Bruce and Walls in their paper On the classification of cubic surfaces state at the end the "final observation" that a number of distinct lines on cubic surface with $k$ isolated du Val singularities is given by formula $$ \binom{8-c}{2} + k - 1 $$ where $c$ is the codimension of space parameterising surfaces of this type in $\mathbb PH^0(\mathcal O_{\mathbb P^3}(3)) \simeq \mathbb P^{19}$. However, if I am understanding correctly, their proof relies on checking directly all possible cases, which is not very convincing.
Could anyone sugest a reference with more intersection-theoretic proof of this fact?