XXVI 



x', y' and a, y transform the equation H = 0, derived from U = T 

 into the equation H' = 0, derived in the same way from U' = 0. 

 Moreover, as in the former case by direct calculation, it is found that 

 H', a specially constructed function of x', y' and the coefficients in U' y 

 is divisible by H, the same function of x, y and the coefficients in U ; 

 the quotient being a quantity depending again only upon the constants 

 in the transforming relations. Consequently it appears that a com- 

 bination of the coefficients and the variables in the original equation 

 exists such that, when the equation is transformed by means of rela- 

 tions of the type indicated, and exactly the same combination of the 

 new coefficients and the new variables is constructed, the two com- 

 binations are equal to one another save as to a factor dependent solely 

 upon the transforming relations. Such a combination of the coeffi- 

 cients and the variables is called a covariant. 



The first notice of such a property appears to have been made by 

 Lagrange. And Gauss discussed the invariance of the discriminants 

 of certain expressions when the latter are subjected to linear trans- 

 formations. Again, Boole in 1841 had shown that this invariantive 

 property belongs to all discriminants, and he gave a method of deduc- 

 ing some other functions of this kind. Boole's paper suggested to 

 Cayley a much more general subject the permanence of invarian- 

 tive form so that he set himself the question of finding " all the 

 derivatives of any number of functions which have the property of 

 preserving their form unaltered after any linear transformation of 

 the variables." The first set of results obtained by his investiga- 

 tions related to invariants ; they appeared in his famous paper,* ' On 

 the Theory of Linear Transformations,' published half a century ago. 

 The second set of results related to covariants ; they appeared in the 

 paper, f ' On Linear Transformations,' published in the succeeding 

 year. In these two papers Cayley demonstrated the general exist- 

 ence of a number of functions, both invariants and covariants (afe 

 first he called them hyperdeterminants), which preserve their form 

 under linear transformation. 



These discoveries of Cayley establish him as the founder of what 

 is called sometimes modern algebra, sometimes invariants and Co- 

 variants, sometimes theory of forms ; the origination of the theory is 

 incontestably his, and it is universally ascribed to him. 



A discovery of this general importance and complete novelty soon 

 attracted the attention of other workers. It is not too much to say 

 that the subsequent investigations long absorbed the active interest 

 of many mathematicians, and, as a result, the theory has influenced all 



* ' C. M. P.,' vol. 1, No. 13; 'Camb. Math. Jour.,' TO!. 4 (1845), pp. 193-209. 



t 'C. M. P.,' vol. 1, No. 14; 'Camb. and Dubl. Math. Jour.,' vol. 1 (1846), pp. 

 104 122. The two papers were rewritten, and appeared in ' Crelle,' vol. 30 

 (1846), pp. 1 37, under the title " Memoire sur les Hyperdeterminants." 



