When Richard R. Arnold ran for another term of the Bowie City Council last year, he placed an advertisement in the weekly newspapers urging his constituents to "be sure to vote on Tuesday." It was not a bad idea, except for the fact that the election was on a Monday.

The blunder made little difference to Arnold. He overcame it to serve once again on the governing body of the largest city (behind Baltimore and Rockville) in Maryland. That, too, made little difference to Arnold, because he is not sure that the governing body in Bowie - which consists of a mayor and six Council members - makes any difference.

"Oh, we have tremendous authority," Arnold said recently, when asked what he and his colleagues do. "We can pick up your trash. That's about the extent of our authority."

That statement, although offered with due explaining the talk of the town in Bowie, a municipality of some 43,000 suburbanities on the northeastern rim of Prince George's County. The talk is about whether Bowie needs a mayor to lead its government and how ambitious that mayor should be.

Debates of this sort appear periodically in Maryland, where the 22 counties, not the municipalities, have control over the traditional functions of government - schools, zoning, police, sewers, parks and social services. In 1976, another Prince George's municipality to the extreme by holding a referendum on whether it should abolish its government altogether.

Arnold started the talk in Bowie at a Monday night Council session a few weeks ago when he porposed that his city eliminate the position of mayor. He said that Bowie's government would function better with a rotating the Council selected from among the Council members.

The mayor, Audrey Scott, took that suggestion as a personal attack; the other Council members took it as an interesting idea, and the two local newspapers, the Bowie Blade and the Bowie News, took it as a great subject for editorials and cartoons.

As issues go in Bowie, which usually makes news only when it blossoms with another acre of ranch houses, this one promised some flair.

"It's quite simple, the way I see it," said Arnold. 'We don't have a mayor in the context of Baltimore, Chicago or Los Angeles. Our charter says that the mayor is just another member of the Council. There is no veto power. It's a part-time job. The only thing the Mayor is required to do is chair the Council meetings."

Mayor Scott, however, had done far more than the charter - or her $5,100-year-a year salary - requires of her. As Arnold put it: "She acts like a real mayor."

What that means is that Scott works at the job full-time, rarely missing a ribbon-cutting ceremony, a dedication, or meetings of the County Council, the Maryland-National Capital Park of Planning Commission or the Washington, all of which have more than the local government.

Arnold does not think the mayor of Bowie should act like a real mayor. "That silutes the effectiveness of the other six Council members, who by charter are supposed to have the same amount of clout," he argued.

Scott, for her part, says that Arnold's reasoning is backward. "The reality is that we have and need a full-time mayor," she said. "Arnold wants to change reality to meet the charter. I say we change the charter to meet reality." Scott said that proposal to eliminate the mayor, she will match him with a proposal to make it a full-time position.

Most of the other Council members side with Arnold. While saying they hold nothing against Mayor Scott or hold nothing against Mayor Scott or her tireless efforts, they contend that she has made more of the job than is necessary. Council member Herbert Sachs, for instance, had this cautious reproach:

Some people tend to emphasize the ceremonial aspects of returning a city, the coffees, the ribbon-cuttings. They tend to politicize it. I tend to think there are more important roles for a mayor. Here in Bowie, that role should be effectively running the Council, nothing more."

The only Council member who supported the retention of the mayor's post was John Cummings, who defended it in political terms. "It provides a power base for an individual who may want to move up in county or state politics," Cummings said. "I dont't think you'd have that base if you just had seven Council members and no mayor."

Scott uses a similar argument. "A mayor is a much more authoritative figure than a Councilman," she said. "The main job of the mayor is to lobby at the other levels of government andyou need some authority to do that. You need someone who can say, 'speak for the 43,000 people of Bowie.' That is vitally important. THe clout is important."

Arnold, a four-year Council veteran, and Scott, who was elected mayor in 1976, take into their debate markedly different perspectives on the workings of municipal government and the incentives for holding office.

Arnold said there was one reason he ran for the City Council. "To stimulate my ego. What the hell, it's an ego trip." Scott said her interests were solely the interests of Bowie, and, to emphasize the point, she said it took to lobby effectively for a full-time mayor.

When asked to expand on his statement that the Council has authority over trash collection and little else, Arnold said: "Let's see, there must be some pretty important things. We give out proclamations - we do a super job on proclamations. And we change the names of street signs. Mostly, we listen to people talk about zoning changes. We'll listen to anyone who wants to talk about zoning. Sometimes we listen to people who are talking about a zoning change that isn't even in our jurisdiction."

In Arnold's view, the Bowie Council is "just another civil association, when you come down to it."

Scott, however, said that the 28 municipalities in Prince George's - they represent 194,911 of the 677,848 county residents - are capturing more power from the county and state every year.

As examples, she cited state legislation last year that allowed municipalities to impose local zoning controls on yard fences, street parking and storage facilities. In the next legislative session, Scott said, the cities will push a bill that would allow them greater control of advertising signs within the municipalities.

Scott's predecessor, William Wildman, who was a retired government worker when he held the mayor's post, said he lobbied as vigorously as Scott to get more autonomy for Bowie and other municipalities.

"I'd spend my days traveling between Annapolis, Hyattsville and Upper Marlboro," he said."That's what the leader of the local government has to do.I spent so much time at Upper Marlboro some people though I was living in the County Courthouse."

To which Arnold replied: "The last two mayors have raised people's expectations too much. The people now expect the mayor to be everywhere. It's gotten so that only a housewife or a retired person could hold the job."