Tokyo — The second Cabinet of Yasuhiro Nakasone takes office today with enhanced prospects for political stability in Japan despite the ruling Liberal Democrats' severe election setback last week.
The new Diet (parliament) elected Mr. Nakasone prime minister Monday after the Liberal Democrats had increased their voting strength by bringing in nine conservative independents and forming a parliamentary alliance with the eight-member New Liberal Club.
The Liberal Democrats now have 259 members in the powerful 511-seat House of Representatives, plus the votes of the eight New Liberals. This enables them to control all the House committees except science and technology. Most important, they have a clear majority on the budget committee.
Mr. Nakasone's political position is thus considered relatively secure, at least until his two-year term as party president expires next November.
The price the outspoken premier had to pay to continue in office - despite the Liberal Democrats' loss of 36 seats in the Dec. 18 election - was to promise to ''eliminate all so-called political influences coming from Mr. (former Prime Minister Kakuei) Tanaka.''
''Shadow shogun'' Tanaka's influence does appear to have been diminished somewhat. The Tanaka faction, the largest in the Liberal Democratic Party with 63 members in the lower house, has lost two key posts - secretary-general of the party and chief Cabinet secretary. But it has been given six Cabinet portfolios, more than any other faction, plus the chairmanship of the party's executive council.
(The chief issue of the Dec. 18 general election was political morality, centered on Mr. Tanaka's refusal to resign his Diet seat after being found guilty of accepting bribes from Lockheed Aircraft Corporation during his 1972-74 premiership.)
The Reagan administration will be watching the new Cabinet's first steps with keen interest. Of particular importance to the United States is whether there will be any dilution of Mr. Nakasone's forthright stance on Japan's responsibilities as a full partner in the Western alliance, or of his efforts to soften economic frictions between Washington and Tokyo as the US enters an election year.
Partly in order to reassure Japan's friends and partners, Mr. Nakasone has retained Shintaro Abe as foreign minister and Noboru Takeshita as finance minister.
He has also appointed as deputy premier and minister in charge of the Economic Planning Agency one of his most forthright rivals and critics, Toshio Komoto.
Mr. Komoto, who heads the smallest of the five major factions, opposes Mr. Nakasone's tight money policies and wants increased government spending to pump the economy, even at the cost of larger deficits.
It remains to be seen to what extent Mr. Nakasone can harmonize views within a Cabinet that rests upon a fragile compromise between the Liberal Democrats' major factions.
Of the 21 Cabinet posts including the premiership, the Tanaka faction has six , with four posts for each faction headed by Nakasone and former premiers Zenko Suzuki and Takeo Fukuda. The Komoto faction has two and the New Liberals have one.
This is the first time since the formation of the Liberal Democratic Party in 1955 that it has given a Cabinet post to another party.
The New Liberals left the Liberal Democrats in 1976 because of anger over the Lockheed affair and over what they called the ''money speaks'' structure of the ruling party. Many of them were dissident members of the Nakasone faction. They have been only a minor party ever since, but the eight seats they won in the Dec. 18 election are a crucial number that spell the difference between the Liberal Democrats controlling all the major committees or having to cede several important chairmanships to the opposition. Their leader, Seiichi Tagawa, himself once a member of the Nakasone faction, now becomes minister of home affairs.