The British constitution still knows all too little of the people. Constitutionally, it is the House of Commons, not the voters, that chooses the prime minister. An incumbent leader, therefore, is entitled to meet parliament even if his opponents have won an overall majority. Most do not bother with this procedure, because they would be voted out of office immediately.

That does not apply when there is a hung parliament. There have been five since 1880. In 1886 and 1892, Lord Salisbury, the Conservative leader, met parliament; so did Stanley Baldwin in 1924. They were, in each case, voted down. Nevertheless, in 1886 and 1892, the Conservatives succeeded in exhibiting that the Liberals were dependent on the Irish Nationalists; and, by rejecting Baldwin in 1924, the Liberals put Britain's first Labour government into office.

In 1929, by contrast with 1924, Baldwin resigned without meeting parliament. In March 1974, Edward Heath sought coalition with the Liberals and with some of the Ulster Unionists. When that failed, he resigned, although he was constitutionally entitled to test his strength in parliament.

But the constitutional position conflicts with modern political reality. For the voters feel that it is they, not the MPs, who should decide who occupies 10 Downing Street. In 1974, there was a feeling that Heath had "lost" the election. He had sought a mandate to deal with the striking miners. The voters, admittedly, returned an inconclusive verdict. But they certainly did not endorse him.

Were the 2010 general election to yield a hung parliament, Gordon Brown, like Heath, would be constitutionally entitled to seek coalition; or he could meet parliament as the head of a minority government and challenge the other parties to vote him down. But he might then seem a bad loser. Even if Labour were to win more seats than any other party, Brown would be thought to have "lost". The imprint of first-past-the-post is so strong that voters see general elections as football matches in which a side has either "won" or "lost". Nuances such as which party has the most seats or which party has the most votes are hardly noticed in the post-election melee.

Therefore, the most likely outcome would be Brown's resignation. The Queen would then call for David Cameron, who could seek a coalition or an arrangement with other parties. But he would perhaps be more likely to meet parliament, challenging the other parties to vote him out. They probably would not do so, given that a defeat for Cameron would mean a second general election (which the minor parties could ill afford). The general feeling in the country would no doubt be that Cameron should be given his chance.

Cameron would probably follow the example of Harold Wilson, who, in 1974, sought the right tactical moment for a second general election in which he could secure an overall majority. During this period, it would be difficult, as it was in 1974, to take the tough measures needed to restore economic confidence. Minority governments have not been particularly conducive to economic stability. Nevertheless, a Cameron minority government could easily inaugurate a long period of Conservative rule, as the economic difficulties could be blamed on Labour.

Labour's position vis-à-vis a hung parliament is contradictory. It would, one assumes, prefer to negotiate an agreement with the Liberal Democrats than submit to Conservative rule, and would perhaps offer electoral reform as bait. The Lib Dems, especially if they have lost seats to the Conservatives, might also be tempted by a post-election agreement. The voters, however, could complain that an alliance between the two parties should have been agreed before the election. A post-election pact is unlikely to succeed.

Thus, Labour's only chance of avoiding a period of Conservative hegemony is to seek an agreement with the Liberal Democrats before the election. Perhaps it is already too late, but what is certain is that it will be too late to begin after the votes have been counted.

Vernon Bogdanor is professor of government at Oxford University. His latest book is "The New British Constitution"