Blogging is probably going to be light today, so I'll pose a hypothetical question that may never be answered.
First, we know that the late William Rehnquist, the former Chief Justice of the United States Supreme Court, did suffer from severe back pain for much of his career. Suppose that during the month of December 2000, Rehnquist's back pain became so severe that, due to surgery or narcotics, he was temporarily unable to fulfill his duties as Chief Justice.
On December 8, 2000, the Florida Supreme Court ordered a statewide manual recount of all ballots cast in the Presidential election held the month before. The Florida recount, according to some, might have taken months to complete and certify. President Bill Clinton was scheduled to leave office on January 20, 2001.
On December 12, 2000, the Supreme Court ordered a halt to the recount, handing an effective victory along with Florida's electoral votes to George W. Bush in the still controversial case of Bush v. Gore. The case was decided by a vote of 5 to 4, with Rehnquist in the majority.
Now, if any Justice in the majority, let's say Rehnquist, had been unable to decide the case for any reason, the vote would be 4 to 4, leaving the lower (Florida Supreme) Court's decision standing. The recount would go on. Let's say until March.
Which leads to another question: Who would be President on January 21, 2001?
You are armed with a copy of the United States Constitution, the most authoritative document which purports to answer this question. And I'll submit that its answer is not at all clear. You may use any school of legal reasoning to decide this question. You will receive extra credit for explaining your answer in detail, whether by reference to text, history, statute, or case law (assuming that, unlike me, you can find an applicable statute or case).