  Let $X$ be a smooth projective surface and let $\mathcal{C}$ be an
arrangement of curves on $X$. The Harbourne constant of $\mathcal{C}$ was
defined as a way to investigate the occurrence of curves of negative
self-intersection on blow ups of $X$. This is related to the bounded negativity
conjecture which predicts that the self-intersection number of all reduced
curves on a surface is bounded below by a constant. We consider a geometrically
ruled surface $X$ over a smooth curve and give lower bounds for the Harbourne
constants of transversal arrangements of curves on $X$.
