She is a consumer of public health care provided by the District of Columbia, and on a chill, damp, November afternoon she sits in the D.C. General Hospital emergency room struggling to get four toddlers into winter coats. "I came here for a test today, but I was misled," says the young other. "The social worker told me to come here, but it's the wrong place, the wrong day and the wrong time. I'm just disgusted about it. When you drag your kids out for nothing..."

An elderly woman who has just been treated for a heart condition sits nearby, waiting fro her son to pick her up. She has been sent D.C. General despite the presence of a city-run geriatric clinic in the basement of her own apartment building, miles away in Northwest Washington.

Such foul-ups are common, according to those familiar with the public health care system in the District of Columbia -- for the system, they say, is scarcely a system at all. Among many examples of inefficiency and confusion in the delivery of health care here, critics cite the following:

Forty-one of 161 physician positions are vacant. City officials say they cannot pay high enough salaries under the civil service system to attract physicians -- creating a situation in which an estimated 250 new cases of veneral disease go untreated every day because there are no doctors to handle them.

Public health clinics routinely refer patients to the public hospital, but virtually never send a patient's clinic records to the hospital, forcing doctors to begin treatment without relevant medical background.

Patients calling some public health clinics are told they will have to wait as long as two weeks to see a doctor. If they say they are sick, they are told to go to a hospital emergency room, an arrangement which costs the city far more money than treating the patient in a clinic, leads to overcrowding of emergency rooms, and negates the value of a patient's becoming known to clinic doctors.

Poor people are often given Medicaid cards without fully understanding what the cards can do for them -- provide at least some access to private physicians and private hospitals. One man brings his elderly sister to D.C. General Hospital "because she has Medicaid," as he explained it. "Those other hospitals only take you if you can pay."

Just two weeks ago, a special task force on health and welfare recommended to Mayor Marion Barry that he consider removing the health and hospitals department from DHR, and that he give immediate attention to solving such chronic health problems in Washington as high infant mortality, tuberculosis, venereal disease and cirrhosis of the liver.

The task force, whose job was to single out problems needing the new mayor's immediate attention, came to many of the same conclusions as a task force appointed 10 years ago by then -Mayor Walter E. Washington.

The earlier group, which made a lengthy studay of the city's public health system, concluded that the District was "health poor, nearly health bankrupt" and demanded: "Something must be done: Decisively, adquately and now."

The 1969 report was entitled "The Crisis of Impotence," and according to Dr. Arthur Hoyt, cochairman of the Barry task force, "We still have a crisis of impotence."

"It's almost beyond conception to have lived in this damn town for 40 years, as I have, and keep seeing the same things the same problems and proposed solutions)," said Dr. Paul Cornely, former chairman of Howard University Medical School's department of community medicine and first black president of the American Public Health Association.

Cornely, an influential member of the 1969 task force and also a member of the Barry group 10 years later, said he can remember "Mrs. (Eleanor) Roosevelt making visits to D.C. Genral and saying how bad things were," And today Rosalynn Carter makes the same visits and says how good things will be.

By all commonly accepted indicators of the s tate of a people's helath, the District of Columbia is almost as health-poor in 1979 as it was a decade earlier.

The city still has the highest urban infant mortality rate in the nation, almost twice as high as the national rate and a rate higher than of any western nation.

The city has the highest cirrhosis of the liver death rate in the nation.

The city has the fourth highest tuberculosis rate in the nation, and the D.C. Lung Association believes there are a larger nember of undetected cases.

The city has one of the nation's highest venereal disease rat es, with thousands of cases going untreated.

The entire southeast area of the city is badly underserved, in terms of the numbers of physicians and hospital beds, while the rest of the city is over-doctored and over-beded. Some residents of Southeast have to travel literally hours by bus to and from medical care.

Public health officials here have traditionally blamed these problems on what they say is a particularly high proportion of poor residents in the city.

However, a 1975 study by the Federal Health Resources Administration showed Washington to have proportionally fewer persons below the poverty line than Atlanta, Baltimore, Chicage, Cleveland and new York City, all cities with lower infant mortality rates.

There is no question that many, if not all, of the Districts's health problems are tied to conditions of poverty, overcrowding, poor sanitation and inferior education. At the same time, experts in the field say a well-organized, properly staffed health department could markedly lower these high rates of death and disease -- just as Boston cut its non-white infant mortality rate by 54 percent between 1970 and 1976, while Washington's dropped only 5 percent during the same period.

Many of those interviewed -- including several dozen present and former city health officials, workers and other professionals who have contact with the public health system -- point to the very structure of the health department and DHR as a major part of the problem.

"DHR as an organizational structure is really not able to cope with health care delivery as such," said one local hospital administrator. "It's too complicated. It's hard to find people with the competence to operate a structure that large."

Because health is just one of the priorities within DHR, critics say, health services must vie for dollars with other DHR programs. If the city is taken to court and ordered to make welfare payments more swiftly, for example, personnel slots that might have been filled with health workers may be filled with welfare workers instead.

"I felt the concept of a Department of Human Resources was wrong." said Donald K. Wallace, now health officer for Prince Geroge's Country, and the man in charge of Washington's disease control, immunization and epidemiology programs, and the dog pound, from 1972 to 1974.

"The communications channels (within DHR) were badly blocked," said Wallace, and "my overall impression of the D.C. system was one of anarchy. No one had to respond to anyone else. My employes could go around me and I could go around my boss."

Wallace is a staunch supporter of Dr. Raymond Standard, the city's health and hospitals administrator. In 1976, Standard was temporarily shifted to a job as head of the city's Bicentennial Commission, and some DHR critics say the transfer was ordered because he had angered then-DHR dirctor Joseph P. Yeldell.

Standard was later reinstated in his health post, adn his defenders, including Wallace, picture him as a well meaning man trapped between the health needs of the city and the budgetary and program constraints placed on him from above.

But mayoral adviser Hoyt, an obstetrician and associate dean at Georgetown University Medical School, does not see a lack of funds as the basic problem. Rather, he said, "I have the feeling this city has more resources per capita than most cities."

One major problem, said Hoyt, speaking for Barry's task force, his been the seeming inability of the city to work cooperatively with the private sector -- the three medical schools, private hospitals and medical societies.

Barry, said Hoyt, has to make it clear to the private medical establishment in the District that if it expects to benefit from city programs, it had better address the needs of all the people of the District, and not just the middle and upper classes.

"Washington is a very unique city because it can be called basically a very stable, middle class community," said D.C. General's Roman. Roman recently came here from Boston, where he was director of out-patient services at Boston City Hospital, a facility linked to the majority of Boston's 32 neighborhood health centers. Washington had 15 such centers -- not linked to any hospital -- and a population about 70,000 larger than Boston's.

Roman advances the theory, accepted by many of those interviewed, that the District is health poor because it lacks a vocal constituency for health services. "Problems like infant mortality and underserved pockets may not be viewed as problems" by the middle class, he said. Noting that the city has one of the largest black middle-class populations in the nation. "A health department is no stronger than the constituency it serves," he added.

DHR director Albert P. Russo disagrees. He argues that a vocal health constituency exists here, and that DHR often responds to its pleas. For evidence, Russo points to the "howl that went up" from residents of the area east of the Anacostia River when the city announced plans to close the Parkside Clinic at 701 Kenilworth Ter. NE.

"Because we respond to client needs, that clinic remained open, said Russo. The city had recommended closing Parkside for economy reasons, arguing that there was another clinic five blocks away. But Parkside was one of the city's busiest clinics, and the other clinic was on the east side of heavily traveled Interstate 295, while those served by Parkside largely lively to the west of the highway.

About a dozen people interviewed for this article were asked to name a D.C. public health official who had a positive impact on the city's health. The only two who could think of such a person named Dr. Murray Grant, the last man to head the city's health department before it was merged into DHR.

"It's no coincidence that the only person anybody could name was the last person to head an independent health department," said Dr. Ozella Webb, director of maternal and child health for the city.

Despite budgetary constraints, despite organizational difficulties, Cornely said the city should be doing better than it has in the health field. "We're working with a population that's markedly better off (than other urban areas) as I see it. And I don't see why in the hell we haven't taken advantage of that particular fact."

The overriding problem is one priorities, according to most of many of those interviewed. I often felt," said Donald Wallace, "the dog pound had a higher priority than public health."