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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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. The nature of the individual circuits (or complete branches) that make up a curve of order n has not received very much attention. VON STAUDT (1847) distinguished between odd and even circuits; MOEBIUS (1852, Ueber die Grundformen der Linien der dritter Ordnung) by projection on to a sphere from the centre, brought out even more clearly the distinction, since the odd circuit is represented on the sphere by a single line, the even circuit by two distinct lines. CAYLEY (1865, On Quartic Curves, Collected Papers, vol. 5, op. 361) applied the Moebius projection to the quartic, and thus proved that not only the individual circuits of any non-singular quartic, but all the circuits at once, can be projected into the finite part of the plane. He pointed out that this conclusion does not hold as regards the non-singular sextic; there exists such a curve, composed of a single circuit, which cannot be projected into the finite part of the plane. On a sphere, this circuit is not confined to one hemisphere. In the concluding paragraph the remark is made that a quartic with one node may consist of two odd circuits; such a quartic is not the projection of any finite curve. CLIFFORD (1870, Synthetic Proof of Miquel's Theorem) mentioned another sextic that is not confined to one hemisphere. ZEUTHEN, in his classical discussion of quartic curves (1874, Mathematische Annalen, vol. 7) emphasized the distinction between odd and even circuits, and proved the important theorem that in one, and only one, of the two regions t into which a plane is divided by a nonsingular even circuit, odd circuits can lie. He showed (p. 426) that a quartic can be composed of a single even circuit, cutting itself twice, of such a form that every straight line meets it in at least two real points. There may be an additional double point of any kind, or the curve may have a simple circuit, necessarily a simple oval. A quartic of this character, with p = 0, can be obtained by quadric inversion from a conic which separates one of the three fundamental points from the other two. If such a circuit be projected on to the sphere, it will be found that the two lines by which it is represented interlace.

*Presented to the Society in the form of two papers February 22, 1902, and (San Francisco) May 3, 1902. Received for publication March 14, 1902. t To facilitate references to my previous papers on this subject, I shall quote them in the text by the initial words of their titles, viz.: Invariants, Geometry, Reciprocal Systems They have all been published in these T ra n sacti o n s: Invariants, vol. 2, p. 1 ; Geometry, vol. 2, p. 343; Reciprocal Systems, vol. 3, p. 60. 423

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

The investigations on the foundation of geometry of RIEMANN, HELMHOLTZ and LIE have as a basal notion the distance between two points of space, and the analytic representation of this distance plays an important r6le. A point in space is determined by means of three independent coordinates; and the distance between two points is a certain function of their six coordinates. Rigid bodies are used and the continuous motions of such, determined by the continuous variation of the co6rdinates of their points. This notion of three independent and continuous co6rdinates of a point is, in reality, not simple, as has been pointed out by LIE and others. In an article entitled Ein Beitrag zur Jllannigfaltigkeitslehre, Borchhardt's Journal, vol. 84 (1877), G. CANTOR describes a space in which the co6rdinates of a point are neither independent nor continuous. It does not seem easy, however, to determine the distance between two points of space if the co6rdinates of a point are neither independent nor continuous. It would not be evident, a priori, that certain loci (as spheres and circles) exist, defined by means of certain equations in the coordinates of their points. Such loci are used in the processes employed by HELMHOLTZ and LIE, and the question arises whether their axioms involving such loci and the motion of points in space, etc., should be modified or replaced by others. One such new axiom, used in the following paper, refers to distance-relations, i. e., it is assumed that certain definite relations connect the mutual distances between any n + 2 different points in space of n dimensions.t One notices that the assumption, used by HELMHOLTZ and LIE and their successors, that the distance between two points in space is a certain function of the

The theory of invariants originally confined itself to forms involving a single set of homogeneous variables; but recent investigations, geometric as well as algebraic, have proved the importance of the study of forms in any number of sets of variables. In passing from the theory of the simple to the theory of the multiple forms, an entirely new feature presents itself: in the latter case the linear transformations which are fundamental in the definition of invariants may be the same for all the variables or they may be distinct, i. e., the sets of variables involved may be cogredient or digredient. Multiple forms thus have two distinct invariant theories, a cogredient and a digredient. The object of this paper is to study the relations between these two theories in the case of forms involving any number of binary variables. Geometrically, such a form may be regarded as establishing a correspondence between the elements of two or more linear manifolds; in the digredient theory the latter are considered as distinct, thus undergoing independent projective transformations, while in the cogredient theory the linear manifolds are considered to be superposed, thus undergoing the same projective transformation. The first part of the paper, ?? 1-5, is devoted to the double forms. The extension of the results is made first, for convenience of presentation, to the triple forms in ? 6, and then to the general case in ? 7. The case of the double binary forms is perhaps the most interesting geometrically. In addition to the general interpretation by means of an algebraic correspondence between two manifolds, such a form may be interpreted as an algebraic curve on a quadric surface, or as a plane algebraic curve from the view point of inversion geometry. In the former of these special interpretations the two binary variables are the (homogeneous) parameters of the two sets of generators on the quadric, while in the latter they are the parameters of the two sets of minimal lines in the plane. These interpretations suggest the

Later Dr. PELL, ? in a paper read at the summer meeting of the Arnerican Matbematical Society, 1902, investigated the same case of a linear system, employing DARBOUX'S method based upon the inverse problem of the calculus of variations. Neither Dr. STECKER nor Dr. PELL arrive at a solution of the problem proposed. As a matter of fact the extension of BELTRAMI'S problem to a linear system (2) is trivial. For by the point of transformation

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

In the last paper published during his lifetime, CLEBSCHt enriched the analvtical geometry of the plane by the introduction of a new form, the connex, which includes as very special cases both the curve considered as point locus and the curve considered as line envelope. The connex (mn, n), of the m-th order and n-th class, is represented by an equation ( Xl ,X2, x3; n,l b U3)= 0, involving a set of point coordinates and a set of line co6rdinates. It may be defined as a manifold of c and with his pupil GODT, the case (1, n). The general connex (m, n) has been only incompletely investigated, principally in conniection with the theory of algebraic differential equations. ? An extension to space was first proposed by KRAUSE, 11 who took for element the combination of a poinlt and plane. An equation

The following paper is a development of Linear Associative Algebra, as distinguished from linear associative algebras. That is to say, not onlly are algebraic systenis of a definite number of units handled, but the foundation is laid for a treatment of associative numbers in general, irrespective of whether they belong to a system of units of one order rather than of another. From the writer's standpoint, a hypercomplex number anid an associative number are extensionis of the field of number beyond the limits of ordinary negative or complex numbers. An associative number is a number defined by some equation, or set of equations, whose terms and their comiiponent factors are associative quantities. Thus, a quaternion may be defined by the identity

* 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

The object of this paper is the study of differential equations possessing certain properties, later defined, which may be called Sturmian properties, since they characterize equations first discussed by STURM in his celebrated 1Inzoire sur les equations in eaires du second ordre, in the first volume of L io u v i11 e's Journal. This memoir of STURM'S is chiefly devoted to the study of the equation d [K( dx/Y(X0x)] ?(x, )y(,X)=-O

It is customary to classify hypercomplex number systems as reducible and irreducible and, owing to the theorem that a reducible systemn can always be built up out of irreducible ones, the latter only are enumerated. In the followlowing paper, in which only systems with modulus are considered, the irreducible systems are further classified by means of their groups as " semireducible of the first kind" (" 1), "semireducible of the second kind" (? 2) and " absolutely irreducible " (? 3).

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.

The group (F) which is the object of the present investigation has already been briefly discussed by HuRWIEZ.t His paper deals chiefly with the groups of signs (0, 3; 2, 4, oc) and (0, 3; 2, 6, cc)4 He proves them "commensurable " with the modular-group (0, 3; 2, 3, oc) and hence derives certain algebraic relations between the simplest functions belonging to his groups on the one hand and the invariant J of the modular group on the other. He also obtains the arithmetic character of the substitutions of his groups. The group r is also mentioned briefly in FRICKE-KLEIN'S treatise, ? where it appears as the " reproducing group " of a certain ternary quadratic form. In Part I, we derive r as the monodromy group of the Riemann surface R (p = 2) defined by the equation

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.

Introduction. 1. In a paper presented to this Society at a previous meeting, t the general structure of linear associative algebra was discussed and certain fundamental theorems of great generality were proved. The present paper is an application of these theorems to the study of what may be called group algebras. By a group algebra is meant that linear algebra whose units are defined to be such that each unit et corresponds to an operator Oi of some given abstract finite group, f and conversely, and such that for each equation of the group 0 0. = 0, corresponds an equation eze. = ek of the algebra. From the syinbolic point of view the algebra differs from the group only in that expressions-for brevity, let us say, numbers-of the form Ex. e. are possible, wherein the coefficients x, are any scalars. ? That this algebra is linear and associative, is obvious from the definition. When no confusion is feared, the notations and terminology of the group and of the algebra will be used interchangeably. 2. In the paper cited as Theory is developed the theorem that the numbers of a linear associative algebra are subject to the laws of matrices, and certain conclusions are drawn from this fact. This method of development enables us to make immediate use of any theorem needed which is true of matrices, and so saves a redevelopment of many such theorems. In the present paper I shall consider the numbers as multiple algebraic entities, referring to the theory of matrices only when some needed theorem is to be translated into an algebraic theorem. 3. In Part 1 of the paper the general form of any grotup algebra is to be dis. cussed and certain general theorems established. In Part 2 a few special cases

The formulke and the theorems developed in my recent paper Studies in the yeneral theory of ruled squrfaces are not directly applicable to the case when 04 =0, i. e., when the flecnode curve intersects every generator in two coincident points. The general notions, employed in that paper, t may however be applied to this special case as well, and give rise to a number of interesting and important considerations.

* 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-.