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(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.

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.

(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.

(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.

§1. It is well known that by a simple transformation of a determinant we may cause a zero to take the place of any one of the elements. The theorems of Hermite and Horner, for example, for depressing the order of a determinant may each of them be viewed as the result of repeated transformations of this kind, the operation being continued until all the elements of a row or column except one are replaced by zeros.

(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).

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

(1) There being n2 independent variables in a general n-line determinant, the Hessian of the determinant with respect to the said variables must be a determinant with n2 lines: and as a general n-line determinant has all its terms linear in the elements involved, it follows that each element of the Hessian being a second differential-quotient cannot be of a higher degree in the variables than the (n − 2)th, and that consequently the degree of the Hessian itself cannot exceed n2(n − 2). The object of the present short paper is to show that this degree is attained by the n(n − 2)th power of the given determinant being a factor of the Hessian.

The rows of a determinant of the nth order having been separated into two sets, one containing the first p rows and the other the rest, if each minor of the pth degree formed from the first set be multiplied by a minor, called its complementary, formed from the second set, and the result have its sign chosen in accordance with a certain law, it is well known as an elementary theorem that the aggregate of the products thus obtained is equal to the original determinant.

§1. As a preliminary to the establishment of the law in question, it is necessary to state and exemplify another law to which I have elsewhere directed attention, viz.,THE LAW OF COMPLEMENTARIES.To every general theorem which takes the form of an identical relation between a number of the minors of a determinant or between the determinant itself and a number of its minors, there corresponds another theorem derivable from the former by merely substituting for every minor its cofactor in the determinant, and then multiplying any term by such a power of the determinant as will make the terms of the same degree.

In Crelle's Journal for 1844, at the end of a paper on Cubic Forms, Eisenstein gives the following results:—where m is any odd number and ρ a primitive root of the equation zm = 1;whereNo demonstrations are given of these identities; but they are said by the author to be only particular cases of a very general equation, the complete theory connected with which he hoped to give on another occasion.At p. 193 of the same volume of Crelle he returns to the subject, not however for the purpose of giving the promised theory, but to add several other results similar to those before given.

(1) This is a subject to which very little study has been directed. The first to enunciate any proposition regarding it was Jacobi; but the solitary result which he reached received no attention from mathematicians,—certainly no fruitful attention,—during seventy years following the publication of it.Jacobi was concerned with a problem regarding the partition of a fraction with composite denominator (u1 − t1) (u2 − t2) … into other fractions whose denominators are factors of the original, where u1, u2, … are linear homogeneous functions of one and the same set of variables. The specific character of the partition was only definable by viewing the given fraction (u1−t1)−1 (u2−t2)−1…as expanded in series form, it being required that each partial fraction should be the aggregate of a certain set of terms in this series. Of course the question of the order of the terms in each factor of the original denominator had to be attended to at the outset, since the expansion for (a1x+b1y+c1z−t)−1 is not the same as for (b1y+c1z+a1x−t)−1. Now one general proposition to which Jacobi was led in the course of this investigation was that the coefficient ofx1−1x2−1x3−1…in the expansion ofy1−1u2−1u3−1…, whereis |a1b2c3…|−1, provided that in energy case the first term of uris that containing xr.

(1) The continuantis conveniently denoted byand, when the elements b1, b2, b3 of the variable minor diagonal are each equal to 1, a further simplification is obtained by leaving them out;—thus,stands fo

(1) The simplest possible case of the development referred to in the title is one which is seen to follow instantly from the determinant definition of a Pfaffian. Thus, by the said definition,and thereforethe result reached being that a Pfaffian with certain zero elements is expressible as a determinant. It should be noted that, as in many instances where this form of definition is used, there is an ambiguity as to sign : it is of less importance to note that here the appropriate sign is –.