The arithmetically reduced indefinite quadratic form in n-variables

ƒ (x 1 ,x 2 , ... ,x n ) = Ʃn r,s = 1a rs x r x s , or for brevity, sayƒ (x ), wherea rs =a sr anda rs is any real number, rational or irrational, be a quadratic form inn -variables. Suppose that the determinant ∆ = │a rs │≠ 0, so thatƒ (x ) cannot be expressed as a quadratic form with fewer thann variables. From (1) can be derived an infinity of formsg (y 1 ,y 2 , ... ,y n ) = Ʃn r,s = 1b rs y r y s , sayg (y ), withb rs =b sr , by means of the linear substitutionsx r = Ʃn s = 1λrs y s , (r = 1, 2, ...,n ), where the λ’s are integers and the determinant | λrs | = 1. We consider throughout only such substitutions. All the formsg (y ) have the same determinant ∆. They are said to be equivalent toƒ (x ) and to define a class of forms, the class including all the forms equivalent toƒ (x ) and only these. The problem of selecting a particular form as representing the class,i. e ., the so-called reduced form, is fundamental.