§ 1. The method by which Hamilton has established the symbolical cubic

and more particularly the relation

is founded on a generalisation of the result

where ϕ represents a linear and vector function, and ϕ' its conjugate according to the definition

In the most general case of ϕρ the expression of this function is reducible to a vector sum of three terms of the form αSα1ρ, and the directions which ϕρ is susceptible of representing are generally not limited in space.