If at any time it becomes necessary or expedient to extend a levee, not exceeding ten miles, in levee districts already organized, or to enlarge any levee district for the purpose of extending any levee therein to be constructed, in order to better protect such district from overflow, and if a petition of not less than five landowners, of the district, be presented to the county commission, requesting the same, the said county commission shall call a meeting of the landowners of the district and of the landowners in the territory sought to be connected with the district, in accordance with the prayer of the petitioner, at some convenient point to the said territory, by giving due notice as required in section 245.460, and at such meeting an estimate of the costs of the proposed new work made by an engineer, must be laid before the meeting, and the landowners present shall then determine, by the majority of the votes cast, whether such proposed new territory shall be added, and also whether the proposed new work shall be done. But said vote may be cast by the landowners present, or by agent or proxy, representing them. And if it is determined to add the new territory or to extend the levee in districts already organized, there shall be a readjustment of the taxes for benefits, and the county assessor shall proceed to assess the benefits to all of the land in the district that will be benefitted by the proposed work; and the board of equalization shall act upon the same, and all the assessments, levying of taxes and the collecting of the same shall be proceeded with in the same manner as in the case of organizing a new district.