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 **quantic**s, and subsequently to make all the **quantic**s 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.