- Symbolic Form.-Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce the symbolic form of Aronhold, Clebsch and Gordan; they write the form Iln n n-1 n-1 n n n aixi+a2x2) = 44+(1) a l a 2 x 1 x2+...+a2.x2=az wherein al, a2 are
**umbrae**, such that n-1 n-1 n a 1, a 1 a 2, ...a 1 a 2, a2 are symbolical respreentations of the real coefficients ï¿½o, ai,... - If we restrict ourselves to this set of symbols we can uniquely pass from a product of real coefficients to the symbolic representations of such product, but we cannot, uniquely, from the symbols recover the real form, This is clear because we can write n-1 n-2 2 2n-3 3 a1a2 =a l a 2, a 1 a 2 = a 1 a2 while the same product of
**umbrae**arises from n n-3 3 2n-3 3 aoa 3 = a l .a a 2 = a a 2 . - 1 1 Hence it becomes necessary to have more than one set of
**umbrae**, so that we may have more than one symbolical representation of the same real coefficients. - = a k; and if we wish to denote, by
**umbrae**, a product of coefficients of degree s we employ s sets of**umbrae**. - We write;L 22 = a 1 a 2 .b 1 n-2 b2s 3 n - 3 3 n-3 3 n-3 3 a 3 = a 1 a 2 .b 1 b 2 .c 1 c2, and so on whenever we require to represent a product of real coefficients symbolically; we then have a one-to-one correspondence between the products of real coefficients and their symbolic forms. If we have a function of degree s in the coefficients, we may select any s sets of
**umbrae**for use, and having made a selection we may when only one quantic is under consideration at any time permute the sets of**umbrae**in any manner without altering the real significance of the symbolism.

» more...