It did not turn out that way. European Union leaders took her short, late-night intervention, and subsequent remarks, as truculent and inflexible. The Prime minister, and Britain's predominantly eurosceptic press, saw the European Union's response as rigid and disrespectful. All this spread distrust between Brussels and London, creating further uncertainty about the outcome of the negotiations, as the date for Brexit, 29 March 2019, now enshrined in British law, approaches. Still, optimists asked whether Salzburg could turn out to be the salutary shock needed to shake the negotiations out of their present torpor.

But the odds on Britain making a smooth exit from the European Union next 29 March have fallen since the Salzburg summit and now stand at around sixty per cent. The British Prime minister's appeal to the 27 leaders failed to move the other Member States, several of whose leaders ruled out her convoluted Chequers plan. The European Council president, Donald Tusk declared himself "sceptical and critical" about key aspects of the plan, while recognizing that it showed movement in the UK's negotiating position. Statements by other European Union leaders, particularly French president Emmanuel Macron, suggested a hardening of the European Union's position. Theresa May received support from the Hungarian Prime minister, Viktor Orbán, though this was cold comfort in light of the recent European Parliament vote in favour of pursuing his country for the violation of fundamental EU principles.

Theresa May and European Union's leaders miscalculated the other side's willingness to compromise on how to reach their shared goal of avoiding a hard border across the island of Ireland. Brexiters and the British euro-sceptic press highlighted Theresa May's discomfort at the European Union's refusal to take her Chequers plan as a basis for the negotiations.

But those ready to countenance a no-deal Brexit may yet be disappointed. Theresa May needed a political showdown in Salzburg to prove her toughness ahead of the annual Conservative Party conference on 30 September - 3 October in Birmingham, where Brexiters may try to stage a coup against her. European Union leaders needed to demonstrate their support for chief negotiator Michel Barnier to discourage their own euro-sceptics and to dispel any illusion that the British government could achieve its objectives through "divide and rule" tactics.

Meanwhile the shape of a settlement, which could be charted at the next European Council meeting on 18 October, or in a possible follow-up meeting in mid-November, is beginning to emerge. Some Irish observers concede that the notion of a legally binding "backstop", in the form of the present draft protocol prepared by the Commission, may have been "oversold" and is not necessarily an immediate requirement. The status quo - including free movement without border controls - will continue during the 21 month transition period after Britain leaves, provided a withdrawal agreement is in place. In practice, the transition period will have to be prolonged beyond 31 December 2020, as negotiations on the future partnership cannot be concluded in such a short period, especially with European Parliament elections and a change of guard at the other EU institutions due in 2019. British officials are working on a "bridge" beyond December 2020 to the time when a new partnership takes effect. If mutual trust can be restored, after Salzburg, a compromise on the Irish backstop may be reached as part of an overall Brexit package.

The other unresolved part of the withdrawal package is the political declaration on future relations between the UK and the European Union. The public debate on the "Chequers" scheme, versus the "Norway" or "Canada" models, assumes that a detailed future framework must be set out in the political declaration. However, there have been hints from Berlin and London that there is no need for this non-binding declaration to specify the precise model for the future partnership. It could be a broad declaration of principle, confirming both sides' willingness to conclude a far-reaching agreement covering such fields as trade, regulatory matters, police and judicial cooperation, foreign policy, security, aspects of defence and so on. Some, including the French President, have decried this prospect as a "blind Brexit" but it seems the most likely way forward.

If Theresa May staves off a challenge from Brexit extremists at the Conservative party conference, she may well succeed in getting a withdrawal agreement through the House of Commons.

Conservative MPs will think twice about voting against the government if there is a risk that this could lead to an election bringing the Labour Party leader, Jeremy Corbyn, to power. Some Labour MPs, dismayed at Corbyn's euroscepticism and mismanagement of the party, may be willing to vote for a withdrawal agreement, if the alternative is the cliff edge.

The real negotiations on the future partnership will begin only after Britain has left the European Union. At that time Michel Barnier will stand down, his job complete; negotiating directives for the future partnership may not be finalized until late in the year when new European Union leaders have taken up their posts. In the UK, there should be more flexibility once Brexit has occurred, as the Brexiters' worst fear is that Britain might not actually leave.

Little progress was made in Salzburg on migration, the original focus of the informal summit, despite the positive gloss given by Donald Tusk, after the meeting. With the sharp decline in the number of migrants reaching the European Union in 2018, the EU's sense of urgency has evaporated, though the number of deaths in the Mediterranean Sea this year is still alarming and there is continuing pressure from "populist" political parties and movements in Member States.

European Union leaders were due to build on the controversial conclusions of the June European Council including proposals for "regional disembarkation platforms", presumably in North Africa, and "controlled centres", to be established on a voluntary basis in Member States, as well as the adaptation of the Common European Asylum System and the upgrading of the European Union's Border Protection and Coast Guard Agency, widely known as FRONTEX. Commission President Jean-Claude Juncker gave priority to this in his "state of the union" speech earlier this month on 12 September .

However, there was no eagerness in Salzburg to establish holding centres in third countries or Member States and no country has volunteered to host these. There was no follow-up to the call by Austrian and Italian ministers for asylum claims to be processed on board rescue ships in the Mediterranean. Instead Donald Tusk announced a series of dialogues with Egypt, other African countries and the Arab League. As in June, legal specialists, human rights groups and non-governmental organizations are likely to question an approach to curbing migration which includes restraint agreements with non-democratic governments.

The conversation among European Union leaders on migration in Salzburg produced neither break-throughs nor break-downs. The summit is more likely to be recalled as delivering a salutary shock on Brexit that might bring new urgency to the negotiations as they enter their final phase.