Any petition for the formation of a public hospital district may describe a less area than the entire county in which the petition is filed, the boundaries of which shall follow the then existing precinct boundaries and not divide any voting precinct; and in the event that such a petition is filed containing not less than ten percent of the voters of the proposed district who voted at the last general election, certified by the auditor in like manner as for a countywide district, the board of county commissioners shall fix a date for a hearing on such petition, and shall publish the petition, without the signatures thereto appended, for two weeks prior to the date of the hearing, together with a notice stating the time of the meeting when such petition will be heard. Such publications required by this chapter shall be in a newspaper published in the proposed or established public hospital district, or, if there be no such newspaper, then in a newspaper published in the county in which such district is situated, and of general circulation in such county. The hearing on such petition may be adjourned from time to time, not exceeding four weeks in all. If upon the final hearing the board of county commissioners shall find that any lands have been unjustly or improperly included within the proposed public hospital district the said board shall change and fix the boundary lines in such manner as it shall deem reasonable and just and conducive to the welfare and convenience, and make and enter an order establishing and defining the boundary lines of the proposed public hospital district: 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 such lands. Thereafter the same procedure shall be followed as prescribed in this chapter for the formation of a public hospital district including an entire county, except that the petition and election shall be confined solely to the lesser public hospital district.