P. Gordan first proved that for any system of forms there exists a finite number of covariants, in terms of which all others are expressible as rational and integral functions.
The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important theorem of expressibility is, directly we exclude unity from the partitions, a theorem concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies.
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.
Occasionally the word " invariants " includes covariants; when this is so it will be implied by the text.
Just as cogrediency leads to a theory of covariants, so contragrediency leads to a theory of contravariants.
-2 _ ab 2an-2bn-2Crz z x () x x x, Each term on the right-hand side may be shown by permutation of a, b, c to be the symbolical representation of the same covariant; they are equivalent symbolic products, and we may accordingly write 2(ac) (bc)ai -1 bi -1 cx 2 =(ab)2a:-2b:-2c:, a relation which shows that the form on the left is the product of the two covariants n (ab) ay 2 by 2 and cZ.
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.
In general we may have any two forms 01/1X1+ 'II ï¿½ Yy + 02x2) p Y'x =, / / being the umbrae, as usual, and for the kth transvectant we have (4)1,,, 4)Q) k = (4)) k 4)2 -krk, a simultaneous covariant of the two forms. We may suppose of, 4, 2 to be any two covariants appertaining to a system, and the process of transvection supplies a means of proceeding from them to other covariants.
- An important method for the formation of covariants is connected with the form f +X4), where f and 4 are of the same order in the variables and X is an arbitrary constant.
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).
The Partial Differential Equations.--It will be shown later that covariants may be studied by restricting attention to the leading coefficient, viz.
We can so determine these n covariants that every other covariant is expressed in terms of them by a fraction whose denominator is a power of the binary form.
And that thence every symbolic product is equal to a rational function of covariants in the form of a fraction whose denominator is a power of f x.
Y1 = x 15+f2n; fï¿½ y2 =x2-f?n, f .a b = ax+ (a f) n, l; n u 2 " 2 22 2 +` n) u3 n-3n3+...+U 2jnï¿½ 3 n Now a covariant of ax =f is obtained from the similar covariant of ab by writing therein x i, x 2, for yl, y2, and, since y?, Y2 have been linearly transformed to and n, it is merely necessary to form the covariants in respect of the form (u1E+u2n) n, and then division, by the proper power of f, gives the covariant in question as a function of f, u0 = I, u2, u3,...un.
Of two or more binary forms there are also complete systems containing a finite number of forms. There are also algebraic systems, as above mentioned, involving fewer covariants which are such that all other covariants are rationally expressible in terms of them; but these smaller systems do not possess the same mathematical interest as those first mentioned.
Further, it is convenient to have before us two other quadratic covariants, viz.
T = (j, j) 2 jxjx; 0 = (iT)i x r x; four other linear covariants, viz.
Remark.-The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C = o, p is the common factor of i and j.
F= ai; the Hessian H = (ab) 2 azbx; the quartic i= (ab) 4 axb 2 x; the covariants 1= (ai) 4 ay; T = (ab)2(cb)aybyci; and the invariants A = (ab) 6; B = (ii') 4 .
Iv.) was the first to remark that the study of covariants may be reduced to the study of their leading coefficients, and that from any relations connecting the latter are immediately derivable the relations connecting the former.
Q 1 The Unreduced Generating Function Which Enumerates The Covariants Of Degrees 0, 0' In The Coefficients And Order E In The Variables.
Besides the invariants and covariants, hitherto studied, there are others which appertain to particular cases of the general linear substitution.
The Hessian A has just been spoken of as a covariant of the form u; the notion of invariants and covariants belongs rather to the form u than to the curve u=o represented by means of this form; and the theory may be very briefly referred to.
The case is less frequent, but it may arise, that there are covariant systems U= o, V = o, &c., and U' = o, V' = o, &c., each implying the other, but where the functions U, V, &c., are not of necessity covariants of u.
The theory of the invariants and covariants of a ternary cubic function u has been studied in detail, and brought into connexion with the cubic curve u = o; but the theory of the invariants and covariants for the next succeeding case, the ternary quartic function, is still very incomplete.
Since +xZ=x x we have six types of symbolic factors which may be used to form invariants and covariants, viz.
The number of different symbols a, b, c,...denotes the the covariants are homogeneous, but not in general isobaric functions, of the coefficients of the original form or forms. Of the above general form of covariant there are important transformations due to the symbolic identities: - (ï¿½b) 2 2)2 = a b - a b; (xï¿½ = as a consequence any even power of a determinant factor may be expressed in terms of the other symbolic factors, and any uneven power may be expressed as the product of its first power and a function of the other symbolic factors.
Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms.