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'Hark an awful voice is sounding' is a translation of the Latin hymn 'En clara vox redarguit' by Edward Caswall. It was intended to impose new and uniform standards on Catholic hymn singing, which at that time had a poor reputation. This chapter focuses on the Catholic musical context in which it appeared; the functions it was expected to fulfil; how the texts reflect the character of the English Catholic church at that time; and the impact of Richard Runciman Terry. The Westminster Hymnal was one of several changes radically affecting the musical heritage of English Catholics early in the twentieth century. The implications for singing were profound, for the private reading and recital of hymns confirmed the separation between those who wrote hymns and those who composed music. In other words, because English Catholics had been a persecuted minority for so long, no tradition of hymn singing had emerged and a Counter-Reformation habit of private devotions had been substituted.

In the early nineteenth century there were three main styles of music performed in the English Catholic Church. First, there was plainchant, mainly that copied by John Francis Wade in the previous century and then refracted through arrangements by Samuel Webbe the elder, Samuel Wesley and Vincent Novello. Second, there was a native and, for its day, a fairly up-to-date style associated again with the two Webbes, Wesley and Novello. Third, there were Continental imports, especially grand masses, composed by Viennese Classical masters such as Mozart and Haydn, or Hummel and Weber. All three styles were developed and remained popular throughout the nineteenth century; but increasingly they were challenged by a revived interest in Renaissance-style polyphony, especially music composed between 1551 and 1650. This paper examines that development looking at, first, the general factors that encouraged it; second, the main stages in its revival; and third, the extent and effects of its influence and achievement.

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.

(1) The persymmetric determinant of the nth orderbeing such that in every case the element in the place r, s is the same as the element in the place r − 1, s + 1, and therefore having only 2n − 1 independent elements, viz., the elements a1, a2, …., a2n−1 forming the first row and last column, is conveniently denoted byAs it is a special form of axisymmetric determinant, any known relation between minors of the latter must of course hold in regard to the corresponding minors of the former.

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.

1. If in a circulant the places (r, s) and (p, q), owing to the cyclical permutation, be occupied by the same element, then the complementary minor of this element in its first place is to its complementary minor in the second as (−1)r+s: (−1)p+q.To prove that the complementaries are as (−1)r+s:(−1)p+q is the same as to prove that the cofactors are identical. And as the cofactor of an element of a determinant is not altered by the transposition of rows and columns provided the determinant itself is not thereby altered, it is evident that all we have to show is that the element in the place (r, s) may by transposition of rows and columns be made to take the place (p, q), and the determinant remain in outward form the same as before. Now it is a known property of the circulant that this can be done by first bringing the element in the place (r, s) into the pth row by cyclical transposition of rows, and thereafter making an exact similar set of transpositions of columns. Thus the theorem is established.

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.

(1) It is known that a continuant whose three diagonals are formed of certain equidifferent progressions is resolvable into linear factors, the earliest specimens placed on record being those of Sylvester and Painvin. The object of the present paper is to show that there are continuants of quite a different type which are also so resolvable, and to expound a general mode of investigating the subject.

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.

(1922). A Budget of Exercises on Determinants. The American Mathematical Monthly: Vol. 29, No. 1, pp. 10-14.

(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

n/a

(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.

(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.