A short note on Cayley Salmon equations.
A Cayley Salmon equation for a smooth cubic surface $S$ in \mathbb P^3$ is an expression of the form $l_1l_2l_3 m_1m_2m_3 0$ such that the zero set is $S$ and $l_i$ $m_j$ are homogeneous linear forms. This expression was first used by Cayley and Salmon to study the incidence relations of the 27 lines on $S$ There are 120 essentially distinct Cayley Salmon equations for $S$ In this note we give an exposition of a classical proof of this fact. We illustrate the explicit calculation to obtain these equations and we apply it to Clebsch surface and to the octanomial model. Finally we show that these $120$ Cayley Salmon equations can be directly computed using recent work by Cueto and Deopurkar.