  We give a proof in modern language of the following result by Paul Gordan and
Max N\"other: a homogeneous quasi-translation in dimension $5$ without linear
invariants would be linearly conjugate to another such quasi-translation $x +
H$, for which $H_5$ is algebraically independent over $\mathbb C$ of $H_1, H_2,
H_3, H_4$. Just like Gordan and N\"other, we apply this result to classify all
homogeneous polynomials $h$ in $5$ indeterminates from which the Hessian
determinant is zero.
  Others claim to have reproved 'the result of Gordan and N\"other in $\mathbb
P^4$' as well, but some of them assume that $h$ is irreducible, which Gordan
and N\"other did not. Furthermore, they do not use the above result about
homogeneous quasi-translations in dimension $5$ for their classifications.
(There is however one paper which could use this result very well, to fix a gap
caused by an error.)
  We derive some other properties which $H$ would have. One of them is that
${\rm deg}\, H \ge 15$, for which we give a proof which is less computational
than another proof of it by Dayan Liu. Furthermore, we show that the Zariski
closure of the image of $H$ would be an irreducible component of $V(H)$, and
prove that every other irreducible component of $V(H)$ would be a
$3$-dimensional linear subspace of $\mathbb C^5$ which contains the fifth
standard basis unit vector.
