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The character of Jacobi's expansion of this form of alternant will be more readily understood if the two simplest special cases be first considered.

Two curious identities have been established regarding certain aggregates of minors of special determinants; the first, which concerns axisymmetric determinants, having been discovered by Kronecker in 1882, and the second, which concerns centrosymmetric determinants, having been published by me in 1888, When we come to think of the possibility of generalising these identities, it is readily seen that there are at least two lines of attack which suggest, themselves on reading the mere description of the kind of identity; for, in saying that the identities deal with “an aggregate of minor determinants of a special determinant,” we are conscious of two points of limitation in the description, the one signalised by the word “minor” and the other by the word “special.” If, therefore, an identity were obtained regarding an aggregate of which the terms were determinantsunrestricted by a family relationship, we might have one form of generalisation; and if, while retaining the family relationship, we succeeded in removing the restriction as to the form of the parent, a generalisation of a different type might be the result.

In the period which we have now reached the interest taken in the study of Alternants shows comparatively little sign of slackening, especially if the years given over to the War be taken into account. Close on fifty (50) writings, in six different languages and of various degrees of importance, fall to be considered.One noteworthy feature of the collection is the exceptional number of the papers occupied with the investigation of Symmetric Functions: approximately a third of the whole may be so classified. It is also somewhat striking in connection therewith to note that this body of work has two quite independent sources, the one centring in a small East-Prussian town, the other confined mainly to a busy city in the State of New York.

(1) As is well known, the simplest form of alternant isand the problem of multiplying it by any symmetric function of a, b, c, d, … has been in a manner fully solved.(2) When the symmetric function is linear in each of the variables—that is to say, when it takes any of the forms Σa, Σab, Σabc, ….—the result is an alternant got from the multiplicand by increasing the last index, the last two indices, the last three indices, …. respectively by 1. Thus, writing for shortness' sake five variables only, we haveThis was first established in 1825 by Schweins in his Theorie der Differenzen und Differentiale, p. 378; but it is also barely possible that it was known to Prony in 1795 (see Journ. de l'Ec. Polyt., i. pp. 264, 265), and Cauchy in 1812 (see Journ. de l'Ec. Polyt., x. pp. 49, 50).(3) When the symmetric function is non-linear, the result takes the form not of one alternant, but of an aggregate of alternants. These cannot be so readily specified, but the mode of obtaining them can be made clear without any difficulty. Let us take the case of the function Σa3b the multiplicand being ∣a0b1c2d3∣.

(1) If we seek to extend the definition of an orthogonant by making the square of each row equal to σ instead of 1, this fixed quantity will naturally be found to appear in the statement of almost all the properties of the determinant in question. It is therefore convenient and appropriate to call it the base of the orthogonant, and to speak of an orthogonant to the base σ or an orthogonant of base σ.It is a simple matter to ascertain what modifications are necessary in the statement of the known properties of the ordinary orthogonant to make them applicable to the extended definition.

The aids which may fairly be expected from a library towards the prosecution of research, mathematical or other, are of two kinds. First, there is the aid given by furnishing the names of previous workers in the same field, accompanied by the names of the publications in which the results of their labours have been preserved or entombed. Second, there is the help in the opportunity given of consulting the said publications themselves and of borrowing certain of them for lengthened study. In other words, what the scientific investigator wants from libraries is books and books about books. One of the two is often obtainable without the other, but in that case its value is immensely lessened. To the so-called “ordinary” reader, a library is a labyrinth which probably would give him all he needs if only he had a guide through its intricacies: to the specialist, on the other hand, with his methodic bibliographies and booksellers' lists, it not unseldom presents itself as a lucky-bag in which the blanks are ultimately more in evidence than the prizes.

So far back as March 1855 Brioschi in effect formulated the theorem that any even-ordered determinant is expressible as a Pfaffian; for example in the case of the fourth order he gave the equalitywhere (h, k) stands forthis expression being obtained by multiplying the hth columnof the given determinant by the kth columnof an equivalent determinant.

Sets of equations of the formwhere the coefficient of xr in the sth equation differs only in sign from the coefficient of xs in the rth equation, had often made their appearance in analytical investigations before the determinant of such a set came to be considered. An instance is to be found in a memoir of Poisson's, read before the Institute in October 1809, and printed in the Journal de l'École Polytechnique, viii., pp. 266 — 344; and similar instances of an earlier date in writings of Lagrange and Laplace therein referred to. The mathematician who first referred definitely to the determinant appears to have been Jacobi.

If we take the set of equationsthe eliminant of which isand from the 1st, 2nd, and 3rd eliminate xyw and xyz, from the 2nd, 3rd, and 4th eliminate xyz and yzw, from the 3rd, 4th, and 1st eliminate yzw and xzw, and from the 4th, 1st, and 2nd eliminate xzw and xyw, we obtain the setthe eliminant of which is clearly

(1) The fundamental propositions in regard to the sum of the signed primary minors of a determinant are—(A) An expression for the negative sum of the signed primary minors of any determinant is got by taking a determinant of the next higher order whose first element is zero with the given determinant for complementary minor, and whose remaining elements are units all positive or all negative.

There now only remain for consideration those special forms which, prior to 1860, had not received any noteworthy attention. These will be found to include: (α) permanents, which are touched on by three authors; (β) determinants with the typical element ars + brsi, which are referred to in four memoirs; (γ) two other forms, which are each dealt with in two papers; and (δ) nine others, which make their appearance only once.

My last communication in reference to the history of Jacobians dealt with the period 1815–1841 (Proc. Roy. Soc. Edin., xxiv. pp. 151–195). The present paper continues the history up to 1860.

A general account has already been given of this interesting paper—interesting as regards the subject, and interesting as being the author's first prentice effort. All that remains to be noticed here is what may be called Cayley's series of vanishing axisymmetric determinants.

In a paper communicated to the Berlin Academy on 27th July 1882, Kronecker pointed out that certain sets of minors of any axisymmetric determinant are connected by a linear relation, or, as I have tried to put it more definitely in the present title, that certain aggregates of minors are equal to zero. This discovery has attracted considerable attention in Germany, as the list of papers herewith given suffices to show.

(1) The main specialty in the construction of the arrays in question is that the elements of each row are the coefficients of the powers of x in the expansion of a product of the formthat is to say, the rows all belong to the typeThe variables from which the elements of any row are formed are n − 1 consecutive members of a series of n(n − 1) members, the member considered to be consecutive to the last being the first member; for example, when n is 3 and the 3(3 − 1) independent variables area, b, c, d, e, f,the rows of the arrays areFurther, these rows in order are taken to form all the first rows of the arrays, then all the second rows, and so on.

The twenty-year period, 1900–1920, shows a marked increase in the number of writings concerned with persymmetric determinants. There is, it is true, a still greater rate of increase of those of them that belong to the class of “solved questions,” but even taking this into account, we find a growth of interest in the subject that is distinctly striking. And then, too, we must guard carefully against accepting the belief that the members of this solved-question class are uniformly of slight importance, and that the others are never trivial.

On 17th March 1879 I gave to the Eoyal Society of Edinburgh an account of some researches on the subject of Alternants. A short descriptive abstract of the communication was published in the Proceedings, vol. x. pp. 102, 103, the eoncluding paragraph of which is as follows

(1) The cubinvariant J of a binary quartic beingits Hessian, H(J), isPerforming on this the operationsand on the result the operationswe obtainThe three-line determinant here is seen to contain c as a factor, because it manifestly vanishes when c is put equal to 0.

The previous history of Wronskians being not at all lengthy, was included in the chapter on “Miscellaneous Special Forms” (History, i., chap, xvi.), and is to be found there under Wronski 1812, Wronski 1815, Wronski 1816–17, and Schweins 1825 (pp. 472–478, 482–485).The name dates only from 1882, being first suggested on p. 224 of my Text-book on Determinants.

The special form of determinant to which we have now come is connected with a problem in coordinate geometry—the problem of transformation from one set of rectangular axes to another set having the same origin. The actual appearance of determinants in any of the attempts to solve the geometrical problem did not take place until comparatively late in its history, probably because the connection between the two subjects was less patent than in other cases, the problem when transformed into algebraical language being not a mere matter of elimination of unknowns from a set of linear equations. The earlier portion of the history of orthogonal substitution, although of considerable interest, is thus not sufficiently germane to our subject to warrant detailed treatment of it. For those interested in this earlier portion it will suffice to give the following chronologically arranged list of papers from 1770 to 1840:—

MY last communication in reference to the history of skew determinants dealt with the period 1827–1857 (Proc. Roy. Soc. Edin., xxiii, pp. 181–217). The present paper continues the history up to the year 1865, but in addition contains an account of two writings belonging to the previous period, namely, by Brioschi (1855) and Bellavitis (1857).

By Cauchy the word “symmetric,” as applied to functions, was used in a more general sense than that in which we ordinarily use it now, or than that in which it was used before his time. He spoke of “fonctions symétriques permanentes” and “fonctions symétriques alternées:” we apply the word symmetric only to certain of the first of these; the second we call simply “alternating functions.” Thus the expressionwas called by him a permanent symmetric function, since if a1, a2, a3 are interchanged in order with b1, b2, b3, the function remains unaltered. It would be more definite to say that it is a permanent symmetric function with respect to a1, a2, a3 and b1, b2, b3.

(1) In his “Recherches sur les déterminants,” published in the Bulletin of the Royal Academy of Belgium in 1846, Catalan gives a proposition in regard to circulants of the typewhich deserves some special attention, as there can be little doubt that it is seriously misleading. Unfortunately, too, it has actually already misled to a considerable extent, the proposition having been reproduced by quite a number of writers without any care being taken to verify it. Even the very latest text-book, that written by Professor Pascal of Pavia and translated into German by H. Leitzmann, gives importance to the proposition, and by omitting the so-called proof diminishes the chance of detection of error.

(1) In a paper in the Proceedings of the London Mathematical Society, vol. xxxiii. p. 229, Professor Elliott has occasion to consider the equationwhich, it is stated, presents difficulties when a direct algebraical solution is attempted, but is seen indirectly to beaccording as p is odd or even.

Determinants whose elements are themselves determinants made their appearance at a very early stage in the history of the subject, the first foreshadowing of them being contained in Lagrange's “équation identique et très remarquable” of 1773, namely,whereThis, viewed as a result in determinants, is a case of Cauchy's theorem of 1812 regardingthe adjugate, and the adjugate of course is an instance of the special form to which we have now come. Jacobi's theorem regarding any minor of the adjugate has a like history and may be similarly classified. Passing from the case of the adjugate, where each element is a primary minor of the original determinant, Cauchy also considered the determinants of other “systèmes dérivés,” that is to say, the determinants whose elements are the secondary, ternary, … minors of the original, and gave the theorem that the product of the determinants of two “complementary derived systems” is a power of the original determinant, the index of the power being where n is the order of the original determinant and p the order of each element of one of the “derived systems.”

(1) If we temporarily denote the product of the rth and sth rows of a determinant Δ by rs, we shall have conveniently

The problem in question with its solution appeared in the Cambridge Mathematical Journal, vol. ii. pp. 232–236, being given by Sylvester to show how his recently discovered “dialytic” method might be applied to ternary quadratics.

The theorem in question appears in his well-known paper of the year 1894, “Ueber die Elementartheiler der Determinanten.” Save that English takes the place of German, the enunciation of it stands exactly as follows:—If ξ ≦ r ≦ s < n, and Dξbe any determinant of the nth order in the arrayDξ′ being its complementary minor of the (n − ξ)th order in the determinant of the nth orderthen

Notwithstanding the generalisations made by Jacobi and Cauchy, the special case with which the whole theory originated continued from time to time to attract attention. In 1843 William Thomson, afterwards known as Lord Kelvin, published under the signature “T.” in the Cambridge Math. Journ., iii. pp. 247–248, a short note in which he proved the detached theorem that if l1, m1, n1, l2, … be nine quantitiessuch thatthen it follows thatThis led to a short paper by A. Göpel in the Archiv d. Math. u. Phys., iv. (1843) pp. 244–246.

Most of the theorems in question occur at the outset of a paper by Professor Cayley, entitled, “Chapters on the Analytical Geometry of n Dimensions” they constitute, in fact, Chapter I. The first theorem I should prefer, for the present, to enunciate as follows:-If m determinants of the nthorder all have the same n − 1 columns in common, and all vanish, then every determinant of the nthorder whose n columns are chosen from the m + n − 1 different columns must vanish likewise.

As has already been pointed out (Hist., i. pp. 485–487), the special form of determinant named “persymmetric” in 1853 by Sylvester came first to light in 1835 in a paper of Jacobi's on the elimination of the unknown from two equations of the nth degree, the fact being that the adjugate of Bezout's condensed eliminant—in other words, the adjugate of the determinant resulting from Bezout's “abridged method“ of elimination—is there shown to be such that the elements of it whose place-numbers have the same sum are equal.

1. The “dialytic” method of elimination, in the case of more than one variable, is not without its drawbacks, as most mathematicians know. The requisite derived equations are not always easily obtained, the difficulty being due as often to the existence of too many as of too few; and, when this has been got over, it not unfrequently happens that the order of the resulting determinant is alarmingly high, and unaccompanied by any hope of a successful guess as to the character of the extraneous factor. The discoverer's original paper affords sufficient testimony of this, and very little has been done since its appearance to put matters on a sounder footing. The most noteworthy improvement, due to Cayley, is more interesting theoretically than practically, his main object being the detection of the extraneous factor when there is an over-plus of equations. Nothing, indeed, will be found more conducive to an understanding of the limitations of the method than a careful comparison of the application of this process of Cayley's to the problem of eliminating x, y, z from the three ternary quadricswith Sylvester's original treatment of the same problem, and the latter especially as commented on in footnotes by the author himself.2. The want of definiteness in the mode of arriving at the exact number of derived equations is not without its compensating advantage, as it leaves that scope for the exercise of ingenuity which is half the charm of mathematical work.

The first of the writings dealt with in the present communication is of a considerably earlier date than the others: its proper place is immediately in front of Sylvester's paper of the same year. The account of it here given should therefore he found between pages 140 and 141 of this volume of the Proceedings.

A first note on this subject, viz., the elimination of x, y, z from the equationswas communicated to the Society in April 1892. By reason of my immediate departure for South Africa, the manuscript was not prepared for the press, and the paper was consequently not printed until this year. In the meanwhile, both Professor Tait and Lord M'Laren had written notes on the subject, under the impression, I fear, that the object of my original paper was to obtain in any way an easy solution of the problem.

(1) If note be taken that, when k in the title is 0, there is only one such determinant, namely the adjugate, it will be seen that what we are about to consider is a family of compound determinants which includes the adjugate as a very special case. Such determinants have not altogether escaped study, but until recently the attention paid to them has been incidental and fragmentary.

The object of the present paper is to make some advance in the theory of those skew determinants that have the elements of the diagonal all equal. A few results on determinants of more general form and on related subjects are given incidentally.

The increased interest taken in Orthogonants during the period 1880–1900 is more than maintained in the period which we have now reached. Out of considerably more than 100 writings sought out and examined, as many as 65 have been found to deal directly with the theory of orthogonal substitution, and 35 to be more closely concerned with the practical applications of it. These numbers, 65 and 35, are what have to be set over against 38 and 12 in the earlier period referred to, note, however, at the same time being taken that the contrast is unfairly heightened by several papers of the first period, chiefly Magyar, having been received only in time to be joined with those of the second. These Magyar papers, it may be added, make also an appreciable contribution in the matter of quality.

1. The given set of equations in question isand the problem was solved by Cayley in the Quarterly Journal of Mathematics, viii, pp. 183–185. The method was peculiar and is still interesting. He introduced three additional unknowns, deduced a fresh set of five equations, from which he then proceeded to eliminate four unknowns, namely, the new three and x/y.

1. Cayley commences his third paper on Skew Determinants (May, 1854) by recalling his development of them in terms of Pfaffians, and then goes on to say:—“J'ai trouve recemment une formule analogue pour le developpement d'un déterminant gauche borde, tel queCette formule est:and he explains that the expressions 12, 1234, etc., are Pfaffians, whose law of formation is—12 = 12,1234 = 12·34 + 13·42 + 14·23,123456 = 12·34·56 + 13·45·62 + 14·56·23 + 15·62·34 + 16·23·45 + 12·35·64 + 13·46·25 +14·52·36 +15·63·42 + 16·24·53 + 12·36·45 + 13·42·56 + 14·53·62 + 15·64·23 + 16·25·34.No proof is given, and the law of formation of the development itself is not explained.

Robert's theorem concerns the circulant C whose elements are the terms of the expansion of and is to the effect (1) that there are no odd powers of t in the development of the circulant, and (2) that if in the said development ξ be put for t2, the equation in ξ

Since the year 1889, when the last of a series of six papers with a title similar to the above appeared, further research has led to the discovery of a number of writings belonging to the period then dealt with, viz., 1693–1844. Of those an account is now given before proceeding to the papers of later date than 1844.

Like Wronskians, and for the same reason, recurrents were at first dealt with among “Miscellaneous Special Forms”: their previous history is thus to be found under Wronski 1812, Scherk 1825, and Schweins 1825 in the chapter so entitled. (History, i. pp. 472–474, 478–481.)The name is quite recent, having been first proposed by E. Pascal in 1907 in a paper published in the Rendiconti …. 1st. Lombardo, (2) xl. pp. 293–305.

The number of writings on continuants belonging to the twenty-year period now reached is 44, being 15 more than for the immediately preceding period 1880–1900. The period bears also three other marks which serve to give it some little additional distinction, (1) the exceptionally full consideration given to the subject of factorisation (factorisation of continuants), (2) the first appearance of an entirely fresh form for special study, the “block continuant” of Simandl, (3) the fact that one of the said 44 writings is a text-book of very considerable extent devoted exclusively to continuants and continued fractions.

The chapter on Jacobians and the chapter on Hessians were the only two for the period 1880–1900 that seemed to indicate that a slackening had begun in the interest taken in their subjects. We now find that, so far as Jacobians are concerned, any movement of the kind must have been temporary, the new period (1900–1920) showing the number of relevant writings listed for report to be almost if not quite double that of its predecessor.

The special form of determinant to which we have now come is connected with a problem in coordinate geometry—the problem of transformation from one set of rectangular axes to another set having the same origin. The actual appearance of determinants in any of the attempts to solve the geometrical problem did not take place until comparatively late in its history, probably because the connection between the two subjects was less patent than in other cases, the problem when transformed into algebraical language being not a mere matter of elimination of unknowns from a set of linear equations. The earlier portion of the history of orthogonal substitution, although of considerable interest, is thus not sufficiently germane to our subject to warrant detailed treatment of it. For those interested in this earlier portion it will suffice to give the following chronologically arranged list of papers from 1770 to 1840:—

The theorem in question arises out of a consideration ol several passages in Jacobi's important memoir of 1833 or orthogonal transformation. Having determined the substitutior which simultaneously changesand

Perhaps the most interesting theorem as yet known in regard to circulants concerns the determinant whose matrix is the sum of two circulant matrices, one of the latter being taken symmetric with respect to the primary diagonal and the other with respect to the secondary diagonal: for example, the determinantwhose matrix is the sum of the matricesthat is to say, the matrices of

My last communication in reference to the history of bigradients (Proc. Roy. Soc. Edin., XXX. pp. 396–406) brought the record up to the year 1859. The present paper continues it to the year 1880.

It is easily seen that each of the equations of the setremains unaltered for each of the three interchangesand that the set as a whole is not altered by the simultaneous performance of the cyclical substitutions

1. If for aα+bβ+cγ+…we putthe the product of |a1b2c3| and |f1g2h3| may be written