If a lease of land, in which rent is reserved, payable at the end of the year or other certain period of time, is determined by the death of any person during one of the periods in which the rent was growing due, the lessor or his personal representative may recover a part of the rent which becomes due after the death, proportionate to the part of the period elapsed before the death, subject to all just allowances; and if any security was given for such rent it shall be apportioned in like manner. (1868-9, c. 156, s. 6; Code, s. 1747; Rev., s. 1987; C.S., s. 2345.)