VII. On the Theory of Determinants. By A. Cayley, Esq. Fellow of Trinity 



College. 



[Read Feb. 20, 1843.] 



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 re- 

 solution 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 Crenner's 

 Introduction a VAjialyse des Lignes ConcJies, 1750. They are afterwards met with in a Memoir 

 On Elimination, by Bezont, Mhnoires de V Acadcmie, 1764. In two Memoirs by Laplace and 

 Verndermonde in the same collection, 1772. In Bezont's Theory of Equations, and in Memoirs 

 by Binet, Journal PolytecJtnique, Vol. ix. ; by Cauchj', ditto. Vol. x. ; by Jacobi, Crelles Journal, 

 Vol. XXII.; Lebcsgue Liouville, Vol. vi. &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 developements.) 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 par- 

 ticular 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. 



In the first section I liave denoted a determinant, by simply writing down in the form of 

 a square the different quantities of which it is made up. This is not concise, but it is clearer 

 than any abridged notation. The ordinary properties of determinants, I have throughout taken 

 for granted ; these may easily be learnt by referring to the Memoirs of Cauchj' and Jacobi, 

 quoted ai)ove. It may however be convenient to write down the following fundamental pro- 

 perty, demonstrated by these authors, and by Binet. 



«',/3' 



p't a 



pa + (7/3 . . , pa' + a ji'. ■ , 

 pa + (t'/3 . . , pa + cr'fi'. ■ 



.(o). 



K2 



