before the element which is unique: thus, I a means a is unique. and U or not . : .: :: etc. punctuation signs; the principal implication of a sentence has its signi accompanied by the largest number of periods, thus a :): b .). c is a statement that a implies that (b implies c) whereas a.). b :): c states that the implication a implies b, implies the fact c. We may also use punctuation to show continued implication, thus a.). b .). c means a.). b and b .). c.

This paper is a study of a new method of enumeration of the partitions of multipartite numbers.Incidentally an algebraic function, which is derived from the repetitional exponents of partitions of unipartite numbers, presents itself. The generating function which enumerates the partitions of unipartite numbers is expressible in terms of these functions and finds in such expression its fullest connection with the divisors of numbers. There are also similarly derived functions connected directly with bipartite, tripartite, etc. numbers. It has not been necessary to study these for the purposes of this paper.

vanishes are known to be infinite in number for a general value of n. Their importance in mathematical physics has led to their calculation for positive integral values of n. They may also be regarded as branches of an infinitely many branched function of a complex variable. From this point of view they have been studied but little, and even for the case of real values of n few results are known. For example, it has been proved that if n > -1 the roots are all real, and that each root is an increasing function of n if n>O. From the recursion relations between the functions Jf(z) for consecutive values of n a set of polynomials is derived which has been considered incidentally in the theory. Hurwitz t made a systematic study of these polynomials regarded as functions of z. These polynomials are also polynomials in n but have not been examined as such. In what follows some properties showing their character as functions of n are deduced, and it is found that these properties are susceptible of some applications. In particular, it is shown that the roots of Jf(z) =0 are increasing functions of n when n lies between -1 and 0, a result that cannot be deduced from Poisson's integral formula employed by Schlafli in deriving the similar conclusion for n > 0. 2. In this paper the notation and results of Hurwitz are used and the following extract gives what is essential for its comprehension. If 1 z

[s] is the greatest integer in s; w is an integer in the field QC(a), a= e2 irFr; [1: r] is the integer i in the relation ri -1 (mod p), and if a fraction f/g occurs as an exponent of a, then that exponent is the integer u in the relation fJgu (mod p). In the present paper I shall develop a new line of attack on the Last Theorem by the introduction of power characters in the field Q(e2i1:Ph), h prime to p, in connection with (2). 1. Let n be a prime 0 0 or 1 (mod p) and suppose that xyz 0 (mod n); then (3) xn-1 yn-1 = 0 (mod n). If : is a primitive (n1)th root of unity then in the field Q3() we have (n) = qlq2. . . qlAn-l) where the q's are distinct prime ideals, and So(n-l) is the indicator of n-1. We may take as one of the q's the ideal q = (3 r,n)

A class K is said to be a field with respect to a pair of operations A, 0 if A, 0 play in K the same role which the operations +, X play in the class of rational numbers. My aim in this paper is to determine in any boolean algebra all pairs of operations expressible in terms of addition, multiplication, and negation for which the elements are a field. Postulates for fields. The conditions that make a class K a field with respect to a pair of operations A, 0 are t given by the following seven postulates :t

near a point solution xi ai (i 1, 1 *, n) when certain conditions are satisfied. In the present paper these conditions are replaced by less stringent ones; the methods of proof of certain existence theorems are very similar to those employed in the first paper, and these proofs are given here in an abbreviated forin. In addition, the asymptotic properties of certain trajectories are discussed by an extension of the methods of Bohl. On account of the more complicated form of certain quadratic forms which occur here, it has been convenient to leave undetermined certain constants which are determined explicitly in the first paper; this procedure, together with the algebraic method of transforming the canonical equations, reduces to a small amount the results common to both papers.

In a forthcoming paper by E. V. Huntingtont a number of sets of postulates or determining conditions for the type of order called "separation of point-pairs" have been given. These sets are selected from a list of general properties which characterize reversible order on a closed line, and each of the sets is shown to imply all the others so that the several selections are equivalent. It is to be shown in the present paper that sets of postulates for separation of point-pairs are characterized by a property which is closely analogous to ordinary completeness. A class of propositional functions will be defined, to be called general laws,tto which any member of a set of postulates for this type of order belongs, and it will be shown that such sets are sufficient to determine the truth or falsity of any general law which can be constructed on the base K, R4, the base for the set. This is a question of deducibility; one or the other of every pair of mutually contradictory general laws on K, R4 must be deducible. The question of deducibility arises here in the following manner. It seems to be true from inductive considerations that each of these sets is a sufficient characterization of the type of order in question and thus that the theorems which follow from any one of them might be held to be exhaustive of the general properties which are understood to attach to systems involving separation of point-pairs. Any such set might then be taken as a set of defining properties for separation of point-pairs in the sense that any theorem which is commonly understood to hold for this type of order is implied by the postulates and no theorem which is recognized as not belonging to this type of order does follow from the postulates. In this sense

In the present paper I offer a new method for solving some questions regarding the distribution of residues and non-residues of powers. The difference between the present method and the methods developed in my papers of 1916-18 lies in its entirely elementary character. The chief idea of this method consists of two different ways of calculating the number of numbers of the form a(ax+b), where a ranges over all the different least positive residues of numbers congruent to Axv (mod p) and where x independently of a assumes all values 0, 1, * , h -1 (h

is uniform, the solution is a rational function, or a rational function of eIIz (u constant), or an elliptic function. It is part of a famous theorem of Weierstrass that every uniform function with an algebraic addition theorem is of one of the three types just enumerated. It is a result of Picard that if two meromorphic functions are algebraically related to one another, either the relation is of genus zero, in which case the two functions are rational functions of a single meromorphic function, or the relation is of genus unity, in which case the two functions are elliptic functions of a single integral function.t The present paper treats a problem which might easily be suggested by any of the above theorems. We seek all meromorphic functions f(z) such that, for some linear X(z), an algebraic relation