In the defeat of the common site picketing bill in the House last week, House Republicans feel they have found a combination of ingredients to cure what ails them.

What ails them is a sheer dificiency of numbers. With 289 Democrats, 114 Republicans and two vacancies in the House, Republicans have had little hope of thwarting the huge Democratic majority. They faced a frustrating session, where they could criticize but were not much of a factor to be reckoned with.

"You just get so darned frustrated, you sometimes wonder whether" it's worth getting up in the morning," said Republican Whip Robert H. Michel of Illinois.

And aide, Mike Johnson, chimed in. "It's like beinga Maytag repair man. All you can do is sit around and wait for the Democrats to break down."

With the top-heavy Democratic margin in Congress and a Democrat in the White House, in most cases Republicans will still be overwhelmed.

But in the defeat of the bill that labor was counting on as a showcase of its strength in this Congress, Republicans not only slowed down the majority steamroller, but found a mix of outside help and inside coalitions that may allow them to win again. Those factors were:

A strong united lobbying effort that involved not only the special interests directly affected by the bill - contractors, home builders and construction, suppliers - but large industries, such as Pittsburgh Plate Glass and Dow Chemical, and the three biggest business lobbying organizations, the Chamber of Commerce, the National Association of Manufactures, and Business Roundtable, comprising the largest corporations in the country.

A method of lobbying that not only used powerful Washington lobbyists but generated a grass-roots campaign that brought more mail to Capitol Hill "than at any time since the Vietnam war"?, as House Speaker Thomas P. (Tip) O'Neill said.

One element of the drives was the House Republican Campaign Committee's mailing to 300,000 election contributors.They were asked to write Democratic leaders saying stop the bill, and to contribute to the Republican cause to stop any future legislation like common situs, which would have allowed a building trade union to shut down an entire construction site although its grievance was with only one subcontractor.

But the most important and effective part of the grass-roots campaign came from the Association of General Contractors. Its members were given postcards urging defeat of the bill, to be distributed with union employees' paychecks and the collected and mailed by supervisors. The result: While national union leaders were lobbying for the bill on the Hill, House members were getting mail from rank and file members of construction trade locals in their districts urging them to vote against it.

They heavy lobbying effort enabled the Republican Whip organization, temporarily at least, to revive the Southern Democrat-Republican coalition that had existed in Congress during the '50s and until the late '60s. Every Democrat from Alabama, Arkansas, Georgia, Mississippi and North Carolina voted against the bill.

Republican Whip Michel, a some what portly, silver haired nuts and, bolts legislator from Peoria, whose job it is to count noses on important pieces of legislation, does not take credit for the win.

He admits, "Had I been asked about this in the first part of January, I would have responded that there was no way we could win this one."

Michael had his doubts because a common site bill had passed the House in 1975 with only 178 votes against it. At the time, President Ford had been giving indications of signing a bill if passed, and the bill was strongly supported by his Secretary of Labor, John T. Dunlop. Ford decided after the bill passed to veto it, Dunlop resigned in protest and Congress did not try to override the veto.

The Republican leadership, which had lost two more seats in the 1976 elections, "weren't overly enthusiastic" about trying to defeat the bill, said Clarence Randall, chief lobbyist for the Associated Builders and Contractors. However, his group began working hard "the day after the election" and by February, when they could show Republican leaders that "the right grid" was coming together, the GOP chiefs buckled down.

Michel does take credit for preaching to the contractors and business lobbies the two important ingredients of legislative vote-getting.

He flew to San Francisco to advise a convention of "political action" groups of the Association of General Contractors on how to put together a grass-roots campaign.

"I told them they didn't need thousands of letters. So often when a member says 'my mail has been so heavy on that' he's talking about a dozen letters from key people in his district." If they could get the rank and file union members to write, that would be best of all.

Last Tuesday morning Michel addressed a joint meeting of lobbyists for the Chamber of commerce, National Association of Manufacturers and the Business Roundtable at the Chamber building.

t was not a gentle speech. Michel gave them hell. The one-third minority would never be effective without substantial outside lobbying support, he warned, and he blasted business lobbies for not hanging together the way labor unions do.

An industry only lobbies for its own narrow interest, he said; beyond that, it tends to walks away because it is not "prudent" to get involved in political causes.

"Insofar as helping friends and punishing enemies is concerned, labor unions make the business community look like a bunch of kindergartners," Michel said. "You don't see organized labor running three ways from Sunday when one of their issues reaches Congress. Labor is organized, well-organized united and committed to one approach."

He warned the business groups the there would be amendments to insulate the home builders from the picketing provisions. "But I would hope the home builders would not get suckered into this one, for that would simply gut our efforts and guarantee the legislation's passage. You've either got to stand united on the principles upon which we oppose the legislation, period, or you're going to lose everything in the process."

Such amendments were offered and adopted, and for a while it appeared the weaker bill would be accepted. But the home builders did not desert, and in the end the bill lost, 217-205.

Michel does not believe that the prescription will work all the time, or that the Republican-Southern Democrat coalition is permanently back together, or that all future labor-backed legislation is in jeopardy. He thinks common situs was a bad bill that would have caused inflation, raised expenses for an already depressed industry, and cost jobs.

"But it won't be the only time we'll play the game," Michel said. "If another issues comes along that lends itself to such widespread support, we'll try again.Labor probably learned something, but so did we, and hopefully so did our business supporters. We all learned the name of the game is votes."