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