Statement - I : Potential of mercury cell remain constant during its life
Statement - II : Reactions involving mercury cell does not involve any change in concentration of ions

$Zn-Hg +\hspace{0.17em}Hg{O}_{\left(s\right)}\to Zn{O}_{\left(s\right)}+\hspace{0.17em}H{g}_{\left(l\right)}$

The cell potential remains constant during its life as the overall reaction does not involve any ion in the solution whose concentration may change during its life time.