The tripolar coordinates of a point P are defined as the squares of its distances from the angular points of the triangle of reference.

The equation of the first degree in x, y, z (x = PA2, etc.) has been considered by E. Lucas (Mathesis, 1889) and by de Poulain (Journal de Mathématiques spéciales, 1889). I am also indebted to Mr. R. F. Davis for a note in accordance with which a portion of this paper has been rewritten.

In what follows x, y, z will denote the tripolar coordinates; α, β, γ the trilinear coordinates of a point with respect to a triangle of reference ABC; while ξ and η will denote Cartesian coordinates referred to B as origin and BC as the axis of x.