Eating cake and licking his fingers Arlyn Turnquist was standing on the steps of the U.S. Capitol telling what it was that got him interested in the first place.

"Mary [his wife] would be talking about solid waste or human resources or the welfare system -- or whatever the issue was," Turnquist was saying, "and I wouldn't even know what she was talking about."

Then six years ago all that changed and Turnquist, a social worker with the Lutheran Social Services of Wisconsin, became one of 3,000 men across the country to sign up for membership.

"I wanted to be able at least to converse on the issues, to know what was going on," he said.

Which is why it is Arlyn Turnquist -- not Mary -- who is the delegate for the League of Women Voters in Wassau, Wis., at the 1980 national convention here this week.

There were banners, band music and a cake yesterday at the U.S. Capitol. On the steps, a ceremonial knot of officialdom looked out over several hundred heads, all but a few of them women.

For 60 years, one speaker said, they had been on a very fast track. And though she did not enumerate, everybody there knew "track" took in six decades of educating voters on the issues affecting their lives, those of the young, the old and deprived.

"I am very gratified to stand here, cake knife in hand, and say that we cannot only cut it but we can have our cake and eat it, too," said Ruth Hinerfeld, national president of the League of Women Voters.

Then, with a flourish, she cut into the blue and white tiered cake commemorating the 60th anniversary of the organization that was started when 20 million women got the vote.

Yet for all the league's reasons to celebrate, membership has dropped around the country so that today there are 117,000 women and 3,500 men who belong. League officers express concern over the trend.

"We're getting more men in leadership positions," says Nancy Neuman, head of the league's action committee, but like almost all voluntary organizations today, including main-line churches, we're having problems attracting volunteers."

She calls it a "built-in paradox" that the league trains its members so well they are able to transfer their volunteer activites to paid jobs.

"Frequently they drop the league because they feel they don't have enough time for it, or they feel guilty because there's not enough time."

Membership grew in 1976 when the league enjoyed the benefits of national attention brought on by its sponsorship of the presidential debates. But since its last national convention two years ago, 5,000 members have dropped out and there have been few new men members.

While not militant on the subject, Arlyn Turnquist, one of four men delegates among 2,000, nonetheless believes that one reason men are reluctant to join is because of the word "women" in the league's title.

"We're a citizens' group, not a women's group. We invited men back in 1974, voted for them to be part of the league," said Turnquist, who led an unsuccessful effort this week to change the league's name.

Eleanor Charwat of Dutchess County, N.Y., said resistance to a change comes from pride and tradition, "But the practicality is that volunteer groups are losing membership." Dropping "women" from the league's name is "something whose time will come but not yet," said Charwat.

Said Nancy Neuman: "I'm of two minds. From the point of view of equality, I feel the name should change, from tradition and name recognition I think it should stay the same. So I voted for a change. I had to decide which takes precedence, principle or tradition. I decided on principle."

There is some thought that membership may raise this year. The league has annonced that it will sponsor three presidential debates and one vice-presidential debate this fall. President Carter told members on Monday that if he is a candidate he will debate his Republican opponent. The first presidential debate is expected to be scheduled sometime after Labor Day.