ALGEBRAIC FORMS - BINARY FORMS The rational invariants of a single binary form of order less than ten have been thoroughly studied and tabulated. First to be noticed was naturally the discriminant or determinant of a quadric form, for it was known to vanish when the quadric has two equal factors; and their equality could not be altered by linear transformations, therefore the determinant after trans formation must be a multiple of that before the transformation. To verify this, take the quadric axi2 2bxix2-1-cx22 , the dis criminant of which is D =4b2-4ac. The transformation is given by the equations:— These four functions, all well known in the theory of equations, are connected by an identical relation or syzygy : RP = o. This shows that Q-1-f•/—R and Q—f1/—R are perfect third powers of the factors of H.
Their explicit formulae are:— This suggests that if ternary forms of higher order are ex pressed symbolically as powers of linear forms, operators of this same type, denoted by a a a , acting on a collection of y (ax a az/ such forms would produce invariants. The property of invariance is now located in the elementary operator. A single form may be represented by several symbols in the same problem; a ternary quadric, by and indifferently. Then the 2 operation — a will yield, when explicated, a This result, if not identically equal to zero, is an invariant of the ternary quadric; in fact, its discriminant. An invariant so symbolized can be translated into terms of the actual coeffi cients of the quadric if we agree that shall mean the factors on the right are perfect squares, a fact which aids in solving the quartic equation. The discriminant R is expressible in terms of i and j: R = — When R = o, f and H have each a double factor, while T has one five-fold factor. If further i and j are both zero, the quartic f has a triple factor, and that factor occurs in 11 to the fourth power. Were H to vanish identically, f must be a perfect fourth power. (If H' is formed from H as 11 is from f, and T' from H and H' as T is from f and H, then T' differs from T only by an invariant factor.) These details show that, for forms of higher order and in more than two variables, invariants would be unwieldy if written out in full. An abbreviated notation is a necessity for the develop ment of the theory. The most generally used are the hyper determinant notation of Arthur Cayley, the Clebsch-Aronhold notation, which is practically the same, and modifications of either. Aronhold, R. F. A. Clebsch and Gordan use the follow ing, some slight improvements having been made by E. Study.