the function F, on the right which multiplies r, is said to be a simultaneous invariant or covariant of the system of quantics. 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 quantics, and subsequently to make all the quantics identical. Moreover, instead of having one pair of variables x1, x2 we may have several pairs y1, y2; z1, z2;… in addition, and transform each pair to a new pair by substitutions, having the same coefficients α11, α12, α21, α22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the above definied invariant property. 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. Such quantics have been termed by Cayley multipartite.
Symbolic Form.—Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce the symbolic form of Aronhold, Clebsch and Gordan; they write the form
(a1x1 + a2x2)n = an
1xn
1 + (n
1)an−1
1a2xn−1
1
x2 + ... + an
2xn
3 = an
x
wherein a1, a2 are umbrae, such that
an
1, an−1
1a2, ... a1an−1
2. an
2
are symbolical representations of the real coefficients a0, a1, ... an−1, an, and in general an−k
1ak
2 is the symbol for ak. If we restrict ourselves to this set of symbols we can uniquely pass from a product of real coefficients to the symbolic representations of such product, but we cannot, uniquely, from the symbols recover the real form, This is clear because we can write
a1a2 = an−1
1a2. an−2
1a2
2 = a2n−3
1a3
2
while the same product of umbrae arises from
a0a3 = an
1.an−3
1a3
2 = a2n−3
1a3
2
Hence it becomes necessary to have more than one set of umbrae, so that we may have more than one symbolical representation of the same real coefficients. We consider the quantic to have any number of equivalent representations an
x ≡ bn
x ≡ cn
x ≡ …. So that an−k
1ak
2 ≡ bn−k
1bk
2 ≡ cn−k
1ck
2 ≡ … = ak; and if we wish to denote, by umbrae, a product of coefficients of degree s we employ s sets of umbrae.
Ex. gr. We write a1a2 = an−1
1a2.bn−2
1b2
2
a2
2 = an−3
1a3
2.bn−3
1b3
2.cn−3
1c3
2,
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.Ex. gr. To express the function a0a2−a2
1, which is the discriminant of the binary quadratic a0x2
1 + 2 a1x1x2 + a2x2
2 = a2
x = b2
x, in a symbolic form we have
2(a0a2 − a2
1) = a0a2 + a1a2 − 2a1 . a1 = a2
1b2
1 + a2
2 b2
1 − 2a1a2b1b2
= (a1b2 − a2b1)2.
Such an expression as a1b2−a2b1 which is
∂ax∂x1∂bx∂x2 − ∂ax∂x2∂bx∂x1,
is usually written (ab) for brevity; in the same notation the determinant, whose rows are al, a2, a3; b1, b2, b3; c1, c2, c3 respectively, is written (abc) and so on. It should be noticed that the real function denoted by (ab)2 is not the square of a real function denoted by (ab). For a single quantic of the first order (ab) is the symbol of a function of the coefficients which vanishes identically; thus
(ab) = a1b2 − a2b1 = a0a1 − a1a0 = 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) = 0.
To find the effect of linear transformation on the symbolic form of quantic we will disuse the coefficients a11, a12, a21, a22, and employ λ1, μ1, λ2, μ2. For the substitution
x1 = λ1ξ1 + μ1ξ2, x2 = λ2ξ1 + μ2ξ2,
of modulus |λ1
λ2μ1
μ2| = (λ1μ2 − λ2μ1) = (λμ),
the quadratic form a0x2
1 + 2ax1x2 + a2x2
2 = 2
x = ƒ(x),
becomes
A0ξ2
1 + 2A1ξ1ξ2 + A2ξ2
2 = A2
ξ = φ(ξ),
where
A0 = a0λ2
1 + 2a1λ1λ2 + a2λ2
2,
A1 = a0λ1μ1 + a1(λ1μ2 + λ2μ1) + a2λ2μ2,
A2 = a0μc+ 2a1μ1μ2 + a2μ2
2.
We pass to the symbolic forms
a2
x = (a1x1 + a2x2)2,A2
ξ = (A1ξ1 + A2ξ2)2,
by writing for
a0, a1, a2 the symbols a2
1, a1a2, a2
2
A0, A1, A2 the symbols„ A2
1, A1A2, A2
2
and then
A0 = a2
1λ2
1 + 2a1a2λ1λ2 + a2
2λ2
2 = (a1λ1 + a2λ2)2 = a2
λ,
A1 = (a1λ1 + a2λ2) (a1μ1 + a2μ2) = aλaμ,
A2 = (a1μ1 + a2μ2)2 = a2
μ;
so that
A2
ξ = a2
λξ2
1 + 2aλaμξ1ξ2 + a2
μξ2
2 = (aλξ1 + aμξ2)2;
whence A1, A2 become aλ, 'aμ respectively and
φ(ξ) = (aλξ1 + aμξ2)2.
The practical result of the transformation is to change the umbrae al, a2 into the umbrae
aλ = a1λ1 + a2λ1,aμ = a1μ1 + a2μ2
respectively.
By similarly transforming the binary nic form an
x we find
A0 = (a1λ1 + a2λ2)n = an
λ + An
1,
A1 = (a1λ1 + a2λ2)n−1 (a1μ1 + a2μ2) = an−1
λaμ = An−1
1A2,
········
Ak = (a1λ1 + a2λ2)n−k (a1μ1 + a2μ2)k = an−k
λak
μ = An−k
1An−k
2,
so that the umbrae A1, A2 are aλ, aμ respectively.
Theorem.-When the binary form
an
x = (a1x1 + a2x2)n
is transformed to
An
ξ = (A1ξ1 + A2ξ2)n
by the substitutions
x1 = λ1ξ1 + μ1ξ2, x2 = λ2ξ1 + μ2ξ2,
the umbrae A1, A2 are expressed in terms of the umbrae a1, a2 by the formulae
A1 = λ1a1 + λ2a2, A2 = μ1a1 + μ2a2,
We gather that A1, A2 are transformed to a1, a2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (λμ). For this reason the umbrae A1, A2 are said to be contragredient to x1, x2. If we solve the equations connecting the original and transformed unbrae we find
(λμ)(−a2) = λ1(−A2) + μ1A1,
(λμ)a1 = λ2(−A2) + μ2A1,
and we find that, except for the factor (λμ), −a2 and +a1 are transformed to −A2 and +A1 by the same substitutions as x1 and x2 are transformed to ξ1 and ξ2. For this reason the umbrae −a2, a1 are said to be cogredient to x1 and x2. We frequently meet with cogredient and contragedient quantities, and we have in general the following definitions:-(1) "If two equally numerous sets of quantities x, y, z, ... x′, y′, z′, ... are such that whenever one set x, y, z,... is expressed in terms of new quantities X, Y, Z, ... the second set x′, y′, z′, ... is expressed in terms of other new quantities X′, Y′, Z′, .... by the same scheme of linear substitution the two sets are said to be cogredient quantities." (2) "Two sets of quantities x, y, z, ...; ξ, ηζ, ... are said to be contragredient when the linear substitutions for the first set are
x = λ1X + μ1Y + ν1Z +
y = λ2X + μ2Y + ν2Z +
z = λ3X + μ3Y + ν3Z +
·····
and these are associated with the following formulae appertaining to the second set,
Ξ = λ1ξ + λ2η + λ3ζ + ,
Η = μ1ξ + μ2η + μ3ζ + ,
Ζ = ν1ξ + ν2η + ν3ζ + ,
····
wherein it should be noticed that new quantities are expressed in terms of the old, as regards the latter set, and not vice versa."
Ex. gr. The symbols ddx, ddy, ddz, ... are contragredient with the variables x, y, z, ... for when
(x, y, z, ...) = (λ1, μ1, ν1, ...)(X, Y, Z, ...)
( x , z, ���) = (A l, �i, VI I ���)
(X, Y, Z, ���), I A 2, / 2 2, Y2, ... I I A S, 1 2 3, Y 3, .... 1
(Tr (T d d d d d d ,.. rd Y' ' ...) = 01, A2, A 3, ...)
(d ' ' z / 2 1, /22, / 1 3, ... Pl, P2, P3, ...
