In 1864 he went as Hessian envoy to Vienna, retiring in 1872 when the post was abolished.
In general for a form in n variables the Hessian is 3 2 f 3 2 f a2f ax i ax n ax 2 ax " ï¿½ï¿½ ' axn and there is a remarkable theorem which states that if H =o and n=2, 3, or 4 the original form can be exhibited as a form in I, 2, 3 variables respectively.
We find that Di must be equal to p x g x for then t x (p x) 3 +, u (g x) 3, Hence, if px, qx be the linear factors of the Hessian 64, the cubic can be put into the form A(p x) 3 +ï¿½(g x) 3 and immediately solved.
This method of solution fails when the discriminant R vanishes, for then the Hessian has equal roots, as also the cubic f.
The quartic has four equal roots, that is to say, is a perfect fourth power, when the Hessian vanishes identically; and conversely.