III. A memoir on the automorphic linear transformation of a bipartite quadric function

Abstract

The question of the automorphic linear transformation of the function x 2 + y 2 + z 2 , that is the transformation by linear substitutions, of this function into a function x 2 + y 2 + z 2 , of the same form, is in effect solved by some formulæ of Euler’s for the transformation of coordinates, and it was by these formulæ that I was led to the solution in the case of the sum of n squares, given in my paper “ Sur quelques propriétés des déterminants gauches. ”A solution grounded upon an à-priori investigation and for the case of any quadric function of n variables, was first obtained by M. Hermite in the memoir “Remarques sur une Mémoire de M. Cayley relatif aux déterminants gauches.” This solution is in my Memoir “ Sur la transformation dune function quadratique en ellememe par des substitutions linéairesj,” presented under a somewhat different form involving the notation of matrices. I have since found that there is a like transform­ation of a bipartite quadric function, that is a lineo-linear function of two distinct sets, each of the same number of variables, and the development of the transformation is the subject of the present memoir. 1. For convenience, the number of variables is in the analytical formulae taken to be 3, but it will be at once obvious that the formulae apply to any number of variables what­ever. Consider the bipartite quadric ( a, b, c )( x, y, z )(x, y, z) | a', b', c' | | a", b", c" | which stands for ( ax + by + cz )x +( a'x + b'y + c'z )y +( a"x + b"y + c"z )z, and in which ( x, y, z ) are said to be the nearer variables, and (x, y, z) the further variables of the bipartite.