I 8 AMERICAN ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE THE REGULAR SPRING MEETING OF THE EXECUTIVE COMMITTEE OF THE COUNCIL THE executive committee of the council of the association met at the Cosmos Club in Washington on Sunday, April 25, with the chairman, Dr. J. McK. Cattell, in the chair and all members present excepting Dr. F. R. Moulton, who was unable to come. Three sessions were held, one in the forenoon, a second in the afternoon and a third in the evening. The committee dined together at 6:30 in the small tea-room in the Cameron House of the club. The following items of business were transacted: (1) The committee expressed its hearty appreciation of the excellent services being performed for the association by the executive assistant, Mr. Sam Woodley, especially in connection with the task of preparing the manuscripts for the recently published directory of members. (2) Certain features of the organization of the Washington office were discussed and the permanent secretary was asked to study the possibilities of such changes as may render the organization more efficient, reporting the results of the study to the committee at its next meeting. (3) One hundred and twenty-seven members were elected to fellowship, on nominations regularly approved by the section secretaries. These newly elected fellows are distributed among the several sections as follows: (4) It was voted that the formation of a new division of the association to include Nebraska, Oklahoma and Kansas, as well as Colorado and Wyoming, is not feasible. (5) A special committee was established to make inquiry and submit further suggestions on the problem of the formation of a new division to include Colorado and Wyoming. This committee is to consist of Dr. H. B. Ward (chairman), Dr. Aven Nelson and a third person to be named by these two. (6) Proposals to form sections on home economics and on cosmology were considered and it was voted. that neither of these proposals is feasible at present. The scientific aspects of these subjects are represented in the present arrangement of sections and it seems generally undesirable to attempt to form new sections that would simply divide the field of science in a different way from that followed by the constitution. It was suggested that such borderline and overlapping fields might well be specially cultivated in the programs of the meetings, by means of joint sessions of two or more sections, the subjects for such joint meetings being stated so as to show the interrelations of the several branches of science thus brought together. (7) Official affiliation with the association was ratified for the American Oil Chemists' Society and the American Veterinary Medical Association. Each of these organizations is to have one representative in the council of the association. (8) Official affiliation, according to the special arrangement for affiliated academies of science, was ratified for the Alabama Academy of Science and the Pennsylvania Academy of Science. (9) In connection with the general aim of the association to aid in the popular dissemination of scientific knowledge, a proposal was considered by which a non-technical scientific weekly publication might be organized under the auspices of Science Service (in the control of which the association takes part) with a special subscription price to members of the association. A committee was appointed, with power, to consider and decide upon this proposal, the association to take no financial responsibility. The committee consists of: Dr. J. McK. Cattell (chairman), Dr. W. J. Humphreys, Dr. Vernon Kellogg, Dr. B. E. Livingston and Dr. Edwin B. Wilson. (10) The executive committee considered again the proposal for a non-technical illustrated monthly magazine under the control of the association, but no action seemed warranted as yet. (11) A special committee (Dr. W. J. Humphreys, chairman, Dr. J. McK. Cattell and Dr. H. B. Ward) was appointed to consider the possibility of strengthening the appeal of SCIENCE, both on the scientific and on the popular side. (12) Dr. M. I. Pupin was nominated to Science Service as the representative of the American Association to succeed Dr. W. W. Campbell. (13) A committee on exhibition was established, with Dr. H. E. Howe as chairman, to have charge of the exhibition at the annual meetings. This committee is to consist of two representatives of the exhibitors, two representatives of the men of science, one representative of the city of the meeting, and, in addition, as ex-officio members, the president, the permanent secretary and the manager of the exhibition. (14) The permanent secretary was instructed to name a member of each section of the association, these to be consulting members of the committee on exhibition. (15) It was voted to appropriate $500 from the permanent secretary's funds for the expenses of preliminary arrangements for the Philadelphia exhibition, the amount thus used being charged against the exhibition. (16) A proposal was received from Major H. S. Kimberly regarding the management of the annual exhibition. A special committee, consisting of Drs. J. McK. Cattell and B. E. Livingston, was named, with power to arrange and close a contract with Major Kimberly as manager of the exhibition. (17) It was voted that the local committee for the annual meeting shall arrange for one or more persons to be present at the exhibition at all times to explain the non-commercial exhibits. (18) It was voted that Dr. J. Playfair McMurrich and Dr. H. B. Ward be official representatives of the American Association at the Oxford meeting of the British Association. (19) President Bailey and Dr. Theodore W. Richards were named to represent the American Association at the inaugural ceremonies at Boston University, May 15. (20) The permanent secretary reported the appointment of four members of the committee on prize award for the fifth Philadelphia meeting and this report was accepted, the remaining member of the committee on awards to be named as soon as possible. (21) Mr. Austin H. Clark, of the U. S. National Museum, was appointed director of publicity for the fifth Philadelphia meeting, to have charge of all publicity work on the part of the association. (22) The problem of gratis or reduced subscriptions for American scientific journals to be sent to European libraries that can not otherwise receive them was discussed and this was referred to Dr. Vernon Kellogg with the request that he study the question and make a subsequent report to the executive committee. (23) In response to an invitation, it was voted that a representative of the association be named to attend the Conference on Narcotic Education, to be held in Philadelphia, June 29 to July 2. (24) Two resolutions were adopted, one on the protection and conservation of the U. S. national forests and public lands and the other on the cooperation of the American Association with the National Academy of Sciences in the academy's plan to further American scientific research. These resolutions are published below. (25) An appropriation of $25 was made from the treasurer's funds, to aid in the preparation of a second volume of the "Naturalists' Guide to the Americas," in charge of the Ecological Society of America. (27) An appropriation of $60 from the permanent secretary's funds was made to aid in the educational work of the American Institute of Sacred Literature. (28) The formal procedure of opening the annual meeting of the association was discussed and this was made a special order of business for the next meeting of the executive committee. (29) The next meeting of the executive committee will be the regular fall meeting, to occur at Philadelphia on Sunday, October 17. Business to be brought before the committee should be in the permanent secretary's hands at least a week before the meeting. BURTON E. LIVINGSTON, Permanent Secretary. TWO RESOLUTIONS ADOPTED BY THE EXECUTIVE COMMITTEE OF THE COUNCIL OF THE AMERICAN ASSOCIATION AT ITS MEETING IN WASHINGTON ON APRIL 25, A resolution on the cooperation of the American Association with the National Academy of Sciences in connec tion with the Academy's plan to further American scientific research. Resolved: That the Executive Committee of the Council of the American Association for the Advancement of Science has learned with much interest and gratification of the organization by the National Academy of Sciences of a special Board of Trustees, composed of a number of distinguished men of science and of public affairs, under the chairmanship of Honorable Herbert Hoover, which is attempting to raise a large national fund for the support of research in pure science; And, That the Executive Committee, on behalf of the Council of the Association, heartily endorses this plan of the National Academy of Sciences and the special Board of Trustees, and expresses the willingness of the American Association for the Advancement of Science to cooperate, in any way feasible to it, with the Academy and Board to help bring success to their effort. A resolution on the protection and conservation of the U. S. national forests and public lands. Resolved: That the Executive Committee of the Council of the American Association for the Advancement of Science, on behalf of the Council and the Association, takes this opportunity to record again the view of the Association that the rights of the nation in public lands and national forests should not be alienated or transferred to private individuals, groups or corporations. The Association is opposed particularly to the provisions of the Stanfield Bill, which would grant grazing rights without proper means of safeguarding public interests and without retaining proper control over public lands and national forests. I SCIENCE VOL. LXIII JUNE 4, 1926 CONTENTS No. 1640 THE ALGEBRAIC NUMBERS AND The Algebraic Numbers and Division: PROFESSOR The Science Exhibition of the American Association; The Fiftieth Anniversary of the Johns Hopkins University; A Survey of Forestry Research; Appointment to the Non-resident Lectureship in Chemistry at Cornell Scientific Notes and News Why the Temperature of the Air decreases with Buller's Researches on Fungi: G. P. C. Zane Grey's Tales of Fishing in Virgin Seas: DR. DAVID STARR JORDAN Scientific Apparatus and Laboratory Methods: Oil Water Models Illustrating Biological Phenomena: V. E. HALL, F. DE EDS and P. J. HANZLIK. A New Type of Electron Spectrograph: KENNETH COLE SCIENCE: A Weekly Journal devoted to the Advancement of Science, edited by J. McKeen Cattell and published every Friday by THE SCIENCE PRESS Lancaster, Pa. Garrison, N. Y. New York City: Grand Central Terminal. Annual Subscription, $6.00. Single Copies, 15 Cts. SCIENCE is the official organ of the American Association for the Advancement of Science. Information regarding membership in the Association may be secured from the office of the permanent secretary, in the Smithsonian Institution Building, Washingon, D. C. Entered as second-class matter July 18, 1923, at the Post Office at Lancaster, Pa., under the Act of March 8, 1879. DIVISION1 As to what should most appropriately be the character of an address delivered on an occasion such as the present I am not quite certain. Whether it were better that one should be somewhat general and discursive in his remarks or whether he would be justified in offering a considerable amount of highly specialized and technical material I hardly know. Possibly a critical or historical survey of some subject would be more in place, or again under circumstances it might perhaps be permitted to the speaker to discuss some phase or aspect of a special field which would afford opportunity to present, among others, results obtained by himself or to develop methods employed in his investigations. On this occasion I shall venture to say a little about a subject in which I have had a special interest, but in which results obtained are of some years' standing, since with the routine of teaching, executive activities in various connections and a vast amount of organizing work have, in recent years, combined to prevent productive effort and have interfered with the formulation for publication of results already there. What I have to say will consist largely in statements. There will be no attempt to give proofs. These will be available elsewhere. A certain amount of recapitulation of more or less familiar facts will be necessary in order to establish connectivity, and as a preliminary to the later statements. The net result, I trust, will be found to contain an element of novelty. An integer can be represented as a product of powers of primes, the exponents being positive integers. A rational fraction can be represented as a product of powers of primes, the exponents being positive or negative integers. The exponent of a given prime in the representation of a rational number as a product of powers of primes we call the order number of the rational number for the prime in question. We say that 0 is the order number of a rational number for a prime which does not appear explicitly in the representation of the number as a product of powers of primes. One rational number we say is divisible by another when the quotient of the first number by the second is integral, otherwise 1 Address of the retiring vice-president and chairman of Section A-Mathematics-American Association for the Advancement of Science, Kansas City, Mo., December, 1925. said, when for every prime p the order number of the former is at least as great as the corresponding order number of the latter. Besides the product representation just referred to there is also for any positive integer r and with reference to a single prime p a unique representation in the form (1) r = a + a1p+ +2x-1 where each of the coefficients a, a,, of the numbers 0, 1, . . ., p-1. pk-1 , ak-1 is one p. , ax-1 1 such We shall for the moment say of a rational number r=s/q represented in its reduced form, as the quotient of two integers, that it is divisible by a power of the prime p if its numerator is divisible by that power of p. We can then make use of the notation of congruences and find coefficients a, a, included among the integers 0, 1, that, however great k may be, we have (2) rao+a, P+ +ak-1 pk-1 (mod pk) provided the denominator q is not divisible by p. This holds whether r is positive or negative. We can multiply both sides of the congruence by a negative power of p, the modulus being at the same time multiplied by this power of the prime. It is then evident that we can include the case where the denominator q is divisible by a power of p on writing (3) 1 -1+1 . p-1+1 + r=a, p1+a +a p-1 +a+a1 p + + ax-1 pk-1 (mod pk) The coefficients in the series on the right-hand side of (4) constitute a perfectly definite sequence. A series in ascending powers of p with a definite sequence of integral rational numbers as coefficients we shall designate as a p-adic number. Hensel defines the p-adic numbers more generally and develops their theory in his book "Theorie der Algebraischen Zahlen." For the purposes of the present paper, however, the more general p-adic numbers are superfluous and the elementary p-adic numbers to which we here restrict ourselves throw light on the subject of the algebraic numbers and suffice for the development of a simple theory of the ideals. A positive integer r might be represented as a polynomial in p with coefficients which are not all included among the numbers 0, 1, , p-1. We then have a simple process by which from such rep resentation we can obtain in succession the coefficients a0, a1, , a-1 in the form of representation given in (1) where these coefficients are included among the numbers 0, 1,, p-1. The same process applied to any p-adic number whose coefficients are not all included under the numbers in question will give us a p-adic number with a sequence of coefficients which are included under these numbers. The p-adic number so obtained is regarded as equivalent to the p-adic number from which it was deduced and is referred to as the reduced p-adic form of that number. The sum, difference, product of two integral p-adic numbers a +,a1 p + and b + b1 p+ spectively, taken as = a + a1 p+· ·・・ (p) and determining the coefficients co, C1, in succession by the aid of congruences (mod p). It is evident what modifications are to be introduced in defining and obtaining the sum, difference, product or quotient of two p-adic numbers where negative powers of p present themselves. We have seen that to every rational number corresponds a definite p-adic number of reduced form which may be said to characterize that number relatively to the prime p or to which the rational number may in a certain sense, with reference to p, be said to be equal. Conversely, however, it does not follow that to every reduced p-adic number there corresponds a rational number, which is represented by it in the p-adic field. By the order number of a p-adic number we shall mean the lowest exponent in its series. In the case of a rational number then this p-adic order number coincides with what we have called the order number of the rational number relative to the prime p. It is evident that the p-adic order number of a sum or difference of two p-adic numbers is equal to the lesser of the order numbers of the p-adic numbers when these order numbers are unequal and that in the case where they are equal the order number of the sum or difference in question is at least equal to the common order number of the two p-adic numbers. The p-adic order number of a product is equal to order mber the nde two the sum of the p-adic order numbers of the factors and the p-adic order number of a quotient is equal to the difference between the order number of the numerator and the order number of the denominator. Consider an algebraic equation = (5) f(x) = x2-α1x2-1+...+(-1)α = 0 in which the coefficients are rational numbers. If x" had a coefficient a + 1 division by a would reduce the equation to the form in question. Equation (5) is said to be integral if the coefficients are all integral. If the equation is not integral a substitution y=ax 7 where a is an integer will give us an equation in y which is integral. Equation (5) may or may not be irreducible in the field of the rational numbers. It is in any case equivalent to a number of equations which are irreducible in the field of the rational numbers. If the equation (5) is integral each of these irreducible equations, it can be shown, is integral. Any "root of an irreducible algebraic equation is called an algebraic number. Such root is called integral if the irreducible equation is integral. All the roots of an algebraic equation then are algebraic numbers and all the roots of an integral algebraic equation are integral algebraic numbers. Any number built up by addition, subtraction, multiplication and division out of the rational numbers and a finite number of algebraic numbers can be shown to be an algebraic number. The quotient of two algebraic numbers is then an algebraic number, k and one algebraic number is said to be divisible by a second algebraic number when the quotient of the former by the latter is an integral algebraic number. Supposing equation (5) to be irreducible in the field of the rational numbers let us consider the body of numbers built up by rational operations out of one of its roots ɛ. The field or corpus of algebraic numbers so obtained we designate by C(ɛ). Any number R (ε) of the corpus can be represented as a polynomial of degree n − 1 in ɛ with rational coefficients. polynomial in X. e= to R(E) is a root of an algebraic equation of degree n (6) F(X) = X"-A,X-1+...+(-1)"A, = 0, where the coefficients A are rational. This equation may or may not be reducible. If it is reducible, however, the left-hand side is a power of an irreducible If it is irreducible its root R(e) is a primitive number of the corpus, that is say the corpus C(e) coincides with the corpus C(E). Within the corpus C(E) we designate A, and A as trace and norm, respectively, of the number e. The trace and norm of ɛ are α, and α, respectively. Though the polynomial on the left-hand side of equation (5) is irreducible in the field of the rational numbers it may so happen that it is reducible in the field of the p-adic numbers. We may then write (i) f¡(x) = x2i the coefficients a," here being p-adic numbers. It is to be understood that there is here no question of equating the p-adic factors to 0. If f(x) is p-adically integral, that is to say, if its coefficients are integral relatively to the prime p, so also are the coefficients in the factors f, (x) p-adically integral, as may readily be shown. These factors we shall assume to be irreducible in the p-adic field. We shall associate the term cycle with these factors and shall speak of the first, second, etc., rth cycles in regard to the polynomial f(x) and with reference to the prime p. As ith p-adic partial trace and ith p-adic partial norm of ɛ with reference to the specific prime p and within the corpus C(e) we shall designate a,() and a(1) respectively-or we may speak of the p-adic partial trace and the p-adic partial norm of ε for the ith cycle. The sum of the r p-adic partial traces of a corresponding to a specific prime p is evidently equal to the trace of ɛ and the product of the r p-adic partial norms of & is equal to the norm of ɛ. Where f(x) splits into r irreducible p-adic factors f(x) it can be shown that F(X) will also split into r p-adic factors F, (X) which may or may not happen to be irreducible. We can then write F1(X) = X";-A (1) X';-1+ ··· +(-1)"'An (1), There is a certain correspondence between the r p-adic factors of F(X) and the r p-adic factors of f(x) which we may assume to be indicated by their suffixes, the factor F(X) of F(X) corresponding to the factor f(x) of f(x). As f1 (x) is irreducible in the p-adic field it can readily be shown that the corresponding factor F(X) of F(X) is either an irreducible p-adic polynomial or a power of such a polynomial. If then F(X) is irreducible in the field of the rational numbers it is clear that the r p-adic factors F(X) must be p-adically irreducible and distinct from one another. It can be shown that the correspondence between the factors F, (X) and f, (x) is independent of the choice of a primitive number to represent the numbers of the corpus. The ith p-adic partial trace a,(1) of ε has a certain order number relative to the prime p, which we shall refer to as the p-adic partial trace order of & corresponding to the ith cycle. Because of the correspondence between the factor F, (X) of F(X) and the factor f1(x) of f(x) one may speak of the order number of A as the p-adic partial trace order of |