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XL. On Contrariants, a New Species of Invariants. By 

 J. J. Sylvester, F.R.S., Savilian Professor of Geometry in 

 the University of Oxford* . 



" Non notationes sed notiones novse desunt." 



ALL algebraists of the new school are familiar with the 

 use of contrariants derived from and, so to say, sub- 

 ordinate to, one or more primordial forms; but I am not aware 

 that, previous to my researches into the theory of the linear 

 equation in matrices, published in recent numbers of the 

 Comptes Renclus, any one has ever had occasion to consider 

 algebraical forms perfectly independent of, and given coordi- 

 nately as contravariantive to, one another. An invariant of 

 two such forms or systems of forms is called a Contrariant. 



It is no new circumstance in Mathematical History for a 

 general doctrine to take its rise in some process applied to a 

 particular investigation. It is a fact, but one not generally 

 known in this country, that the Cartesian method of coordi- 

 nates originated in a happy idea which occurred to Descartes, 

 how to solve completely a geometrical problem bequeathed to 

 posterity by Pappus, apparently without any design on the 

 part of its immortal author to create a new geometrical cal- 

 culus. So much is this the case, that some who have studied 

 that enigmatic treatise Descartes' Geomdtrie have come to 

 the conclusion that it was designed rather with the view of 

 applying geometry to algebra and obtaining graphical solu- 

 tions of equations, than with that of reducing geometrical 

 analysis under the dominion of algebraical methods. With 

 the object of familiarizing my fellow workers with the novel 

 and (as the researches above alluded to demonstrate) im- 

 portant conception, which at one stroke doubles the area of 

 invariantive theory and its geometrical applications, I propose 

 to consider the complete system of irreducible contrariants 

 to two quadratic forms with any number of variables. 



Another simple and interesting problem will be to show how 

 to transform simultaneously two contravariantive quadratics 

 (of course by contragredient substitutions) into one and the 

 same sum of powers of the substituted variables ; but this 

 must be reserved for some future occasion. 



The fundamental or irreducible invariants of two covarian- 

 tive quadratics, say/ and <£, of n variables we know are n-\- 1 

 in number (being the coefficients of the binary Quantic in X, p, 

 obtained by supplying the same system of variables to the two 

 forms /and <fi and then taking the discriminant of X/"+^0), 



* Communicated by the Author. 



