If one of the critic centres be taken for origin, we can readily see that such a point is also a critic centre on each of its systems of polar curves. It is thus at a vertex of the self-polar triangle of its system of polar conics and the opposite side of the triangle is the common polar line of the critic centre with respect to each of the systems of curves. The tangents at the node of the nodal quartic coincide with those of its polar cubic and these we know coincide with the lines which constitute its polar conic. If two of the sixteen basal points coincide, such a point is a critic centre. The argument is the same as for a system of cubics. We can also see that two of the basal points of each of its systems of polar curves coincide at the critic centre. The sixteen basal points of the system of quartics may unite two and two so that it is possible to draw a system of quartics touching eight given lines each at a fixed point. If three of the basal points of our system of quartics coincide, all the quartics have at such a point a common point of inflection and a common inflectional tangent. The demonstration is the same as that already given for cubics. The system of polar cubics of such a point also have this point for a common point if inflection and the same tangent for a common inflectional tangent. I prefer to show this analytically for the sake of the method. The equation of the system of quartics having the origin for a common point of inflection and the axis of y for a common inflectional tangent may be written u₄ + u₃ + {(B + kB₁)xy + (C + kC₁)y²} + (A + kA₁)y = 0. The equation of the polar cubics of the origin is therefore, u₃ + 2{(B + kB₁)xy + (C + kC₁)y²} + 3(A + kA₁)y = 0, which proves the proposition. The properties of the system of polar conics of such a point are therefore the same as those already proved for cubics. One quartic of the system has a double point at the common point of inflection of the others. When four basal points coincide they give rise either to a common point of undulation or a common double point on all the quartics of the system. The equation of the system having a common point of undulation may be written u₄ + (A + kA₁)x²y + (B + kB₁)xy² + (C + kC₁)y³ + (D + kD₁)xy + (E + kE₁)y² + (F + kF₁)y = 0. There is one value of k for which the last term vanishes, and hence the origin is a critic centre. The polar cubics of the point of undulation break up into a system of conics through four points and the common tangent at the common point of undulation. For the equation of the polar cubics is y{(A + kA₁)x² + (B + kK₁)xy + (C + kC₁)y² + 2(D + kD₁)x + 2(E + kE₁)y + (F + kF₁)} = 0. The system of polar conics of the origin consequently breaks up into the line y = 0 and a pencil meeting in a point. The common tangent at the common point of undulation is also the common polar line of the point of undulation.