The procedure for the establishment of an island library district shall be as follows:

(1) Petitions signed by at least ten percent of the registered voters of the island, outside of the area of incorporated cities and towns, asking that the question, "Shall an island library district be established?" be submitted to a vote of the people of the island, shall be filed with the board of county commissioners.

(2) The board of county commissioners, after having determined that the petitions were signed by the requisite number of qualified petitioners, shall place the proposition for the establishment of an island library district on the ballot for the vote of the people of the island, outside incorporated cities and towns, at the next succeeding general or special election.

(3) If a majority of those voting on the proposition vote in favor of the establishment of the island library district, the board of county commissioners shall forthwith declare it established.