# Covariant Sentence Examples

- X i, x 2) is said to be a
**covariant**of the quantic. The expression " invariantive forms " includes both invariants and, and frequently also other analogous forms which will be met with.**covariants** - From these formulae we derive two important relations, dp4 = or the function F, on the right which multiplies r, is said to be a simultaneous invariant or
**covariant**of the system of quantics. - And we may suppose such identities between the symbols that on the whole only two, three, or more of the sets of umbrae are not equivalent; we will then obtain invariants of two, three, or more sets of binary forms. The symbolic expression of a
**covariant**is equally simple, because we see at once that since AE, B, Ce,... - It is always an invariant or
**covariant**appertaining to a number of different linear forms, and as before it may vanish if two such linear forms be identical. - In general it will be simultaneous
**covariant**of the different forms n 1 rz 2 n3 a, b x, ? - +P=n1, j+k+...+T =n3, It will also be a
**covariant**if the symbolic product be factorizable into portions each of which satisfies these conditions. - If the forms be identical the sets of symbols are ultimately equated, and the form, provided it does not vanish, is a
**covariant**of the form ate. - For the cubic (ab) 2 axbx is a
**covariant**because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous**covariant**of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic**covariant**, of the single cubic, commonly known as the Hessian. - By simple multiplication (al b l b2 -24a2bib2+ala2b;)xi +(aibz -ala214b2-aia2blb2+a2b2)xlx2 + (aia 2 b2 - 2a l a2b l b2 +a2/4b 2)x2; and transforming to the real form, (aob 2 - 2a1b,+a2bo)xi (aob 3 -a l b 2 - alb,+a3bo)xlx2 + (aib3 - 2a2b2+a3b1)x2, the simultaneous
**covariant**; and now, putting b = a, we obtain twice. - The degree of the
**covariant**in the coefficients is equal to the number of different symbols a, b, c, ... - The order of the
**covariant**) is P+P+T+... - It will be apparent that there are four numbers associated with a
**covariant**, viz. - 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. - -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 twon (ab) ay 2 by 2 and cZ.**covariants** - (ab)(ac)bxcx = - (ab)(bc)axcx = 2(ab)c x {(ac)bx-(bc)axi = 1(ab)2ci; so that the
**covariant**of the quadratic on the left is half the product of the quadratic itself and its only invariant. - (a m b n) k (ab) kamkbn-k x, x - x it is clear that the k th transvectant is a simultaneous
**covariant**of the two forms. - The two forms ax, bx, or of, 0, may be identical; we then have the kth transvectant of a form over itself which may, or may not, vanish identically; and, in the latter case, is a
**covariant**of the single form. - For the quadratic it is the discriminant (ab) 2 and for ax2 the cubic the quadratic
**covariant**(ab) 2 axbx. - In particular, when 4) is a
**covariant**of f, we obtain in this mannerof f.**covariants** - In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a
**covariant**which is called the first evectant of the original invariant. - D x the first evectant; and thence 4cxdi the second evectant; in fact the two evectants are to numerical factors pres, the cubic
**covariant**Q, and the square of the original cubic. - Similarly regarding 1 x 2 as additional parameters, we see that every
**covariant**is expressible as a rational function of n fixed.**covariants** - We can so determine these n
that every other**covariants****covariant**is expressed in terms of them by a fraction whose denominator is a power of the binary form. - Every
**covariant**is rationally expressible by means of the forms f, u 2, u3,... - To exhibit any
**covariant**as a function of uo, ul, a n = (aiy1+a2y2) n and transform it by the substitution fi y 1+f2 y where f l = aay 1, f2 = a2ay -1, x y - x y = X x thence f . - 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 thein respect of the form (u1E+u2n) n, and then division, by the proper power of f, gives the**covariants****covariant**in question as a function of f, u0 = I, u2, u3,...un. - If the
**covariant**(f,4) 1 vanishes f and 4 are clearly proportional, and if the second transvectant of (f, 4 5) 1 upon itself vanishes, f and 4) possess a common linear factor; and the condition is both necessary and sufficient. - The .sextic
**covariant**t is seen to be factorizable into three quadratic factors 4 = x 1 x 2, =x 2 1 - 1 - 2 2, 4) - x, which are such that the three mutual second transvectants vanish identically; they are for this reason termed conjugate quadratic factors. - -, reduce s x2ax1 -x10x2 to the form j Oz ON 2 1 1 j 2 i The Binary Quintic.-The complete system consists of 23 forms, of which the simplest are f =a:; the Hessian H = (f, f') 2 = (ab) 2axbz; the quadratic
**covariant**i= (f, f) 4 = (ab) 4axbx; and the nonic co variant T = (f, (f', f") 2) 1 = (f, H) 1 = (aH) azHi = (ab) 2 (ca) axbycy; the remaining 19 are expressible as transvectants of compounds of these four. - We will write the cubic
**covariant**(f, i) 2 =j, and then remark that the result, (f,j) 3 = o, can be readily established. - The form j is completely defined by the relation (f,j) 3 =o as no other
**covariant**possesses this property. - On this principle the
**covariant**j is expressible in the form R 2 j =5 3 + BS 2 a+4ACSa 2 + C(3AB -4C)a3 when S, a are the above defined linear forms. - Now, evidently, the third transvectant of f, expressed in this form, with the cubic pxgxrx is zero, and hence from a property of the
**covariant**j we must have j = pxgxrx; showing that the linear forms involved are the linear factors of j. - X x x To form an invariant or
**covariant**we have merely to form a product of factors of two kinds, viz. - Such a symbolic product, if its does not vanish identically, denotes an invariant or a
**covariant**, according as factors az, bz, cz,... - We can see that (abc)a x b x c x is not a
**covariant**, because it vanishes identically, the interchange of a and b changing its sign instead of leaving it unchanged; but (abc) 2 is an invariant. - The complete
**covariant**and contravariant system includes no fewer than 34 forms; from its complexity it is desirable to consider the cubic in a simple canonical form; that chosen by Cayley was ax 3 +by 3 + cz 3 + 6dxyz (Amer. - One more
**covariant**is requisite to make an algebraically complete set. - When R =0, and neither of the expressions AC - B 2, 2AB -3C vanishes, the
**covariant**a x is a linear factor of f; but, when R =AC - B 2 = 2AB -3C =0, a x also vanishes, and then f is the product of the form jx and of the Hessian of jx. - It has been shown above that a
**covariant**, in general, satisfies four partial differential equations. - Two of these show that the leading coefficient of any
**covariant**is an isobaric and homogeneous function of the coefficients of the form; the remaining two may be regarded as operators which cause the vanishing of the**covariant**. - Let a
**covariant**of degree e in the variables, and of degree 8 in the coefficients (the weight of the leading coefficient being w and n8-2w = ï¿½), be Coxl -}- ec l l 1 x 2 -{-... - - zn +9 1 -z2.1 -z3....1-z8; and since this expression is unaltered by the interchange of n and B we prove Hermite's Law of Reciprocity, which states that the asyzygetic forms of degree 0 for the /t ie are equinumerous with those of degree n for the The degree of the
**covariant**in the variables is e=nO-2w; consequently we are only concerned with positive terms in the developments and (w, 0, n) - (w - r; 0, n) will be negative unless nO It is convenient to enumerate the seminvariants of degree 0 and order e=n0-2w by a generating function; so, in the first written generating function for seminvariants, write z2 for z and az n for a;. - Again, for the cubic, we can find A3(z) - -a6z6 1 -az 3.1 -a 2 z 2.1 -a 3 z 3.1 -a4 where the ground forms are indicated by the denominator factors, viz.: these are the cubic itself of degree order I, 3; the Hessian of degree order 2, 2; the cubi-
**covariant**G of degree order 3, 3, and the quartic invariant of degree order 4, o. - A Similar Theorem Holds In The Case Of Any Number Of Binary Forms, The Mixed Seminvariants Being Derived From The Jacobians Of The Several Pairs Of Forms. If The Seminvariant Be Of Degree 0, 0' In The Coefficients, The Forms Of Orders P, Q Respectively, And The Weight W, The Degree Of The
**Covariant**In The Variables Will Be P0 Qo' 2W =E, An Easy Generalization Of The Theorem Connected With A Single Form. - If now the nti c denote a given pencil of lines, an invariant is the criterion of the pencil possessing some particular property which is independent alike of the axes and of the multiples, and a
**covariant**expresses that the pencil of lines which it denotes is a fixed pencil whatever be the axes or the multiples. - The general form of
**covariant**is therefore (ab) h i (ac) h2 c) (b h3 a i bb2c'e3...abia?2b?3... - It may denote a simultaneous orthogonal invariant of forms of orders n i, n2, n3,...; degree 0 of the
**covariant**in the coefficients. - The number of different symbols a, b, c,...denotes the the
are homogeneous, but not in general isobaric functions, of the coefficients of the original form or forms. Of the above general form of**covariants****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. - Hence in the above general form of
**covariant**we may suppose the exponents h 1, h2, h3,...ki, k2, k3,... - For the linear forms aoxi+aix2=ax = b x there are four fundamental forms ax=a:,x i --+a i x 2 of degree-order (1, 1), x7-1--4_ ï¿½ (0, 2), a i x, 1), a b =a2+ai ï¿½ (2, 0), (iii.) and (iv.) being the linear
**covariant**and the quadrinvariant respectively. - There is no linear
**covariant**, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x . - We have in the Hessian the first instance of a
**covariant**of a ternary form. - The Hessian A has just been spoken of as a
**covariant**of the form u; the notion of invariants andbelongs 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.**covariants** - 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 necessityof u.**covariants** - M - i) have the
**covariant**property.