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(1845 and 1846) in the Cambridge and the Cambridge and Dublin 

 Mathematical Journals, and under a different title in * Crelle.' The 

 antecedent state of the problem was as follows : The theory of the 

 linear transformations of binary and ternary quadratic functions had 

 been established by Gauss, the same being in fact the foundation of 

 his researches upon quadratic forms, as developed in the ' Recherches 

 Arithmetiques ; ' and that of the linear transformations of quadratic 

 functions of any number of yariables had been considered by Jacobi 

 and others. A very important step was made by Mr. Boole, who 

 showed that the fundamental property of the determinant (or, as it is 

 now commonly called, discriminant) of a quadratic form applied to 

 the resultant (discriminant) of a form of any degree and number of 

 variables, the property in question being, in fact, that of remaining 

 unaltered to a factor pres, when the coefficients are altered by a linear 

 transformation of the variables, or as it may for shortness be called, 

 the property of invariancy : the theorem just referred to, suggested 

 to Mr. Cayley the researches which led him to the discovery of a 

 class of functions (including as a particular case the discriminant), 

 all of them possessed of the same characteristic property. These 

 functions, called at first hyperdeterminants, are now called invariants ; 

 they are included in the more general class of functions called co- 

 variants, the difference being that these contain as well the va- 

 riables as the coefficients of the given form or forms. The theory 

 has an extensive application to geometry, and in particular to the 

 theory of the singularities of curves and surfaces. This theory for 

 plane curves was first established (1834) by Pliickerupon geometrical 

 principles ; the analytical theory for plane curves is the subject of a 

 memoir by Mr. Cayley in * Crelle,' and of his recent memoir in the Phi- 

 losophical Transactions, " On the Double Tangents of a Plane Curve," 

 based upon a Note by Mr. Salmon. The corresponding geometrical 

 theory for curves of double curvature and developable surfaces, was 

 first established in Mr. Cayley's memoir on this subject in ' Liouville' 

 and the ' Cambridge and Dublin Mathematical Journal.* The theory 

 for surfaces in general, is mainly due to Mr. Salmon. Among Mr. 

 Cayley's other memoirs upon geometrical subjects, may be mentioned 

 several papers on the Porism of the in-and-circum scribed polygon, 

 and on the corresponding theory in solido ; a memoir on the twenty- 

 seven right lines upon a surface of the third order, and the memoir 



