Let U and V represent a system of quartics through sixteen points.
Since the discriminant of quartic is of the 27th degree in the
coefficients it follows that there are 27 values of k for which the
discriminant vanishes, and hence 27 quartics of the system which
have double points. As in case of cubics these 27 points are called
the critic centres of the system. Let the equation of the system of
quartics be written
u₄ + u₃ + u₂ + u₁ + u₀ = 0.
In a manner similar to that employed for cubics, we find the equation
of the polar cubics of the origin with respect to the system to be
u₃ + 2u₂ + 3u₁ + 4u₀ = 0.