Then 4, =o over the cylinder r = a, which may be considered a fixed post; and a stream line past it along which 4, = Uc, a constant, is the curve (r - ¢2) sin 0=c, (x2 + y2) (y - c) - a 2 y = o, (3) a **cubic curve** (C3).

The cartesian parabola is a **cubic curve** which is also known as the trident of Newton on account of its three-pronged form.

The whole theory of the inflections of a **cubic curve** is discussed in a very interesting manner by means of the canonical form of the equation x +y +z +6lxyz= o; and in particular a proof is given of Plucker's theorem that the nine points of inflection of a **cubic curve** lie by threes in twelve lines.

It may be noticed that the nine inflections of a **cubic curve** represented by an equation with real coefficients are three real, six imaginary; the three real inflections lie in a line, as was known to Newton and Maclaurin.

The theory of the invariants and covariants of a ternary cubic function u has been studied in detail, and brought into connexion with the **cubic curve** u = o; but the theory of the invariants and covariants for the next succeeding case, the ternary quartic function, is still very incomplete.