If upon the hearing provided for in Sections 51-29-5 through 51-29-9 a petition is presented to the chancery court, or the chancellor in vacation, signed by a majority of the landholders owning one third of the land, or by one third of the landholders owning a majority of the land, praying that the improvements be made, it shall be the duty of the court or chancellor to make the order establishing the district without further inquiry, if it appear that the establishment thereof be necessary for the promotion of public health and for agricultural purposes. However, if upon that day a petition signed by a majority of the landowners owning one third of the land, or by one third of the landowners owning a majority of the land, be presented praying that the improvements be not made, it shall be the duty of the court or chancellor to so order; but if no such petition is filed, it shall be the duty of the court or chancellor to investigate as provided in the aforesaid sections and to establish such drainage districts as he is of the opinion the establishment thereof will be to the advantage of the owners of real property therein, and is for the public benefit. The petition provided for therein may be signed by women, whether married or single, owning land in the proposed district; guardians may sign for their wards; and trustees, executors, and administrators may sign for the estates represented by them; and if the signature of any corporation thereto is attested by the corporate seal, the same shall be sufficient evidence of the assent of the corporation to said petition.