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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,...

00If 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 .

001 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.

00= a k; and if we wish to denote, by umbrae, a product of coefficients of degree s we employ s sets of umbrae.

00We 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.

00A 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.

00By similarly transforming the binary n ic form ay we find Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1ï¿½1 +a2m2) = aa a ï¿½ - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (alï¿½1+a2ï¿½2) = a A ï¿½ =A 1 A2, so that the umbrae A1, A 2 are a A, a ï¿½ respectively.

00= (A11+A22)n by the substitutions 51 = A l, E1+ï¿½1 2, 52 = A2E1+ï¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = ï¿½la1 +ï¿½2a2ï¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c).

00For this reason the umbrae A1, A 2 are said to be contragredient to xi, x 2.

00For this reason the umbrae -a 2, a l are said to be cogredient to 5 1 and x 2.

00may be the same or different, it is necessary that every product of umbrae which arises in the expansion of the symbolic product be of degree n, in a l, a 2; in the case of b,, b 2 of degree n 2; in the case of c 1, c 2 of degree n3; and so on.

00Kircher (Ars Magna Lucis et Umbrae), who notes elsewhere that Porta had taken some arrangement of projecting images from an Albertus, whom he distinguished from Albertus Magnus, and who was probably L.

00Kircher (Ars Magna Lucis et Umbrae, 1646); J.

00But they seem to be more " nominis umbrae " than real men; they serve the purpose of enabling the satirist to aim his blows at one particular object instead of declaiming at large.

00Symbolic 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 Ã¯¿½o, ai,...

00If 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 .

001 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.

00= a k; and if we wish to denote, by umbrae, a product of coefficients of degree s we employ s sets of umbrae.

00We 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.

00A 1, A2 Ã¯¿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) (alÃ¯¿½l +a2Ã¯¿½2) = aAaÃ¯¿½, A 2 = (alÃ¯¿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 Ã¯¿½ = a1/Ã¯¿½1 + a21=2 respectively.

00By similarly transforming the binary n ic form ay we find Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1Ã¯¿½1 +a2m2) = aa a Ã¯¿½ - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (alÃ¯¿½1+a2Ã¯¿½2) = a A Ã¯¿½ =A 1 A2, so that the umbrae A1, A 2 are a A, a Ã¯¿½ respectively.

00= (A11+A22)n by the substitutions 51 = A l, E1+Ã¯¿½1 2, 52 = A2E1+Ã¯¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = Ã¯¿½la1 +Ã¯¿½2a2Ã¯¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c).

00For this reason the umbrae A1, A 2 are said to be contragredient to xi, x 2.

00For this reason the umbrae -a 2, a l are said to be cogredient to 5 1 and x 2.

00may be the same or different, it is necessary that every product of umbrae which arises in the expansion of the symbolic product be of degree n, in a l, a 2; in the case of b,, b 2 of degree n 2; in the case of c 1, c 2 of degree n3; and so on.

00and we may suppose such identities between the symbols that on the whole only two, three, or more of the sets of umbrae are not equivalent; we will then obtain invariants of two, three, or more sets of binary forms. The symbolic expression of a covariant is equally simple, because we see at once that since AE, B, Ce,...

00(I.) Introduce now new umbrae dl, d 2 and recall that +d 2 -d 1 are cogredient with x, and x 2.

00In general we may have any two forms 01/1X1+ 'II Ã¯¿½ Yy + 02x2) p Y'x =, / / being the umbrae, as usual, and for the kth transvectant we have (4)1,,, 4)Q) k = (4)) k 4)2 -krk, a simultaneous covariant of the two forms. We may suppose of, 4, 2 to be any two covariants appertaining to a system, and the process of transvection supplies a means of proceeding from them to other covariants.

00a n) in which only its scalar coefficients occur; in fact, the special units only serve, in the algebra proper, as umbrae or regulators of certain operations on scalars (see Number).

00Kircher (Ars Magna Lucis et Umbrae), who notes elsewhere that Porta had taken some arrangement of projecting images from an Albertus, whom he distinguished from Albertus Magnus, and who was probably L.

00Kircher (Ars Magna Lucis et Umbrae, 1646); J.

00But they seem to be more " nominis umbrae " than real men; they serve the purpose of enabling the satirist to aim his blows at one particular object instead of declaiming at large.

00

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