JOHNS HOPKINS UNIVERSITY. Opened for Instruction in 1876. PLAN OF THE CIRCULARS. The Johns Hopkins University Circulars are published at convenient intervals during the academic year for the purpose of communicating intelligence to the various members of the University in respect to work which is here in progress, as well as for the purpose of promulgating official announcements from the governing and teaching bodies. During the current academic year, successive circulars may be expected in the months of January, February, March, April, The foundation of the University is a capital, May, and June, to be followed at the close of the The Johns Hopkins University was founded by the munificence of a citizen of Baltimore, Johns Hopkins, who bequeathed the most of his large estate for the establishment of a University and a Hospital. It was intended that these institutions should cooperate in the promotion of medical education. The Hospital buildings are approaching completion. in land and stocks, estimated in value at more than $3,000,000; the capital of the Hospital is not less in amount. The University was incorporated under the laws of the State of Maryland, August 24, 1867. Power to confer degrees was granted by the Legislature in 1876. Suitable buildings have been provided in Baltimore at the corner of Howard and Little Ross Sts., and are furnished with the necessary apparatus and books. ACADEMIC STAFF, 1883-4. DANIEL C. GILMAN, LL. D., President of the University. CHARLES D. MORRIS, A. M., Collegiate Professor of Latin and IRA REMSEN, M. D., PH. D., Professor of Chemistry and Director of the Chemical Laboratory. HENRY A. ROWLAND, PH. D., Professor of Physics and Director of the Physical Laboratory. J. J. SYLVESTER, F. R. §., D. C. L., Professor of Mathematics. JOHN S. BILLINGS, M. D., Lecturer on Municipal Hygiene. JAMES BRYCE, D. C. L., Lecturer on Roman Law. J. THACHER CLARKE, Lecturer on Classical Archaeology. HIRAM CORSON, A. M., LL. D., Lecturer on English Literature. G. STANLEY HALL, PH. D., Lecturer on Psychology. J. RENDEL HARRIS, A. M., Lecturer on New Testament Greek. H. VON HOLST, PH. D., Lecturer on History. GEORGE S. MORRIS, A. M., PH. D., Lecturer on the History of CHARLES S. PEIRCE, A. M., S. B., Lecturer on Logic. erature. WILLIAM TRELEASE, S. B., Lecturer on Botany. HERBERT B. ADAMS, PH. D., Associate Professor of History. MAURICE BLOOMFIELD, PH. D., Associate Professor of Sanskrit. WILLIAM K. BROOKS, PH. D., Associate Professor of Morphology and Director of the Chesapeake Zoological Laboratory. THOMAS CRAIG, PH. D., Associate Professor of Applied Mathe matics. CHARLES S. HASTINGS, PH. D., Associate Professor of Physics A. MARSHALL ELLIOTT, A. M., Associate in Romance Languages. RICHARD T. ELY, PH. D., Associate in Political Economy. GEORGE H. WILLIAMS, PH. D., Associate in Mineralogy. CHARLES F. RADDATZ, Examiner in German. G. THEODORE DIPPOLD, PH. D., Instructor in German. EDWARD M. HARTWELL, M. D., PH. D., Instructor in Physical Culture. HUGH NEWELL, Instructor in Drawing. HENRY A. TODD, A. B., Instructor in Romance Languages. H. H. DONALDSON, A. B., Assistant in Biology. OTTO LUGGER, Assistant in Biology. HARRY F. REID, A. B., Assistant in Physics. EDWARD H. SPIEKER, PH. D., Assistant in Latin and Greek. year by an Index. HOPKINS HALL LECTURES. Notice in Respect to the Admission of the Public. In answer to inquiries, and in correction of some current misapprehensions, the following statements are made in respect to these courses of lectures annually given in the Johns Hopkins University. 1. These courses are academic lectures, designed primarily for the members of the University, and supplementary to the regular class-room work of the students. 2. As the members of the University rarely require the the Trustees have taken great pleasure in inentire room, viting other persons, not connected with the University, to attend. 3. As these lectures are not intended for popular entertainment, but for the instruction of students, those persons first receive tickets, in most cases, who are known to be especially interested in a particular course,-ladies as well as gentlemen. Preference is thus given according to the character of the course, to teachers in other institutions, public and private; students of medicine, law, etc.; professional men and others. If any tickets remain undistributed, they are given out to those who may have applied for them, in order of application. 4. The hall is full when 200 hearers are present; it is uncomfortable if more are admitted. Not infrequently two or three times that number of persons apply for admission, and often applications for tickets cannot be granted. To give the lectures elsewhere would alter their character as a part of the ordinary academic work of the University. 5. There is no general course ticket issued. Applications should state specifically the course for which tickets are desired. Programmes and other current information pertinent to university work may be found in the University Circulars, sent to subscribers on the payment of one dollar per annum, either by Messrs. Cushings & Bailey, Messrs. John Murphy & Co., or the University. The usage of giving personal notification is not likely to be continued, and those therefore who have been accustomed to receiving such announcements, should hereafter consult the Circulars. 6. It will save much delay if applications for tickets and inquiries on these and other routine matters are addressed not to individuals but to the JOHNS HOPKINS UNIVERSITY, by postal card, and answers will be promptly returned by mail. Personal applications consume time needlessly. The lectures begin at 5 o'clock punctually. The doors of the hall are opened at fifteen minutes before 5, and the lectures do not exceed an hour in the delivery. PUBLICATIONS ISSUED UNDER THE AUSPICES OF THE UNIVERSITY. 1. American Journal of Mathematics. The publication of this journal commenced in 1878, under the editorial direction of Professor Sylvester. Five volumes of about 400 pages each have been issued, and the sixth is in progress. It appears quarterly, in the quarto form. Subscription $5 per year. Single numbers $1.50. II. American Chemical Journal. This journal was commenced in 1879, with Professor Remsen as editor. Five volumes of about 450 pages each have been issued, and the sixth is in progress. It appears bi-monthly. Subscription $3 per year. Single numbers 50 cts. III, American Journal of Philology. The publication of this journal commenced in 1880, under the editorial direction of Professor Gildersleeve. Three volumes of about 570 pages each have been issued, and the fourth is in progress. It appears four times yearly. Subscription $3 per volume. Single numbers $1.00. IV. Studies from the Biological Laboratory. [Including the Chesapeake Zoological Laboratory.] The publication of these papers commenced in 1879, under the direction of Professor Martin, with the assistance of Dr. W. K. Brooks. Two volumes of about 500 pages, octavo, and 40 plates each, have been issued, and the third is in progress, V. Studies in Historical and Political Science. The publication of these papers was begun in 1882, under the editorial direction of Dr. H. B. Adams. The first volume of 470 pages is now completed, and the second is in progress. Subscription $3 per volume. The following publications are also issued by the University: The UNIVERSITY CIRCULARS. Subscription $1 per year. The ANNUAL REPORT presented by the Presi dent to the Board of Trustees reviewing the operations of the University during the past academic BALTIMORE, JANUARY, 1884. CURRENT INTELLIGENCE. The University Circulars are intended to include notes of progress in the work of the Johns Hopkins University as a whole, and in the several laboratories, seminaries, and associations which are directed by the principal teachers. They are not designed to be a magazine for long articles. The University contributes liberally to the maintenance of six journals devoted to the study of Mathematics, Chemistry, Biology, Physiology, Philology, and History-and there are many other publications in this country through which extended memoirs may reach the public. Under these circumstances it is deemed best by the authorities to reserve these pages for official announcements, and for concise statements respecting researches which have here been made in the different branches of science and literature. There is ample space for such communications. DEPARTURE OF PROFESSOR SYLVESTER. Professor Sylvester, who has been the Professor of Mathematics in this University since its opening in 1876, has left Baltimore to take the chair of Savilian Professor of Geometry in the University of Oxford, to which he was elected December 5, 1883. A farewell assembly of gentlemen and ladies was held in Hopkins Hall December 20, when addresses were made by the President of the University, Judge Brown, Professor Gildersleeve, and Professor Sylvester. Among the guests were Mr. Matthew Arnold, Professor Newcomb, Professor Asaph Hall, Professor Hilgard, of Washington, Mr. Scudder, the Editor of Science, and other p distinguished persons from a distance, besides a large number of the literary, scientific, and professional men of Baltimore. Before the addresses which pertained to the departure of Professor Sylvester, Mr. Matthew Arnold (who had at an earlier hour given a public lecture in the Academy of Music), was presented to the assembly, with a complimentary allusion to his former connection with the University of Oxford, and to the fact that Professor Sylvester had dedicated to him the "Laws of Verse," published in 1870. Mr. Arnold made a friendly response. Mr. Sylvester's remarks were full of pleasant allusions to his work at the Johns Hopkins, his esteem for his colleagues, and his good wishes for the foundation. The address was not written and occupied nearly an hour in delivery. - On the afternoon of December 20, the academic staff of the University met in Hopkins Hall, by invitation of the President, and after a brief review by Dr. Story of the mathematical lectures here given from 1876 to 1883, and a like review by Dr. Craig of the contributions printed in the American Journal of Mathematics, Professor Gildersleeve read the following paper, which, on motion of Professor Rowland, was adopted by the meeting as an expression of their respect and good will. "The teachers of the Johns Hopkins University, in bidding farewell to their illustrious colleague, Professor Sylvester, desire to give united expression to their appreciation of the eminent services he has rendered the University from the beginning of its actual work. To the new foundation he brought the assured renown of one of the great mathematical names of our day, and by his presence alone made Baltimore a great center of mathematical research. "To the work of his own department he brought an energy and a devotion that have quickened and informed mathematical study not only in America, but all over the world; to the workers of the University, whether within his own field or without, the example of reverent love of truth and of knowledge for its own sake, the example of a life consecrated to the highest intellectual aims. To the presence, the work, the example of such a master as Professor Sylvester, the teachers of the Johns Hopkins University all owe, each in his own measure, guidance, help, inspiration, and in grateful recognition of all that he has done for them and through them for the University, they wish for him a long and happy continuance of his work in his native land; for themselves the power of transmitting to others that reverence for the ideal which he has done so much to make the dominant characteristic of this University." -The Savilian Professorship of Geometry in the University of Oxford, was founded and endowed in the year 1619, by Sir Henry Savile, Knight, Warden of Merton College. The Professors may be chosen from any part of Christendom, provided they are persons of good character and repute, well skilled in mathematics, and twenty-six years of age; if Englishmen, they must be M. A., at the least. The following list gives the names of the holders of the Professorship since its foundation: 1619. Henry Briggs. 1631. Peter Turner. 1649. John Wallis. 1704. Edmund Halley. 1742. Nathaniel Bliss. 1765. Joseph Betts. 1766. John Smith. 1797. Abram Robertson. 1827. Baden Powell. (From the Honors Register of the University of Oxford, 1883). The construction of a new building, in order to provide the members of the University with facilities for gymnastic exercises, was authorized by the Board of Trustees early in June last. Work in accordance with that vote was actively pushed forward from the middle of June until December 7, when the completed building, with its fittings and furniture, was opened to the public for inspection. The gymnasium has been planned to meet the requirements of 250 persons, and especial pains have been taken to secure an abundance of light and air in the main hall and in the dressing rooms connected with it. Until further notice it will be open to all members of the University from 9 a. m. till 9 p. m., daily, except Sunday. Matriculate students will hereafter be required, in order to secure the degree of Bachelor of Arts, to take the course in Physical Culture. Announcements in detail concerning the nature and requirements of this course will be made at an early day. The ground plan of the building in which the gymnasium and dressing rooms are contained resembles in shape a letter turned thus. The main building of the abutting on Garden street is 104 feet in length, includes the gymnasium proper and a vestibule, and the wing of the extending from Garden street to the rear of Bentley Hall is nearly 85 feet in length. The entrance to the building is on Garden street at the junction of the main building and the wing of the г. The entrance is through a vestibule, out of which, upon the first floor, doors open into the main hall and into the private dressing rooms, while on the second floor at the head of a flight of steps, is the door of the Director's rooms, in which the physical examinations are made and recorded. The vestibule and gymnasium hall are in the new building; while the dressing and bath rooms, and the offices of the Director are in the wing. The main hall comprises a single room, open to the roof, and has a total height from floor to ridge pole of 43 feet. It has upwards of 3400 square feet of flooring; its walls, of painted brick, are 25 feet high and 18 inches thick; and each of its side walls contains 7 high and wide windows whose sills are seven feet from the floor. A very complete set of Dr. Sargent's developing appliances is ranged against the walls on the four sides of the room, and a variety of apparatus such as is usually found in gymnasiums has also been provided. In the dressing rooms a large number of private lockers are placed, besides bath tubs and set bowls supplied with hot and cold water. The gymnasium is a place for "body-building" and a place for recreation. It will be administered as far as possible in accordance with the plan followed by Dr. Sargent (in the Hemenway Gymnasium of Harvard Uni versity). The purpose is to give to each individual guidance and counsel based upon and determined by a careful examination of his physique. It is held that the end and aim of physical training is to enable the body to do with pleasure and ease all the work of which as a mechanism it is capable. E. M. H. 31 During the latter part of this academic year, after the return of Dr. G. Stanley Hall, a course of lectures will be given to graduate students and others who are preparing themselves for the work of teachers in colleges and high schools. The particulars in regard to this course are not fully arranged but the topics presented will be nearly as follows: 1. The Present State of University and Collegiate Instruction in this country, by D. C. GILMAN. 2. Recent Observations on Educational Foundations in Europe, by D. C. GILMAN. 3. The Educational Value of Grammar, by B. L. GILDERSLEEVE. 4. The Future Sphere of Classical Philology, by B. L. GILDERSLEEVE. 5. Educational Value of the Study of Chemistry, by IRA REMSEN. 6. What to Teach in Biology, by H. NEWELL MARTIN. 7. One Lecture by H. A. RoWLAND. 8. The Observational Element in Mathematics, by C. S. PEIRCE. 9. The a priori Element in Physics, by C. S. PEIRCE. 10. The naïve in Education, by H. WOOD. 11. Modern Methods in the Study of History, by H. B. ADAMS. 12. Methods of Comparative Philology as pursued to-day, by M. BLOOM FIELD. 13. The New Impetus given to the Study of Latin by the application of the Historical Method and by the Study of Inscriptions, by MINTON WARREN. 14. Hygiene in Collegiate Training, by E. M. HARTWELL. 15. Rhythm and Education, by G. STANLEY HALL. 16. The Educational Value of Specialization and Original Work, by G. STANLEY HALL. 17. On the Uses of Libraries in Education, by D. C. GILMAN. Assemblies of the matriculated students have been held on successive Fridays during the current session, under the direction of the President of the University. Informal lectures on subjects pertinent to collegiate life have been given by various members of the academic staff, and on one occasion the members of the class in History brought forward the results of some of their studies. These assemblies will be continued during the remainder of the academic year. The Matriculated Students of this university including those who are candidates for the degree of B. A. are more than fifty in number, and are most of them from the State of Maryland. Several of them live in the neighborhood of Baltimore and are absent from their homes during the whole day. For the benefit of all this company of students, two rooms have been set apart by the Trustees and appropriately furnished,—one to be a place for quiet study, where dictionaries and other reference books may be found; and the other to be a parlor for conversation, and the reading of current newspapers and magazines. To promote the interests of this important body of students, and to secure the proper care of these rooms, an Association has been formed under the name of "The Matriculate Society." Its officers are: President, W. L. Glenn; Vice President, E. G. Miller, Jr.; Secretary, G. D. Penniman; Executive Committee, the President, Vice-President, Secretary, A. C. Woods, C. H. Howard, L. Williams, and J. Hinkley; Committee on Rooms, E. G. Miller, Jr., W. R. Orndorff, J. M. Horner; Committee on Literary Exercises, A. C. Woods, W. K. Williams, J. S. Hodges; Committee on Entertainments, G. D. Penniman, G. G. Carey, Jr., R. H. Bayard. Dr. CHARLES S. HASTINGS, Associate Professor of Physics, who has been connected with this institution since 1876, has accepted an invitation to the chair of Physics in the Sheffield Scientific School of Yale College, and will leave Baltimore at the close of the current academic year. His resignation was presented to the Trustees at their meeting in December, and a resolution was adopted expressing their acknowledgments for his past services and their regrets at his prospective departure. Mr. W. P. DURFEE, who received the degree as Doctor of Philosophy here in June last, has been appointed Instructor in charge of the department of Mathematics, in Hobart College, Geneva, N. Y. Mr. E. W. BEMIS has resigned his graduate scholarship here, to accept a position upon the editorial staff of the St. Paul (Minnesota) Pioneer Press. Dr. WALTER B. PLATT, after passing the usual examinations was admitted to the degree of Fellow in the Royal College of Surgeons in London, November 24, 1883. The American Journal of Mathematics for December, 1883 (Vol. VI, No. 2), contains papers: On Compound Determinants, by C. A. Van Velzer;-Note on Weierstrass' Methods in the Theory of Elliptic Functions, by A. L. Daniels ;— On Quadruple Theta-Functions, by Thomas Craig;-Principles of the Solution of Equations of the Higher Degrees, with Applications, (continued), by George Paxton Young;-Resolution of Solvable Equations of the Fifth Degree, by George Paxton Young;-On Certain Possible Abbreviations in the Computation of the Long-Period Inequalities of the Moon's Motion due to the Direct Action of the Planets, by G. W. Hill;Semi-Invariants and Symmetric Functions, by Capt. P. A. MacMahon, R. A;-A Graphic Method of Solving Spherical Triangles, by Charles H. Smith;-Extract from a letter to Mr. Sylvester from M. Hermite. The American Chemical Journal for November, 1883, (Vol. V, No. 5,) contains the following papers: On the Use of Mercury Thermometers with Particular Reference to the Determination of Melting and Boiling Points, by J. M. Crafts;-On the Action of Aldehydes on Phenols, by Arthur Michael and A. M. Comey; Laboratory Notes, III, by Arthur Michael;-Researches on the Complex Inorganic Acids, by Wolcott Gibbs ;-Contributions from the Chemical Laboratory of Harvard College;-Communications from the Chemical Laboratory of Rutgers College. No. XII. of Studies in Historical and Political Science (Local Government and Free Schools in South Carolina, by B. J. Ramage), has just appeared. This completes the first series, which comprises 470 pages and twenty distinct papers collected in twelve special groups. Subscribers are furnished with a complete Index to the first volume of the Studies and with a general title-page, including the special sub heading, Local Institutions, which will serve to characterize the contents of the first volume. A second Series of University Studies, comprising about 500 pages in twelve monthly monographs, devoted to Institutions, Economics, and Politics, is offered to subscribers at the former rate, $3. As before, a limited number of Studies will be sold separately, although at higher rates than to subscribers for the whole set. The first number of the New Series will shortly be published. It is entitled "New Methods of Study in History," and was read before the American Social Science Association, at Saratoga, September 4, 1883. The very limited number of complete sets of the First Series now remaining in the hands of the Publication Agency of the University compels the announcement that no further subscriptions for this volume can be received at the original rate. The few sets on hand, bound in cloth, will be sold at $5 net, by the Publication Agency only. The future interests of the work represented by this journal will require the Agency to give preference, in disposing of the remainder of the First Series, to libraries, specialists, and other patrons who are likely to prove continuous subscribers to the Studies. A synopsis of the contents of American Journal of Philology, for October, 1883, (Vol. IV, No. 3,) is given on page 44. SCIENTIFIC COMMUNICATIONS Most or all of which have been read before Societies or Classes in this University. MATHEMATICS. On the Three Laws of Motion in the world of Universal Algebra. By PROFESSOR SYLVESTER. [Substance of a lecture given on December 7, 1883, to the Mathematical Fellows at the Johns Hopkins University]. In the preceding Circular allusion was made to the three cardinal principles or conspicuous landmarks in Universal Algebra; these may be called, it seems to me (without impropriety), its Laws of Motion, on the ground that as motion is operation in the world of pure space, so operation is motion in the world of pure order, and without claiming any exact analogy between these and Newton's laws, it will be seen that there is an element in each of the former which matches with a similar element in the latter, so that there is no difficulty in pairing off the two sets of laws and determining which in one set is to be regarded as related by affinity with which in the other. They may be termed the law of concomitance or congruity, the law of consentaneity and the law of mutuality or community. The law of congruity is that which affirms that the latent roots of a matrix follow the march of any functional operation performed upon the matrix, not involving the action of any foreign matrix; it is the law which asserts that any function of a latent root to a matrix is a latent root to that same function of the matrix; in so far as it regards a matrix per se, or with reference solely to its environment, it obviously pairs off with Newton's first law. The law of consentaneity, which is an immediate inference from the rule for combining or multiplying substitutions or matrices, is that which affirms that a given line (or parallel of latitude) can be followed out in the matrices resulting from the continued action of a matrix upon a fixed matrix of the same order, i. e. in the series M, mM, M2m, m3M, (which may be regarded as so many modified states of the original matrix) without reference to any other of the lines or parallels of latitude in the series, or again any column or parallel of longitude in the correlated series M, Mm, Mm2, . . . without reference to any other such column or parallel of longitude. An immediate consequence of this obvious fact (a direct consequence for the rule of multiplication) obtained by dealing at will with either of the systems of parallels referred to, is that a system of simultaneous linear equations in differences may be formed for finding each term in any given line or in any given column at any point in the series, and the integration of these equations leads at once to the conclusion that any term of given latitude and longitude in the ith term of either series is a syzygetic function of the ith powers of the latent roots of m. If, then, M be made equal to multinomial unity, this at once shows that supposing to be the order of m, on substituting m for the carrier (or latent invariable) in the latent function to m, and multiplying the last term by the proper multinomial unit, the matrix so formed is an absolute null, which proves the proposition concerning the "identical equation first enunciated by Professor Cayley in his great paper on Matrices in the Phil. Trans. for 1858. This proposition admits of augmentation, 1°, from within, as shown in a former note, by applying to it the limiting law of the nullity of a product (a branch of the 3d law), which leads to the very important conclusion that the nullity of any factor of the function of a matrix which is an absolute null, or more generally of any product of powers of its linear factors, is exactly equal to the number of distinct linear factors which such factor or product contains, at all events, in the general case where the latent roots are all unequal; and 2°, from without, by substituting for m, m + en where n is any second matrix whatever and e is an infinitesimal. This leads to the catena of identities, to which allusion has been made in the preceding Circular. Then, again, the endogenous growth of the theorem (that which determines the exact nullity of any factor of the left hand side of the identical equation) in its turn seems to lead to a remarkable theorem concerning the form of the general term of any power of m into M. Observe that every such term is expressed as a syzygetic function of powers of the latent roots, and contains, therefore, a constants, so that the total number of syzygetic multipliers is ; but the number of variables in m and M together is 22; and, consequently, apart from the w arbitrary latent roots the number of independent constants in miM should be 2w2 ω. The 3 syzygetic multipliers ought then to contain only (21) arbitrary constants, and such will be found to be the case by virtue of the following hypothetical theorem: Calling 2 any one of the λ latent roots, the multipliers of λi in miM will form a square of 2 quantities; the theorem in question is that every minor of the second order in such square is zero, so that the w terms in the square is given when the bounding angle containing 2-1 terms is given; and the same being true for the multipliers of each latent root (which restore themselves into squares) the number of arbitrary quantities in all is w(2w—1) as has to be shown. The law of consentaneity in so far as it relates to the decomposition of the motion of a matrix into a set of parallel motions, has an evident affinity with Newton's second law.† Remains the law of mutuality which is concerned with the effect of the mutual action upon one another of two matrices and so claims kindred with Newton's third law. This law branches off into two, one which may be termed the law of reversibility, the other that of co-occupancy or permeability. The law of reversibility affirms that the latent function of the product of two matrices is independent of the sense in which either of them operates upon the other, i. e., is the same for mn as for nm, just as the kinetic energy developed by the mutual action of two bodies is not affected by their being supposed to change places. As regards the second branch of the third law, the word co-occupancy refers to the fact that although the space occupied by two similarly shaped figures (say two spheres) is not absolutely determined (in the absence of other data) by the spaces occupied by them each separately (for they may intersect or one of them coincide with or contain the other) a superior as well as an inferior limit to such joint occupation is so determined; the inferior limit being the space occupied by either such figure, i. e., the dominant of these two given spaces, and the superior limit their arithmetical sum. So the nullity resulting from the action in either sense of two matrices upon one another is not given when their separate nullities are assigned but has for an inferior limit the dominant of these two nullities and for a superior limit their sum; the nullities of the two component matrices may also be conceived under the figure of two gases or other fluids which are mutually permeable and capable of occupying each other's pores. Although the limits spoken of are independent of the sense in which the two matrices act on one another, it must not however be supposed that the actual resultant nullity is unaffected by that circumstance; thus, ex. gr., if the latent roots of a ternary matrix m are 2, 27, 2, the nullity resulting from (m — 2) (m — 2′) acting sinistrally upon (m-2'')n, i. e., of (m 2)(m — 2') (m —2)n is 3 but from the same acting dextrally upon the same, i. e., of (m2)n(m2) (m—2') need not necessarily exceed 2. Such then are the three primary Laws of Algebraical Motion; but as Conservation of arcas, Vis viva, D'Alembert's Principle, the principle of Synchronous Vibrations of Least action, and various other general laws may be deduced from Newton's three ground laws, so, of course, various subordinate but very general laws may be deduced from the interaction of the above stated three ground laws, viz., the law of Congruity, the law of Consentaneity, and the law of Mutuality. The deduction of the catena of identical equations connecting two matrices m and n from the second and third laws combined, affords an instance of such derivative general laws. Another instance of the same is * I have not had leisure of mind, being much occupied in preparing for my departure, to reduce this theorem to apodictic certainty. I state it therefore with all due reserves. For another and closer bond of affinity between the two laws see concluding paragraph of this note. the theorem that when the product resulting from the action upon one another of two matrices, is the same in whichever of the two senses the action takes place, the matrices must be functionally related, unless one of them is a scalar, i. e. a multiple of multinomial unity. This very important and almost fundamental law (seemingly so simple and yet so hard to prove) may be obtained as an immediate inference from that identical equation in the catena of such equations connecting the matrices m and n, in which one of the two enters only singly at most in any term. As ex. gr. of m and n are of the 3d order, the identical equation m3-3bm2 + 3dm —ƒ=0 gives immediate birth to the identity m2n+mnm+nm2 — 3b (mn + nm) + 3cm2 + 3dn+3em—3g=0. But if nmmn then mnm= = m2n, nm2 = mnm = m2n, so that this equacm2 em +9 * tion becomes m3n — 3bmn + dn = m2c — em +g, or n= m2 - 2bm + d' The first branch of the third law, viz., the law of reversibility is an almost immediate inference from the rule for the multiplication of matrices, and becomes intuitively evident when the process of multiplication in each of the two senses between m and n is actually set out. The second branch, viz., the law of co-occupancy or permeability, as it is the most far-reaching so it is the most deep seated (the most caché) of all the primary laws of motion. I found my proof of it upon the fact that the value of any minor determinant, say of the ith order, in either product of m and n (two matrices of the order ) may be expressed as the quantitative product of a certain couple of rectangular matrices (in Cauchy's sense of the term), of which one is formed by i columns and the other by i lines in the two given matrices respectively. Such rectangle as shown by Cauchy (and as may be intuitively demonstrated by the simplest of my π(W) umbral theorems on compound determinants) is the sum of the π(w — i)ñi complete determinants of the one rectangle multiplied respectively by the corresponding complete determinants of the other rectangle. This shows at once the truth of the proposition in so far as relates to the lower limit, i. e., that if mn = p, and m, n have the nullities ɛ, 5, and p the nullity 0, then @ must be at least as great as e and at least as great as 5. As regards the superior limit the proof is also founded on the theorem in determinants already cited, and the form of it is as follows. If ε be any number r, it may be shown that must be at least as great as 0. -r; hence giving r all values successively from 0 to 5-1, it follows that ɛ+5 cannot be less than 6, i. e., that @ cannot be greater than & +5. 1 The proof of the first law, that of concomitance or congruity, I ought to have stated antecedently, is a deduction from the theory of resultants and the well-known fact that the determinant of a product of matrices is the product of their determinants. Thus each of the three laws of motion is deduced independently of the two others. As another example of a derivative law of motion, I may quote the very notable one which results from the interaction of the first and second fundamental laws upon one another and which gives the general expression for any function whatever of a matrix in the form of a rational polynomial function of the same and of its latent roots, to wit, the magnificent theorem that whatever the form of the functional symbol ø, and whether it be a single or many valued function, if 21, 22, ... 2w be the latent roots of m, 4m = £67, (m— ??) (m — 73) . . . (m —7w) As ex. gr. if om = m2 (21 — 72 (21 — 2.3) . . . (^1 — 2w)" ... mi will have q roots which are completely determined by the above formula. The first law, as already stated, regards a single body or matrix, uninfluenced by the action of any external force. The second law regards the effect upon a single matrix, subject to external impulses, taking their rise in an external source; whilst the third law has regard to the mutual action or joint effect of two bodies or matrices simultaneously operating upon one another. * Whence it follows that a must be a function of m convertible into an integral polynomial form, unless the numerator and denominator of the fraction to which a is equated vanish simultaneously, which is what happens when m is scalar. If the numerator exactly contains the denominator n becomes a scalar. Seeing that a constant e is a specialized case of a function of a variable x although the converse is not true, we may say that whenever nm = mn, one at least of the two matrices m and n is a function of the other, and that each is a function of the other unless that other is a scalar. Compare Clifford's "Fragment on Matrices" in the posthumous edition of his collected works. On Involutants and other allied species of Invariants to Matrix Systems. By PROFESSOR SYLVESTER. [Continued from page 12 of Circular No. 27]. By the theorem proved at the beginning of this note, the nullity of M and that of N are each — 1, hence the nullity of MN and consequently à fortiori its vacuity cannot be less than 1, and accordingly the identical equation to MN may be written under the form (MN) (MN)-1=0, where H is the sum of the product of each element in the Matrix M or the Matrix N multiplied by its altruistic opposite in the other. Suppose now that I= 0 then for some one system of 2, μ out of the systems given by the equations F= 0, G= 0, H must vanish (for the nullity and à fortiori the vacuity of MN in that case becomes w) hence the double norm of H, i. e., the product of the values of H, or which comes to the same thing, the resultant of F, G, H must vanish when I vanishes and must therefore contain I; in like manner because the nullity of NM and à fortiori its vacuity is w when J=0, it follows that the same resultant, say R, must contain also J; R will therefore contain IJ, from which it may readily be concluded that it can differ from IJ, if it differ at all, only by a numerical factor. I need hardly pause to defend the assumption that I, J have no common factor, and that it is the first and not necessarily any higher power of R 100 0k 0 Op 0, n=00 k, of I, k00 00 p2 which contains IJ; the single instance, when m= J being respectively the cubes of k3 — p and k3 — p2 which have no common factor, settles the first part of this assumption at all events for the case of w= 3, and as regards the second, it is only necessary to show that neither I nor J is equal to, or contains a square or higher power of a function of the letters in m and n as may be done easily enough when w=3 by another simple instance.* We may then at once proceed to compare the dimensions of R with those of I and J. R being the product of values of 2-1 μ-1+ etc., where 21. μ are codimensional with the elements in m and ʼn respectively is obviously of the degree w3. (w — 1) in regard of each set of elements, i. e., of the degree 2w2 (w — 1) in regard of the two sets taken together. Consider now the degree of I; this is the topical resultant of matrices of the form m3.n3, where i = 0, 1, 2, . . . w, j = 0, 1, 2, . . . w, so that each term in I will consist of a combination of elements selected respectively ... 2 from these matrices. If w is even, there will be pairs of matrices, one of any such pair of the form mini, the other of form mw-1-i, nw-1-), and the combination of elements taken from any such pair will be of the collective degree 2(-1) in the two sets of elements, so that the total degree of the Involutant will be. 2 (−1) or (w—1). If again w is odd, there will be w2+1 such pairs, and one factor (unpaired) belonging 2 2 *Limiting ourselves to the case of matrices of the third order, if we take for mn the 060 OBO matrices d 0f, DOF, it may be shown by direct computation that one of the Invol 040 оно utants becomes (bH — AB)2( ƒD — dF) 2 (bd+fh}{BD — FH)(dB—ƒH) {(h F + 6D) 2 — (bd +fh)(BD+FH)} and consequently if there were any square factor in either involutant such factor would contain the elements belonging to the two sets indecomposably blended, but on the other 100 0fF hand, if we take for mn the matrices 0 p 0, G0g, either involutant to m, n may 00p hHO easily be shown (also by direct computation) to be made up of three factors, each of which is an indecomposable cubic function of f, g, h, F, G, H. Hence it follows that neither involutant can in its general form contain any square factor. As a matter of fact, not only for ternary matrices but for matrices of any order, there can be no reasonable doubt whatever in any sane mind that every Involutant is absolutely indecomposable. One must try, however, to obtain a strict proof of this upon the general principle of crushing every logical difficulty regarded as a challenge to the human reason, which falls in our way; it is in overcoming the difficulties attendant upon the proof of negative propositions that the mind acquires new strength and accumulates the materials for future and more significant conquests. To prove that involutants in their general form are indecomposable may possibly, I imagine, prove to be a hard nut to crack, or it may be exceedingly easy. |