THE ending of the round-table talks among the leaders of the main political parties in Northern Ireland might seem, from afar, to be yet another breakdown in the apparently endless search for peace in this province. For nearly 10 weeks the political representatives of Northern Ireland's 1 million Protestants, who favor retaining the links with Britain, and its half-million Roman Catholics, a majority of whom favor Irish unity by peaceful means, met at Stormont, near Belfast. The talks were chaired by Peter Brooke, Britain's cabinet secretary for Northern Ireland, whose diplomatic skills and apparently inexhaustible patience had brought the parties to the table, for the first time in 16 years. It had taken Mr. Brooke 18 months to persuade the Ulster politicians to try to find some common ground, and many people in Northern Ireland fervently hoped this might lead to a breakthrough on the road to peace. Unfortunately, most of the 10 weeks were taken up by procedural wrangling. The politicians took a long time to agree on a venue that would not be seen as a loss of face. They also took a long time to agree on a neutral outsider who would chair the second stage of the talks, involving the Dublin government. The parties had only two weeks for what were described as "meaningful talks." The main stumbling block was the scheduled July 16 meeting under the Anglo-Irish Agreement, effected in 1985 between the British and Irish governments. The agreement guaranteed a limited advisory role for Dublin in the affairs of Northern Ireland in exchange for Dublin's recognition of the legitimacy of Northern Ireland. This agreement was bitterly resented by Northern Ireland's Protestant Unionists, who favor the link with Britain. They felt that it was the beginning of a process leading to Irish unity. The round-table talks were regarded by some Unionists as an opportunity to negotiate something better than the Anglo-Irish Agreement. They refused to accept that the scheduled July 16 meeting of British and Irish representatives, one of a series of regular meetings under the terms of the agreement, should go ahead. They argued that this meeting could have been postponed to let the parties make up for lost negotiating time. However, the nationalists in Dublin and Belfast feared that the Unionists were using the prospect of the July 16 meeting as yet another delaying tactic and that, if successful, they would seek ways to undermine the agreement itself. Equally the Social Democratic and Labor Party (SDLP), which represents mainly Northern Roman Catholics, was not prepared to put its substantive proposals on the table if the Unionists were going to walk out before they could be fully discussed. BROOKE, despite his powers of diplomacy, was unable to break that deadlock and, with the agreement of the party leaders, he brought the process to an end July 3. It was felt that he did so to prevent the process dwindling into an acrimonious public exchange. Yet there are indications that something useful was learned and that the parties might have built a tentative base for further exchanges. The public pronouncement on the ending of the talks was relatively muted, by Northern Ireland standards. Brooke concluded that, while he was disappointed, "foundations have been laid for progress in the future which neither the cynics nor the men of violence will be able to undermine." Charles Haughey, the Irish premier said, "The present opportunity for progress has not easily come about and cannot lightly be allowed to go by." The Rev. Ian Paisley, leader of the Democratic Unionists, predictably blamed the British and Irish governments for failing to give way on a postponement of the July 16 meeting, and said that new talks "which would not be brought down by the Anglo-Irish Agreement" should be envisaged. James Molyneaux, leader of the Official Unionists, praised Mr. Brooke, who, he said, had "retained the confidence and respect of all who ... participated in the talks." John Hume, leader of the SDLP, regretted the ending of the talks, but stressed that the exchanges that did take place were "very valuable indeed" and were very different from the exchanges which had taken place in the past. He added, "We hope that they can be built on in the future." All of which is more positive than the bitter public wrangling of previous years. There is a feeling, and no more than that, about the possibility of future talks in the autumn or early next year, provided the delicate groundwork can be laid. Austin Currie, a Dublin politician with strong Northern roots, summarized succinctly: "The talking will resume because there is no sane alternative."