THE THEORY OF ALGEBRAIC FORMS.
12. This part of modern algebra deals with the invariants of algebraic forms with respect to the group of linear transformations. It has been variously termed the theory of forms, the algebra of quantics, the algebra of linear trans formations, the linear or projective invariant theory. Traces of the idea may be found in papers by Lagrange, Gauss and Cauchy; and sample results on quadric forms were given by G. Boole (1841); but the construction of the systematic theory begins with the memoirs of A. Cayley (1845-) and J. J. Sylvester* (1851-). The recent developments are due mainly to German and Italian mathematicians.
13. The objects considered are algebraic forms. A form or quantic is a rational integral homogeneous function of any number of varia bles. Forms are classified according to the number of variables and the degree in which they are involved. A binary form involves two variables; a ternary, three; a quaternary, four; in the general case of p variables the adjective p-ary is employed. The degree in which the variables enter as termed the order of the form; we denote it in general by n. If n is one, the form is linear; if two, quadric (or quadratic); if three, cubic; if four, quartic (biquadratic) ; etc.
Thus the binary cubic form is where x, y are the variables and a, b, c, d the coefficients. It is usual, however, to distinguish the variables and the coefficients by subscripts, a notation which has the advantage of generality and symmetry; also binomial coefficients are introduced. Thus the above form is written
The ternary quadric is The general binary form is (1) aain -I- xi 1) . 14. The group employed consists of the totality of linear transformations of the varia bles. A linear transformation from the original variables xs. x,, . . . x to the new variables 2(1, X,, .. is defined by p equations of the first degree, as follows: . . . 4-4 pXp, . . . +1,pXp, xp = ?pal IppXp.
The determinant 4 = whose elements are the p' coefficients in these equations, is termed the modulus of the trans formation. It is assumed that 4 does not vanish; for in the latter case the X's cannot be expressed in terms of the x's, that is, the transformation is not reversible.
We shall deal mainly with binary variables and then employ the simpler notation (2) • The modulus is I —limr—lorh.
15. When a linear transformation is carried out on a form, the latter is converted into a form of the same order containing the new variables. The coefficients of this transformed quantic depend of course upon the transforma tion employed. Thus if in the quadric f we make the substitution (2), the result is a new quadric,