and Dp = 1/p! (δα1 + α1δα2 + α2δα3 + ...)p, the multiplication being symbolic, so that Dp is an operator of order p, the function is
aλ1aλ2aλ3 ... aλm,
and the operator Dp1Dp2Dp3 ... Dpn. The number Dp1Dp2 ... Dpnaλ1aλ2aλ3 ... aλm enumerates the solutions. For the mode of operation of Dp upon a product reference must be made to the section on “Differential Operators” in the article [Algebraic Forms]. Writing
aλ1aλ2 ... aλm = ... ΑΣ αp11 αp22 ... αpnn + ...,
or, in partition notation,
(1λ1) (1λ2) ... (1λm) = ... + Α(p1p2 ... pn) ... + Dp1Dp2 ... Dpn (1λ1) (1λ2) ... (1λm) = Α
and the law by which the operation is performed upon the product shows that the solutions of the given problem are enumerated by the number A, and that the process of operation actually represents each solution.
Ex. Gr.—Take
| λ1 = 3, λ2 = 2, λ3 = 1, p1 = 2, p2 = 2, p3 = 1, p4 = 1, D²2D²1 a3a2a1 = 8, |
and the process yields the eight diagrams:—