# 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. - F(a ' a ' a, ...a) =r A F(ao, a1, a2,ï¿½ï¿½ï¿½an), 0 1 2 n the function F(ao, al, a2,...an) is then said to be an invariant of the
**quantic**gud linear transformation. - X i, x 2) is said to be a covariant of the
**quantic**. The expression " invariantive forms " includes both invariants and covariants, and frequently also other analogous forms which will be met with. - Instead of a single
**quantic**we may have several f(ao, a1, a2...; x1, x2), 4 (b o, b1, b2,...; x1, x2), ... - 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, and subsequently to make all the**quantics**identical.**quantics** - 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. - We consider the
**quantic**to have any n number of equivalent representations a- b n -c n So that a 1 -k a 2 = b l -k b 2 - c 1 -k c 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 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. - To find the effect of linear transformation on the symbolic form of
**quantic**we will disuse the coefficients a 111 a 12, a21, a22, and employ A1, Iï¿½1, A2, ï¿½2. - If u, a
**quantic**in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities contragredient to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ... - As between the original and transformed
**quantic**we have the umbral relations A1 = A1a1 d-A2a2, A2 = /21a1+/22a2, and for a second form B1 =A 1 b 1+ A 2 b 2, B 2 =/21bl +ï¿½2b2ï¿½ The original forms are ax, bi, and we may regard them either as different forms or as equivalent representations of the same form. - The orders of the
**quantic**and covariant, and the degree and weight of the leading coefficient; calling these 'n, e,' 0, w respectively we can see that they are not independent integers, but that they are invariably connected by a certain relation n9 -2w = e. - For, if c(ao i ...x l, x 2) be a covariant of order e appertaining to a
**quantic**of order n, t (T. - 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. - If the invariants and covariants of this composite
**quantic**be formed we obtain functions of X such that the coefficients of the various powers of X are simultaneous invariants of f and 4). - 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. - If the variables of the
**quantic**f(x i, x 2) be subjected to the linear transformation x1 = a12Et2, x2 = a21E1+a2252, E1, being new variables replacing x1, x 2 and the coefficients an, all, a 21, a22, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed**quantic**f% 1tn n n-1 n-2 52) = a S +(1)a11 E 2 + (2)a2E1 E 2 +ï¿½ï¿½ï¿½ in the new variables which is of the same order as the original**quantic**; the new coefficients a, a, a'...a are linear functions 0 1 2 n of the original coefficients, and also linear functions of products, of the coefficients of substitution, of the nth degree.