The Common Council shall have power to appoint and constitute a Board of Health consisting of one reputable physician and two of its own members, which said Board of Health under the supervision and with the approval of said Common Council shall have full power to make all needful rules and regulations for the keeping and maintaining of the said Town in proper healthful condition, and when such Board of Health shall have been elected and organized it shall have exclusive control of all matters relating to the public health within the limits of the said Town.