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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.

(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The communication on this subject, as originally made to the society, consisted of a series of theorems, giving (1) expressions for the radii of a great many sets of circles, (2) identities connecting several sets of these radii, and (3) miscellaneous identities closely related thereto. As, however, the paper culminated in a general theorem which may be looked upon as fundamental, and the proof of which makes evident the mode of arriving at the said expressions for radii, and as the relations connecting sets of radii are easily found when attention has been directed to their existence, I have thought it best to print little more than the fundamental theorem and a few auxiliary notes.

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.

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.

(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

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(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).

(1) The theorem in question concerns the finding of an expression for a determinant of composite order in terms of compound determinants or permanents of lower order; or, more definitely, the expression of a determinant of the mnth order as an aggregate of determinants or permanents of the mth order, each of whose elements is a determinant of the nth order.

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.

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 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

(1) It is well known that Cayley effected the transformation ofby introducing an intermediary set of variablesconnected with each of the other sets by means of a linear substitution of a peculiar type. The substitutions in fact wereandwhere the determinant of the first substitution is unit-axial and skew, and the determinant of the second substitution is got from the preceding determinant by changing rows into columns, and where, therefore, the number of arbitrary quantities introduced by the two substitutions is only ½n(n−1).

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.

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.

(1922). NOTE ON THE PRODUCT OF ANY DETERMINANT AND ITS BORDERED DERIVATIVE. Transactions of the Royal Society of South Africa: Vol. 10, No. 1, pp. 89-93.

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.

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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.

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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.

A glance at the expressionis sufficient to verify the fact that it is symmetric with respect to a, b, c, d, and also with respect to α, β, γ, δ. It is likewise, although not quite so evidently, an alternating function with respect to the interchangethat is to say, if a and α be interchanged, and at the same time b and β, c and γ, d and δ, the function is not altered in magnitude, but merely changes sign. With a little trouble, indeed, the expression can be transformed intoor say