Introduction. Most of the results in this chapter are classical—some dating back to the time of Gauss and earlier—and can be derived independently of the previous general theory. But viewing the binary forms as special cases of our previous results illuminates the general theory on the one hand and economizes labor on the other. Furthermore certain problems, such as the determination of all automorphs, inaccessible in the general case, can be completely solved for binary forms.

Since much of the theory of binary forms was developed in advance of the general theory there is a wide divergence in the use of the term “determinant” as applied to a form. Gauss wrote the binary form a s f = ax2 + 2bxy + cy2 and defined the determinant of f to be b2 − ac. Kronecker preferred f = ax2 + bxy + cy2 and called b2 − 4ac its determinant. These expressions or their negatives have been variously referred to as the “discriminant” of the form. The confusion of terminology is so great that, in reading the literature, one must take great care to inform himself of the meaning of the author. We shall in this book make a clean break with tradition and define the determinant of a binary form just as it was denned for forms in more variables. That is, the determinant of ax2 + 2b0xy + cy2 shall be ac − b02 and that of ax2 + bxy + cy2 shall be ac − b2/4.