CHAPTER II - REGRESSIVE MULTIPLICATION

Summary

Progressive And Regressive Multiplication. (1) According to the laws of combinatorial multiplication just explained the product of two extensive magnitudes S_ρ and S_σ respectively of the ρth and σth order must necessarily be null, if ρ + σ be greater than ν, where the original manifold is of ν - 1 dimensions. Such products can therefore never occur, since every term of any equation involving them would necessarily be null.

In the case ρ + σ > ν we are accordingly at liberty to assign a fresh law of multiplication to be denoted by the symbol S_ρS_σ. Let multiplication according to this new law (to be defined in § 100) be termed ‘regressive,’ and in contradistinction let combinatorial multiplication be called progressive.

Thus if ρ + σ < ν, the product S_ρS_σ is formed according to the progressive law already explained. Such products will be called progressive products. If ρ + σ > ν the product S_ρS_σ will be formed according to the regressive law. Such products will be called regressive products. If ρ + σ = ν, the product S_ρS_σ may be conceived indifferently as formed according to the progressive or regressive law.

(2) In this last case it is to be noted that S_ρS_σ, must necessarily be of the form α(e₁e₂ … e_ν), where e₁e₂ … e_ν are ν independent reference elements of the original manifold.