# Quantic Sentence Examples

In the theory of forms we seek functions of the coefficients and variables of the original

**quantic**which, save as to a power of the modulus of transformation, are equal to the like functions of the coefficients and variables of the transformed**quantic**. We may have such a function which does not involve the variables, viz.This notion is fundamental in the present theory because we will find that one of the most valuable artifices for finding invariants of a single

**quantic**is first to find simultaneous invariants of several different quantics, and subsequently to make all the quantics identical.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 a single

**quantic**of the first order (ab) is the symbol of a function of the coefficients which vanishes identically; thus (ab) =a1b2-a2bl= aw l -a1ao=0 and, indeed, from a remark made above we see that (ab) remains unchanged by interchange of a and b; but (ab), = -(ba), and these two facts necessitate (ab) = o.It will be a useful exercise for the reader to interpret the corresponding covariants of the general

**quantic**, to show that some of them are simple powers or products of other covariants of lower degrees and order.For the unipartite ternary

**quantic**of order n he finds that the fundamental system contains a (n+4) (n -1) individuals.If the form, sometimes termed a

**quantic**, be equated to zero the n+I coefficients are equivalent to but n, since one can be made unity by division and the equation is to be regarded as one for the determination of the ratio of the variables.A particular

**quantic**of the system may be of the same or different degrees in the pairs of variables which it involves, and these degrees may vary from**quantic**to**quantic**of the system.