Any petition for the formation of a public hospital district may describe an area lying in more than one county, the boundaries of which shall follow the then existing precinct boundaries and not divide a voting precinct; and if a petition is filed with the county auditor of the respective counties in which a portion of the proposed district is located, containing not less than ten percent of the voters of that area of each county of the proposed district who voted at the last general election, certified by the said respective auditors in like manner as for a countywide district, the board of county commissioners of each of the counties in which a portion of the proposed district is located shall fix a date for a hearing on the petition, and shall publish the petition, without the signatures thereto appended, for two weeks prior to the hearing, together with a notice stating the time of the meeting when the petition will be heard. The publication required by this chapter shall be in a newspaper published in the portion of each county lying within the proposed district, or if there be no such newspaper published in any such portion of a county, then in one published in the county wherein such portion of said district is situated, and of general circulation in the county. The hearing before the respective county commissioners may be adjourned from time to time not exceeding four weeks in all. If upon the final hearing the respective boards of county commissioners find that any land has been unjustly or improperly included within the proposed district they may change and fix the boundary lines of the portion of said district located within their respective counties in such manner as they deem reasonable and just and conducive to the welfare and convenience, and enter an order establishing and defining the boundary lines of the proposed district located within their respective counties: PROVIDED, That no lands shall be included within the boundaries so fixed lying outside the boundaries described in the petition, except upon the written request of the owners of the land to be so included. Thereafter the same procedure shall be followed as prescribed for the formation of a district including an entire county, except that the petition and election shall be confined solely to the portions of each county lying within the proposed district.