The anharmonic ratio here considered is that of the pencil (P) formed by the four lines (lying in a plane perpendicular to the zonal axis) drawn from any point M perpendicular to four faces of the zone. The anharmonic ratio of the pencil P is evidently independent of the position of M.

Let four planes be drawn through M perpendicular to the four lines of the pencil P; these are parallel to the crystal faces and meet in a line (the line through M perpendicular to the plane of the pencil P and parallel to the zonal axis).

These four planes are cut by the plane of the pencil in a pencil having the same angles, and therefore the same anharmonic ratio, as the pencil P.