12] 
63 
12. 
ON THE THEORY OF DETERMINANTS. 
[From the Transactions of the Cambridge Philosophical Society, vol. vin. (1843), pp. 1 16.] 
The following Memoir is composed of two separate investigations, each of them 
having a general reference to the Theory of Determinants, but otherwise perfectly 
unconnected. The name of “ Determinants ” or “ Resultants ” has been given, as is well 
known, to the functions which equated to zero express the result of the elimination 
of any number of variables from as many linear equations, without constant terms. 
But the same functions occur in the resolution of a system of linear equations, in the 
general problem of elimination between algebraic equations, and particular cases of them 
in algebraic geometry, in the theory of numbers, and, in short, in almost every part 
of mathematics. They have accordingly been a subject of very considerable attention 
with analysts. Occurring, apparently for the first time, in Cramer’s Introduction à 
VAnalyse des Lignes Courbes, 1750 : they are afterwards met with in a Memoir On 
Elimination, by Bezout, Mémoires de VAcadémie, 1764; in two Memoirs by Laplace 
and Vandermonde in the same collection, 1774; in Bezout’s Théorie générale des 
Equations algébriques [1779] ; in Memoirs by Binet, Journal de l’Ecole Polytechnique, 
vol. ix. [1813]; by Cauchy, ditto, vol. x. [1815]; by Jacobi, Crelle’s Journal, vol. xxii. 
[1841] ; Lebesgue, Liouville, [vol. II. 1837], &c. The Memoirs of Cauchy and Jacobi 
contain the greatest part of their known properties, and may be considered as constituting 
the general theory of the' subject. In the first part of the present paper, I consider 
the properties of certain derivational functions of a quantity U, linear in two separate 
sets of variables (by the term “ Derivational Function,” I would propose to denote 
those functions, the nature of which depends upon the form of the quantity to which 
they refer, with respect to the variables entering into it, e.g. the differential coefficient 
of any quantity is a derivational function. The theory of derivational functions is 
apparently one that would admit of interesting developments). The particular functions 
of this class which are here considered, are closely connected with the theory of the 
reciprocal polars of surfaces of the second order, which latter is indeed a particular case 
of the theory of these functions. 
In the second part, I consider the notation and properties of certain functions 
resolvable into a series of determinants, but the nature of which can hardly be explained 
independently of the notation.