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1~ Iathematics. Equations in Matrices. by JJ SYLVESTER

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1~ Iathematics. Equations in Matrices. by JJ SYLVESTER JOHNS HOPKINS UNIVERSITY CI RCULARS Pub/is/zed wit/i t/ie approbatiou of tAe Board of Trustees VOL. IlL—No. 32.] BALTIMORE, JULY, 1884. [PRIcE, 10 CENTS. CALENDAII, 1884-8~. Tuesday, September 23. Academic Year Begins. Tuesday, September 23. Examinations for Matriculation Begin. Tuesday, September 30. Instructions Resumed. Friday, June 12. Academic Year Closes. CONTENTS. PAGE NOTES AND CO1~D1UNLCATIONS: Equations in Matrices. By J. J. Sy1vester,~ - - - - - - - - - - - - Note on Peirce’s Linear Associative Aigebra. By A. Cayley, - - - - - - - - - - - 122 On the Equations which Determine the Axes of a Qnadric Surface. By W. E. Story, - - - - - - - 122 Some Remarks on Unicursal Curves. By E. W. Davis, - - - - - - - - - - - 123 Note on Cycles. By A. S. ilatliaway, - - - - - - - - - - - - - 123 Note on Lines of Curvature. By &~. Bissing, - - - - - - - - - - - - - 124 On K. Br~igmann’s recent Grammatical Studies. By C. D. lilorris, - - - - - - - - - - 124 A Study of Dinarchus. By E. G. Sililer, - - - - - - - - - - - - - - 124 Parallelism in Beowuif. By C. B. Wright, - - - - - - - - - - - - 124 On the Dialectic Equivalence of a to ~ in Proto-Babylonian. By C. F. Lehmann, - - - - - - - - 125 Rhythmical Pronunciation of Greek and Latin Prose, etc. By C. W. E. Miller, - - - - - - - - 125 On Inchoative or a-Verbs in Gothic and other Germanic Dialects. By A. E. Egge, - - - - - - - 12~ The City of Ilarran. By C. Adler, - - - - - - - - - - - -. - - - 12~i Christian Mosaic-painting in Italy. By A. L. Frothingliani, Jr., - - - - - - - - - - 127 The Significance of the Larval Skin in Decapods. By II. W. Coan, - - - - - - - - - - 127 On the Life History of Thalassema. By H. W. Conn, - - - - - - - - - - - - 127 The Coagulation of the Blood. By W. H. Howell, - - - - - - - - - - - - 128 On the Molluscan Gill. By H. L. Oshorn, - - - - - - - - - - - - - 128 Congressional Government. By Woodrow Wilson, - - - - - - - - - - - - 128 County Government in Virginia. By E. Ingle, - - - - - - - - - - - - 129 Samuel Adams, the Man of the Town-Meeting. By I. K. Hosmer, - - - - - - - - - - 129 Sir George Culvert, Baron of Baltimore. By L. W. Wilhelm, - - - - - - - - - - 129 State and Local Taxation in Kentucky. By Arthur Yager, - - - - - - - - - - - 130 On the Syllogism. By J. Rendel Harris, - - - - - - - - - - - - - - 130 On the so-called Quartz Porphyry at Hollins Station, Md., - - - - - - - - - - - - 131 RECENT APPOINTMENTS, - - - - - - - - - - - - - - - - 131 LECTURES ON CLASSICAL ARCII9EOLOGY, - - - - - - . - - - - - - 132 On the Excavations at Ass6s. By J. Tliaelier Clarke, - - - - - . - - - - - - 132 On Archuology aud Art, etc. By W. J. Stillman, - - - - - . - - - - - - 134 On Olympia, etc. By A. Emerson, - - - - - - - . - - - - - - 134 The Relations of Literary and Plastic Art. By B. L. Gildersleeve, - - - . - - - - - - 137 PROCEEDINGS OF SOCIETIES, - - - - . - - - - - - - - 137 The Johns Hopkins Unioersity Gircalars are printed by Messrs. JOHN ATURPHY ~ CO., 18fi? West Baltimore Street, Balti- more, from whom single copies may be obtained. They may also be procured, as soon as published, from Messrs. CUSHLNUS & BAILEY, No. 26~ West Baltimore Street, Baltimore. 122 JOfINS hOPKINS [No. 32. NOTES AND COMMUNICATIONS. By 1~IATHEMATIcs. Note on Peirce’s Linear Associative Algebra. A. CAYLEY. Equations in Matrices. By J. J. SYLVESTER. I find that to the double systems given by Peirce, viz., [Extract of a letter from Professor Sylvester to Dr. F. Franklin]. ~c p x y x y I have been lately considering the subject of equations in matrices. Sir (Os) xx y (b William Hamiltors in his.” Lectures on Quaternions” has treated the case 2) X x~y (cm) x y 0 of what I call unilateral equations of the form ~A+ px + g — 0, or sA + up y y 0 yO~0~ y~0 01 I ______ + q — 0, where we may, if we please, regard r, p, q as general matrices of the second order. He has found there are six solutions, which may be obtained should bejoined by the solution of an ordinary cubic equation. In a paper now in print and xy which will probably appear in the May number of the Philosophical Maga- (d5) x x 0 zine, I have discussed by my own methods the general unilateral equation, ‘yy 0. say To show that this is really distinct fromm~(btm2+), observecfl(xy +thatyx) +startingfltmym —fromotmx where x, p, q... 1, are quaternions or matrices of the second order, and have +(b2afly,) and writin~o(ox +z —fly):ax whence+ Ily, weinhave(he),z~z— o + fly, fi arbitrary, is the only shown, by a method satisfactory if not absolutely rigorous, that the number idempotent symbol, and y is the only nilpotent symbol. And z having this 3 — A + a, that is to say, the nearest superior integer to the of solutions is a general maximum number of roots (a4) divided by the augmented degree value — x + fly, we have / — z zy — y, yz —. 0, ym — 0; viz., in (bin) intro- (a-j-1). ducing the general symbol zin place ofx, the system in z, y retains its original But after I had done this it occurred to me that there were multitudinous form that is, we cannot in anywise transform (b failing cases of which neither Hamilton nor myself had taken account, as 2) into (d2). I have further considered the question in a paper communicated to the London Mathe- ex. yr. x~ + px — 0, besides the solutions ‘r — 0 r — — p, will admit of a solu- tion containing an arbitrary constant, I think; but that is a matter which I matical Society but not yet published. shall have to look further into before committing myself to a positive asser- Cambridge, England, May 12, 1884. tion about it. I have only had time to pass in review the more elementary case of a unilateral simple equation, say px q, where p, q are matrices of any order a. On the Equations which determine the Axes of a Ifp is non-vacuous there is one solution, viz.; x — p ‘q; but suppose p is vacuous: what is the condition that the equation may be soluble? Quadric Surface. By W. E. STORY. 10. Suppose q 0, p being vacuous has for its identical equation pP~ 0, [Abstract ofa communication to the University Mathematical Society, May 21, 1884]. and consequently we may make u XP where 2u is an arbitrary constant. 20. Suppose q is finite and that r is one solution, then obviously the If u1, um, sue, ..., u,, are homogeneous point (or tangential) coordinates in general solution is x r + 2~P. an (n— 1)-fold space, and We have now to inquire what is the condition that r may exist. I find Qua 2 •~; buscsscc 0 (ba bkt) from the mere fact of x being indeterminate (and confirm the result by is the eqm¶mtion of the absolute, where tIme summation extends to all integral another order of considerations) that the determinant of q + tp must vanish values of i and k from 1 to n, inclusive; the coordinates of the centres (or I/c principal sections) of the quadric surface whose equation is identically; so that for instance when p, q are of the second order and d ef Paw ~— 2521.amkusuc 0 (al/c Oki) are the parameters to the corpus (p, q), we must have when d — 0 which is are determined by the equations implied in the vacuity of p, f~ 0 and e 0. The first of these conditions is known dpriori immediately from my third law ofmotion; but not so, without (1) 21ai,cu; introducing a slight intervening step, the intermediate one (I mean the being n equations corresponding to the values k 1, 2, 3, .. ., ii respectively, in which 2~ is a parameter whose value is determined by the condition that connective to d andf, viz.) e — 0. So in general in order that px ±q 0 may be soluble, i. e. in order that equations (1) shall be coexistent for some set of values of the u’s, i. e. by time p’q wherep is simply vacuous may be Actual and not Ideal, q must satisfy as vanishing of time determinant of the n-th order whose constituents are many conditions as there are units in theor(ier ofp or q, all implied inthe fact a~k — 2.bg~. Let Fcc — 212i~ccmcrimcc, ~ that the determinant to p -1- ~q, where ?L is an arbitrary constant, vanishes identically. When these conditions are satisfied p 1q becomes actual but then Qc,c — 0 is tIme condition that the two points (or planes) v and sc sIsal] indeterminate. (This,by theway, shows the disadvantageof callinga vacuous be mutually perpendicular and QcwQincr — QcrQinrr 0 is the condition tlmat matrix indeterminate, as was done in the infancy of tIme theory by Cayley tIme lines joining the points me and ar respectively (or the intersections of and Clifford—for we want this word as you ~ee to signify a combination of the planes1m, ste..., andv err respectively) shmcll be mutually perpendicular. Now the inverse of a vacuous matrix with another which takes the combination let p, r,, 5 and v, S’~~ win, ..., w5 be any two different sets of solutions out of the ideal sphere and makes it actual). of equations (1), so that ~‘ and61kvs,v are different,2 then So in general in order that p mq where p is a null of the ith order (i. e. (2) 21ai,cci — y21 1cma.zci — v21bacmcs; where all the (i 1)th but not all the ith minors ofp are zero) shall be an + operating2/c~k onwetheobtain,first seton accountof these ofequationsthe symmetrywith 2/c/cof IA~andandon theQ,v,second set actual (although indeterminate) matrix, it is necessary and sufficient that with(3) Pew aQuc, ~ — m’Qcw, p + ?~q, where 2b is arbitrary, shall be a null of the same (ith) order. What and hence, since je and a are difibrent, will be the degree of indeterminateness in p ‘q, i. e. how many arbitrary (4) Pc,c~0, Qc,c~0, constants are contained in the value bf which satisfies the equation px — 0 ‘c.
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