Other theorems on such a system of quartics will be given in the next section.
SYSTEMS OF QUARTICS THROUGH
SIXTEEN POINTS.
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.
The polar conics of the origin are given by
u₂ + 3u₁ + 6u₀ = 0 ;
and the polar lines of the origin, by
u₁ + 4u₀ = 0 .