In case of the absence or disability of the Mayor, the President of the Council shall perform the duties of the Mayor during the period of such absence or disability; provided, however, that he shall not have the power of veto. He shall not have the power to remove the director of any department during any period in which he shall act in the capacity of Mayor until he has so acted for a period of thirty consecutive days during such period, and no appointment made by him shall be permanent until such appointee shall have served for a period of sixty days following his appointment.