Page:EB1911 - Volume 01.djvu/669

vanish. From the above D_pq is an operator of order pq, but it is convenient for some purposes to obtain its expression in the form of a number of terms, each of which denotes pq successive linear operations: to accomplish this write

d_pq ＝ $\sum$a_rsd/a_p+r,q+s

and note the general result^[1]
⁠exp (m₁₀d₁₀ + m₀₁d₀₁ + … + m_pqd_pq + …)
⁠= exp (M₁₀d₁₀ + M₀₁d₀₁ + … + M_pqd_pq + …);
where the multiplications on the left- and right-hand sides of the equation are symbolic and unsymbolic respectively, provided that m_pq, M_pq are quantities which satisfy the relation
⁠exp (M₁₀ξ + M₀₁η + … + M_pqξ ^pη^p + …)
⁠= 1 + m₁₀ξ + m₀₁η + … + m_pqξ ^pη^q + …;
where ξ, η are undetermined algebraic quantities. In the present particular case putting m₁₀ = μ, m₀₁ = ν and m_pq = 0 otherwise
⁠M₁₀ξ + M₀₁η + … + M_pqξ ^pη^q + … = log (1 + μξ + νη)
or
⁠M_pq = ( − )^p+q−1(p + q − 1)!/p! q!μ^pν^q;
and the result is thus
⁠exp (μd₁₀ + νd₀₁)
⁠= exp { μd₁₀ + νd₀₁ − 1/2 (μ²d₂₀ + 2μνd₁₁ + ν²d₀₂) + … }
⁠= 1 + μD₁₀ + νD₀₁ + … + μ^pν^qD_pq + …;
and thence
⁠μd₁₀ + νd₀₁ − 1/2 (μ²d₂₀ + 2μνd₁₁ + ν²d₀₂) + …
⁠=⁠log (1 + μD₁₀ + νD₀₁ + … + μ^pν^qD_pq + …).
From these formulae we derive two important relations, viz.
 ( − )^p+q−1(p + q − 1)!/p!q!d_pq = $\sum _{\pi }$ ( − )^Σπ−1 (Σπ − 1)!/π₁! π₂! …Dπ₁
p₁q₁Dπ₂
p₂q₂…,
 ( − )^p+q−1D_pq = $\sum _{\pi }$$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$(p₁ + q₁ − 1)!/p₁!q₁!$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$^π₁$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$(p₂ + q₂ − 1)!/p₂!q₂!$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$^π₂ …
⁠… ( − )^{Σπ − 1)!}/π₁! π₂! …dπ₁
p₁q₁dπ₂
p₂q₂…,
the last written relation having, in regard to each term on the right-hand side, to do with Σπ successive linear operations. Recalling the formulae above which connect s_pq and a_pq, we see that d_pq and D_pq are in co-relation with these quantities respectively, and may be said to be operations which correspond to the partitions (pq), (10^p 01^q) respectively. We might conjecture from this observation that every partition is in correspondence with some operation; this is found to be the case, and it has been shown (loc. cit. p. 493) that the operation
⁠1/π₁!1/π₂!…dπ₁
p₁q₁dπ₂
p₂q₂…⁠ (multiplication symbolic)
corresponds to the partition (/p₁q₁^π₁ /p₂q₂^π₂…). The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula. This fact is of extreme importance in the theory of algebraic forms, and is easily representable whatever be the number of the systems of quantities.

We may remark the particular result
⁠( − )^p+q−1(p + q − 1)!/p!q!d_pqs_pq = D_pq(pq) = 1;
d_pq causes every other single part function to vanish, and must cause any monomial function to vanish which does not comprise one of the partitions of the biweight pq amongst its parts.

Since
⁠d_pq = ( − )^p+q−1(p + q − 1)!/p!q!d/ds_pq
the solutions of the partial differential equation d_pq = 0 are the single bipart forms, omitting s_pq, and we have seen that the solutions of D_pq = 0 are those monomial functions in which the part pq is absent.

One more relation is easily obtained, viz.
⁠d/da_pq = d_pq − h₁₀d_p+1,q − h₀₁d_p,q+1 + … + ( − )^r+sh_rsd_p+r,q+s + … .

References for Symmetric Functions.—Albert Girard, Invention nouvelle en l’algèbre (Amsterdam, 1629); Thomas Waring, Meditationes Algebraicae (London, 1782); Lagrange, Mém. de l’acad. de Berlin (1768); Meyer-Hirsch, Sammlung von Aufgaben aus der Theorie der algebraischen Gleichungen (Berlin, 1809); Serret, Cours d'algèbre supérieure, t. iii. (Paris, 1885); Unferdinger, Sitzungsber. d. Acad. d. Wissensch. i. Wien, Bd. lx. (Vienna, 1869); L. Schläfli, “Ueber die Resultante eines Systemes mehrerer algebraischen Gleichungen,” Vienna Transactions, t. iv. 1852; MacMahon, “Memoirs on a New Theory of Symmetric Functions,” American Journal of Mathematics, Baltimore, Md. 1888–1890; “Memoir on Symmetric Functions of Roots of Systems of Equations,” Phil. Trans. 1890.

III. The Theory of Binary Forms

A binary form of order n is a homogeneous polynomial of the nth degree in two variables. It may be written in the form
⁠axn
1 + bxn−1
1x₂ + cn−2
1x2
2 + …;
or in the form
axn
1 + (n
1)bxn−1
1x₂ + (n
2)cn−2
1x2
2 + …;
which Cayley denotes by
⁠(a, b, c, …) (x₁, x₂)ⁿ
(n
1), (n
2)… being a notation for the successive binomial coefficients n, 1/2n (n − 1), …. Other forms are
⁠axn
1 + nbn−1
1x₂ + n(n − 1)cxn−2
1x2
2 + …,
the binomial coefficients (n
s) being replaced by s!(n
s), and
⁠axn
1 + 1/1!bn−1
1x₂ + 1/2!cxn−2
1x2
2 + …,
the special convenience of which will appear later. For present purposes the form will be written
⁠/a₀xn
1 + (n
1)/a₁xn−1
1x₂ + (n
1)/a₁xn−1
1x2
2 + … + /a_nxn
2,
the notation adopted by German writers; the literal coefficients have a rule placed over them to distinguish them from umbral coefficients which are introduced almost at once. The coefficients /a₀, /a₁, /a₂, … /a_n, n + 1 in number are arbitrary. If the form, sometimes termed a quantic, be equated to zero the n + 1 coefficients are equivalent to but n, since one can be made unity by division and the equation is to be regarded as one for the determination of the ratio of the variables.

If the variables of the quantic 𝑓(x₁, x₂) be subjected to the linear transformation
⁠x₁ = α₁₁ξ₁ + α₁₂ξ₂,
⁠x₂ = α₂₁ξ₁ + α₂₂ξ₂,
ξ₁, ξ₂ being new variables replacing x₁, x₂ and the coefficients α₁₁, α₁₂, α₂₁, α₂₂, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed quantic
⁠𝑓 (ξ₁, ξ₂) = a′
0ξn
1 + (n
1)a′
1ξn−1
1ξ₂ + (n
2)a′
2ξn−2
1ξ₂ + … + a′
nξn
2
in the new variables which is of the same order as the original quantic; the new coefficients a′/0, a′/1, a′/2 . . . a′/n are linear functions of the original coefficients, and also linear functions of products, of the coefficients of substitution, of the nth degree.

By solving the equations of transformation we obtain
⁠rξ₁ = ⁠ α₂₂x₁ − α₁₂x₂,
⁠rξ₁ = − α₂₁x₁ − α₁₁x₂,
⁠where r = |α₁₁α₁₂
α₂₁α₂₂ | = α₁₁α₂₂ − α₁₂α₂₁;
r is termed the determinant of substitution or modulus of transformation; we assure x₁, x₂ to be independents, so that r must differ from zero.

In the theory of forms we seek functions of the coefficients and variables of the original quantic which, save as to a power of the modulus of transformation, are equal to the like functions of the coefficients and variables of the transformed quantic. We may have such a function which does not involve the variables, viz.
⁠F(a′
0, a′
1, a′
2, … a′
n) = r ^λ F(/a₀, /a₁, /a₂, … /a_n),
the function F(/a₀, /a₁, /a₂, … /a_n) is then said to be an invariant of the quantic quâ linear transformation. If, however, F involve as well the variables, viz.
⁠F(a′
0, a′
1, a′
2, … ; ξ₁, ξ₂) = r ^λ F(/a₀, /a₁, /a₂, … ; x₁, x₂),
the function F(/a₀, /a₁, /a₂, … ; x₁, x₂) 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. Invariantive forms will be found to be homogeneous functions alike of the coefficients and of the variables. Instead of a single quantic we may have several
⁠𝑓 (/a₀, /a₁, /a₂, … ; x₁, x₂), φ(/b₀, /b₁, /b₂, … ; x₁, x₂), …
which have different coefficients, the same variables, and are of the same or different degrees in the variables; we may transform them all by the same substitution, so that they become
⁠𝑓 (a′
0, a′
1, a′
2, … ; ξ₁, ξ₂), φ(b′
0, b′
1, b′
2, … ; ξ₁, ξ₂), …
If then we find
⁠F(a′
0, a′
1, a′
2, … b′
0, b′
1, b′
2, …, …; ξ₁, ξ₂),

⁠= r ^λ F(/a₀, /a₁, /a₂, … /b₀, /b₁, /b₂, …, …; x₁, x₂),

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1. Phil. Trans., 1890, p. 490.