S OF PAPERS SUBMITTED FOR PRESENTATION TO THIS SOCIETY The following papers have been submitted to the Secretary and the Associate Secretaries of the Society for presentation at meetings of the Society. They are numbered serially throughout this volume. Cross-references to them in the reports of the meetings will give the number of this volume, the number of this issue, and the serial number of the abstract. 192. Dr. W. I. Miller: Fundamental regions for the simple group of order 168 in S±. Fundamental regions have been determined for certain groups in 6*4 (Price, American Journal of Mathematics, vol. 40 (1918), pp. 108-112). In this paper fundamental regions for the ternary G\^ are obtained by the use of 21 forms of degree four in which every term is of degree two in the variables x\, #2, #3, and of degree two in the conjugate imaginary variables. These forms are written as the differences of seven positive definite forms, so that their behavior under the group may be studied by means of the permutation group of degree seven. If we set xi/xz — x+iu, x\ ABSTRACTS OF PAPERS 501 " 3 ," the "relation of implication" of Principia Mathematica. The number of postulates is thus considerably smaller than the number (eight, including an inadvertently omitted existence postulate) in Huntington's set of postulates expressed in terms of " 3 " (Proceedings of the National Academy of Sciences, vol. 18(1932), p. 180). The author also shows that there is a very close relation between the operation "D " and the operation " — " of "exception," by virtue of which relation any set of postulates in terms of " — " is essentially also a set of postulates in terms of "D ," and vice versa. Finally, the author derives from his postulates the "theory of deduction " of Principia Mathematica. This derivation brings out the fact, perhaps more directly than has been done before, that the propositions of the theory of deduction are all propositions in the general logic of classes. (Received May 16, 1933.) 196. Mr. A. M. Tut t le : A system of independent mathematical postulates f or Dirac's quantum mechanics. This paper exhibits a system of postulates which are sufficient to furnish a rigorous formal foundation for a type of mathematics used by Dirac in his quantum mechanics (Dirac, Principles of Quantum Mechanics, Oxford, 1930). The first twelve postulates furnish a calculus for the system. The notion of a general linear process has been introduced instead of the processes of summation and integration which Dirac uses. In the first twelve postulates this process is not so restricted as to exclude an ordinary integration and many of Dirac's results are obtained upon this basis. However, in order to derive all of Dirac's expansion theorems, it has been necessary to add two more postulates which so restrict the general linear process as to exclude an ordinary integration. The postulates are shown to be consistent and independent. (Received May 19, 1933.) 197. Professor H. S. Vandiver: Summary of results and proofs on Fermât's last theoem. Seventh paper. The main result obtained is as follows. If / is an odd prime and x+y-\-z=*0 is satisfied in rational integers, none zero, and prime to the odd prime I, then the second factor of the class number of the algebraic field defined by e is divisible by /. Other criteria are also found and the proofs indicated. (Received May 19, 1933.) 198. Professor A. A. Albert: Integral domains of rational generalized quaternion algebras. In recent years several papers on the integers of generalized quaternion algebras over R have appeared and various special cases have been considered. But the general case has not yet been completed; in fact the question as to whether it would not be possible to eliminate many of these special cases has not been discussed. The present paper utilizes simple algebraic transformations as well as the theory of ternary quadratic null forms to obtain a remarkable canonical form for any generalized quaternion algebra over R. Moreover the author shows that for algebras in the canonical form there is a single domain of integrity containing the basal units, and determines this domain. (Received May 20, 1933.) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 502 ABSTRACTS OF PAPERS [July, 199. Professor H. L. Olson: Bilinear correspondences in two and in three dimensions. Every bilinear form in n-\-n variables xi, • • • , xn, yi, • * * , yn can be interpreted geometrically as determining a correspondence, in terms of homogeneous coordinates, between points and hyper-planes (linear subspaces of n — 2 dimensions) in a flat space of » —1 dimensions. This paper classifies, projectively, such correspondences in 3 + 3 and 4 + 4 variables, and presents a canonical form for each class. (Received May 17, 1933.) 200. Miss A. V. Newton: Consecutive covariant configurations at a point of a space curve. The purpose of this paper is to make some contributions to the projective differential geometry of space curves. There are many configurations covariantly connected with a point P of such a curve, such as tangent line, osculating cone, and Halphen point. As the point P moves along the curve each covariant point describes a curve, each covariant line describes a surface, and each covariant surface has an envelope. On considering a point Q infinitesimally near to the point P and setting up a transformation of coordinates between local coordin* ate systems at P and Q, it becomes possible to obtain considerable information about these curves and envelopes. In particular, one can find the tangent lines of the curves, the characteristics of the envelopes, and the foci of the edges of regression of the envelopes. (Received May 20, 1933.) 201. Professor A. H. Copeland: Consistency of the conditions determining Kollektivs. A rigorous basis for the statistical theory of probability is obtained by the establishment of the consistency of the conditions determining the set of Kollektivs. The author shows that , in a certain sense, the physical situation must be in agreement with this theory. In this respect the theory differs from all other physical theories. Moreover, the Kollektiv establishes a closer relationship between physical situations and their mathematical formulations. The conditions determining Kollektivs are due to von Mises. Consistency can be established by means of the construction of sequences of points satisfying the conditions. The consistency of the conditions for Kollektivs of a very special type can be shown by means of the nombres normaux of Borel. For this type the conditions assume a much simpler form. In this paper it is shown that a slight modification of the conditions for the more general type is essential to achieve consistency. I t is further shown by a method similar to that of Borel that these modified conditions can be satisfied. (Received May 17, 1933.) 202. Mr. T. N. E. Greville: Invariance of the property of admissibility under certain general types of transformations. This paper is concerned with a set of transformations on certain sequences, which include the type of Kollektiv called by von Mises "die einfachste alternative" and the admissible number of Copeland. I t is possible to formulate completely in terms of the transformations of this set a large class of problems arisLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1933-] ABSTRACTS OF PAPERS 503 ing in the classical theory of probability for which the operations given by von Mises are not adequate. The consistency of the assumptions of the classical theory is investigated by means of certain properties of the sequences which are invariant under these transformations. The method used in demonstrating the invariance of the properties in question is similar to that employed by Borel in proving the existence of nombres normaux. (Received May 17, 1933.) 203. Dr. Francis Regan: The application of the theory of admissible numbers to time series with variable probability. A time series is a sequence of occurrences which may be represented by a set of points on a time axis. In this paper the author is interested in the time series whose points are determined with relation to a distribution function which assigns a definite probability to every interval of the time axis. The probability varies according to the length and position of the interval. The concept of admissibility is extended to this type of time series by s