ALGEBRAIC FORMS - TERNARY FORMS If three ternary linear forms vanish for the same values of the variables, not all zero, i.e., if a1x1 + a2x2+ a3x3 = O b1 xl + b2x2+ b3x3 = o cixi+c2x2+c3x3=O, then their determinant, denoted by or merely (abc), is zero. No linear transformation will disturb this relation, hence (abc) is an invariant of the three linear forms. It is the type of all invariant combinations of symbols. If the forms are abbreviated to (ax), (bx) and (cx) respectively, and written with different sets of variables, e.g., (ax), (by), (cz), then (abc) is, in another expression : and that aik = aki . Further, any symbolic product means and means . Similar conventions yield, for a ternary cubic or the invariant: (abc) (abd) (acd) (bcd), which is one of the two well-known invariants of the cubic. It is obvious also that (ax) (bx) (cx) will be a symbol of a covariant of the cubic form; it is commonly called the Hessian of the cubic.
Such a structure might, however, prove to signify identically zero when symbols are translated into actual coefficients. Such cases are and (ade) (bde) (cde) . Again, the difference of two supposed invariant forms may be a zero form, in which case the expressions, apparently distinct, signify actually the same invariant. Ambiguity of expression results from the ex istence of zero forms, and these in turn from a few elementary identities which are essentially alike. For binary covariants containing not more than two sets of variables, these identities are combinations of the following three: whatever the symbols may represent.
Equally recognizable is the possible ambiguity of symbolic formulae in invariants of ternary forms. The typical identity is this: (abc) (dx) (abd) (cx) + (acd) (bx)(bcd) (ax) = o , or this: (abc) (def) (abd) (cef) + (abe) (cdf) (abf) (cde)=o .