SECTION 11. Contribution by lessee; rights and remedies of lessee

If an assessment is made upon land the whole or part of which is leased, the owner shall pay the assessment, and may collect of the lessee an additional rent for the portion so leased equal to ten per cent per annum on that proportion of the amount paid which the value of the leased portion bears to that of the whole estate, after deducting from the whole amount any money received for damages to such land in excess of what he has necessarily expended thereon by reason of such damages. A lessee aggrieved by the imposition of this burden may, within six months from the time demand is made upon him for such additional rent, file a petition in the superior court for the county in which the land is situated, to determine the proportion of the assessment which he ought to bear, and the proportion determined upon the petition shall be substituted for the proportion provided by this section. If such proportion is reduced the lessee shall recover costs from the owner; otherwise the owner shall recover costs from the lessee.