x, y, z)m = 0, or general curve of the order m, has double tangents and inflections; (2) presents itself as a singularity, for the equations dx(*
x, y, z)m = 0, dy(*
x, y, z)m = 0, dz(*
x, y, z)m = 0, implying (*
x, y, z)m = 0, are not in general satisfied by any values (a, b, c) whatever of (x, y, z), but if such values exist, then the point (a, b, c) is a node or double point; and (1) presents itself as a further singularity or sub-case of (2), a cusp being a double point for which the two tangents becomes coincident.
In line-co-ordinates all is reversed:—(1) and (2) are not singularities; (3) presents itself as a sub-case of (4).