When a tenant for life who shall have demised lands, shall die on or after the day when any rent became payable, his executor or administrator may recover from the under-tenant the whole rent due; and if he die before the day when any rent is to become due, he may recover the proportion of the rent which accrued before the time of the death of the tenant. The tenant for the life of another, his executor or administrator, in case of the death of the person for whose life the estate is held, on or before the day when any rent shall become due shall have like remedy; and a like apportionment shall be made in the case of annuities.