The petition, after the filing thereof, shall be taken up and considered by the county legislative authority at the next regular or special meeting thereof, or as soon thereafter as may be convenient, and if the petition be defective in any particular it may be amended and an adjournment of the matter may be had to permit of the amendment, for a time not exceeding thirty days. If the petition be defective and be not sufficiently amended within the adjournment taken, it shall be dismissed. But if the petition is sufficient, or if by amendment it be made sufficient, it shall be the duty of the county legislative authority to enter an order setting a time for a public hearing thereon within thirty days from the date of the order, and directing the clerk of the county legislative authority to give notice of the time and place of the hearing in the official newspaper of the county by publication therein at least once each week for three successive weeks before the time of hearing. The notice shall be addressed to the owners of lands not petitioning, as shown by the petition or as may be ascertained to be the fact, and to all other persons known and unknown having or claiming an interest in the lands in the district, and shall state the pendency of the proceeding, its object, the names of the signers of the petition, the number of acres of land they claim to own, the whole number of acres proposed to be improved, the boundaries of the lands to be included in the improvement district, and the time and place of hearing. And notice shall also be given that at the time and place named, or at such time as the same may be adjourned to, the board will consider the petition under the provisions of this chapter, and will hear all objections offered by interested parties and grant or refuse the petition as it may be advised. The clerk of the board shall keep a record of all orders, hearings and proceedings of the board in reference to the waterway district in a separate bound book, designated as the record of proceedings as to such district.