# How to use *Umbrae* in a sentence

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.Advertisement