Justice William Douglas (who served 36 years, the most in Supreme Court history)suffered a stroke during his next-to-last term, the Fall 1974 Term.

He nonetheless continued to perform what Supreme Court duties he could (much as the late Chief Justice, when ill, reviewed oral argument tapes and transcripts when he could not sit for argument).

However, the other Justices became gravely concerned about his competence. They therefore entered into (a fact which was at the time not publicly revealed until years later, I believe) an “inter-office” agreement/pact that dictated that, in the event a case had already been argued (but not yet decided upon), where the vote was 4-4 (and thus Douglas’ vote would make the difference), the case would be put off until the following term.
No such cases were put off, as far as I know, and Douglas retired early into the next term (Fall 1975).

On a separate issue — the speculation about when and why Justices recuse themselves suggests that there should be more transparency in this area. Justices should be required, or at least strongly encouraged, to explain publicly why they have chosen to do so.

In answer to your question, there don’t appear to be any rules regarding how to dispose of equally divided cases caused by a retirement, but the practice seems to depend on whether there is a replacement available to take up the case.

For instance, Justice O’Connor helped to avoid this situation when she made her decision two years ago to stay on until her replacement could be confirmed; she thus ensured that there would never be a time when only eight justices were on the Court. Hence, while there were those three aforementioned cases for which Justice O’Connor was on the Court at oral argument but the Court was unable to issue an opinion by the time she retired, Justice Alito was able to immediately step in.

Likewise, when Justice Marshall retired effective October 1, 1991 and Justice Thomas did not replace him until later that month, the cases in the October 1991 sitting were argued in front of an eight member Court. But, in that circumstance, the Court did choose to hold two closely divided cases and reargue them with Justice Thomas on the Court.

On the other hand, looking back at the situation in 1987 (when Justice Powell retired immediately in June of that year), for much of OT87, there genuinely were only eight members on the Court, and it was unclear when the ninth would be confirmed (indeed, Justice Kennedy did not take his commission until February of 1988). Perhaps that’s why the Court decided to issue an equally divided affirmance rather than wait for a ninth member to break the tie.

-Jason