If the authority and the trustees, directors or managers of any nonprofit or charitable hospital in a city should agree upon and consummate a transaction whereby the nonprofit or charitable hospital should thereafter be included within the hospital project or projects of the authority, the number of commissioners of such authority shall be increased to not exceeding fifteen. The additional commissioners shall be appointed by the mayor from nominations of the commissioners then in office, and the terms of the additional commissioners shall be arranged by the mayor in making such appointments as follows:

The terms of one-third of the commissioners shall expire in two years or less, one-third in four years or less, and one-third in six years or less, concurrently with the expiration of the terms of the commissioners then in office.