At first glance, it undoubtedly seems very odd.

Last September, the Scottish National Party (SNP) invited people in Scotland to vote to leave the United Kingdom and become an independent country. The proposition was clearly rejected, with just over 55 percent voting against and fewer than 45 percent in favor.

Yet far from heralding a period of nationalist anguish and introspection, little more than six months later, the SNP is looking forward to the prospect of winning an electoral landslide in the U.K. general election scheduled for May 7. According to the latest polls, the party could win as many as 47 of the 59 Scottish seats in the 650-seat House of Commons.

Moreover, because neither the Conservative nor the Labour Party appears likely to win an overall majority, and because support for the Liberal Democrats has imploded throughout the U.K., the SNP not only could be the third-largest party in the next House of Commons but also could play a decisive role in determining the partisan color of the next government.

How can a party that so recently failed to persuade voters to back its principal raison d’être now be apparently heading for what would be a stunning success — and perhaps significant influence on the U.K.-wide stage?

There are three main parts to the answer. The first is that May’s election is being held under very different rules from the referendum. Second, the question of what Scotland’s constitutional status should be now matters more to voters than it did in the past. And third, the SNP has come to be regarded by many voters as the party keenest on creating a more equal society.

At 45 percent, the average level of support for the SNP in current Scotland-wide polls matches the percentage that the “yes” side won in the referendum. Indeed, it also equals the 45 percent that the SNP won in the election to the devolved Scottish Parliament held in May 2011. The crucial difference is that whereas 45 percent is always insufficient to win a referendum, it can be enough to win a landslide in an election that is held under the first-past-the-post electoral system and in which a multitude of parties are in contention.

However, that still means that we need to explain why the SNP has been able to retain the 45 percent level of support that the “yes” side secured in last year’s referendum. After all, the party has until now performed relatively poorly in elections to the House of Commons. Such elections invite voters to focus on which party is best able to provide effective government for the U.K. as a whole, a position to which, given that Scotland constitutes a relatively small part of the U.K., the SNP can never aspire.

As a result, many voters who back Scottish independence have hitherto not voted for the SNP in a U.K. general election. According to the Scottish Social Attitudes survey, only 55 percent did so in the last election in 2010, for example. Consequently, the SNP won only 20 percent of the total vote in Scotland (and six seats) on that occasion, well below the 45 percent it was to achieve in the devolved election just 12 months later. Labour, in contrast, maintained its dominant position in the 2010 election, with 42 percent of the vote and 41 seats.

However, the referendum has served to make support for independence and backing the SNP more or less synonymous, even in the context of a U.K.-wide election. Last September’s ballot focused voters’ minds — and especially the minds of those who voted for independence — on the future of Scotland. The question of who might be best able to govern Britain as a whole has been put in the shade. Consequently, supporters of independence see little reason why they should not follow up their “yes” vote with a vote for the SNP in May’s general election.

Recent polling evidence suggests that as many as 84 percent of those who voted “yes” in September (and who are willing to indicate how they will vote in May) now say they intend to vote for the SNP on May 7. Included among these “yes” supporters are the 40 percent or so of 2010 Labour voters who voted for independence, over three-quarters of whom are now backing the SNP. Conversely, only around one in 10 of those who voted “no” in September are now backing the nationalists — though that is still more or less enough to compensate for the limited number who voted “yes” in September but are now not backing the SNP.

However, the increased salience of the independence debate in voters’ minds is not the only reason that many former Labour supporters have switched to the SNP. During the referendum campaign, the SNP was also able to lay out its vision for the country, claiming that an independent Scotland could be a more equal Scotland. This argument directly challenged Labour’s position that it was better able to bring about greater social justice, not least by using the resources and the institutions of the U.K.

It is an argument that seems to have hit home.

Those who have switched from backing Labour in 2010 to supporting the SNP now are not only in favor of independence, they are also politically on the left. For example, according to the British Election Study, three-quarters of them want the government to engage in greater efforts to make people’s incomes more equal, a markedly higher proportion than the equivalent figure of three-fifths among Scots as a whole.

At the same time, these voters are more likely to believe that the SNP shares their outlook on this issue than they are to think that Labour does. Whereas slightly less than half (48 percent) believe Labour wants to make incomes more equal, no less than three-quarters reckon the SNP does. For this crucial group of voters, the SNP has won not only the argument about how Scotland should be governed but also the debate about which party has the stronger commitment to equality.

Of course, also central to the SNP’s apparent ability to win a landslide is the fact that those voters who want to stay in the U.K. are divided in their political loyalties. Only 45 percent of them propose to vote Labour, while 27 percent are currently backing the Conservatives. But could a division of the anti-SNP vote be avoided? Might Conservative supporters be willing to vote Labour in those constituencies where Labour is better placed to defeat the SNP — and vice versa — thereby perhaps denying the SNP victory in some constituencies?

It seems unlikely that many voters will take this path. Given the relative strength of the two parties, the suggestion is more or less tantamount to inviting Conservative voters to back Labour in most seats in Scotland. Yet, of course, across the U.K. as a whole, it is Labour that is the Conservatives’ principal opponent in the battle for power. It seems unlikely that many Conservative voters will be willing to do anything that might assist Labour in that endeavor, however much they might view the prospect of SNP electoral success with trepidation and distaste.

When four years ago the U.K. government accepted that the SNP’s success in the 2011 Scottish Parliament election had given the SNP the moral right to hold a referendum on independence, it did so with the expectation that a clear vote in favor of staying in the U.K. would stop the nationalist bandwagon in its tracks. Scotland may have voted to stay in the U.K., but the nationalist bandwagon seems to have just as much momentum as ever.