OH S S Pub/is/ied wit/i t/ie appro/iation of t/ie Board of Trustees VOL. 11.—No. 20. I BALTIMORE, DECEMBER, 1882. [PRICE 10 CENTS. CALENDAR, 1882-83. Tuesday, September 19. Current Academic Year Began. Friday, December 22—Tuesday, Jan. 2. Christmas Recess. Tuesday, January 2. Mr. F. Seymour Haden’s Lectures on Etching Begin. (See p. 36). Wednesday, January 3. Meeting of Scientific Association, 8 P. M. Friday, January 5. Meeting of Philological Association, 12 M. Monday, January 8. Professor G. S. Morris’s Course on Philosophy and Christianity Begins. (See p. 36). Tuesday, January 9. Meeting of Metaphysical Club, 8 P. M. Wednesday, January 10. Professor Corson’s Course on Poetry Begins. (See p. 36). Saturday, January 13. M. Rabillon’s Course in French Literature Begins. (See p. 36). Wednesday, January 17. Meeting of Historical and Political Science Association, 8 P. M. Friday, January 19. Meeting of Mathematical Society, 8 P. M. Friday, June 8. Term of Instruction Closes. CONTENTS. PAGE. PAGE. Scientific Notes : — University Studies in Historical and Political Science. Mcdhemalics. Mr. Freeman’s Visit to Baltimore, by H. B. ADAMS, • - . 80—82 On the Fundamental Theorem in the New Method of Partitions, Prospectus of the Series 82 by J. J. SYLVESTER; The Maximum Value ofa Certain Deter- minant, by E. W. DAVIS; Notes on Symmetric Functions, by Synopsis of Recent Scientific Journals: W. P. DURFEE; On the Number of Self-Opposite Partitions, American Journal of Philology. by W. P. DURFEE; Note on the last-named paper, by J. J. On the Culex of Vergil, by R. ELLIS; On the Creole Patois of SYLVESTER; On Crocchi’s Theorem, by F. FRANKLIN, . 22—24 Louisiana, by J. A. HARRISON; The English Perfect Parti- Chemisiry. ciple used Infinitivally, by FITZEDWARD HALL; The Athe- On White Phosphorus, by IRA REMSEN,. 24 nian Naval Arsenal of Philon, by T. W. LUDLOW; Etymo- logical Studies, by J. P. POSTOATE 8884 Biology. Observations on the Pulse Wave in the Coronary Arteries of the Contents of American Journal of Mathematics Studies from the Heart, by H. N MARTIN and W. T. SRDGWICK; Influence of Biological Laboratory, and Journal of Physiology, . 34 Digitaline on the Heart of the Slider Terrapin, by H. H. DON- Mr. Peirce’s Preface to Contributions to Logic, 84 ALDSON and M. WARFIELD; On a New Form of Pilidium, by E. B. WILSON; Polar Effects upon Nerves of Weak Induction Professor Cayley’s Introduction to a Memoir on the Abelian and Currents, by H. SEWALL; Researches on the Growth of Starch Theta Functions 85 Grains, by A. F. W. SOHIMPER; Form of the Pulse-wave and Mean Arterial Pressure in a Dog with Patent Duclus crteriosus, Meeting of German Philologians, 85 by W. H. HOWELL and F. DONALDSON, JR.; Variations of Re- Recent Publications. flex-Excitability in the Frog, by W. T. SEDOWIcK; Influence of Variations of Intracardiac Pressure upon the Action of the Brooks’s Development of Luczfer; Browne’s Glarendon Diclionary; Gildersleeve’s Lalin Primer 86 Vagus Nerve, by H. S WALL and F. DONALDSON, JR.; Dis- cussion of Recent Researches on the Lower Organisms and their Hopkins Hall Lectures, . 86 Connection with Diseases, by’W. T. COTJNOLLMAN, . 24, 25 Special Courses for Advanced Students: Philology. Lectures on Physiology; Historical Lectures on Chemistry, Functional or Dialectic Differences in the Present Systems of the 87 Veda, by M. BLOOMFIELD; On Matthew Arnold’s Poetry Test, Current Information: by W. J. ALEXANDER; On the Use of iirt (or 4g) in Direct Recent Appointments, etc., 87 Quotations, by E. H. SPIEKER, . 26 Ilislorical and Polis~ieal Science. Proceedings of Societies: Scientific; Philological; Historical and Political Science; Meta- Introduction to American Institutional History, by EDWARD A. physical; Mathematical; Naturalists’ Field Club ;—Maryland FREEMAN; The Past and Present of Political Economy, by R. Historical Society . 88 T. ELY; Political Economy in Germany in 1882, by R. T. ELY; P1ymouth Rock Restored, by H. B. ADAMS; Local Hours for Lectures and Recitations, . 89 Government in the Northwest, by E. W. BEMIS The Dis- trict System of South Carolina, by B. J. RAMAGE, . 26—29 Standing Announcements: Physics. University Publications 40 Admission to Hopkins Hall Lectures, 40 On Concave Gratings for Optical Purposes, by H. A. ROWLAND; Calendar, . 40 The Transit of Venus ;—International Commission of Electri- Appointments of Societies cians, 29 40 22 JOHNS HOPKINS [No. 20. SCIENTIFIC NOTES On the work of members of this University in Mathematics, Physics, Chemistry, Biology, Ancient and Modern Languages, History, Political Science, Psychology, Ethics, and Logic. ascending powers of x is the same as the Residue of ~F(2uex)[?u= a, b,.. 1]. MATHEMATICS. To prove this it is only necessary to suppose the fraction Fx separated into simple partial fractions with constant numerators and the theorem On the Fundamental Theorem in the New Method of Par- becomes self~evident.* titions, by J. J. SYLVESTER. It follows, therefore, writing n in place of i that the coefficient of x~ in [Abstractof a paper read before tbe University MathematicalSociety, November 15, 1882]. ascending-power series for the fraction Gx = — )J (x — l)~) The new method of partitions which I gave to the world more than a quarter of a century ago is an application of a theorem which, I think it will be the Residue with its sign changed of ~(a—sse—slx)G(aex),or which must be conceded, is, after Newton’s Binomial Theorem, the most impor- is the same thing is the Residue of ~a—~e~ G(ae—X) which theorem we tant organic theorem which exists in the whole range of the Old Algebra. now see is true not merely for the case where G is a proper-fraction, i. e., What Newton’s theorem effects for the development of radical—that is a function of x whose degree is a negative integer but remains true theorem accomplishes for the development offraclional forms of algebral- when the degree of G is any number inferior to n, for when that condition cal functions. G One (but not the most perfect) form in which it can be presented is the is satisfied — is a proper fraction, which is all that is required in order following. If I~c be any proper algebraical fraction in x, whose infinity for the parent theorem to apply. roots (i. e., the values of x which make Fx infinite) are a, b, . 1, quan.. tities all supposed to differ from zero then the coefficient of x~ for any The MaximumValue of a Certain Determinant, by E. W. DAVIS. 1 value of n will be the residue, i. e., the coefficient of— in [Abstractof a paper read before theUniversity Mathematical Society, November 15, 1582]. The elements of a determinant being restricted to a variation between ~(2unenX)F(2ueX) [2u=a,b,. ,ll. By supposing Fx broken up into proper simple fractions of the form the limits — a and + a, it is found that for all determinants whose order fx is greater than 2, a numerical maximum will be obtained by putting it is very easy to see that the theorem will be true in general the elements (— a) in the principal diagonal and making all the other (a — elements of the determinant —f-- a. If we denote such a determinant of if true for - and from this it is but a step to see that the theorem order n when a = 1 by D~, and the minor of an element in the ith row (a—x)i will be true in general if true for the simplest form of rational function, and kth column by Ask, we have always Au =D~—i, ~ so 1 that (l—x)i~ Dfl=— (1 ~). D~~=+ (1 ~) (1 D~2 All then that remains to do is to show that the coefficient of nfl in this + ~ +~j). 1 eflx fraction is the same as the coefficient of — in which may be w (l~e—x) done as follows: —4- (n—2)2n—i, since D 3==’ 4. The sign will be ~ according as n is odd even lessen the greatest factor The effect of a change in any element is to leaving the rest the same. 1a~, we have merely to make A 1 2B + 1.2.SG-YY--+.. .}-j--positsvepowersofx. allForthethestratamaximumidenticalcubicand equaldeterminantto D$ 2~.D$~Its value is j n I D$Ya~. The 4-dimensional determinant may5 be looked upon as a sort of deter- 1±nx+ ~ ___ 1. minant of plane determinants, the terms of the development being cubic Therefore the coefficient of — us x (t c—b)i determinants of which the plane determinants are the strata. In this 2+Gn3+...) quasi-determinant the same rule of signs applies as in the ordinary deter- ==(1+An+Bn minant and we get for the maximum value accordingly ~ ~ —4- — (n+l)(n—I—’2) ... (n—j--i—l) ffi~tfei 1 Similarly, = -i-. This method of proof, however, is not the simplest or best; as soon as we mould the theorem into a form most easily admitting of being ex- In any of these determinants we have at once a formula for the maxi- pressed in general terms that very form itself suggests a simpler (nay, so mum number of positive terms in. the development. This is for D$~~: to say, an instantaneous) proof, and moreover relaxes an unnecessarily stringent condition in the previous statement of the theorem. 4- Of course by a finite infinity root of a functidn no one can fail to under- 2 stand a value of the variable differing from zero which makcs the function We use the h sign according as D$,~’1 is ±, infinite.