Render unto Arlington that which is Arlington's and unto Falls Church that which is Falls Church's. At least that's what two jurisdictions recently told residents whose property straddles the county-city border.

The idea may seem logical enough, but in this case it's not that simple. The two jurisdictions don't agree on where the border falls, leaving an estimated 31 properties in limbo.

Take Betty and Joe Harris, who have been paying taxes to Arlington County on their two-bedroom brick house at 1010 N. Roosevelt St. since they moved into the Whistling Pine subdivision in 1970.

On Jan. 8, the Harrises got a "Dear Taxpayer" letter from Falls Church Assessor William W. Delanoy, informing them that the chunk of their property that falls within the Falls Church boundary--2,000 of the 9,733-square-foot lot--was going to be reassessed.

But, said Betty Harris, since they never before had been assessed by Falls Church and rely on no city services, they were perplexed that they were to be "reassessed."

Arlington appraised its portion of the Harris property at $90,300 and Falls Church appraised its share at $7,700. But Betty Harris contends that she never received notice of the Falls Church assessment.

Arlington has a tax rate of 96 cents per $100 of valuation, set to increase this year to 98 cents; Falls Church taxes at $1 per $100 of assessed valuation.

The Arlington assessor, James R. Vinson, told the Harrises in a letter that Falls Church indeed was entitled to tax its portion of her lot--but that, by Arlington maps, only 1,000 square feet were in Falls Church.

Vinson said, however, that the disputed area "cannot be determined exactly since the civil engineer who made the plat map did not precisely set the area."

In any case, officials in both Falls Church and Arlington agree that the Harrises will not have to pay taxes twice on the same piece of property. The problem, said Delanoy, is that the city and county based their boundary upon lines drawn by the subdivision developer's private surveyor, which apparently were not precise.

"In the good old days, I'm afraid they weren't very careful," Delanoy said. "It didn't matter much. You'd see a survey which said the lot line went from the tree over to the road and down to the barn and back."

Until this year, Falls Church had not assessed the small scraps of Arlingtonians' land within the city. Delanoy said they started it this year because Arlington had begun the practice last year. Arlington started, said Vinson, in response to a legal inquiry about land titles. Similar problems have not cropped up along the city's border with Fairfax County, officials agree.

At one point, William R. Hicks, Falls Church assistant city attorney, and Charles G. Flinn, Arlington's acting county attorney, discussed the possibility of readjusting the border by using the old stone markers that originally established the Arlington boundary when the state deeded it to the federal government in the 1790s to be the southwestern portion of the District of Columbia triangle. Arlington, then called Alexandria County, was returned to the state in 1846 and became Arlington County in 1915, Flinn said.

Another option, which Vinson said Falls Church had suggested, was for Arlington to do a new survey of the border since the county has its own surveying department.

But both jurisdictions agree the cost of such a survey--which Delanoy estimated at more than $50,000--would be prohibitive and probably not cost-effective because of the small amount of land involved.

The border battle may be resolved soon, however. Last week Flinn discovered that the county had surveyed the disputed area about 30 years ago. That survey, Vinson added, was apparently "just sitting around in someone's desk" and no one knew about it.

"If Falls Church agrees to it," Vinson said, the problem could be resolved in a matter of weeks. "I don't think the people on the border should be put in the middle of some petty squabble."