Specifications Ofa Group. (1) If S1, S2, … S6 be any six independent [cf. § 96 (2)] systems of forces, then any system can be written in the form λ1S1+λ2S2+ … λ6S6. Let λ1, λ2, … λ6 be called the co-ordinates of S as referred to the six systems.

Definitions. The assemblage of systems, found from the expression λ1S1+λ2S2 by giving the ratio λ1: λ2 all possible values, will be called a ‘dual group’ of systems. The assemblage of systems, found from the expression λ1S1+λ2S2+ λ3S3 by giving the ratios λ1: λ2: λ3 all possible values, will be called a ‘triple group’ of systems.

The assemblage, found from λ1S1+λ2S2+ λ3S3 + λ4S4 by giving the ratios λ1: λ2: λ3: λ4 all possible values, will be called a ‘quadruple group.’ The assemblage, found from λ1S1+λ2S2+ λ3S3 + λ4S4 + λ5S5 by giving the ratiosλ1: λ2: λ3: λ4: λ5 all possible values, will be called a ‘quintuple group’

(2) A dual group will be said to be of one dimension, a triple group of two dimensions, and so on.

It is obvious that a group of ρ - 1 dimensions (ρ = 2, 3, 4, 5) can be defined by any ρ independent systems belonging to it; and also that not more than ρ independent systems can be found belonging to it.