A 1, A2 ï¿½ Ai, A 1 A 2, A2 and then Ao = al Ai+2a1a2AIA2+a2 A2 - (a1A1+a2A2) 2 = a?, A l = (a 1 A 1 +a2A2) (alï¿½l +a2ï¿½2) = aAaï¿½, A 2 = (alï¿½l +a2/-12) 2 = aM; so that A = aa l +2a A a u 152+aM5 2 = (aA6+a,e2)2; whence A1, A 2 become a A, a m, respectively and ?(S) = (a21+a,E2) 2; The practical result of the transformation is to change the umbrae a l, a 2 into the umbrae a s = a1A1 +a2A2, a ï¿½ = a1/ï¿½1 + a21=2 respectively.0 | 0 |
The two quadratic forms f, 4); the two discriminants (f, f')2,(0,4')2, and the first and second transvectants of f upon 4, (f,, >) 1 and (f, 402, which may be written (aa)a x a x and (aa) 2 .0 | 0 |
= wj, aa 1 aa 2 a a 3 the complete system of equations satisfied by an invariant.0 | 0 |
+a"aa"-1 have been much studied by Sylvester, Hammond, Hilbert and Elliott (Elliott, Algebra of Quantics, ch.0 | 0 |
= 0; and such functions satisfy the differential equation aoaa i +2a0a 2 +3a 2 aa 3 +...0 | 0 |
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