we find
(ddX, ddY, ddZ, …) = (λ1, λ2, λ3, …) (ddx, ddy, ddz, …)
| μ1, | μ2, | μ3, | … | ||
| ν1, | ν2, | ν3, | … | ||
| . | . | . | . |
Observe the notation, which is that introduced by Cayley into the theory of matrices which he himself created.
Just as cogrediency leads to a theory of covariants, so contragrediency leads to a theory of contravariants. If u, a quantic in x, y, z, …, be expressed in terms of new variables X, Y, Z …; and if, ξ, η, ζ, …, be quantities contragredient to x, y, z, …; there are found to exist functions of ξ, η, ζ …, 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 Ξ, Η, Ζ, … of the transformed coefficients of u; such functions are called contravariants of u. There also exist functions, which involve both sets of variables as well as the coefficients of u, possessing a like property; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form.
As between the original and transformed quantic we have the umbral relations
A1 = λ1a1 + λ2a2, A2 = μ1a1 + μ2a2,
and for a second form
B1 = λ1b1 + λ2b2, B2 = μ1b1 + μ2b2.
The original forms are an
x, bn
x, and we may regard them either as different forms or as equivalent representations of the same form. In other words, B, b may be regarded as different or alternative symbols to A, a. In either case
(AB) = A1B2 − A2B1 = (λμ)(ab);
and, from the definition, (ab) possesses the invariant property. We cannot, however, say that it is an invariant unless it is expressible in terms of the real coefficients. Since (ab) = a1b2 − a2b1, that this may be the case each form must be linear; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, its real expression being a0b1 − a1b0. This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets a1, a2; b1, b2 are alternative, are ultimately put equal to one another and (ab) vanishes. A single linear form has, in fact, no invariant. When either of the forms is of an order higher than the first (ab), as not being expressible in terms of the actual coefficients of the forms, is not an invariant and has no significance. Introducing now other sets of symbols C, D, …; c, d, … we may write
(AB)i(AC)j(BC)k… = (λμ)i+j+k+…(ab)i(ac)j(bc)k…,
so that the symbolic product
(ab)i(ac)j(bc)k…,
possesses the invariant property. If the forms be all linear and different, the function is an invariant, viz. the ith power of that appertaining to ax and bx multiplied by the jth power of that appertaining to ax and cx multiplied by &c. If any two of the linear forms, say px, qx, be supposed identical, any symbolic expression involving the factor (pq) is zero. Notice, therefore, that the symbolic product (ab)i(ac)j(bc)k… may be always viewed as a simultaneous invariant of a number of different linear forms ax, bx, cx, …. In order that (ab)i(ac)j(bc)k… may be a simultaneous invariant of a number of different forms an1
x, bn2
x, cn3
x,…, where n1, n2, n3, … may be the same or different, it is necessary that every product of umbrae which arises in the expansion of the symbolic product be of degree n1 in a1, a2; in the case of b1, b2 of degree n2; in the case of c 1, c2 of degree n3; and so on. For these only will the symbolic product be replaceable by a linear function of products of real coefficients. Hence the condition is
i + j + … = n1,
i + k + … = n2,
j + k + … = n3,
....
If the forms an
x, bn
x, cn
x, … be identical the symbols are alternative, and provided that the form does not vanish it denotes an invariant of the single form an
x.
There may be a number of forms an
x, bn
x, cn
x, … 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 Aξ, Bξ, Cξ, … are equal to ax, bx, cx, … respectively, the linear forms ax, bx, cx, … possess the invariant property, and we may write
(AB)i(AC)j(BC)k…Aρ
ξBσ
ξCτ
ξ…
= (λμ)i+j+k+…(ab)i(ac)j(bc)k…aρ
xbσ
xcτ
x…,
and assert that the symbolic product
(ab)i(ac)j(bc)k…aρ
xbσ
xcτ
x…,
possesses the invariant property. 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 an1
x, bn2
x, cn3
x, … if
i + j + … + ρ = n1,
i + j + … + σ = n2,
i + j + … + τ = 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 an
x.
The expression (ab)4 properly appertains to a quartic; for a quadratic it may also be written (ab)2 (cd)2, and would denote the square of the discriminant to a factor près. For the quartic
(ab)4 = (a1b2 − a2b1)4 = a4
1b4
2 − 4a4
1a2b1b3
2 + 6a2
1a2
2b2
1b2
2
− 4a1a3
2}b3
1b2 + a4
2b4
1 = a0a4 − 4a1a3 + 6a2
2 − 4a1a3 + a0a4
= 2(a0a4 − 4a1a3 + 3a2
2),
one of the well-known invariants of the quartic.
For the cubic (ab)2axbx 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
(a3
1b1b2
2 − 2a2
1a2b2
1b2 + a1a2
2b3
1)x2
1
+(a3
1b3
2 - a1a2
2b2
1b2 - a2
1a2b1b2
2 + a3
2b3
1)x1x2
+ (a2
1a2b3
2 - 2a1a2
2b1b3
2 + a3
2b2
1b2)x2
2;
and transforming to the real form,
(a0b2 − 2a1b1 + a2b0)x2
1 (a0b3 − a1b2 − a2b1 + a3b0)x1x2
+ (a1b3 − 2a2b2 + a3b1)x2
2,
the simultaneous covariant; and now, putting b = a, we obtain twice. the Hessian
(a0a2 − a2
1)x2
2 + (a0a3 − a1a2)x1x2 + (a1a3 − a2
2)x2
2.
It will be shown later that all invariants, single or simultaneous, are expressible in terms of symbolic products. The degree of the covariant in the coefficients is equal to the number of different symbols a, b, c, … that occur in the symbolic expression; the degree in the variables (i.e. the order of the covariant) is ρ + σ + τ … and the weight[1] of the coefficient of the leading term xρ + σ + τ …
1 is equal to i + j + k + …. 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, ε, θ, w respectively we can see that they are not independent integers, but that they are invariably connected by a certain relation nθ − 2w = ε. For, if φ(a0,…x1, x2) be a covariant of order ε appertaining to a quantic of order n,
φ(A0,…ξ1ξ2) = (λμ)w φ(a0,…λ1ξ1 + μ1ξ2, λ2ξ1 + μ2ξ2)
we find that the left- and right-hand sides are of degrees nθ and 2w + ε respectively in λ1, μ1, λ2, μ2, and thence nθ = 2w + ε.
Symbolic Identities.— For the purpose of manipulating symbolic expressions it is necessary to be in possession of certain simple identities which connect certain symbolic products. From the three equations
ax = a1x1 + a2x2, bx = b1x1 + b2x2, cx = c1x1 + c2x2,
we find by eliminating x1, and x2 the relation
ax(bc) + bx(ca) + cx(ab) = 0...(I.)
Introduce now new umbrae d1, d2 and recall that +d2 −d1 are cogredient with x1, and x2. We may in any relation substitute for any pair of quantities any other cogredient pair so that writing +d2, −d1 for x1 and x2, and noting that gx then becomes (gd), the above-written identity becomes
(ad)(bc) + (bd)(ca) + (cd)(ab) = 0... (II.)
Similarly in (I.), writing for c1, c2 the cogredient pair -y2, +y1, we obtain
axby − aybx = (ab)(xy). ... (III.)
Again in (I.) transposing ax(bc) to the other side and squaring, we obtain
2(ac)(bc)axbx = (bc)2a2
x + (ac)2bx − (ab)2c2
x. (IV.)
and herein writing d2, −d1 for x1, x2,
2(ac)(bc)(ad)(bd) − (bc)2(ad)2 + (ac)2(bd)2 − (ab)2(cd)2. (V.)
As an illustration multiply (IV.) throughout by an−2
x bn−2
x cn−2
x so that each term may denote a covariant of an nic.
2(ac)(bc)an−1
xbn−1
xcn−1
x
= (bc)2an
xbn−2
xcn−2
x + (ac)2an−2
xbn
xcn−2
x − (ab)2an−2
xbn−2
xcn
x,
- ↑ The weight of a term ak0
0ak1
1…akn
n is defined as being k1 + 2k2 + … + nkn.
