we may write in general

⁠D_s c = ΣD(p₁p₂p₃…) 𝑓,
the summation being for every partition (p₁p₂p₃…) of s, and D(p₁p₂p₃…) 𝑓 being = Σ(Dp₁ 𝑓₁ )(Dp₂ 𝑓₂ )(D p₃ 𝑓₃ ) 𝑓₄…𝑓_m.

Ex. gr. To operate with D₂ upon (21³)(21⁴)(1⁵), we have
⁠D₍₂₎ 𝑓 = (1³)(21⁴)(1⁵) + (21³)(1⁴)(1⁵),
⁠D_(1²) 𝑓 = (12²)(21³)(1⁵) + (21³)(21³)(1⁴) + (21²)(21⁴)(1⁴),
and hence
⁠D₍₂₎ 𝑓 = (21⁴)(1⁵)(1³) + (21³)(1⁵)(1⁴) + (21³)(21²)(1⁵) + (21³)²(1⁴) + (21⁴)(21²)(1⁴).

Application to Symmetric Function Multiplication.—An example will explain this. Suppose we wish to find the coefficient of (52⁴1³) in the product (21³21⁴)(1⁵).

Write
⁠(21³)(21⁴)(1⁵) = … + A(52⁴)(1³) + …;
then
⁠D₅D4
2D3
1(21³)(21⁴)(1⁵) = A;
every other term disappearing by the fundamental property of D_s. Since
⁠D₅(21³)(21⁴)(1⁵) = (1³)(1⁴)(1⁴),
we have:—
⁠D4
2D3
1(1⁴)(1⁴)(1³) = A
⁠D3
2D3
1{(1³)(1³)(1³) + 2(1⁴)(1³)(1²)} = A
⁠D2
2D3
1{5(1³)(1²)(1³) + 2(1⁴)(1²)(1) + 2(1³)(1³)(1)} = A
⁠D₂D3
1{12(1²)(1²)(1) + 7(1³)(1)(1) + 2(1⁴)(1) + 6(1³)(1²)} =A
⁠D3
112(1)³ = A,
where ultimately disappearing terms have been struck out. Finally A = 6 · 12 = 72.

The operator d₁ = a₀∂a₁ + a₁∂a₂ + a₂∂a₃ + … which is satisfied by every symmetric fraction whose partition contains no unit (called by Cayley non-unitary symmetric functions), is of particular importance in algebraic theories. This arises from the circumstance that the general operator
⁠λ₀a₀∂a₁ + λ₁a₁∂a₂ + λ₂a₂∂a₃ + …
is transformed into the operator d₁ by the substitution
⁠(a₀, a₁, a₂, … a_s, … ) = (a₀, λ₀a₁, λ₀λ₁a₂, … , λ₀λ₁ … λ_s−1a_s, … ),
so that the theory of the general operator is coincident with that of the particular operator d₁. For example, the theory of invariants may be regarded as depending upon the consideration of the symmetric functions of the differences of the roots of the equation
⁠a₀xⁿ − (n
1) a₁x^{n − 1} + (n
2) a₂x^{n − 2} − … = 0;
and such functions satisfy the differential equation
⁠a₀∂a₁ + 2a₁∂a₂ + 3a₂∂a₃ + … + na_n−1∂a_n = 0.
For such functions remain unaltered when each root receives the same infinitesimal increment h; but writing x − h for x causes a₀, a₁, a₂ a₃, … to become respectively a₀, a₁ + ha₀, a₂ + 2ha₁, a₃ + 3ha₂, … and 𝑓 (a₀, a₁, a₂ a₃, …) becomes
⁠𝑓 + h(a₀∂a₁ + 2a₁∂a₂ + 3a₂∂a₃ + …) 𝑓,
and hence the functions satisfy the differential equation. The important result is that the theory of invariants is from a certain point of view coincident with the theory of non-unitary symmetric functions. On the one hand we may state that non-unitary symmetric functions of the roots of a₀xⁿ − a₁x^{n − 1} + a₂x^{n − 2} − … = 0, are symmetric functions of differences of the roots of
⁠a₀xⁿ − 1!(n
1) a₁xⁿ⁻¹ + 2!(n
2) a₂xⁿ⁻² − … = 0;
and on the other hand that symmetric functions of the differences of the roots of
⁠a₀xⁿ − (n
1) a₁xⁿ⁻¹ + (n
2) a−
2xⁿ⁻² − … = 0;
are non-unitary symmetric functions of the roots of
⁠a₀xⁿ − a₁/1!xⁿ⁻¹ + a₂/2!xⁿ⁻² − … = 0.

An important notion in the theory of linear operators in general is that of MacMahon's multilinear operator ("Theory of a Multilinear partial Differential Operator with Applications to the Theories of Invariants and Reciprocants," Proc. Lond. Math. Soc. t. xviii. (1886), pp. 61–88). It is definied as having four elements, and is written
⁠(μ, ν; m, n)
⁠=1/m [ μam₀
0∂_αn + (μ + ν)m!/(m − 1)! 1!am−1
0a₁∂_αn+1
⁠+ (μ + 2ν) { m!/(m − 1)! 1!am−1
0a₂ + m!/(m − 2)! 2!am−2
0a2
1 } ∂_αn+2
⁠+ (μ + 3ν) { m!/(m − 1)! 1!am−1
0a₃ + m!/(m − 2)! 1! 1!am−2
0a₁a₂
⁠+ m!/(m − 3)! 3!am−3
0a3
0 } ∂_αn+3
⁠+ … ] ,
the coefficient of ak₀
0ak₁
1ak₂
2 … being m!/k₀!k₁!k₂!…. The operators a₀∂_α1 + a₁∂_α2 + …, a₀∂_α1 + 2a₁∂_α2 + … are seen to be (1, 0; 1, 1) and (1, 1; 1, 1) respectively. Also the operator of the Theory of Pure Reciprocants (see Sylvester Lectures on the New Theory of Reciprocants, Oxford, 1888) is
(4, 1 ; 2, 1) = 1/2{ 42
0∂_α1 + 10a₀a₁∂_α2 +6(2a₁a₂ + a2
1)∂_α2 + … } .
It will be noticed that
⁠(μ, ν; m, n) = μ(1, 0; m, n) + ν(0, 1; m, n).
The importance of the operator consists in the fact that taking any two operators of the system
⁠(μ, ν; m, n) ; (μ¹, ν¹; m¹, n¹)
the operator equivalent to
⁠(μ, ν; m, n)(μ¹, ν¹; m¹, n¹) − (μ¹, ν¹; m¹, n¹)(μ, ν; m, n),
known as the "alternant" of the two operators, is also an operator of the same system. We have the theorem
⁠(μ, ν; m, n)(μ¹, ν¹; m¹, n¹) − (μ¹, ν¹; m¹, n¹)(μ, ν; m, n) = (μ₁, ν₁; m₁, n₁);
where
⁠μ₁ = (m¹ + m − 1) { μ¹/m¹(μ + n¹ν) − μ/m(μ¹ + nν¹) } ,
⁠ν₁ = (n¹ − n)ν¹ν + m − 1/m¹μ¹ν − m¹ − 1/mμν¹.
⁠m₁ = m¹ + m − 1,
⁠n₁ = n¹ + n,
and we conclude that quâ "alternation" the operators of the system form a "group." It is thus possible to study simultaneously all the theories which depend upon operations of the group.

Symbolic Representation of Symmetric Functions.—Denote the elementary symmetric function a_s by αs
1/s!, αs
2/s!, αs
3/s!, … at pleasure; then, taking n equal to ∞, we may write
⁠1 + a₁x + a₁x² + … = (1 + ρ₁x) (1 + ρ₂x) … = e^α₁x = e^α₂x = e^α₃x = …
where
⁠a_s = Σρ₁ρ₂…ρ_s = αs
1/s! = αs
2/s! = αs
3/s! = … .
Further, let
⁠1 + b₁x + b₂x² + … + b_mx^m = (1 + σ₁x)(1 + σ₂x)…(1 + σ_mx);
so that
⁠1 + a₁σ₁ + a₂σ2
1 + … = (1 + ρ₁σ₁)(1 + ρ₂σ₁)… = e^σ₁α₁,
⁠1 + a₁σ₂ + a₂σ2
2 + … = (1 + ρ₁σ₂)(1 + ρ₂σ₂)… = e^σ₂α₂,
⁠⁠⋅⁠⋅⁠⋅⁠⋅⁠⋅
⁠1 + a₁σ_m + a₂σ2
m + … = (1 + ρ₁σ_m)(1 + ρ₂σ_m)… = e^σ_mα_m;
and, by multiplication,
⁠${\displaystyle \prod _{\text{σ}}}$(1 + a₁σ + a₂σ² + … ) = ${\displaystyle \prod _{\text{ρ}}}$ (1 + b₁ρ + b₂ρ² + … + b_mρ^m),
⁠= e^{σ₁α₁ + σ₂α₂ + … + σ_mα_m}.
Denote by brackets () and [ ] symmetric functions of the quantities ρ and σ respectively. Then
⁠1 + a₁[1] + a2
1[1²] + a₂[2] + a3
1[1³] + a₁a₂[21] + a₃ + …
⁠ + a_p1a_p2a_p3…a_pm[p₁p₂p₃…p_m] + …
⁠= 1 + b₁(1) + b2
1(1²) + b₂(2) + b3
1(1³) + b₁b₂(21) + b₃(3) + …
⁠+ bq₁
1bq₂
2bq₃
3…bq_m
m(m^q_mm⁻−1^q_m−1…2^q₂1^q₁) + …
⁠= e^{σ₁α₁ + σ₂α₂ + … + σ_mα_m}.
Expanding the right-hand side by the exponential theorem, and then expressing the symmetric functions of σ₁, σ₂, …σ_m, which arise, in terms of b₁, b₂, … b_m, we obtain by comparison with the middle series the symbolical representation of all symmetric functions in brackets () appertaining to the quantities ρ₁, ρ₂, ρ₃, … To obtain particular theorems the quantities σ₁, σ₂, σ₃, … σ_m are auxiliaries which are at our entire disposal. Thus to obtain Stroh's theory of seminvariants put
⁠b₁ = σ₁ + σ₂ + … + σ_m = [1] = 0;
we then obtain the expression of non-unitary symmetric functions of the quantities ρ as functions of differences of the symbols α₁, α₂, α₃, …

Ex. gr. b2
2(2²) with m = 2 must be a term in
⁠e^{σ₁α₁ + σ₂α₂} = e^{σ₁(α₁−α₂)} = … + 1/4!σ4
1(α₁ − α₂)⁴ + …,
and since b2
2 = σ4
1 we must have
⁠(2²) = 1/24(α₁ − α₂)⁴ = 1/24(α4
1 + α4
2) − 1/6(α3
1α₂ + α₁α3
2) + 1/4α2
1α2
2
⁠= 2a₄ − 2a₁a₃ + a2
3
as is well known.

Again, if σ₁, σ₂, σ₃…σ_m be the m, m^th roots of − 1, b₁ = b₂ = … = b_m−1 = 0 and b_m = 1, leading to
⁠1 + (m) + (m²) + (m³) + … = e^{σ₁α₁ + σ₂α₂ + … + σ_mα_m}
and

⁠∴ (m^s) = 1/ms!(σ₁α₁ + σ₂α₂ + … + σ_mα_m)^sm,