Fibred surfaces with general pencils of genus 5

Let $f:S \fr B$ be a surface fibration with fibres of genus 5. We find a linear relation between the fundamental invariants of the surface. Namely $K_f^2=\chi_f+N$ where $N$ is the number of trigonal fibres. Our proof is based on the analysis of the relative canonical algebra $\cal{R}(f)$.