The HOME SECRETARY and the FOREIGN SECRETARY of the ACADEMY The CHAIRMAN and the PERMANENT SECRETARY of the NATIONAL RESEARCH COUNCIL WILLIAM DUANE, '23 A. L. DAY, '22 J. M. CLARKE, '21 INFORMATION TO SUBSCRIBERS SUBSCRIPTIONS TO THE PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES at the rate of $5.00 per annum should be made payable to the National Academy of Sciences, and sent either to Easton, Pa., or to C. G. ABBOT, Home Secretary, National Academy of Sciences, Smithsonian Institution, Washington, D. C. Single numbers, $0.50. PAST publications of the NATIONAL ACADEMY OF SCIENCES are listed in these PROCEEDINGS, Volume III, pp. 743-753, December, 1917. In good part these publications are no longer available for distribution. Inquiries with regard to them should be addressed to the Home Secretary, National Academy of Sciences, Smithsonian Institution, Washington, D. C. PAST volumes of the PROCEEDINGS may be obtained at five dollars per volume unless the sale of the volume would break a complete set of Volumes I to V. Single numbers may be obtained for fifty cents except where the sale of such numbers would break up a complete volume. Only two hundred complete sets of the Proceedings are available for sale-Volumes I to. VI, price $30.00. Orders should be sent to the Home Secretary, National Academy of Sciences, Smithsonian Institution, Washington, D. C. The following publications are issued by the NATIONAL RESEARCH COUNCIL. Orders and inquiries should be addressed: Publication Office, National Research Council, 1701 Massachusetts, Ave., Washington, D. C. (A) THE BULLETIN OF THE NATIONAL RESEARCH COUNCIL published at irregular intervals; Price $5.00 per volume of about 500 pages; individual numbers priced variously. (B) THE REPRINT AND CIRCULAR SERIES OF THE NATIONAL Research COUNCIL, individual numbers, variously priced. QUATERNIONS AND THEIR GENERALIZATIONS DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO 1. The discovery of quaternions by W. R. Hamilton in 1843 has led to an extensive theory of linear algebras (or closed systems of hypercomplex numbers) in which the quaternion algebra plays an important rôle. Frobenius' proved that the only real linear associative algebras in which a product is zero only when one factor is zero are the real number system, the ordinary complex number system, and the algebra of real quaternions. A much simpler proof has been given by the writer.2 Later, the writer3 was led to quaternions very naturally by means of the fourparameter continuous group which leaves unaltered each line of a set of rulings on the quadric surface x; + x; + x; + x; = 0. The object of the present note is to derive the algebra of quaternions and its direct generalizations without assuming the associative or commutative law. I shall obtain this interesting result by two distinct methods. 2. The term field will be employed here to designate any set of ordinary complex numbers which is closed under addition, subtraction, multiplication, and division. Thus all complex numbers form a field, likewise all real numbers, or all rational numbers. = Just as a couple (a, b) of real numbers defines an ordinary complex number a + bi, where i2 — 1, so also an n-tuple (x1,.., x) of numbers of a field F defines a hypercomplex number x = x1е1 + x2€2 + .... + xnen, (1) where the units e1, en are linearly independent with respect to the field F and possess a multiplaction table n which the y's are numbers of F. Let x' = Ex,e; be another hypercom when ƒ is in F, so that multiplication is distributive. Under these assumptions, the set of all numbers (1) with coördinates in F shall be called a linear algebra over F. = 3. We assume that e, is a principal unit (modulus), so that ex = xe1 = x for every number x of the algebra, and write 1 for e1. We assume that every number of the algebra satisfies a quadratic equation with coefficients in F. If e2 + 2ae + b 0, (e + a)2 = a2-b, so that we may take the units to be 1, E2, En, where E Sii, a number of F. For i and j distinct and >1, E; ± E; satisfies a quadratic, so that (E; ± E;)2 = Sii + S;; ± (E¡E; + E;E;) is a linear function of E; ± E;. Thus E¡E; + E¡E¡ is a linear function of E; + E; and of E¡−E;, and hence is a number 2sji of F. 2sij = Let u2, = ..., u, be arbitrary numbers of F and write U It is a standard theorem that Q can be reduced to Σc,v,2 by a linear transformation uk = Zak with coefficients in F and of determinant ‡0. Write Then 1, 2, k,l =2 (l = 2, . . ., n). en are linearly independent and may be taken as the new units of our algebra over F. Then |