When it is deemed advisable by the board of directors of district to enlarge the boundaries thereof and the conditions mentioned in section 31-401 apply to such enlarged territory, a petition for the enlargement of the district, signed by a majority of the board of directors of the district and by ten owners of land within the territory proposed to be added to the district or, if there are less than twenty such owners, then by at least one-fourth of such owners, may be filed with the county clerk of the county where the original petition was filed. Upon the filing of a petition for the enlargement of a district, the county board, county surveyor, and county clerk of the county shall proceed in all respects as provided in sections 31-402 to 31-408, so far as applicable. The board of directors of the district, at the time of filing such petition for enlargement with the county clerk, may prescribe the conditions on which the additional territory is to be added, which conditions shall be based upon the work previously done by the district and with a view to equalizing assessments according to benefits, and such conditions shall be binding on the enlarged district if formed. Any person may appeal to the district court from the imposing of such conditions in the manner provided for appeals by section 31-412. The appeal shall be taken within thirty days from the time of completing the canvass of the votes of such election. The additional territory shall be deemed added to the district only if a majority of the votes voted thereon in the original district and a majority of the votes voted thereon in the proposed new territory shall each be in favor of such enlarged district. The board of directors of the original district shall constitute the board of directors of the enlarged district.