Whenever a sanitary drainage district shall have been formed under sections 31-501 to 31-523, and trustees thereof have been elected, if such trustees find that municipalities, including cities of the second class and villages, in whole or in part outside of the boundaries of such sanitary district and depending upon the same watercourse or its tributaries for drainage, are inadequately supplied with sewage systems or sewage disposal plants, and that the absence thereof is detrimental to the sanitary conditions of the district or to such municipalities within the same watershed or lands adjacent thereto, the trustees may by resolution determine the existence of such conditions and shall thereupon cause a survey to be made of the territory affected by said conditions and the boundaries of such territory. The trustees shall thereupon file with the county board of the county wherein such district is situated a definite description of the territory affected by said conditions and the boundaries thereof.