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s of papers presented at the meeting are given below. Papers read by title are indicated by the letter "/." Paper number 278 was read by Mr. Hinds, and paper number 284 was read by Professor Roberts. Mr. Hopper was introduced by Professor Wallace Givens, Mr. Butz by Professor Tomlinson Fort, Messrs. Conner and Keesee by Mr. R. E. Allan, and Mr. Derr by Professor A. D. Wallace. ALGEBRA AND THEORY OF NUMBERS 260. R. K. Butz: On 2-commutative matrices. The methods used by W. V. Parker for writing the matrix X such that AX — XB (Bull. Amer. Math . Soc. Abstract 55-7-287) are extended to solve the problem of finding a matrix X with respect to which a given matrix A, with elements in any field F, is 2-commutative. When A is in rational canonical form and X is partitioned properly, these methods display X explicitly in terms of parameters. (Received March 9, 1950.) 261/. Leonard Carlitz: A theorem on higher congruences. This note contains a proof of the theorem tha t if the congruence ƒ (x) s 0 (mod p) is solvable for r = 5 + 1 , where p is the highest power of p dividing the discriminant of f(x), then the congruence is solvable for all r. While the theorem is contained in a more general theorem of Hensel a direct proof seems of interest. (Received March 6, 1950.) 262/. Leonard Carlitz: Some special functions connected with GF(p«, x). This paper is concerned with the properties of the function g(u, t) = ÜCo *i(u)4n(t)/Fi defined in Duke Math . J . vol. 15 (1948) p. 1008, and some related functions. (Received March 6, 1950.) 263/. Leonard Carlitz: Sums of kth powers in GF[p, x]. In this paper an asymptotic formula is obtained for the number of solutions of A f » « i ^ î + • • • -\-asX k s, where <*jÇi.GF(p )\ M, X3^GF[p , x], deg M^tnk, deg Xj~m, Xj p r i m a r y , 7 ^ / ; deg Xi<m, i>l, and X^lSs; finally s^so(k). The method is tha t developed in Duke Math . J . vol. 14 (1947) pp. 1121-1137; vol. 15 (1948) pp. 795-801. The case / = 0 is excluded. (Received March 6, 1950.) 264. J. C. Currie: Unitary-canonical matrices. Preliminary report. I t is known (I. Schur, Über die charakteristischen Wurzeln einer linearenSubstitution, Math . Ann. vol. 66 (1909) pp. 488-510) tha t for any matrix A there exists a unitary matrix U such tha t VA JJ' is triangular. However, the matrix U which performs this transformation is not unique, and the resulting triangular form is not unique. In this note it is established tha t from the set of matrices which are unitary transforms of a given matrix A a particular triangular matrix can be designated as the canonical representative of the set. Let the characteristic roots of A be ordered so tha t if w?is the nullity of \jI-A, then w , ^ w J + i , j — 1 , • • • , n — 1. Then A is unitarily transformed into a triangular matrix in which square submatrices of the form \jlj (where Ij 322 AMERICAN MATHEMATICAL SOCIETY [July is of order ntj) appear in the principal diagonal. Additional zeros are introduced above the principal diagonal in an orderly procedure such that the end result is a triangular matrix with a maximum number of zeros above the principal diagonal, the zeros being systematically placed so that the form is unique for a given A. (Received March 6, 1950.) 265. J. R. Garrett: Normal equations and resolvents infields of characteristic p. The general equation of degree not less than 5 with coefficients in a field F of characteristic p is reduced to the so-called principal and normal forms by means of Tschirnhaus transformations. In addition, a sextic resolvent is obtained for the normal quintic when p — 2, 3, and 5. For corresponding results when F is of characteristic zero see Dickson's Modern algebraic theories, chap. 12. (Received March 6, 1950.) 266. Wallace Givens: A Hadamard bound theorem for the determinant of a real quaternion matrix. The determinant of a regular matrix A of order n with elements in a division ring K is defined by Dieudonné [Bull. Soc. Math. France vol. 71 (1943) pp. 27-45] to be an element A(̂ 4) of the quotient group of the multiplicative group of nonzero elements of K with respect to its commutator subgroup. If one specializes to real quaternions, the group of determinantal values is isomorphic to the non-negative real numbers (with A(^4)=0 if and only if A has no inverse). Normalizing by requiring the determinant of a diagonal matrix to be the square root of the products of the norms of its elements (which gives the square root of Study's determinant) the Hadamard bound theorem is proved in the form: A(A) ^na, where the elements of A are real quaternions an such that Norm (at/) £*a. The stated bound is shown to be attained by specializing the a^ to be complex numbers. If the bound can also be attained by a matrix every element of which is one of the sixteen numbers ( ± l ± * ± J ± £ ) / 2 , then the Hadamard bound for matrices of order 4w is also attained by a real matrix. (Received March 7, 1950.) 267. E. H. Hadlock: A method of construction of a properly primitive form. A method is derived and stated, whereby, given the integral invariants Î2, A and possible sets of progressions involving 2 and the odd prime factors of £2 and A, the integral coefficients a, b, c, r, and S are determined in the properly primitive form ƒ ~ax-\-by-{~cz-\-2ryz-{-2Sxz with determinant d=* OA?^0 so that ƒ will have associated with it the given invariants and sets of progressions. (Received March 6, 1950.) 268. E. H. Hopper: The decision problem in the free modulator lattice. Let J!, [4], jPlf[4], and FD [4] be the free, the free modular, and the free distributive lattice, respectively, on four generators. Polynomials equal in iPL[4] are necessarily equal in ilf<F[4] and FD[4] while inequality in .FJ9[4] implies inequality in the other two lattices. The unsolved decision problem in .Fif[4] is thus reduced to a question of the equality in FM [4] of pairs of polynomials having equal canonical i95o] THE APRIL MEETING IN OAK RIDGE 323 forms in -F^[4] and unequal canonical forms in -PL [4]. This method is used to obtain canonical forms in T̂ ikTfé] for polynomials of length not greater than 5. Inclusion relations between these polynomials are also determined. (Received March 6, 1950.) 269/. Ralph Hull: An algebraic proof of a theorem on the full linear group. The theorem is: every rational integral representation of the full linear group of dimension n is either homogeneous or else is expressible as a sum of homogeneous rational integral representations. (Cf. Murnaghan, The theory of group representations, Johns Hopkins Press, 1938, p. 45.) (Received March 20, 1950.) 270. R. J. Levit: Divided differences and some extremal properties of functions on finite sets. Let An be a. finite set consisting of elements a0i • • • , an of an ordered field rw, and consider f unctions ƒ (#) on^L with values in rw. For any nonzero element c(Erw let $nk be the class of all functions f(x) such that f (do, • • • ,a n )=c , where f (ao, • • • , an) denotes the nth divided difference off(x) with respect to the indicated arguments. This paper has the dual purpose of deriving certain formal algebraic properties of divided differences and then applying them to the determination of the function of <!>„& which deviates least from zero on An and also the functions whose divided differences of any given order k Sn deviate least from zero on An> (Received March 8, 1950.) 271. Nathaniel Macon: Some theorems on the approximation of irrational numbers by the convergents of their continued fractions. Let £ be any positive irrational number. Denote by An/Bn the convergents of its expansion as a regular continued fraction, and set |£—An/Bn\ = l/XnJ5*. Recently Brauer and Macon (Amer. J. Math. vol. 71 (1949) pp. 349-361 and vol. 72 (1950)) obtained the best possible lower bound for the sum of k consecutive X„, for every k. In this paper, the best possible lower bounds for the products of two, three, and four consecutive Xn are obtained. For k—3, this generalizes a well known theorem of Borel (J. Math. Pures Appl. (5) vol. 9 (1903) pp. 329-375). Moreover, Brauer and Macon proved that in any sequence of five consecutive Xn either at least two are greater than 5 or one is greater than 3. They obtained similar results for every sequence of 3m+2 consecutive Xn. These results are now improved for eight and eleven consecutive X's. (Received March 6, 1950.) 272/. Leo Moser: On sets of integers which contain no three in arithmetical progression. Let g(n) be the maximum number of integers under n, no three of which form an arithmetical progression. F. A. Behrend (Proc. Nat. Acad. Sci. U.S.A. (1946) pp. 331333) showed that g(n) >w , where c is a fixed constant. In Behrend's method the existence of appropriate sets is proved, but these differ for different n. An infinite sequence is here constructed, which has no three elements in arithmetical progression, and which has under n at least n~n)12 integers. Upper bounds for g(n) were given by P. Erdös and P. Turân (Proc. London Math. Soc. (1936) pp. 261-264). Their results are here improved to g{n) <4w/ l l+5 . (Received March 6, 1950.) 324 AMERICAN MATHEMATICAL SOCIETY [July 273. O. M. Nikod^m: Remarks on the cartesian product of Boolean measure algebras. Kakutani's theorem, stating the possibility of extension of the finitely additive measure of the cartesian product of denumerably additive <r-fields of sets provided with denumerably additive measures to a denumerably additive measure on the Borelian extension of this product, can be generalized to abstract Boolean measure algebras. Circumstances relating to this extension are discussed. (Received May 8, 1950.) 274. E. B. Shanks: An elementary proof of the fundamental theorem of algebra. Let N(t) be the net number of complete revolutions made by a line segment (of variable length) joining the origin to the point representing ƒ(z)^z-\-aiz"-\r • • • +an (where n is a positive integer and a«5^0) when z traces out the circle \z\=*t one tim

The April meeting in Oak Ridge