In a relaxed atmosphere of modest optimism, powerfully induced by Jimmy Carter's multiplying political problems, the corporal's guard of Republican governors that met here Oct. 10 tried hard to extract the ideological hatchet that is stuck deep in the party's hide.

That they did not succeed goes without saying. But the fact the effort was made, led by Illinois Gov. Jim Thompson, one of the party's biggest vote-getters>a nd by long-time Govs. William Milliken of Michigan and Robert Ray of lowa, points to this conclusion: Sparked by an obviously premature conviction that the Carter presidency may be doomed to one four-year term, the Republicans would be fools if they did not attempt to end suicidal warfare over conservative demands for ideological purity.

The setting was a Victorian relic, the 1902 White Mountain Hotel, but the symbolism seemed misplaced. Although Thompson invoked the name of Republican President William McKinley, it was to memorialize his 1900 presidential campaign slogans of "Employment for Labor" and "Sound Money." Translated into 1977 rehetoric, Thompson said that stands for what should be the universal Republican themes, particularly between now and the 1980 election: more jobs and less inflation. Thompson's warning: Don't get hung up on "ideological testing."

The effort to smother ideological conflict also had a strong advocate in Delaware's new governor. Peter duPont, a political moderate who huddled Sunday night with Mississippi's Charles Pickering. Pickering, a strong conservative, succeeded the irrepressible Clarke Reed as state party chairman but lacks Reed's lust and flair for doctrinal warfare. Du Pont promised to help raise campaign funds for both Pickering and Ray Hutchison of Texas, bot possible gubernatorial nominees in the future.

Along with conspicuously reduced emphasis on ideology, a desire for the political retirement of both former President Gerald Ford and conservative leader Ronald Reagan was much in vogue. This privately expressed hope was particularly evident among gubernatorial candidates, invited here by William Brock. Republican national chairman, to be guests of the incumbent govenors at the two-day session and fill in empty places at the conference table.

One top party operative, moreover, said that both Ford's quck decision to prop up President Carter's Panama Canal treaties and Reagan's to do the opposite, both without formal effort to tap party sentiment, smacked of a papa-knows-best attitude that had rankled rank-and-file Republicans - even though Reagan's stand far overshadowed Ford's within the party.

THe quietly opimistic mood underlined new efforts by Brock to take advantage of Carter's disabilities, starting with a drive for campaign contributions from the American Jewish community, once a private Democratic preserve. Brock met with the most powerful Jewish leadership group in the country during the summer. Subnect: Carter's Mideast policies are supsects, so give us a chance.

Conservative California State Chairman Mike Montgomery, here as an observer, ahs invited Sen. Jacob Javits to be the main speaker at a January political dinner in San Francisco and then tour rich Jewish suburbs in West Los Angeles - a font of Democratic contributions - and San Diego's well-endowed Jewish districts. Brock plans similar campaigns on a national basis, backed by the latest Harris survey, which showed Carter with a 60 per cent negative rating from Jewish voters - by far the largest negative percentage in any voting category.

The modest optimism clearly rested more on the vicissitudes of Carter's popularity than anything else. Moreover, attempts by new, young governors - Thompson, du Pont and Richard Snelling of Vermont - to soften ideology as a litmus test of party purity ran into predictable grumbling.

David Keene, Ronald Reagan's skilled Southern manager in the Ford-Reagan preconvention wars last year, pointedly recalled that at last winter's National Governors' Convention, Thompson was the sole dissenter when Republicans agreed to make the 1976 party platform serve as the party's beacon until 1980. Thompson said then he did not want to be pinned down on party policy.

Likewise, Thompson's refusal to take a position for or against the Panama Canal treaties on grounds that the issue is beyond the horizon of any state governor was duly noted by ideological conservaties.

Nevertheless, the emphasis was on the new govenors and a new pragmatism. As such, the affair was a modest success, the first Republican gathering since 1973 in which the word Watergate was not heard.