Twenty-second Amendment to the United States Constitution
Some argue that the 22nd Amendment and 12th Amendment bar any two-term president from later serving as vice president as well as from succeeding to the presidency from any point in the presidential line of succession. Others contend that the original intent of the 12th Amendment concerns qualification for "service" (age, residence, and citizenship), while the 22nd Amendment, concerns qualifications for "election", and thus (strictly applying the text) a former two-term president is still eligible to "serve" as vice president (neither amendment restricts the number of times an individual can be elected to the vice presidency), and then succeed to the presidency to serve out the balance of the term (though prohibited from running for election to an additional term).
Twelfth Amendment to the United States Constitution
The experiences of the 1796 and 1800 presidential elections spurred legislators to amend the presidential election process, requiring each member of the Electoral College to cast one electoral vote for president and one electoral vote for vice president. Under the new rules, a contingent election is still held by the House of Representatives if no candidate wins a presidential electoral vote from a majority of the electors, but there is no longer any possibility of multiple candidates winning presidential electoral votes from a majority of electors. The Twelfth Amendment also contained other provisions, lowering the number of candidates eligible to be selected by the House in a presidential contingent election from five to three, establishing that the Senate would hold a contingent election for vice president if no candidate won a majority of the vice presidential electoral vote, and providing that no individual constitutionally ineligible to the office of president would be eligible to serve as vice president.