Certain cinematical and physical questions lead to the problem: to determine the directions in which a given linear vector-function, ϕρ, assumes a direction parallel to that of the vector, ρ, on which it depends.

The condition of parallelism is expressed by

and it is translated into the equation

where g represents a certain scalar, on whose determination the whole problem depends.

Following the method traced out by Hamilton, we treat this equation successively by

α,β,γ, being any system of vectors not coplanar between each other; but we will state at once that throughout the whole of this paper we shall assume α,β,γ, to form a system of treble rectangular unit-vectors, of which hypothesis the justification is evident.