Covariants Sentence Examples

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.

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.

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.

The Partial Differential Equations.--It will be shown later that covariants may be studied by restricting attention to the leading coefficient, viz.

Similarly regarding 1 x 2 as additional parameters, we see that every covariant is expressible as a rational function of n fixed covariants.

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.

Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants.

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.

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.

Hence, in both cases, contragrediency and cogrediency are identical, and contravariants are included in covariants.

Since +xZ=x x we have six types of symbolic factors which may be used to form invariants and covariants, viz.

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.

An important fact, discovered by Cayley, is that these coefficients, and also the complete covariants, satisfy certain partial differential equations which suffice to determine them, and to ascertain many of their properties.