Where the offices of tax collector and assessor are separate, the collector shall assess and collect taxes on land liable to taxation left unassessed by the assessor, and on land that has become liable to taxation since the last assessment. He shall also assess such other persons and personal property as he may find unassessed by the assessor. He shall report to the board of supervisors on making each monthly settlement, under oath, additional assessments made by him, a copy of which the clerk shall transmit to the auditor within ten (10) days, and he shall charge the amount of the state tax thereon to the collector; provided, however, that the collector's right to assess and collect taxes on lands liable to taxation, left unassessed by the assessor, shall not be for a longer period than seven (7) years from the date when his right so to do first accrued.