LONDON — A BATTLE is being fought to win the hearts and minds of one-quarter of British voters. The result of the contest may determine whether Britain turns left at the next general election or holds to a right-wing course for another five years. How the struggle will eventually turn out promises to be clearer after Britain's three small center parties have displayed their political wares at a series of annual conferences being held during the next four weeks in Britain.
This week it is the turn of the Democrat Party meeting at Brighton to try to impress voters that they remain a credible political force. Later the Social Democrats and the Green Party each will argue that Britons should shun the ruling Conservatives and the opposition Labour Party and throw their support to the center.
Neil Kinnock, the Labour Party leader, and Margaret Thatcher, Tory prime minister for the past 10 years, find common ground on one thing at least: growth of the center parties should be discouraged. Third parties refuse, however, to go away, even though they suffer persistent identity crises and quarrel among themselves.
Crucially, the results of recent elections and voter surveys suggest that one voter in four feels alienated from the Conservative and Labour Parties and would prefer to vote for someone else.
Polls carried out on the eve of the Democrats' annual conference showed them enjoying the support of under 10 percent of the electorate, with the Social Democrats and the Greens sharing about 15 percent.
The problems of Paddy Ashdown, leader of the Democrats for the past 15 months, exemplify how easily these floating voters could be tempted, out of exasperation, to support either Mrs. Thatcher or Mr. Kinnock at the general election likely to be held in about two years.
Mr. Ashdown's party - a powerful force in the 19th and early 20th centuries - these days cannot even be certain what to call itself. It used to be called the Liberal Party, but when in 1987 it merged with one wing of the Social Democrat party, it fell to bickering whether to call itself the Democrats, the Liberal Democrats, or the Social and Liberal Democrats. Former members of the Social Democrats prefer it to be called Democrats, while former Liberals argue that it should be called Liberal Democrats.
In addition, David Owen - leader of the original Social Democrats, which was formed in 1981 - refused in 1987 to merge his party with the Liberals at all. This meant that while Ashdown and his followers were quarreling about their party's name, Owen and Ashdown were quarreling with each other. Both parties did badly at the European elections, while the Green Party, offering a conservationist message, attracted some 15 percent of the nationwide vote.
Ashdown, a former marine commando, warned his followers Sept. 9 that a failure to resolve their differences over the party's name and what it stands for could lead to oblivion. His rank and file, however, pressed for a vote on a motion calling for the resignation of Ian Wrigglesworth, the party's president.
Then Owen sent a message to Brighton, proposing an electoral alliance between his party and Ashdown's. The two leaders deeply distrust each other, and Ashdown promptly turned the offer down.
Labour and Conservative strategists are hoping the confused state of the smaller parties will persuade Britons that support for the center parties at the next general election would be wasted.
Kinnock is deliberately leading his party away from left-wing policies such as nationalization and unilateral nuclear disarmament.
The Tories, under their new chairman, Kenneth Baker, are beginning to put the stress on environmental issues in the hope that they will be able to win back support that went to the Greens last June.