If it is found, by the county commission, after examining the expenditures of the levee district, that the cost of constructing the work will exceed the estimates made by the engineer, the commission may call a meeting of the landowners, upon giving the notice as before required in the case of organizing a new district; there shall be laid before them an estimate of the cost of the completion of the work, made by an engineer, and a vote shall be taken, at such meeting, by the landowners present, voting either by themselves or by proxy, whether an additional tax shall be levied for the purpose of completing the work, and if a majority of the votes cast are in favor of completing such work, the board shall proceed to complete the same and the additional taxes that may be levied and collected as herein provided.