Neb. Rev. Stat. § 46-182: Neb. Rev. Stat. § 46-182

If the land excluded from any district shall embrace the greater portion of any division or divisions of such district, then the office of director for such division shall become and be vacant at the expiration of ten days from the final order of the board excluding the lands, and such vacancies shall be filled by appointment by the county board of the county where the office of such board is situated from the district at large. A director appointed as above provided shall hold his office until the next regular election for the district, and until his successor is elected and qualified.