and we see further that (σ₁a₁ + σ₂a₂ + … + σ_ma_m)^k vanishes identically unless k ≡ 0 (mod m). If m be infinite and

⁠1 + b₁x + b₂x² + … = (1 + σ₁x) (1 + σ₂x) … = e^β₁x = e^β₂x = …,
we have the symbolic identity
⁠e^{σ₁α₁ + σ₂α₂ + σ₃α₃ + …} = e^{ρ₁β₁ + ρ₂β₂ + ρ₃β₃ + …,}
and
⁠(σ₁a₁ + σ₂a₂ + σ₃a₃ + …)^p = (ρ₁β₁ + ρ₂β₂ + ρ₃β₃ + …)^p.
Instead of the above symbols we may use equivalent differential operators. Thus let
⁠δ_a = a₁∂_a0 + 2a₂∂_a1 + 3a₃∂_a2 + …
and let a, b, c, … be equivalent quantities. Any function of differences of δ_a, δ_b, δ_c, … being formed, the expansion being carried out, an operand a₀ or b₀ or c₀ … being taken and b, c, … being subsequently put equal to a, a non-unitary symmetric function will be produced.

Ex. gr.⁠ (δ_a − δ_b)²(δ_a − δ_c) = (δ2
a − 2δ_aδ_b + δ2
b)(δ_a − δ_c)
⁠= δ3
a − 2δ2
aδ_b + δ_aδ2
b − δ2
aδ_c + 2δ_aδ_bδ_c −δ2
bδ_c
⁠= 6a₃ − 4a₂b₁ + 2a₁b₂ − 2a₂c₁ + 2a₁b₁c₁ − 2b₂c₁
⁠= 2 (a3
1 − 3a₁a₂ + 3a₃) = 2(3).
The whole theory of these forms is consequently contained implicitly in the operation δ.

Symmetric Functions Several Systems Quantities.—It will suffice to consider two systems of quantities as the corresponding theory for three or more systems is obtainable by an obvious enlargement of the nomenclature and notation.

Taking the systems of quantities to be
⁠α₁, α₂, α₃, …
⁠β₁, β₂, β₃, …
we start with the fundamental relation
⁠(1 + α₁x + β₁y)(1 + α₂x + β₂y)(1 + α₃x + β₃y)…
⁠= 1 + a₁₀x + a₀₁y + a₂₀x² + a₁₁xy + a₀₂y² + … + a_pqx^py^q + …
As shown by L. Schläfli^[1] this equation may be directly formed and exhibited as the resultant of two given equations, and an arbitrary linear non-homogeneous equation in two variables. The right-hand side may be also written
⁠ 1 + Σα₁x + Σβ₁y + Σα₁α₂x² + Σα₁β₂xy + Σβ₁β₂y² + …
The most general symmetric function to be considered is
⁠Σαp₁
1βq₁
1αp₂
2βq₂
2αp₃
3βq₃
3…
conveniently written in the symbolic form
⁠(/p₁q₁ /p₂q₂ /p₃q₃…)
Observe that the summation is in regard to the expressions obtained by permuting the n suffixes 1, 2, 3, …n. The weight of the function is bipartite and consists of the two numbers Σp and Σq; the symbolic expression of the symmetric function is a partition into biparts (multiparts) of the bipartite (multipartite) number Σp, Σq. Each part of the partition is a bipartite number, and in representing the partition it is convenient to indicate repetitions of parts by power symbols. In this notation the fundamental relation is written
⁠(1 + α₁x + β₁y)(1 + α₂x + β₂y)(1 + α₃x + β₃y)…
⁠= 1 + (10)x + (01)y + (10²)x² + (10 01)xy + (01)²y²
⁠+ (10³)x³ + (10²01)x²y + (1001²)xy² + (01³)y³ + …
where in general a_pq = (10^p 01^q).

All symmetric functions are expressible in terms of the quantities a_pq in a rational integral form; from this property they are termed elementary functions; further they are said to be single-unitary since each part of the partition denoting a_pq involves but a single unit.

The number of partitions of a biweight pq into exactly μ biparts is given (after Euler) by the coefficient of aμx^py^q in the expansion of the generating function
⁠1/1 − ax. 1 − ay. 1 − ax². 1 − axy. 1 − ay². 1 − ax³. 1 − ax²y. 1 − axy².1 − ay³…

The partitions with one bipart correspond to the sums of powers in the single system or unipartite theory; they are readily expressed in terms of the elementary functions. For write (pq) = s_pq and take logarithms of both sides of the fundamental relation; we obtain
⁠s₁₀x + s₀₁y = Σ(α₁x + β₁y)
⁠s₂₀x² + 2s₁₁xy + s₀₂y² = Σ(α₁x + β₁y)²
and
⁠s₁₀x + s₀₁y − 1/2(s₂₀x² + 2s₁₁xy + s₀₂y²) + …
⁠= log (1 + a₁₀x + a₀₁y + … a_pqx^py^q + …).
From this formula we obtain by elementary algebra
⁠(−)^p+q−1(p + q − 1)!/p! q!s_pq = ${\displaystyle \sum _{\text{π}}}$(−)^Σπ−1(Σπ − 1)!/π₁! π₂!…aπ₁
p₁q₁aπ₂
p₂2₁ …
corresponding to Thomas Waring's formula for the single system. The analogous formula appertaining to n systems of quantities which expresses s_pqr… in terms of elementary functions can be at once written down.

Ex. gr.⁠We can verify the relations
⁠s₃₀ = a3
10 − 3a₂₀a₁₀ + 3a₃₀,
⁠s₂₁ = a2
10a₀₁ − a₂₀a₀₁ − a₁₁a₁₀ + a₂₁.
The formula actually gives the expression of (pq) by means of separations of
⁠(/10^p /01^q),
which is one of the partitions of (pq). This is the true standpoint from which the theorem should be regarded. It is but a particular case of a general theory of expressibility.

To invert the formula we may write
⁠1 + a₁₀x + a₀₁y + … + a_pqx^py^q + …
⁠= exp {(s₁₀x + s₀₁y) − 1/2s₂₀x² + 2s₁₁xy + s₀₂y²) + …},
and thence derive the formula—
⁠(−)^p+q−1a_pq
⁠= Σ { (p₁ + q₁ − 1)!/p₁!q₁! } ^π₁ { (p₂ + q₂ − 1)!/p₂!q₂! } ^π₂ … (−)^Σπ−1/π₁! π₂! …sπ₁
p₁q₁sπ₂
p₂q₂…,
which expresses the elementary functions in terms of the single bipart functions. The similar theorem for n systems of quantities can be at once written down.

It will be shown later that every rational integral symmetric function is similarly expressible.

The Function h_pq.—As the definition of h_pq we take
⁠1 + n₁₀x + n₀₁y + … + n_pqx^py^q + …
⁠=1/(1 + α₁x − β₁y)(1 + α₂x − β₂y)…;
and now expanding the right-hand side
⁠h_pq = Σ ( p₁ + q₁/p₁) (p₁ + q₁/p₁) … (/p₁q₁ /p₂q₂…)
the summation being for all partitions of the biweight. Further writing
⁠1 + h₁₀ + h₀₁y + … + h_pqx^py^q + …
⁠= 1/1 − a₁₀x − a₀₁y + … + (−)^p+qa_pqx^py^q + …,
we find that the effect of changing the signs of both x and y is merely to interchange the symbols a and h; hence in any relation connecting the quantities h_pq with the quantities a_pq we are at liberty to interchange the symbols a and h. By the exponential and multinomial theorems we obtain the results—
⁠(−)^p+q−h_pq = ${\displaystyle \sum _{\text{π}}}$(−)^Σπ−1(Σπ)!/π₁! π₂! …aπ₁
p₁q₁ aπ₂
p₂q₂ …
and in this a and h are interchangeable.
⁠(p + q − 1)!/p! q!s_pq = ${\displaystyle \sum _{\text{π}}}$(−)^Σπ−1(Σπ − 1)!/π₁! π₂! …hπ₁
p₁q₁ hπ₂
p₂q₂ …;
⁠h_pq = Σ { (p₁ + q₁ − 1)!/p₁! q₁! } ^π₁ { (p₂ + q₂ − 1)!/p₂! q₂! } ^π₂ … 1/π₁! π₂! …sπ₁
p₁q₁sπ₂
p₂q₂ … .

Differential Operations.—If, in the identity
⁠(α₁x + β₁y)(α₂x + β₂y) … (α_nx + β_ny)
⁠=1 + a₁₀x + a₀₁y + a₂₀x² + a₁₁xy + a₀₂y² + …,
we multiply each side by (1 + μd₁₀ + νd₀₁), the right-hand side becomes
⁠1 + (a₁₀ + μ)x + (a₀₁ + ν)y + … + (a_pq + μ)a_p−1,q + νa_p,q−1)x^py^q + …;
hence any rational integral function of the coefficients a₁₀, a₀₁, … a_pq, … say 𝑓 (a₁₀, a₀₁, …) = 𝑓 is converted into
⁠exp(μd₁₀ + νd₀₁) 𝑓
⁠where d₁₀ = Σa_p−1,q d/da_pq, d₀₁ = Σa_{p, q−1} d/da_pq,
The rule over exp will serve to denote that μd₁₀ + νd₀₁ is to be raised to the various powers symbolically as in Taylor's theorem.

Writing ⁠D_pq = 1/p! q!dp
10dq
01,
⁠exp (μd₁₀ + νd₀₁) = (1 + μD₁₀ + νD₀₁ + … + μ^pν^qD_pq + … ) 𝑓 ;
now, since the introduction of the new quantities μ, ν results in the addition to the function (/p₁q₁ /p₂q₂ /p₃q₃ …) of the new terms
⁠μ^p₁ν^q₁(/p₂q₂ /p₃q₃…) + μ^p₂ν^q₂(/p₁q₁ /p₃q₃…) + μ^p₃ν^q₃(/p₁q₁ /p₂q₂…) + …,
we find
⁠D_p1q1(/p₁q₁ /p₂q₂ /p₃q₃ …) = (/p₂q₂ /p₃q₃ …) ;
and thence
⁠D_p1q1D_p2q2D_p3q3…(/p₁q₁ /p₂q₂ /p₃q₃ …) = 1;

while D_rs 𝑓 = 0 unless the part rs is involved in 𝑓. We may then state that D_pq is an operation which obliterates one part pq when such part is present, but in the contrary case causes the function to

1. ↑ Vienna Transactions, t. iv. 1852.