definition
The fully shaded inner region of a shadow cast by an opaque object.
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The central region of a sunspot.
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A shadow.
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An uninvited guest brought along by one who was invited.
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One of the family Umbridae of mudminnows.
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A sciaenoid fish, the umbrine.
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 .
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.
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.
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.
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