by or sit astride of the vetebral processes, thus giving firm protection to the structures within and replacing the ribs. During embryonic life an anal fin is present, which however does not develop completely and disappears entirely in the adult. The gills are in reality not tufted but their appearance is due to the leaflets, of which there are four rows to each rachis, gradually increasing in size from below upwards until near the extremity when they suddenly decrease. The degeneration of the arches and the peculiar arrangement of the leaflets I imagine to be due to the almost complete closing in of the gill-cavity, which is evidently an old ancestral character, since even in the youngest stages the branchial-cavity was completely closed by a membrane. No communication exists between the intestine and the yolk-sack, and a valve is early formed by a constriction of the walls of intestine, which separates the rectum from the intestine proper. An Instance of Sexual Color Variation in Crustacea, by H. W. CONN. [Note from the Chesapeake Zoological Laboratory, 1883]. Differences in the color of the two sexes among Crustacea are of very rare occurrence. Darwin in The Descent of Man, chap. ix, refers to this fact and says he is acquainted with but two instances of this peculiarity. One in the case of Squilla stylifera, and a second in a species of Gelasmus, or fiddler crab, described by Fritz Müller as occurring in Brazil. A third and very striking instance is found in Callinectes (Neptunus) hastata, the common edible crab of our southern coast. There are a number of differences in the shape of the two sexes, but besides these they present a marked difference in color. This color variation is confined to the first pair of thoracic appendages, the pair bearing the large chelae. These appendages are of a yellowish brown on the upper surface, a whitish yellow on the outside and of a brilliant blue on the inside and particularly at those parts which are protected from the light when the appendage is folded. It would seem therefore that this blue coloration was enhanced by not being exposed to light. The color of different individuals is tolerably constant and uniform. Between the colors of the male and female appendage considerable differences are discernable. The most noticeable difference is that the male appendage appears remarkably blue when compared with the female. This is due partly to the fact that the amount of blue surface in the male is much greater than in the female, and partly to the fact that the blue color is of a much more brilliant hue. The blue color in the male extends nearly to the tips of the two fingers of the chelae, both the finger-like process of the propodite and the dactylopodite being largely colored blue. The very tips are, however, of a brilliant purple. In the female these parts are of an orange hue with not a trace of blue about them. Its tips are also colored purple but not so brilliant a purple as is found in the male. In the male the blue color extends partly upon the outer surface. In the female it is confined to the inner surface and only extends to the base of the dactylopodite. The outer surface of the dactylopodite and of the finger-like process of the propodite are in the male white, while in the female they are reddish orange. Upon the male appendage there is no orange color as a rule. These differences in color are in all cases very marked, and will always serve to distinguish a male from a female appendage. No color differences are seen in any part of the crab except upon the first pair of appendages, and it is interesting to note that this sexual difference does not make its appearance till the crab reaches maturity. The chelae of immature males and females cannot be distinguished from each other. Fritz Müller says that the same is true of the Gelasmus species observed by him. On the other hand, considering the habits of Crustacea, these sexual differences can hardly be considered as the results of sexual selection. Experiments upon the Heart of the Dog with reference to the Maximum Volume of Blood sent out by the Left Ventricle in a Single Beat, and the Influence of Variations in Venous Pressure, Arterial Pressure, and Pulse Rate upon the Work done by the Heart, by W. H. HOWELL and F. DONALDSON, JR. [Abstract reprinted from the Proceedings of the Royal Society of London, No. 226, 1883]. Owing to the indirectness of the methods hitherto used for estimating the quantity of blood pumped out from the left ventricle at each systole, this important factor in all calculations of the work done by the heart has never been satisfactorily determined. Volkmann, and afterwards Vierordt, from calculations based upon the mean velocity of the stream of blood in the unbranched aorta, obtained the fraction as representing the ratio between the weight of blood thrown out at each systole and the bodyweight. Fick, from data obtained by placing the arm in a plethysmograph, arrived at a much smaller fraction, roo. In our investigation we have made use of the dog's heart, completely isolated from all other organs of the body, with the exception of the lungs, after the method devised by Professor Martin. By this method it is possible to estimate directly the quantity of blood ejected from the left ventricle at each systole. With regard to the maximum quantity of blood which can be thrown out from the left ventricle at a single systole, the general result of the experiments may be stated as follows: With a mean pulse rate of 180 per minute, the mean ratio of the maximum weight of blood pumped out from the left ventricle at each systole to the body weight is or .0017. In one experiment in which the pulse rate was 120 per minute, about the normal rate, the ratio obtained was or .0014. In applying these results to the normal dog, we believe that the average quantity of blood pumped out from the left ventricle at each systole in the living dog, is approximated most closely in the experiments given by the maximum outflow obtained from the isolated heart. Variations of arterial pressure, from 58 to 147 millims. of mercury, were found to have no direct effect whatever upon the quantity of blood sent out from the left ventricle at each systole. Since the pulse rate is not altered, the work done by the left ventricle varies directly as the arterial pressure against which it works, within the limits named. For how much wider limits than those given this may hold true was not determined. There is every reason to believe that under normal conditions the force of the systole is more than sufficient to completely empty the ventricular cavity, and since, with arterial pressures from 58 to 147 millims., the quantity of blood ejected at each systole remains constant, it seems probable that within these limits, at least, the force of the ventricular contraction is not influenced by variations in arterial pressure, but remains maximal throughout. Variations of venous pressure on the right side of the heart influence in a marked manner the outflow from the left ventricle. As the general result of the experiments it was found that the outflow from the left ventricle, and consequently the work done by it, increases with the venous pressure, but not proportionally, up to the point of maximum work. Variations in the rate of beat of the heart were obtained by heating or cooling the blood supplied to it. The general result may be stated as follows: A diminution of pulse rate, brought about by lowering the temperature of the blood flowing into the heart, causes an increase in the quantity of blood thrown out from the ventricle at each systole, and consequently an increase in the work done at each systole, and vice versa. The changes in the outflow from the ventricle at each systole are not, however, inversely proportional to the changes in the pulse rate. The total outflow, and, therefore, the total work done during any given period of time, decreases with a diminished pulse rate, and increases with an increased pulse rate. NOTES IN PHILOLOGY. On the Exemplar of Cod. C in the Apocalypse, by J. R. HARRIS. [Abstract of a paper read before the University Philological Association, October 5, 1883]. In the Introduction to Westcott's and Hort's New Testament, the following remarks are made: "The transition from small portable MSS. of limited contents is strikingly illustrated by a fortunate accident in the transcription of one of the four great comprehensive MSS. which are the earliest now extant. In the MS. of the Apocalypse from which C was taken some leaves had been displaced, and the scribe of C did not discover the displacement. It thus becomes easy to compute that each leaf of the exemplar contained only about as much as ten lines of the present edition; so that this one book must have made up nearly 120 small leaves of parchment, and accordingly formed a volume to itself or without considerable additions." I propose to see how far the above suggestions will enable us to transfer the text of U into the pages of the exemplar. The manuscript of C runs on smoothly enough to the end of the eighth page, (pages 5 and 6 being lost), after which another leaf is absent containing pages 9 and 10. Both of these lost leaves seem to correspond in compass with the omitted text. Page 11 then follows, and after some lines of page 12, we are abruptly carried back from c. x. 10 to c. vii. 17 so that the text runs as follows: καὶ ἦν ἐν τῷ στόματί μου ὡς μέλι γλυκύ . καὶ ὅτε ἐ δάκρυον ἐκ τῶν ὀφθαλ μῶν αὐτῶν κτέ The scribe continues on the new strain until the close of viii. 4: when he returns as abruptly as he departed; but not, be it observed, to the place he left, but to c. xi. 3, more advanced by about 10 lines of Westcott and Hort's text, so that the manuscript runs ἐκ χειρὸς τοῦ ἀγγέλου ἐνώπιον τοῦ θεοῦ χιλίας After this the text runs on smoothly, with the loss of certain of its leaves to the end of the book. Evidently the passage inserted and the passage omitted are each of them pages of the exemplar. Moreover, since the lost leaves in which the transposed passage vii. 17-viii. 4. ought to have been found, correspond in compass to the missing text, it is evident that the two leaves of the exemplar with which we are concerned have been transposed. Now with regard to the size of these leaves, let us employ as our measuring line, for convenience, a sixteen-syllabled hexameter, and let the text have the abbreviations that commonly belong to the words cos Ἰησοῦς χριστὸς κύριος πνευμα and οὐρανός. The two leaves of the exemplar will then be found to be 11 and 10.5 verses respectively. If the writing had been perfectly even we should be able at once to form the roll of the exemplar by the process of subdivision; for from i. 1 to vii. 17 is evidently an exact number of pages, and from viii. 4 to x. 10 is also an exact number of pages. But the pages not being perfectly uniform, we must adopt the method of averages. We have, then, from i. 1 to vii. 17 by actual measure 373.5 verses, and from viii. 4 to x. 10 we have 134.3 verses; and if the average page be just over 11 verses, this would give 33 pages up to the first transposed leaves and 12 pages between the two variable leaves. The total would, upon the same supposition be 1213.3 verses, or about 110 complete pages. Now there is a very curious piece of transcriptional evidence which seems to show that not only C, but the first four or five verses of the Sinaitic codex were copied in the Apocalypse from a roll of this character. For Tischendorf notes that the first few verses of the book down to κаì áлò tv xũ in c. i. 5 were copied by a different scribe to the rest of the book, the same scribe. viz., who wrote six other conjugate leaves in the New Testament. The evidence for this statement consists in peculiar forms of the letters and in the use of the apostrophus at the close of lines. Assuming, as I think we may, the accuracy of Tischendorf's judgment, we can hardly avoid the conclusion that the portion of the book copied by the scribe was a single page of the exemplar, since he stops in the very midst of a sentence in the text, and at an irregular point in his own column. Examination of the portion of the text in question shows it to be 11.3 verses, or almost exactly the same as we were led to adopt as the average page in Codex C. It seems, therefore, that both these copies are derived from similarly written exemplars, and I think we may say that towards the restoration of this exemplar we have the 1st, 34th and 47th pages accurately defined, and the remaining pages to a close degree of approxi mation. I see no reason to assume that this was a parchment book; it has every appearance of being an ordinary paper-roll. I can see nothing in the text to account for the transposition of the two pages. The reconstructed roll may perhaps, by the study of its parallel columns, help to explain some of the many curious variants in the text of the Apocalypse. On Certain Irregular Vedic Subjunctives or Imperatives, by MAURICE BLOOMFIELD. [Abstract of a paper read before the University Philological Association, October 5, 1883]. The Vedic forms referred to are those of the types stota, guhóta, krnóta; étana; kárta, íyarta; gánta; dádāta; unátta — appearing at first sight to be so-called 'improper subjunctives,' but having irregularly strong stemforms with accent on the stem, instead of on the ending. No real attempt has been previously made to explain them. In looking over the subjunctives with mode-sign a, and secondary endings, we shall be struck by the fact, that the 2d and 3d dual and especially the 2d plural, are entirely unrepresented. The exclusion of so important a case as the 2d plural from an otherwise well-developed category is a priori improbable. It is very probable that just here must be sheltered such forms as punata and dádata, which are in all respects regular subjunctives, save that their ending is secondary. And this furnishes the key for the others also. If we look over the subjunctive forms possible from a stem crnu, for example, we find crnutá, çṛnávatha and *çṛnávata, it is seen at once why an additional form was needed: none of these, being of three or four light syllables, is fit for use in strictly iambic cadence. Nothing is more natural than that the least usable among them, *crnávata (since th at least occasionally makes position), should be remodelled. Accordingly forms like crnóta, juhóta, étana, kárta, etc., are to be regarded as contracted or apostrophized from the hypothetical çṛṇávata, juhávata, etc. This may be urged with especial emphasis for the o-forms; for the metrical correlation of ava and o is established by many instances, which cannot be impugned. Thus the weak stem maghon- for maghavan-, gávas to be read for gós; stonte to be read for stavante, etc. Compare the contraction of aya in the causative to e in Präkrit; also ava at the beginning of words in Pali. Cases in which original etymological ara and ana become ar and an are not, to be sure, found; but extensions of ar or ra to two syllables are wellknown; and cases have been pointed out, in which na is to be read as ana. MATHEMATICAL NOTES. On Quaternions, Nonions, Sedenions, etc., by J. J. SYL VESTER. 1o. Suppose that m and n are two matrices of the second order. the necessary and sufficient conditions for the subsistence of the equation 10 01 = 10 and I of the form 01' the necessary and sufficient condition is that the determinant of x+my+ nz shall be equal to x2 + y2+z2. The simplest mode of satisfying this condition is to write m= ΟΙ i o' n= i meaning 1, which gives mn = 0 i i0 and nm = 0 i i o' i 0 0-i and since the equations M2 = Ï, N2 = 1, MN —— NM imply if we make MNP that P2 : — 1, and MP=— PM, and NP- PN, it follows that M, N, P, are connected with m, n, p, in the same way as the coordinates of a point referred to one set of rectangular coordinates in space are connected with the coordinates of the same point referred to any other set of the same.* Herein lies the ground of the geometrical interpretation to which quaternions lend themselves and it is hardly necessary to do more than advert to the fact that the theory of Quaternions is one and the same thing as that of Matrices of the second order viewed under a particular aspect.† 2o. Let m, n now denote matrices of the third order. We might propose to solve the equation mn — — nm. The result of the investigation is that we must have m2 = = n2, m3 = = 0, n3 = 0, and writing mn = = p, m2 = n2 = q, there results a set of quinions, 1, m, n, p, q, for which the multiplication is that marked (a) p. 144 of the late Prof. Peirce's invaluable memoir in Vol. IV of the American Journal of Mathematics. But instead of this let us propose the equation mn = pnm, where p is one of the imaginary roots of unity; if now we write the determinant of x+my+nz under the form = ƒ= 0, h= 0, k = 0, x2+36x2y+3cx2z+3dx2z+bexyz+3fy3z+gy3 +3hy2z+3kyz2+lz3, it may be shown that we must have b=0, c = 0, d= 0, e=0, and if we superadd the conditions m3 1, n3 1, we must also have g=1,l=1, or in other words the determinant to x+my+nz must take the form 3 + y3 + z3; but this condition (or system of conditions) although necessary is not sufficient (a point which I omitted to notice in my article entitled "A Word on Nonions" inserted in a previous Circular). It is obviously necessary that we must have (mn)3 — 1. Now if the identical equation to mn, be written under the form E = 0, (mn)3 — 3B(mn)2 + 3Dmn = B may be shown to be a linear homogeneous function of b, c, and e; also Egl= 1; but D is not a function of b, c, d, e, f, g, h, k, l, and will not in general vanish (as it is here required to do) when b, c, d, e, f, h, k vanish. Its value is the sum of the products obtained on multiplying each quadratic minor of m by its altruistic opposite in n: [the proper opposite to a minor of m means the minor which is the reflected image of such minor viewed in the Principal Diagonal of m regarded as a mirror; and the altruistic opposite is the minor which occupies in n a position precisely similar to that of the proper opposite in m]. There are, therefore, 10 equations in all to be satisfied between the coefficients of m and n when m3 = n3 = 1 and nm = pmn. These ten conditions I have demonstrated are sufficient as well as necessary. There remains then 18-10 or 8 arbitrary constants in the general solution. If m, n is a particular solution we may take for M, N (the matrices of the general solution), M=a+ẞm+ym2 + a'n + ß'mn + y'm2n +a''n2 +ẞ''mn2+y''m2n3, N=a1+ß1m + Y1m2 + a2 ̧n + B'1mn + y' ̧m3n +alin2 + B'1mn2 + y''1m2n2, and 10 relations between the 18 coefficients must be sufficient to enable to be satisfied the equations M3 = N3 = 1, NM=pMN: but what these relations are and how they may most simply be expressed I am not at present in a condition to state.* I showed in "A Word on Nonions" that the 9 first conditions are satisfied by taking m = 100 Ορθ 00 p2, 00 1 n = p 0 0 0 p2 0. The 10th condition is also satisfied; for the only quadratic minors (not 10 p0 10 having a zero determinant) in m are Op' Op2' 0p2; the altruistic opposites matics. 1o. As regards the equation nm = mn where m, n, are matrices of the second order. *The solution of this problem would seem to involve some unknown expansion of the idea of orthogonalism. Unless MN= NM0, a solution to be neglected, it may be proved that a = 0, a1 = 0. As before let the determinant of (x + ym+zn) = x2+2bxy + 2cxz +dy2+2eyz+fz2. [I may observe here parenthetically that the Invariant of the above Quantic is equal to the determinant of mn—1 nm, and that when it vanishes 1, m, n, mn, as also 1, n, m, nm are linearly related-or, as I express it, this Invariant is the Involutant of the system m, n or n, m, When m, n are of higher than the second order, the Involutant of m, n, say I, is that function whose vanishing implies that the 9 matrices (1, m, m2(1, n, n2) are linearly relatent, and the Involutant of n, m, say J, that function whose vanishing implies that the 9 quantities (1, n, n2(1, m, m2) are so related (I, J being two distinct functions) and so for matrices of any order higher than the second]. By virtue of a general theorem for any two matrices m, n of the second order, the following identities are satisfied: mn + nm- -2bn-2cm + 2e=0, n2 - 2cn+f=0. If then mn+nm= = 0, since m and n cannot be fractions of one another (for then mn=nm), the second equation shows that b=0, c=0,e= =0, and conversely if b=0, c=0, e=0, mn+nm= = 0, and m2+d=0, n2 +ƒ=0, where, if we please, we may make d=1,ƒ=1. 2o. Let m, n be matrices of the third order, and write as before, Det. (x+ym zn) = x3 + 2bx2y +2cx2z + dxy2 +2exyz+fxz2 + gy3 +3hy2z + 3ky2z+lz3. Then by virtue of the general theorem last referred to there exist the identical equations m3 — 3bm2 + 3dm · g=0, From the 1st and 2d of the 4 identical equations combined it may be proved that b=0, d=0; [I do not produce the proof here because to make it rigorous, the theory of Nullity would have to be gone into which would occupy too much space] and in like manner from the 3d and 4th it may be shown that c= =0, f=0*. Hence returning to the two middle equations it follows that e= · 0, h = 0, k = 0, and from the two extremes that g=1,l=1. = If then mn= pnm, m3 = 1, and n3 1, it is necessary that b=0, c= =0, d=0, e=0, ƒ=0, g=1, h=0, k=0, l=1. But these equations although not necessary are manifestly insufficient; for they lead to the equations m3 — 1 = 0, n3 — 1 = 0, and (1) m2n +mnm +nm2 = =0; (2) mn2+nmn+n2m = 0, but not necessarily to nm == pmn. In fact the supposed equations between m and n involve as a consequence the equation (mn)3 = 1. Now the general identical equation to (mn) is (mn)3 +3B(mn)2 + 3D(mn) + F= 0, where B = The sum of each term in m by its altruistic opposite in n= 3bc — 2e=0, F= gl=1, and D= The sum of each first minor in m by its altruistic opposite in n which sum does not necessarily vanish when b, c, d, e, f, h, k, all vanish. Hence there is a 10th condition necessary not involved in the other 9, viz., D=0. These 10 conditions I shall show are sufficient as well as necessary. For when they are satisfied since (mn)3 = 1, mn, mn = = n2m2. Hence from (1) m2n2 + n2m2 + nm2n = 0, m2n2 + n2 m2 + mn2m = 0. and from (2) Hence nm. mn = mn. nm,† and consequently nm is a function of mn. Hence we may write nm = · A+ Bmn + C(mn)2. * Except when m, n are functions of one another, so that mn and nm are identical and consequently are each of them zero. This equation is independent of the equation (mn)3 = 1; for nm2n—mn2m — (m2n + mnm + nm2) n—m (mn2 + nmn + n2m) = 0 by virtue of equations (1) and (2) above: accordingly these equations taken alone imply the equations nm = A+ B1mn + C(mn)2, AC 2 mn = − A + B ̧nm — C(nm)2 where B1, B2 are the roots of B2 + B +1-4C = 0; 4, C being arbitrary and independent except that each vanishes when and only when the cube of mn and (as a consequence) of nm, is a scalar matrix. Hence the only available hypothesis remaining is the equation nm = v.mn, where v is one of the imaginary cube-roots of unity as was to be proved. 3°. It remains to say a few words on the general equation nm= kmn, where m, n are matrices of any order w. To avoid prolixity I shall confine my remarks to the general case, which is, that where the determinants (or as I am used to say the contents) of m and n are each of them finite; with this restriction, the proposed equation is impossible for general values of k as will be at once obvious from the fact that the totalities of the latent roots of min and of nm are always identical, but the individual latent roots are by virtue of the proposed equation in the ratio to one another of 1: k, which, since by hypothesis no root is zero, is only possible when k =1. 3w 2 When the above equation is satisfied the w2 equations arising from the identification of nm with kmn cease to be incompatible and (as is necessary or at all events usual in such a contingency) become mutually involved. Thus, ex. gr., when w=1 and k=1, the number of independent equations is 0, i. e. 1-1, when w=2 and k=-1 the number is 3, i. e. 4 — 1, when w=3 and k= por p2 the number is 8, i. 9-1; it is fair therefore to presume (although the assertion requires proof) that for any value of w when k is a primitive wth root of unity the number of conditions to be satisfied when nm kmn is w2-1. Of these the condition that the content of x+my+nz shall be of the form 3 + cy3 + c'z3 will supply (+1)(w+2) w24-3w 3, i. e. 2, and there will therefore be 2 (w−1)(w—2) + 1 or to be supplied from some other source. 2 When k is a non-primitive wth root of unity, the number of equations of condition is no longer the same. Thus when k1 we know that n may be of the form A + Bm + Cm2 + . . . Lmw-1, where A, B, . . . L, and all the terms in m are arbitrary, and consequently the number of conditions for that case is 2 — (w2+w) or w2 – - W. It seems then very probable that if k is a qth power of a primitive wth root of unity the number of conditions required to satisfy nm= kmn is 2. -d where d is the greatest common measure of q and w: but, of course, this assertion awaits confirmation. 2 When 4 besides the case of nm = mn, i. e. of n being a function of m of which the solution is known there will be two other cases to be considered, viz. nm=— mn and nm=imn: the former probably requiring 14 and the latter 15 conditions to be satisfied between the coefficients of m, the coefficients of n and the two sets of coefficients combined. It is worthy of notice that the conditions resulting from the content of x+my+nz becoming a sum of 3 powers are incompatible with the equation nm vmn when v is other than a primitive wth root of unity (w being of course the order of m or n). Thus suppose w= 4; the conditions in question applied to the middle one of the 5 identical equations give m2n2 + n2m2 + mn2m + nm2n + mnmn+nmnm = :0; when nm imn the left hand side of this equation becomes (1 + i1 + ¿2 + i2 + i + i3)m2n2, i. e. is zero, but when nm — — nm, the value is (1+1—1—1-1-1)m22 which is not zero, and so in general. Thus the pure power form of the content of x+my+nz is a condition applicable to the case of being a primitive root of unity and to no other. nm mn *By virtue of the general theorem that the latent roots of any function of a matrix are the like functions of the latent roots of the original matrix. The case of nm being a primitive root of ordinary unity is therefore the one which it is most interesting to thrash out. w2+3w 2 4 There are in this case we have seen simple conditions expressible by the vanishing of that number of coefficients in the content of (w — 1) (w — 2) +my+nz and supplemental ones. What are these last? 2 I think their constitution may be guessed at with a high degree of probability. For revert to the case of w=3 in which there is one such found by equating to zero the second coefficient in the identical equation (mn)3 — 3B(mn)2 + 3Dmn — G = 0. Suppose now (m3n2)3 — 3B′(m2n2)2 + 3D′m2n — G'0 is the identical equation to m2. By virtue of the 8 conditions supposed to be satisfied we know that nm = pmn as well as m3 = 1, n3 = 1, and consequently that (m2,2)3 = 1. Hence B'=0, D'=0, by virtue of the 7 parameters in the oft-quoted content and of D being all zero, and thus the evanescence of B' or D' imports no new condition. Now suppose w= 4, and that (mn) 1 — 4B (mn) 3 + 6D(mn)2 —4Gmn+M=0, (m2n)1 — 4B′(m2„2) 3 +6D′(m2„2 )2 — 4G'm2n2 + M' = 0. Here we know that B vanishes by virtue of bc and e vanishing, but D=0, G=0, which must be satisfied if nm = imn will be two new conditions not implied in those which precede. It seems then although not certain, highly probable that B' = 0, D' = 0, will be implied in the satisfaction of the antecedent conditions but that G'=0 will be an independent condition, so that D=0, G=0, G'=0, will be the three supplemental conditions: and again when w= 5 forming the identical equations to mn, m2n2, m3n3, and using an analogous litteration to what precedes, the supplemental conditions will be D=0, G=0, M=0, G' = 0, M'= 0, M'0, and so in general for any value of w. The functions D, G, M, etc., above equated to zero are known from the following theorem of which the proof will be given in the forthcoming memoir.* ... If (mn) + k1(mn)w-1+. +ki(mn)w-it O is the identical equation to mn, then k; is equal to the sum of the product of each minor of order i in m multiplied by its altruistic opposite in n. The annexed example will serve to illustrate in the case of w=3 that unless the supplemental condition is satisfied we cannot have nm = pmn. Write 10 0, n = 0 ck, Ορ 0, k 0 cp, 00 på, cp2 k 0, m = then the determinant to x + my + nz will be easily found to be x3 + y3 +(c3 + k3)z3; but D becomes 3pck, and does not vanish unless c=0 or k=0, and accordingly we find Referring to the equation MN=— NM, and to the eight equations expressing M and N in terms of the combinations of the powers of m with those of n [in which it is to be understood that M and N are non-vacuous], we know that the sums of the latent roots of M and of N must each vanish and consequently, as may be proved, that a=0, a'=0, leaving 8—2 or 6 conditions to be satisfied. If we further stipulate that M3 = 1, N3 = 1, there will be 8 relations connecting the coefficients b, c, k and b', c', .. k', so that the 64 coefficients in the 8 equations connecting M, M2; N, N2; MN, M2N2; M2N, MN2, or say rather M, M2; N, N2; p2MN, p2M2N2; pM2N, pMN2, with like combinations or multiples of combinations of powers of m, nt will be connected together by 56 equations; the coefficients in the expression for any one of the above 8 terms may ... This theorem furnishes as a Corollary the principle employed to prove the stability of the Solar System. (See Lond. and Edin. Phil. Mag., October, 1883.) † It is easy to see that the sum of the latent roots of M1Ni must be zero for all values of i,j so that it is a homogeneous linear function of the 8 quantities m, m2, . . ., mn, m2n2. then be arranged in pairs fi, fi'; gi, gi' ; hi, hi' ; ki, ki'; and in the expression for its fellow by Fi, F; G1, G;'; H1, H;'; K1, K;'; so that the Matrix is resolved as it were into 4 sets of paired columns and 4 sets of paired lines; the 4 different sets of paired lines being found by writing successively i 1, 2, 3, 4. It is then easy to see that there will be 4 equations of the form (faGa'+faGa') = 1, and 6 quaternary groups of (i. e. 24) equations of the form ElfaGB'+faGs') = 0 [with liberty to change f into For G into g or each into each]: together then the above are 28 of the 56 conditions required. But inasmuch as the 8 [m, n] arguments may be interchanged with the 8 [M, N] ones; we may transform the above equations by substituting for each letter ƒ its d log A conjugate (where A is the content of the Matrix) and thus obtain df 28 others, giving in all (if the two sets as presumably is the case) are independent the required 56 conditions: the latter 28, however, may be replaced by others of much simpler form.* To me it seems that this vast new science of multiple quantity soars as high above ordinary or quaternion Algebra as the Mécanique Céleste above the "Dynamics of a Particle" or a pair of particles, (if a new Tait and Steele should arise to write on the Dynamics of such pair,) and is as well entitled to the name of Universal Algebra as the Algebra of the past to the name of Universal Arithmetic. If to each term in the principal diagonal of a matrix 2 be subtracted, the content of the resulting matrix is a function of degree in 2; the w values of 2 which make this content vanish are called its latent roots, and if i of these roots are zero, the vacuity (treated as a number) is said to be i. This comes to the same thing as saying that the vacuity is i when the determinant, and the sums of the determinants of the principal minors of the orders 1, w—2, . . . (w —i+1) are each zero. W A principal minor of course means one which is divided into 2 triangles by the principal diagonal of the parent matrix. Again the nullity is said to be i when every minor of the order (w—i+1) [and consequently of each superior order] is zero. It follows therefore that it means the same thing to predicate a vacuity 1 and a nullity 1 of any matrix, but for any value of i greater than 1, a nullity i implies a vacuity i but not vice versâ; the vacuity may be i, whilst the nullity may have any value from 1 up to i inclusive. *I am still engaged in studying this matrix, which possesses remarkable properties Is it orthogonal? I rather think not, but that it is allied to a system of 4 pairs of somethings drawn in four mutually perpendicular hyperplanes in space of 4 dimensions. In the general case of MN= pNM where p is a primitive wth root of unity, there will be an analogous matrix of the order 2-1 where each line and each column will consist of a + 1 groups of ∞ - 1 associated terms. The value of the cube of any one of the 8 matrices M, M2; ...; MN, M2N2 may be expressed as follows: It is Pinto ternary unity. [Such a quantity may be termed by analogy a Scalar.] To find P, I imagine the 8 letters corresponding to MN (but without powers of p attached) to be set over 8 of the 9 points of inflexion to any cubic curve the paired letters being made suitably collinear with the missing 9th point. Then among themselves the 8 letters may be taken in 8 different ways to form collinear triads and the product of the letters in each triad may be called a collinear product; P1, ¿ (which is identical with the Determinant to MN) will be the sum of the cubes of the 8 letters less 3 times the sum of their 8 collinear products, and its 8 values will be analogous to the 3 values of the sum of 3 squares in the Quaternion Theory. Each of these 8 values is assumed equal to unity. It may be not amiss to add that the product of four squares by four is representable rationally as a sum of four squares, so if we place (not now 8 specially related but) nine perfectly abitrary letters over the nine points of inflexion of a cubic curve the sum of their 9 cubes less three times their 12 collinear products multiplied by a similar function of 9 other letters may be expressed by a similar function of 9 quantities lineo-linear functions of the two preceding sets of 9 terms. By the 8 letters of any set as ez. gr. b, . . ., h' being "specialized," I mean that they are subject to the condition bb' + dd' + ff' + h!' = 0. When this equation is satisfied, and not otherwise, M3 will be a Scalar, and it must be satisfied when MN = pNM. |