# Umbrae Sentence Examples

- 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. - 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. - By 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. - = (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). - For this reason the
**umbrae**A1, A 2 are said to be contragredient to xi, x 2. - For this reason the
**umbrae**-a 2, a l are said to be cogredient to 5 1 and x 2. - May 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. - And 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,... - Kircher (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. - Kircher (Ars Magna Lucis et
**Umbrae**, 1646); J. - But 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.