Aronhold, C. Hermite, Francesco Brioschi, R.F.A.
Two methods of treatment have been carried on in parallel lines, the unsymbolic and the symbolic; both of these originated with Cayley, but he with Sylvester and the English school have in the main confined themselves to the former, whilst Aronhold, Clebsch, Gordan, and the continental schools have principally restricted themselves to the latter.
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,...
The Aronhold process, given by the operation a as between any two of the forms, causes such an invariant to vanish.
The ternary cubic has been investigated by Cayley, Aronhold, Hermite, Brioschi and Gordan.
By the x process of Aronhold we can form the invariant S for the cubic ay+XH:, and then the coefficient of X is the second invariant T.
To assist us in handling the symbolic products we have not only the identity (ab) cx + (bc) a x + (ca) bx =0, but also (ab) x x+ (b x) a + (ax) b x = 0, (ab)a+(bc)a s +(ca)a b = 0, and many others which may be derived from these in the manner which will be familiar to students of the works of Aronhold, Clebsch and Gordan.
This peculiarity is not an accident or a special property of the fourbar chain, but is an illustration of a general law regarding the subject discovered by Aronhold and Sir A.