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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 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

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.

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.

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.”

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.

§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.

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.

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

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.

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 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.

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

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.

1. SCHROETER'S t classic work on the general cubic leaves little to be desired in point of symmetry and generality. It is nevertheless interesting to build up the theory of the unicursal cubic, the curve being defined as the locus of the intersection of corresponding rays of two projective pencils, one of the first and the other of the second order. This has in fact been done by DRASCH. t The following discussion, based likewise on this definition, is materially simplified by the use of the properties of the point designated in ? 7 by E. Incidentally the investigation brings to light a remarkable one-to-one correspoindence between the points of the plane and the line elements on the cubic. 2. The locus described above has at least one and at inost three points in common with any line in the plane. We assume the truth of this theorem, a proof of which may be found in the eleventh chapter of REYE's Geometrie der Lage. Notations.-Throughout this paper we shall use the following notations: The pencil of the first order will be denoted by s, its center by S, and its rays by a, 6, c, etc. The pencil of the second order (and also the conic enveloped by it) will be denoted by K, and the rays by a, /3, 'y, etc. The cubic itself will be denoted by C. 3. THEOREM.-NO point of the cubic C lies within the conic K. 4. THEOREM.-The cubic C touches the conic K in at least one point, and at most in three. To prove this take S', a point on K, for the center of a pencil s' of the first order perspective to K. This pencil generates with s a conic which cuts KC in at least one and at most three other points besides S'. These are easily seen to be points on C.

BURNSIDE has considered the groups of order pm (p being any prime) which contain an invariant cyclic subgroup of order pv,-2.t Those in which a cyclic subgroup of order pm-2 iS transformed into itself by an abelian group of order pm-1 anld of type (nz 2, 1) have also been studied. t The main object of the present paper is to deternmine the remaining groups of order pm (i > 4 when p is o4d, and n > 5 when p = 2) which contain a cyclic subgroup of order ptm-2. As such a subgroup must be transformed into itself by pn-1 operators of the group of order pm, ? each of these groups which does not come under one of the cases already considered must include the non-abelian group H of order pr-n which tontains p cyclic subgroups of order pm-2 . The group of isomorphisms (I) of H is of order pt-m' (p 1) and contains invariant operators of order pm3 when p is odd and of order pm-4 when p = 2. Let P1 and P2 represent two independent operators of H whose orders are pm2 and p respectively and let PI3 = p3. Suppose also that P2 has been so chosen that P-'P P2 = P3 P1. The group of cogredient isomorphisnms (12) of H is of order p2 and of type (1, 1). When p is odd I includes an operator (tl) of order p such that t Pt = P2P1, t1 P2t = P2. Since t1 permutes the p cyclic subgroups of order pfm-2 in H cyclically, while some of the operators of 12 are commutative with each operator of only one of these subgroups, the group generated by 12 and t, is the non-abelian group of * Presented to the Society (Chicago) January 3, 1902. Received for publication December 2, 1901. t BURNSIDF, Theory of groups of finite order, 1897, p. 75. tTransactions of the American Mathematical Society, vol. 2 (1901), p. 259. ?BURNSIDE, Proceedings of the London Mathematical Society, vol. 26 (1895), p, 209. Also, FROBENIUS, Berliner Sitzungsberichte (1895), p. 173. 11 With respect to the non-cyclic group of order p2, when p is odd, or p3, when p is even, all the operators of a division have the same pth power or p2th power respectively. Cf. B u 1 etin of the American,Mathematical Society, vol. 7 (1901), p. 350; J. W. YOUNG, Transactions of the American Mathematical Society, vol. 3 (1902), p. 189.

1. Denote by G the group of the equation upon which depends the determi. nation of the curves of order n 3 having simple contact at 1 n ( n -3 ) points with a given curve C of order n having no double points. The case in which n is odd was discussed in the former paper (Transactions, January, 1902) and G was shown to be a subgroup of the group defined by the invariants 43, 04, , , * *, the latter group being holoedrically isomorphic with the first hypoabelian grouip on 2p indices with coefficients taken modulo 2. For n even, G is contained in the group H defined by the invariants 04' 069 * with even subscripts. JORDAN has shown (Traite, pp. 229-242) that H is holoedrically isomorphic with the abelian linear group A on 2p indices with coefficients taken modulo 2. The object of the present paper is to establish the latter theorem by a short, elementary proof, which makes no use of the abstract substitutions [al, 1 ., p, p1] of JORDAN, and which exhibits explicitly the correspondence t between the substitutions of the isomorphic groups.

It is known that the holomorph (K) of a cyclic group (G) is a complete group and that its commutator subgroup is G whenever the order (g) of G is odd. WAThen g > 2 is even, the commutator subgroup of K is the subgroup of G whose order is g/2, and K is never complete.t The main object of this paper is to determine additional useful properties of Kwhose subgroups are of such fundamental importance. In particular, we shall determine the orders of all the operators of Kand some of the properties of its group of isomnorphisms when g is even. It will be observed that the generalized FERMAT'S theorem follows directly from some of the properties of the group of isomorphisms of G. Let g 2a0palpa2 pa,. (r1 2, ., p being any odd prime numbers) and let Ko Xj, K2, K2 , K7 represent the holomorphs of the cyclic groups (GO, G, G .. , G,) of orders 2ao,plal ,pa, ,pam respectively. As K is evidently the direct product of these holomorphs j the orders of all the operators of K can be directly obtained from the orders of the operators in these holomorphs. We shall first consider the operators of Ko (6aO > 2 ) whose order is known to be 22ao1 The group of isomorphisms (I,,) of G. is known to be the direct product of an operator of order two and a cyclic group which may be so chosen that it is composed of all the operators of ]o which transform ? an operator of order 4 in G into itself. ? The orders of the two independent generators (S, 82) of ]o are therefore 2ao-2 and 2 respectively.** We shall first determine the orders of all the

* Presented to the Society, under a slightly different title, December 29, 1902. Received for publication February 2, 1903. [For an abstract of an unpublished paper on the same subject, presented to the Society by the writer on April 26, 1902, see Bulletin of the American Mathematical Society, vol. 8 (1901-02), p. 371.] tCf. D. HILBERT, Ueber den Zahlbegriff, Jahresbericht der deutscben Mathematiker-Vereinigung, vol. 8 (1900), pp. 180-184. -The axioms for real numbers enumerated by HILBERT in this note include many redundancies, and no attempt is made to prove the uniqueness of the system which they define. (Cf. Theorem Il below.) The main interest of the paper lies in his new Axiom der Vollstiindigkeit, which, together with the axiom of Archimedes, replaces the usual axiom of continuity. The other axioms are given also in his Grundlagen der Geometrie (1899), ? 13. Complete sets of postulates for particular classes of real quantities (positive integral, all integral, positive real, positive rational) can be found in the following papers: G. PEANO, Sul coneeito di numero, Rivista di Matematica, vol. 1 (1891), pp. 87-102, 256-267; Formulaire de Mathematiques, vol.3 (1901), pp. 39-44.-Here onlythepositive integers, or the positive integers with zero, are considered. An account of these postulates is given in the Bulletin of the American Mathematical Society, vol. 9 (1902-03), pp. 41-46. They were first published in a short Latin monograph by PEANO, entitled Arithtmetices principia nova mnethodo e.xposita, Turin (1889). A. PADOA: 1) Essai d'une theorie algebrique des nombres entiers, pr-ecede d'une introduction loypque Aunetheorie deductive quelconque, Bibliotheque du congres international de philosop h i e, Paris, 1900, vol. 3 (published in 1901 ), pp. 309-365; 2) Numeri interi relativi, R i v i s ta di Matematica, vol. 7 (1901), pp. 73-84; 3) Un nouveau systeme irreductible de postulats poutr I'algebre, Compte rendu du deuxiieme congr6s international des math6mati ciens, Paris, 1900 (published in 1902), pp. 249-256.-The second of these papers is an ideographical translation of the first; the third reproduces the principal results. E. V. HUNTINGTON: 1) A complete set of postulatees for the theory of absolute continuous maignitude; 2) Complete sets of postulats for the theories of positive integral and positive rational numbers; Transactions, vol. 3 (1902), pp. 264-279, 280-284.-The first of these papers will be cited below under the title: Magnitudes. [ln the fifth line line of postulate 5, p. 267, the reader is requested to change "one and only one element A " to: at least one element A -a typographical correction which does not involve any further alteration in the paper.] Among the other works which may be consulted in this connection are: H. B. FINE, The numsber system of algebra treated theoretically and historically, Boston (1891). 0. STOLZ und J. A. GMEINER, Theoretische Arithmetik, Leipzig (1901-02). Two sections of this work have now appeared. G. PEANO, Aritmetica generale e algebra elementare, Turin (pp. vii + 144, 1902). 358

1. In 1878 JORDAN proved a theorem concerning linear homogeneous groups which may be enunciated as follows: Every such group G in n variables has an abelian self-conjugate subgroup F of orderf, and the order of G is )Xf, where X is inferior to a fixed number which depends only upon n.t The proof of this theorem is quite remarkable, the more so since the limit of X is not determined. The writer of the present article is not aware of any attempts that have been made since 1878 to find a limit to Xaside from the cases n 2, 3 4 -and he presents herewith some theorems which, in connection with some given by him in these Transactions, vol. 4 (1903), pp. 387-397, can be utilized to determine a number that X must divide, at least in the case of "primitive" groups. However, the chief object of the present paper is not this, but rather the presentation of some methods and theorems that are useful in the construction of the groups considered. As an illustration, the primitive groups in three variables are enumnerated at the end of the paper. The technical terms and phrases defined in the paper On the order of linear homogeneous groups, these T r a n s a c t i o n s, vol. 4, already referred to, will be retained here. As the present article is considered a continuation of this earlier paper -to which we shall hereafter refer by Linear groups I we shall begin with Theorem 8, meaning by the Theorems 1-7 those of Linear groups I. Unless otherwise stated, the substitutions used are linear and homogeneous in the variables concerned, and of determinant 1.

* 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

The groups in which every two conjugate operators are cominutative have recently been considered by BURNSIDE.t In the first section of the present paper, the converse limitation is imposed on a group of operations. It is assumed that every two commutative operations are conjugate, provided neither is identity, t and the groups which are possible under this hypothesis are determined. It results that the group of order 2 ? and the symmetric group of order 6 are the only grou.ps which have the property in question. In sections 2-6, somewhat similar but smaller limitations are imposed on the group. The condition is imposed in sections 2-5 that every two commutative operations of the same order are conjugate, and in section 6 that every two commutative operations (identity excluded) are so related that each of them is conjugate to some power of the other. Some of the chief properties of the groups which are possible under these limitations are derived. The sections 7-8 deal with problems closely related to the precedilng. If it is assumed that a certain simple relation exists between the number X of complete sets of conjugate operations, and the number n of distinct prime factors in the order of the group, certain commutative operations are conjugate. Much use is made of this fact in showing what groups are possible under the given hypothesis. The symbol Ga (S1= 1), S2, S3, * , * g will be used to represent the group of order g under consideration, and p1, P2, * p Pn to represent distinct primes in ascending order of magnitude so that gp a1pj2. pa-.

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. The presenit paper is devoted first to the proof of a theorem fundamental in the construction of imprimnitive linear homogeneous groups t in a given niumber of variables. Then, by means of this and earlier theorems given by the author on the subject of linear groups, t JORDAN'S theorem, j to the effect that the order of a linear homogeneous group G in n variables is of the form Xf, where f is the order of an abelian self-conjugate subgroup of G, and X is less than a fixed number depending onlv upon n, is proved for imprimitive groups, a number being found that X must divide. Finally, the principal imprimitive collineation-groups in 4 variables are found and their generating substitutions given. THEOREM. Either an imprinzitive linear homogeneous group G can be written in monomial tfrm, ? or the n variables of the group can be so selected that they fall into k sets qf imiprinit'ivity of m variables each (n = kin), permuted according to a permutation-group K in k letters, which group is transitive (in the sense of transitivity of permutation-groups). The subgroup ( G') Of G, corresponding to the subgroup (K') of K which leaves one letter unchanged, is primitive (in the sense used in linear homogeneous groups) in the m variables of the set corresponding to the letter that K' leaves unchanged. In order that G may be transitive (as a linear homogeneous group, i. e., " irreducible"), it is plainly necessary that its sets of imprimitivity contain the same number of variables, and that the permnutation-group K, permuting these sets, is transitive (as a permutation-group). We shall prove that, if the

The totality formed by all the operators of any group (G) which are common to all the invarianlt subgroups of prinme index (p) constitutes a characteristic subgroup, and the corresponding quotient group is the abelian group of order pA and of type (1, 1, 1, t. ) The number of the invariant subgroupsof index p is therefore p-/p 1. The given totality includes all the operators of G which are pth powers, and it is composed of such operators whenever G is abelian. In this case X is clearly equal to the number of the invariants in the Sylow subgroup of order fr. containied in G. This fact follows also directly from the theory of reciprocal groups, since pA is the order of the subgroup generated by all the operators of order p contained in G. For instance, the abelian group C& contains only one subgroup of index p whenever its Sylow subgroup of order p)2l, mn> 0, is cyclic, it contains p + 1 such subgroups whenever this Sylow subgroup involves two invariants, p2 + ) + I whenlever' there are three invariants, etc. When G is non-abelian the determination of X is much more difficult. Its value can clearly Inot exceed the value of X for a Sylow subgroup of order ptm containied in G, but it may be less. Hence it is of fundamental importance to deterimine the values of X for the groups of order p"', and we shall assume that this is the order of G in what follows. Instead of determiniing the value of X for given types of groups, it seems much more desirable to determine the possible types of groups when the value of X is given. This problem soon becomes very difficult. When X= 1, G is cyclic; and when X = n, G is the abelian group of type (1, 1, 1, *. ). These extreme cases relate to fundanmental groups whose elementary properties are well known. The case when X = m 1 includes the Hamiltonian groups. All the groups which belong to this case have recently been determined. : The main object of the present paper is the study of an interesting category of groups which belong to the case when

Every group besides the abelian group of order 2a and of type (1, 1, 1, ***) contains at least two operators which are squares of other operators in the group. If there are only two such operators they constitute an invariant subgroup and the corresponding quotient group contains only one operator which is a square. All the groups which satisfy this condition have recently been determined.t The present paper is devoted to a complete determination of the groups involving just three operators which are squares. As the idenitity is its own square such a group ( G) contains only two operators besides the identity which have the property in question. If the order (g) of G is divisible by an odd number, this number is 3 and G includes only one subgroup of this order, since every operator of odd order is the square of some power of itself. As suich a group cannot contain an operator of order 4 and its order is 3.2a, it niust be the direct product of the abelian grouip of order 2a and of type (1 , 1, 1, * * *), and the cycle group of order 3; or one of its subgroups of order 3.2a-1 must be of this type. In the latter case, G is evidently the direct product of the symmetric group of order 6 and the abelian group of order 2a-1 and of type ( 1, 1, 1, *) . In what follows, these trivial cases will not be considered. Hence g will always be of the form 2a and G contains only operators of orders two and four in addition to the identity. Moreover, each of the two operators (t,, t2) which are squares but are not the identity is of order two and hence it is the square of some operators of order four. We shall first prove that t1, t2 are invariant under G.

If a grotup (G) of order 1pcontains only one subgroup of order p, a > 0, it is known to be cyclic unless both p = 2 and c = 1.4 In this special case there are two possible groups whenever m> 2. The number of cyclic subgroups of order pa in G is divisible by p whenever G is non-cyclic and p> 2. I In the present paper we shall consider the possible types of G when it is assumed that there are just p cyclic subgrouips of order pa in G. That is, we shall consider the totality of groups of order prn which satisfy the condition that each group contains exactly p cyclic subgroups of order pa. It is evident that

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

In the first part of this paper there are found the numbers of points, lines, etc., in a finite projective geometry of n dimensions. The substance of this has already been worked out by O. Veblen and W. H. Bussey. The second part is concerned with the arrangements of the numbers representing the points in a finite projective plane desarguesian geometry.

One of the most important systems of groups is that which is composed of all the groups which may be defined by the orders of two generators (s1, s2) and the order of their product. It is known that these three orders determine a finite group only when one of them is unity, when two of them are equal to 2, or when one is 2 while the other two are one of the following three pairs of numbers: 3, 3; 3, 4; 3, 5.t The first of these sets of orders define a cyclic group and may be regarded as trivial. When the order of two among the three operators si, 82, 81 92 is equal to 2, the group { s1, 82 }, generated by s, , 82, is the dihedral rotation group whose order is twice that of the third operator. This useful category has recently been generalized under the headling, The groups generated by two operators which have a common square4t This generalized category could also be defined as the set of groups generated by two operators such that the square of one of them is equal to the square of their product. The object of the present paper is to complete this kind of generalization for the dihedral rotation group and to extend it to the other groups of genus zero. The resulting groups have a two-fold interest in view of their close contact with the important system of groups of genus zero and their elementary structures. It is believed that a complete list of these groups will prove very useful in many investigations. Our object is to study all the groups which result when one of the three conditions which may be used to define a group of genus zero is preserved while the other two are replaced by a single one of an elementary type. For instance, the dihedral rotation groups are defined by the conditions ?

In August, 1904, SCHEFFERS presented to the Interllatiollal Mathematical Collgress in Heidelberg a papel elltitled Ueber IsoyonalDuevelr, A equttangentialk7trven, tGnd 1com7?tesee Zahlen. This is published ill the official account of the Ccungress t and again in much greater detail, but under tlle san}e title, in the M a t h e ln a t i s c h e A n n a l e n of the followingr year. t In these papers he has discussed a type of transformation of the plane which he llas called ;; equilong " alld which carries straight lirles into straight lines, keepirlg illvariant the distallce betweell the points of contact of a line with ally two of its envelopes. He has exhibited § the beautiful duality existing between these transformations and those of tlle collformal groul; evell as the latter depend upon an arbitrary functioll of the usual complex variable, so are the equilollg transformations of the plane expressed by an arbitrary fllnction of a complex variable of a differerlt type. There seems no a )riori reasoll why this close analogy of conformal alld equilong transformations should not stlbsist in three or more dimellsiolls. Such is lOt, however, the case. Irl December, 1904, SIUDY stated, in a short article, Uebee7 mehe ere Probleme der Geoiiletrie, die dem Problem der konfornlewa Abbildung analog sind: 11 ;;Alle diese Aufgaben? derell beide letzte Analoga zum Problem der konformen Abbildullg sind, lassen sich durch explizite Formeln losen, . . . Man findet in allen F>illen llnelldliche Gruppell. Die Mitteilung der ztlgehorigen Formeln, die nicht in der Art mit hyperkomplexen Glossen susamrllenzuhangen scheinell . . . wollen wir bis zu einer ausfuhrlicheren Darstellllng aufschieben." Since these words were written, three years ago, nothing further has been published concerning equilollt, trallsformations of space. The obJect of the presellt paper, which, incidentally, was originally written in ignorallce of Professor STUDY'S work, is to give an analytic discussion of the problem and to demonstrate the remarkable theorem which he indicates, namely, that

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

The present note aims to exhibit some elementary relations between wellknown methods of finding the primitive roots of a number and the properties of the cyclic group. Incidentally we arrive at a fundamental theorem relating to the primitive roots of a special class of numbers. A corollary of this theorem gives the primitive roots of all the prime numbers of the form 2p + 1, p being a prime, while it has been customary in the works on the theory of numbers to devote two theorems to the primitive roots of such prime numbers.* The note has close contact with the paper published in this JOURNAL under the title "Some Relations between Number Theory and Group Theory" and may be regarded as a continuation of this article.t It is known that the necessary and sufficient condition that a number g has primitive roots is that the cyclic group G of order g has a cyclic group of isomorphisms I. The numbers which are less than g and prime to it may be made to correspond to the operators of I, unity corresponding to the identity, in such a way that I and the group formed by these numbers, when they are combined by multiplication and the products reduced with respect to modulus g, are simply isomorphic. The orders of the operators of I are the indices of the exponents to which the corresponding. numbers belong. In particular, g -1 corresponds to the operator of order 2 and the primitive roots of g correspond to the operators of highest order in L Hence the method of finding the primitive roots of a number is equivalent to that of finding the operators of highest order in a cyclic group. One of the most instructive methods for finding all the primitive roots of g is analogous to the method known as the "Sieve of Eratosthenes" for finding

Cubic varieties of 003 poillts in space of four dimensions have been extensively studied by CASTELNUOVO t and by SEGRE, t showing the close connection between the section of the enveloping cone by ordinary space (apparent contour) and the well knowll congruences of the second and third orders, with their focal surfaces. The treatment is exclusively synthetic and the methods there employed do not lend themselves readily to the discussion of particular cases. In the following paper I wish to call attention to two particular projections of the variety having nine double points. This variety is given a three page discussion in SEGRE'S first paper, containing most of the results of Articles 1, 2 3, 4, 6 of the present paper. The result of Art. 5 is stated by CASTELNUOVO in a footnote without proof, but no use is there made of it. The results in the remaining articles are new. I have made extensive use of all the papers cited, but as I shall proceed analytically, the development will be along diflerellt lines, and no knowledge of their contents will be assumed. @ Let r 1Z2z3 + \z4z5z6 ° define a cubic variety in space of four dimensions 4, where Ewt O. § Every plane defined by wi ° ( i 1, 2, 3 ) and wk-O (k 4, 5, 6 ) will lie entirely on r. These nine planes may be designated by av. Any two of these having neither subscript in common will intersect in just one point which is a double point on r. Since two values of i and of k have been used to define each point, it ulay be denoted by A with the remaining pair as subscripts. It follows immediately that r contains nine double ?oints; through every double point pass four planes and ill every plane lie four double points.

t. The present paper is devoted to a geometrical application of certain kinds of syzygies connected with binary forms of odd order. If we represent the forms with which we deal by sets of points on a rational curve, the application is to the problem of determining other curves of the same type whose intersections with the given curve are either original sets themselves, or new sets having simple invariant relations to the old. The type of rational curve which we shall use is that having but one multiple pOillt, which then, for a curve of order m, must be an ( m 1 )-fold point, and equivalent to 21(m-1 )(m 2) double points. Such a cllrve we shall designate for brevity as a JONQUIARES curve, and the symbol J(7n) will be used in referring to it. We proceed to consider the intersections distinct from the ulultiple point itself of two such curves having the same multiple point. Let the order of the second curve be n; then through the multiple point pass R 1 branches. The further points of -intersection are then mn ( xn 1 ) ( n 1 ) xn + n 1 in number. On the other hand, if the first curve is given, the number of conditions which the second can fulfil is 2n. This may be seen readily by choosing the multiple point as a vertex of the reference triangle, whereupon the equation of the J(n) reduces to 2n + 1 terms. The ulost interesting case occurs when the llumber of free intersections, m + n 1, is greater by one than the number of conditions that can be imposed on the second curve; that is, when n nb 2. In this case we have an involution, Ilm-3, set up on the first curve. Interpreting two well-known theorems relating to involutions, we have the following geometrical properties, which will be found usefu] in what follows: t) There eacist on a J(m) eacactly 2rn 3 points having the property that a J(m-2)s with the same rnultiple point as the J(m)s can be drawn to have (2m 3 )-point contact in each with the J(m) a 2) These 2m 3 points all lie on a J(m-2) with the same multiple point.

Ever since I first became acquainted with Wilkinson's method of establishing the existence of the nine-points circle of a triangle (see Mackay's “Euclid,” Appendix to Bk. IV., Prop. 2, the lettering of which I have followed in the first three sections of this paper), its simple and fundamental character has pleased me. I propose to point out first that this method yields probably the most elementary proof of the concurrence of the perpendiculars from the vertices, and then, after restating the investigation of the nine-points circle, to sketch some generalizations.

To the theory of multiply transitive groups Bochert has contributed formulae for the upper limit of the degree of a group when the number of letters displaced by a substitution in it is known, f But »for simply transitive primitive groups a corresponding theorem of such elegance does not exist. J However, in the present paper is offered a theorem with some claims to simplicity. It may be stated thus : If a simply transitive primitive group contains a substitution of prime order p and of degree pq ( q less than 2p 43 ), its degree is not greater than the larger of the two numbers qp 4q2 — ?, %q2 —p2Also when the given substitution of prime order has more than 2p 42 and less than p2 cycles or when p is 2 or 3, our results are capable of fairly concise expression ; but when q exceeds p2—1 and p is greater than 3 the corresponding upper limit is unfortunately a more complicated function of p and q. The subject matter of this study is not restricted to the simply transitive primitive groups, but it is proposed to find an upper limit for the degree of some transitive subgroup of a primitive group known to contain a substitution of order p on qp letters. It seems unnecessary to state that a simply transitive primitive group has no transitive subgroup of a lower degree.

The occurrence of rational curves in pairs is a well-known fact: thus, given a rational curve pp, of order n, in a space of p dimensions, there is uniquely determined, to within a collineation, a curve pn_p-l n of order n, in a space of n-p-1 dimensions, by requiring all hyperplane sections of either curve to be apolar to the hyperplane sections of the other.t We call two curves associated in this way conXugate curves. If the rational curve pp is regarded as the projection of the norm-curve pn in a space Sn of n dimensions, from an Sn_p_ln the interpretation of this fact is immediate. An Sn_l in Stl meets Pn in n points which may be regarded as given by a binary form of order n: dually, a point of Sn determines on pn a set of n points, which may be given by a second binary form. The condition of apolarity of the two forms is precisely the condition of incidence of point and Sn--l * All Sn lns having n-point contact with ptl meet Sn_p_l in the hyperplanes of a curve rn_p_l of class n. The curves obtainable by projection from S7t_ and section by Sn_l are conjugate curves. In this paper we shall deal with the case n = 5, p-2, the rational plane quintic. If our curve r5 is given parametrically by

The chief purpose of this paper, carried out in its second part, is the determination of numerical limits for certain constants, hitherto undetermined, which figure in the principal theorems of the first two parts (Abschnitte) of the author's thesis,t concerning the degree of approximation to a given continuous function f (x) that can be attained uniformly in an interval by a polynomial of the nth degree in x, or for all values of x by a trigonometric sum of the nth order. By a " trigonometric sum of the nth order at most " is meant an expression of the form

The list of the primitive groups of the first thirteen classes prepared by JORDAN has long been known to be defective in the case of class 12; in fact, JORDAN states explicitly that the calculations for that case, having been gone through but once, may very well contain errors.* In spite of the considerable advances in substitution theory in recent years, the complete a priori determination of all the groups of this class still involves a good deal of labor. In order therefore not to intrude too far upon the space of this journal, the writer avoids a redetermination of the primitive groups of class 12 on less than 21 letters.1 In a series of memoirs and in his lectures Professor MILLER has gone over this ground without use of the list of the groups of class 12. He has thus checked the results of MiSS MARTIN, of JORDAN, and of Mliss BENNETT on the degrees 18, 19, and 20, respectively, according to whiclh there is no primnitive gr'oup of class 12 on either of these three degrees. The method here followed will be much the same as that used by the author in treating the classes 6, 8, and 10, with the advantage of having at hand the processes employed in the paper entitled "On the Limit of the Degree of Primitive Groups." This last paper, as well as that "On Mlultiply Transitive Groups," and that "On the Order of Primitive Groups," will be used freely without specific reference.t In addition to the 25 primitive groups of class 12 of degree less than 18, there are four others, one of degree 27, two of degree 28, and one of degree 36.

1. Itf n elements 1, 2, 3, *** n can be distributed in triads in such a way that every pair of elements appears in one and only one triad, the totality of triads forms a triple-system of n elements. Reissf has shown that it is possible to form a triple-system of n elements provided n is of the form 6m + 1 or 6m + 3. DiiEerent methods for constructing triple-systems have been given by Reiss,f Netto,: HeiEter,§ and E. H. Moore;ll but methods for testing the non-congruency of these systems when formed are lacking. The group of substitutions that transform a triple-system into itself has hitherto been adopted as the abstract mark of the system, and ZulauflT has shown, by a consideration of the groups, that the four systems on 13 elements of Kirkman, Reiss, De Vries, and Netto are reducible to two incongruent systems whose groups are diiEerent. In the present paper it is shown however that for n = 15 two incongruent triple-systems, 515, may have the same group. By a simple process non-congruent triple-systems are constructed. These are used to illustrate a new method of comparison, by means of a new sort of abstract marks. This method requires no knowledge of the group but incidentally facilitates its determination. No exhaustive determination of every 515 which may be obtained by this process is here undertaken, but the 24 systems discussed include 12 apparently not hitherto constructed.

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. The general theorem underlying the subject of this paper is as follows:—If through c1, c2, c3, … points on AB, a side of ΔABC, rays be drawn from two vertices O1, O2, the former meeting AC in b1, b2b3, …, and the latter meeting BC in a1, a2, a3, …, then the lines a1b2, a2b2, … envelope a conic touching the sides AC and BC. This follows since the ranges a1a2a3 …, b1b2b3 …, are homographic.

In the geometry based on the infinite group of conformal transformations of the plane (or on the equivalent theory of analytic functions of one complex variable), two types of problems must be carefully distinguished: those relating to regions and those relating to curves or arcs. Two regions of the plane are equivalent when there exists a conformal representation of the one on the other, the representation to be regular at every interior point. The classic Riemann theory shows that all simply connected regions are equivalent, any one being convertible into say the unit circle. The difficulties connected with the behavior of the boundary (which may be a Jordan curve or a more general point set) have been cleared up in the recent papers of Osgood, Study, and Caratheodory, Logically simpler problems relating to curves or arcs have received very scant attention. Two arcs are equivalent provided the one can be converted into the other by a conformal transformation, the transformation to be regular at the points of the arcs, and therefore in some (unspecified) regions including the arcs in their interiors. The main problem hitherto discussed by the writer in his papers on conformal geometry is the invariant theory of curvilinear angles.t Such a configuration (which may be designated also as an analytic angle) consists of two arcs through a common point, both arcs being real, analvtic, and regular at the vertex.: In this theory it is necessary to distinguish rational and irrational angles. If 0 denotes the magnitude of the angle (invariant of first order), then when 0/uis rational there exists a unique conformal invariant

In a paper with the above title published in these Transactions* there appeared a statement that "B(x) becomes infinite as x approaches x". In a letter, dated October 18, 1921, Professor Hans Hahn called attention to the fact that the proof of this statement, as implied by the context, although not explicitly mentioned, was insufficient, because it depended upon the nonvanishing of the quadratic form ,5 jij Cj cj, which had been proved to be definite by the aid of the fact that Z(x) + 0, and this condition does not hold at x1. It is the purpose of this note to supply a proof of the statement quoted above, upon which the validity of the concluding theorem of the earlier paper depends. I acknowledge my indebtedness to Professor Hahn for having pointed out the need for this supplemenitary proof. Starting with the definition of B(x) as given on page 435 and putting rij (x) = ikP (xI) Z(kj) (x), we have B (x) = j rij cj cj/Z. We shall show first that not every rij can vanish at x1 of the same order as Z(x). For, suppose that Z(x) = (x-xl)s Z1 (x), Z, (x1) + 0. If now every ij (x) had the factor (xx)s, then, since IRk((Xi)l Ot it would follow that every Z(Jk) (x) had it also. From the definition of the functions Z(Jk) (x), we derive the following system of equations

The continued fractions treated in this paper are of the general formwhere s0, s1, s2, … are real integers (positive, negative, or zero). An arbitrary real number can, of course, be developed in such a fraction in an infinite variety of ways. The continued fractions discussed here have a number of striking properties and present numerous contrasts with the ordinary continued fraction usually employed.