ON THE THEORY OF NUMBERS. 29/ 



and are expressed in a terminology which has been adopted by most sub- 

 sequent writers. The second part (arts. 223-307) is occupied, after some 

 preliminary disquisitions (arts. 223-233), with the ulterior researches of 

 Gauss himself. We proceed then to give a summary of the definitions and 

 theorems contained in the first of these two portions. 



85. Elementary Definitions. — The quadratic form ax^ + ^bxy + cy"^ is 

 symbolized by the formula (a, h, c) (x, yf, or, when it is not necessary to 

 specify the indeterminates, by the simpler formula (a, b, c). The second 

 coefficient is always supposed to be even ; and an expression of the form 

 px- + qxy-\-ry" (in which q is uneven) is not considered by Gauss as itself a 

 quadratic form, but as the half of the quadratic form (2p, q, 1r). The 

 discriminant F—ac of the form (a, b, c) is called by Gauss the determinant 

 of the form ; an expression which at the present time it would be neither 

 possible nor desirable to alter. When two forms are equivalent, they are 

 said to he properly equivalent if the modulus of transformation is +J, but 

 improperly equivalent if it is —1. Only those forms which are properly 

 equivalent to one another are considered to belong to the same class ; two 

 forms which are only improperly equivalent are said to belong to opposite 

 classes. This distinction between proper and improper equivalence is due to 

 Gauss, and is of very great importance. In what follows, unless the con- 

 trary is expressly specified, we shall use the terms equivalence and auto- 

 morphism to denote proper equivalence and proper automorphism. It is readily 

 seen that the greatest common divisors of «, 26, c, and of a, b, c are the same 

 for (a, b, c) and for every form equivalent to («, b, c) ; if each of those greatest 

 common divisors is unity, (a,b,c) is a properly primitive form, and the class 

 of forms equivalent to {a, b, c) a properly primitive class ; if the first greatest 

 common divisor be 2, and the second 1, the form, and the class of forms 

 equivalent to it, are termed improperly primitive. Every form which is not 

 itself primitive, is a numerical multiple of some primitive form of a less de- 

 terminant, and is therefore called a derived form. Thus x"+Sy' is a pro- 

 perly primitive form of det.— 3, 2x"-\-2xy + 2y' is an improperly primitive 

 form of the same determinant ; while 2x'^ + 6y^, 4*^ -j- 4x2/ + 4^^ are derived 

 forms of det. — 12. 



In all questions relating to the representation of numbers, or the equiva- 

 lence of forms, it is sufficient to consider primitive forms, as the solution of 

 these problems for derived forms is immediately deducible from their solution 

 for primitive forms ; but in certain investigations connected with the trans- 

 formation of forms the consideration of derived forms is indispensable. (The 

 problem of art. 82 coincides with that of the representation of numbers, in the 

 case of binary forms of any order.) 



The nature of the quadratic form («, b, c) depends very mainly on the 

 value of its determinant, which we shall symbolize by D. (1) If D=0, the 

 form (a, b, c) reduces itself to an expression of the type m(px+qyy, 

 p and q denoting two numbers relatively prime, and m being the greatest 

 common divisor of a, b, c. The arithmetical theory of such expressions, 

 which are not binary forms at all, since they are adequately represented by 

 a formula such as wX", is so simple, and at the same time diverges so much 

 from that of true binary quadratic forms, that we shall not advert to it 

 again in this Report, and in all that follows the determinant is supposed to 

 be different from zero. (2) When D is a perfect positive square, the form 

 (a, b,c) reduces itself to an expression of the type m(p^x + q^y) {pj: + qoy), 

 i. e. it becomes a product of two linear forms. Owing to this circum- 

 stance the theory of forms of a square determinant is so much simpler 

 than that of other quadratic forms, that we shall not enter into any details 



