200 Prof. H. T. S. Smith on Quadratic Forms [April 31, 



(a( ), its 2th derived matrix, i. e. the matrix of order rrr= — =Ij the con- 



\i\n — i 



stituents of which are the minor determinants of order i of the matrix 

 ; and lastly, by fi, a form of any order containing I indeterminates, 

 the coefficients of which depend on the coefficients of f^. When is 

 transformed by (a^^)), let fi be transformed by (a^^O) ; if, after division or 

 multiplication by a power of the modulus of transformation, the meta- 

 morphic of fi depends on the metamorphic of f, in the same way in 

 which fi depends on f, f i is said to be a concomitant of the zth species 

 of /j. Thus : a concomitant of the 1st species is a covariant ; a con- 

 comitant of the {n—\)ih. species is a contravariant ; if n=2 there are 

 only CO variants ; if n=3 there are only covariants and contravariants ; 

 but if n>3, there will exist in general concomitants of the intermediate 

 species. 



There is an obvious difference between covariants and contravariants on 

 the one hand, and the intermediate concomitants on the other. The 

 number of indeterminates in a covariant or contravariant is the same as in 

 its primitive ; in an intermediate concomitant, the number of indeterminates 

 is always greater than in its primitive. Again, to every metamorphic of a 

 covariant or contravariant, there corresponds a metamorphic of its primi- 

 tive 'y whereas, in the case of a concomitant of the intermediate order z", a 

 metamorphic of the primitive will correspond, not to every metamorphic 

 of the concomitant, but only to such metamorphics as result from trans- 

 formations the matrices of which are the ^th derived matrices of matrices 

 of order n. 



It is also obvious that, besides the n— 1 species of concomitance here 

 defined, there are, when ?z is >3, an infinite number of other species of 

 concomitance of the same general nature. For from any derived matrix 

 we may form another derived matrix, and so on continually ; and to 

 every such process of derivation a distinct species of concomitance will 

 correspond. 



The notion of intermediate concomitance appears likely to be of use in 

 many researches ; in what follows, it is employed to obtain a definition of 

 the ordinal and generic characters of quadratic forms containing more than 

 3 indeterminates. (The case of quadratic forms containing 3 indeterminates 

 has been considered by Eisenstein in his memoir, " Neue Theoreme des 

 hoheren Arithmetik," Crelle, vol. xxxv. pp. 121 and 125.) Let 



represent a quadratic form of 7i indeterminates ; let (A^i)) be the sym- 

 metrical matrix of this form, and (A*^')) the ith. derived matrix of (AO)) ; 

 (A(*)) will also be a symmetrical matrix, and the quadratic form 



will be a concomitant of the 2th species of f. It is immaterial what 



