XVIII. On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure

In the first section of the ensuing memoir, which is divided into five sections, I consider the nature and properties of the residues which result from the ordinary process of successive division (such as is employed for the purpose of finding the greatest common measure) applied to f(x) and ϕ(x), two perfectly independent rational integral functions of x. Every such residue, as will be evident from considering the mode in which it arises, is a syzygetic function of the two given functions; that is to say, each of the given functions being multiplied by an appropriate other function of a given degree in x, the sum of the two products will express a corresponding residue. These multipliers, in fact, are the numerators and denominators to the successive convergents to ϕx/fx expressed under the form of a continued fraction. If now we proceed à priori by means of the given conditions as to the degree in (x) of the multipliers and of any residue, to determine such residue, we find, as shown in art. (2.), that there are as many homogeneous equations to be solved as there are constants to be determined; accordingly, with the exception of one arbitrary factor which enters into the solution, the problem is definite; and if it be further agreed that the quantities entering into the solution shall be of the lowest possible dimensions in respect of the coefficients of f and ϕ, and also of the lowest numerical denomination, then the problem (save as to the algebraical sign of plus or minus) becomes absolutely determinate, and we can assign the numbers of the dimensions for the respective residues and syzygetic multipliers. The residues given by the method of successive division are easily seen not to be of these lowest dimensions; accordingly there must enter into each of them a certain unnecessary factor, which, however, as it cannot be properly called irrelevant, I distinguish by the name of the Allotrious Factor. The successive residues, when divested of these allotrious factors, I term the Simplified Residues, and in article (3.) and (4.) I express the allotrious factors of each residue in terms of the leading coefficients of the preceding simplified residues of f and ϕ. In article (5.) I proceed to determine by a direct method these simplified residues in terms of the coefficients of f and ϕ. Beginning with the case where f and ϕ are of the same dimensions (m) in x, I observe that we may deduce from f and ϕ m linearly independent functions of x each of the degree (m - 1) in x, all of them syzygetic functions of f and ϕ (vanishing when these two simultaneously vanish), and with coefficients which are made up of terms, each of which is the product of one coefficient of f and one coefficient of ϕ. These, in fact, are the very same (m) functions as are employed in the method which goes by the name of Bezout’s abridged method to obtain the resultant to (i. e. the result of the elimination of x performed upon) f and ϕ. As these derived functions are of frequent occurrence, I find it necessary to give them a name, and I term them the (m) Bezoutics or Bezoutian Primaries; from these (m) primaries m Bezoutian secondaries may be deduced by eliminating linearly between them in the order in which they are generated, —first, the highest power of x between two, then the two highest powers of x between three, and finally, all the powers of x between them all: along with the system thus formed it is necessary to include the first Bezoutian primary, and to consider it accordingly as being also the first Bezoutian secondary; the last Bezoutian secondary is a constant identical with the Resultant of f and ϕ. When them times m coefficients of the Bezoutian primaries are conceived as separated from the powers of x and arranged in a square, I term such square the Bezoutic square. This square, as shown in art. (7.). is symmetrical above one of its diagonals, and corresponds therefore (as every symmetrical matrix must do) to a homogeneous quadratic function of (m) variables of which it expresses the determinant. This quadratic function, which plays a great part in the last section and in the theory of real roots, I term the Bezoutiant; it may be regarded as a species of generating function. Returning to the Bezoutic system, I prove that the Bezoutian secondaries are identical in form with the successive simplified residues. In art. (6.) I extend these results to the case of f and ϕ being of different dimensions in x. In art. (7.) I give a mechanical rule for the construction of the Bezoutic square. In art. (8.) I show how the theory of f(x) and ϕ(x), where the latter is of an inferior degree to f may be brought under the operation of the rule applicable to two functions of the same degree at the expense of the introduction of a known and very simple factor, which in tact will be a constant power of the leading coefficient in f(x). In art. (9.) I give another method of obtaining directly the simplified residues in all cases. In art. (10.) I present the process of successive division under its most general aspect. In arts. (11.) mid (12.) I demonstrate the identity of the algebraical sign of the Bezoutian secondaries with that of the simplified residues, generated by a process corresponding to the development of ϕx/fx under the form of an improper continued fraction (where the negative sign takes the place of the positive sign which connects the several terms of an ordinary continual function). As the simplified residue is obtained by driving out an allotrious factor, the signs of the former will of course be governed by the signs accorded by previous convention to the latter ; the convention made is, that the allotrious factors shall be taken with a sign which renders them always essentially positive when the coefficients of the given functions are real. I close the section with remarking the relation of the syzygetic factors and the residues to the convergents of the continued fraction which expresses ϕx/fx and of the continued fraction which is formed by reversing the order of the quotients in the first named fraction.