Upon the establishment of such library district the county clerk of the county within which such library district is located, shall cause a publication notice to be published once each week for two consecutive weeks in a newspaper of general circulation in said library district, stating that a meeting of the qualified voters of said library district will be held at the time and place fixed in said notice, for the purpose of electing seven (7) directors to act as the governing body of said library district. The last publication of such notice shall be made not more than seven (7) days prior to the date fixed for the holding of said meeting. The costs of such publication shall be paid by the city of the third class located within said library district from the general funds of said city, or, from the library funds of said city. At the time and place fixed for the holding of said meeting the county clerk shall call such meeting to order and the electors of such district shall proceed to elect a chairman and a secretary for said meeting. Thereupon, the qualified voters shall proceed to elect, by ballot, seven (7) directors for said library district, one for a term expiring the first Tuesday in April of the year following the date of such election; two (2) for a term expiring the first Tuesday in April of the second year following the date of such election; two (2) for a term expiring the first Tuesday in April of the third year following the date of such election; and two (2) for a term expiring the first Tuesday in April of the fourth year following the date of such election. Upon the expiration of the terms of the first board of directors, their successors shall be elected in the manner provided for electing members of library boards.